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Regression analysis of failure time data
Li, Yihwei, Ph.D. The Ohio State University, 1991
UMI 300 N. Zeeb Rd. Ann Altar, MI 48106
REGRESSION ANALYSIS OF FAILURE TIME DATA
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University
By
Yi-Hwei Li, B.A.
The Ohio State University 1991
Dissertation Committee: Approved by
John P. Klein Melvin Moeschberger Timothy Costigan Adviser Program of Biostatistics Grateful Memory of my Mother ACKNOWLEDGMENTS
I am most grateful to Dr. John P. Klein for his constant guidance and encouragement. Thanks go to the other members of my advisory committee, Drs. Melvin Moeschberger and Timothy Costigan, for their suggestions and comments. I am indebted to my schoolfellows, S. T. Wang and Steve Naber, who have given their time to read the manuscript and offer suggestions. I wish to express my gratitude to many of my friends for their sympathetic support. I wish to thank the Framingham Heart Study for providing the data in this research. This woik was supported by the International Life Sciences Institute. I also wish to thank the Ohio Supercomputer Center for the use of the Cray Y-MP machine. To my father and sisters, I deeply thank you for your understanding and support. VITA
Aug. 27, 1964 ...... Bom - Taipei, Taiwan, R.O.C.
June, 1986 ...... B.A.in Economics, National ChengChi Univ.,Taipei, Taiwan
Aug. 1986 - June, 1989 ...... Teaching Assistant, Department of Statistics, The Ohio State University, Columbus, Ohio
Aug. 1989 - June, 1991 ...... Research Assistant, Department of Statistics, The Ohio State University, Columbus, Ohio
FIELDS OF STUDY
Major Field: Biostatistics Studies in Survival Analysis. Professor John P. Klein TABLE OF CONTENTS
DEDICATION...... ii ACKNOWLEDGMENTS ...... iii VITA ...... iv LIST OF TABLES ...... vii LIST OF FIGURES...... viii INTRODUCTION...... 1 CHAPTER PAGE
I. Regression Analysis of Univariate Failure Time D ata ...... 4
1. Introduction ...... 4 2. Fully Parametric Approach ...... 5 3. Semi Parametric Approach ...... 16
II. A Monte Carlo Study of Univariate Regression Models for Survival D a ta ...... 29
1. Introduction ...... 29 2. M ethods ...... 31 3. Monte Carlo Results ...... 40 4. Summary and Discussion ...... 57
v III. Multivariate Regression Models Induced by Random Effects 60
1. Introduction ...... 60 2. Frailty M odels ...... 62 3. Semiparametric Estimation for Conditional Proportional Hazards Models...... 68 4. Discussion ...... 77
IV. Parametric and Semiparametric Estimation for the Inverse Gaussian Frailty Model ...... 79
1. Introduction ...... 79 2. Models and Semiparametric Estimation ...... 83 3. Generalized Inverse Gaussian Distribution ...... 89 4. Calculation of the Observed Information M atrix ...... 91 5. Parametric Estimation Based on a Weibull Regression Model 93 6. Examples from Framingham Heart S tudy ...... 101
V. Discussion of Future W ork ...... 113
APPENDIXES
A. FORTRAN Program for the Monte Carlo Study ...... 115
B. FORTRAN Program of Semiparametric Estimation for Model I 142
C. FORTRAN Program of Semiparametric Estimation for Model I I 153
D. FORTRAN Program of Parametric Estimation for Model I ...... 162
E. FORTRAN Program of Parametric Estimation for Model I I ...... 177
LIST OF REFERENCES...... 192
vi LIST OF TABLES
TABLE PAGE
2.1 The value of X under the independent exponential censoring for some failure time distributions ...... 39 2.2 Observed sizes of the likelihood ratio tests of single-covariate effects with nominal size a = .0 5 ...... 42
2.3 Empirical powers of the likelihood ratio test (a = .05 and N = 40) of dichotomous-covariate effects of 0% and 40% censoring ...... 47
2.4 Empirical powers of the LR test (a = .05 and N = 40) of discrete (D) and continuous (C) covariate effects for 0% and 40% censoring 51
2.5 Empirical powers of the LR, Score and Wald tests (a=.05 and N=40) of single-covariate effects for 0% censoring ...... 54 4.1 Summary statistics for covariates and events of interest ...... 103 4.2 Semiparametric estimates of parameters based on model I for times to first evidence of coronary heart disease and cerebrovascular accident .. 106 4.3 Parametric estimates of parameters based on model I for times to first evidence of coronary heart disease and cerebrovascular accident 107 4.4 Summary information on sibling group size and number of CHD events in groups for each size group for 45 year old disease-free individuals . 109 4.5 Semiparametric estimates of parameters for time to first evidence of coronary heart disease adjusted for sibling effects ...... 109 4.6 Parametric estimates of parameters for time to first evidence of coronary heart disease adjusted for sibling effects ...... 110
vii LIST OF FIGURES
FIGURE 2.1 Log logistic hazard function ...... 33
2.2 Log normal hazard function ...... 33
2.3 Log Laplace hazard function ...... 35
4.1 Inverse Gaussian probability function (for Kendall's x =.05,. lOand. 13) 81
4.2 Gamma probability function (for Kendall's t =.05, .10and.l3) ...... 81
4.3 Inverse Gaussian probability function (for Kendall's x =. 15,.27and.30) 82
4.4 Gamma probability function (for Kendall's x =.15, .27and .3 0 ) ...... 82
4.5 Relative risk of smoking for males ...... 112
4.6 Relative risk of smoking for fem ales ...... 112
viii INTRODUCTION
Of primary interest in medical research is the investigation of the relationship between certain covariates, such as treatment, patient characteristics or risk factors, and the time to occurrence of an event such as death, disease recurrence or cure. Through a regression model, in which lifetime has a distribution depending upon the covariates, we can examine the association between concomitant variables and lifetime. Two major approaches to the regression problem have been suggested. These are the parametric approach, where both a baseline distribution holding under standard conditions and the dependence structure of covariates on the failure time are fully specified, and the semi-parametric approach in which we construct a regression model without the specification of a baseline distribution. The first objective of this study is to examine these two approaches under different univariate regression models for survival data. In addition, we will also study the small sample properties of the likelihood ratio, the score, and the Wald statistics, and compare the performances of these three test statistics as applied to survival models. Many clinical and epidemiologic studies are concerned with multiple survival times of individuals who are naturally or artificially grouped. For example, in familial research of disease incidence, the times to the event of interest among related individuals such as parents and offspring, siblings, or husbands and wives may be associated because they have shared a common unobservable risk factor, either genetic or environmental. In epidemiology, multiple events within the same subject may also tend to associated. A
1 2 natural approach to the multivariate lifetime regression analysis is to incorporate an unobservable random covariate into the regression model. That is, conditional on the random effect, say W, the life times are independent But W is unknown and if it is averaged out, the life times become dependent. This model has been called a "frailty model" by Vaupel, Man ton, and Stallard (1979). They defined the unobserved random effect as a "frailty". The second objective of this study is to extend the parametric and semi-parametric approaches to the multivariate lifetime regression analysis in which we assume that the random covariate, W, follows an Inverse Gaussian distribution. As an illustration, the procedures developed are applied to a large scale longitudinal risk assessment study, the Framingham Heart Study. Finally, comparisons between the performances of these parametric and semi-parametric analyses, and the fits of Gamma and Inverse Gaussian frailty models are discussed. This dissertation is divided into five chapters. Chapter I introduces the parametric and semi-parametric approaches to regression analysis of univariate failure time data. Chapter II includes a Monte Carlo Study to evaluate the robustness properties of Cox’s semi parametric approach and standard parametric approaches to model misspecifrcation. The small sample performances of the likelihood ratio, the score, and the Wald test statistics based on these models are discussed. In addition, the effects of an independent censoring mechanism on these inference procedures are examined. Chapter III introduces and studies several frailty models and their dependence properties. We also examine three semiparametric estimation schemes for the gamma frailty model assuming conditional proportional hazards. In Chapter IV, we present the methodology to perform parametric and semiparametric estimation for the inverse Gaussian frailty model. We consider a set of 3 data from the Framingham Heart Study to illustrate our methods and to compare the numerical results of parametric and semiparametric approaches. Finally, Chapter V briefly sketches our future work. CHAPTER I
Regression Analysis of Univariate Failure Time Data
1. Introduction
The accelerated failure time and proportional hazards classes of regression models are commonly used in the analysis of survival data. For the proportional hazards model, the hazard rates of individuals with different explanatory variables are proportional to each other. We assume that there is a baseline hazard, say ho(x), corresponding to a standard condition and that the explanatory variables, z, act multiplicatively on the baseline hazard. That is, the effect of the covariates is to increase or decrease the hazard. Thus, the proportional hazards model is defined as
h( x; z )» ho(x) c(z), where c is a specified functional form. For the accelerated failure time model, the explanatory variables act multiplicatively on a baseline time Xo so that their effect in this case is to accelerate or decelerate that failure time X, namely, X=Xoc(z). If c(z) has the form of exp(P z), then the log lifetime follows a linear model: logX = (5 z + E, where E=logXo is a disturbance random variable.
4 5
Both the proportional hazards and accelerated life models require two model assumptions, namely, (1) a baseline distribution where the standard condition holds; and (2) a functional form for the dependency of the lifetime on the covariates, often in terms of some parametric model. In this chapter, we examine two approaches to regression models: a fully parametric approach where those two parts are explicitly specified and a semi-parametric approach where only (2) is identified. Section 2 reviews several parametric regression models where some well-known survival distributions are generalized to the accelerated failure time and proportional hazards models. In Section 3, we will investigate the semi-parametric approach to the regression models.
2. Fully Parametric Approach
We begin in section 2.1 with the exponential and Weibull regression models. Section 2.2 investigates the maximum likelihood method of inference for these models. In section 2.3, we examine the log logistic and log normal regression models. Section 2.4 contains some discussion about parametric regression models. It is convenient to introduce some notation that is used in survival analysis. Throughout this study we will use these notations without further explanation. Let X be a nonnegative continuous random variable representing the "failure" time and z =(zi,....,zp)’ be a vector of p covariates. Note that in medical research the failure time can be defined as the time to occurrence of an event such as death, disease relapse or cure. The explanatory variables, 6
z, may include both quantitative and qualitative variables such as demographic (age, sex, race,etc.) characteristics and treatment group. Let S (x), f (x), and h (x) denote the survivor function, the probability density function and the hazard function of the failure time, respectively.
2.1. Exponential and Weibull Regression Models
Fiegl and Zelen (1965) investigated the exponential distribution with concomitant information. They generalized the exponential distribution to a regression model by specifying the hazard at time x for an individual with covariates z as
h(x;z) = Xc(z) ; X>0
where X is a constant, and c is a specified functional form. Thus, given z, an individual has a constant hazard rate characterizing an exponential distribution, but the lifetime distribution depends on z. Fiegl and Zelen proposed three functional forms for c: (1) c(z) = (5 z, (2) c(z) = (P z)-1, and (3) c(z) = exp(P z), where p = (Pi,....,Pp) is a vector of regression parameters. Although one can arbitrarily choose a specific functional form, die third form seems more natural than others since the requirement c(z) > 0 is automatically satisfied for all z and p. Throughout this study c(z) = exp(p z) is considered explicitly.
Similar inference procedures can be applied to the model where some other function of z is chosen. In die same way, die Weibull regression model can be generalized. Prentice (1973) and Kay (1977) considered the model: 7
hCxjzJsXpxP^eP* ; X>0, p>0, x>0 (1.1) Note that exponential regression model is a special case of the Weibull regression model with p = 1 where the baseline hazard is constant over time x. The hazard function is monotone increasing with time if p > 1, and decreasing if p < 1.
The model (1.1) is a special case of the family of proportional hazards models. The proportional hazards model (Cox, 1972) specifies that h (x ; z) = ho (x) eP z (1.2) where ho(x) is an arbitrary baseline hazard function for X holding in the situation of P = 0.
The survivor function and density are thus S(x; z) = [S0(x)]«P[P *1, f(x; z) = exp(P z)[S0(x)]«P(P f0(x).
Hence, the survivor functions form a family of Lehmann alternatives. The proportional hazards family has the property that different individuals in the study have proportional hazard rates. That is, the ratio h( x; zi) / h( x; Z 2 ) of the hazard functions for two individuals with different regressors zi and Z 2 does not depend on the survival timex. The probability density function of x given z corresponding to (1.1) is f(x;z) = XpxP-1ePzexp(-XxPePz) x>0 (1.3)
Transforming Y = log X, the model (1.3) is also the accelerated failure time model Y = o + P*z + o E (1.4) where a = -logX / p, a = 1/p, P* = -p-1p, and E has the standard extreme value distribution with p.d.f fie) = exp(e - e8). Consider the linear model in terms of hazard function. Let Xo denote a baseline failure time which has a baseline hazard rate of form ho(x), then the explanatory variables, z act multiplicatively on X q. That is, X = X q exp(P*z). Thus, the effect of covariates in the 8 accelerated lifetime model is to accelerate or decelerate lifetime. Then the hazard function of X given the covariates z can be written as h( x; z) = ho{x exp(-p*z)} exp(-P*z). (1.5) Here the hazard rates of different subjects are not necessarily proportional, in fact, they depend on the failure time x.
If die Weibull hazard function is specified, then (1.5) becomes h( x; z) = XpxP‘1exp(-pP*z). This relates to the equation (1.1), where p = -pP*. Note that the only continuous distribution belonging to both the accelerated lifetime and proportional hazards families is the Weibull (and exponential) model. Thus, the Weibull regression model is the unique model where the covariates act multiplicatively both on the hazard function and the lifetime itself.
2.2. Inference Procedure - Maximum Likelihood Method
The standard approach to inference for regression models is the maximum likelihood method. In this section, we briefly introduce the theory of the construction of likelihood function for lifetime data. We shall consider the Weibull regression model as an illustration of the general approach. Censoring is an important feature often encountered in lifetime data analysis. It may due to the limitations on observation period imposed by loss to follow-up, death from causes other than that under investigation, or survival to the study's end (withdrawal). It is 9 assumed here and throughout this study that the censoring process is independent of the cause of death; that is, non-informative censoring. We now suppose that a continuous lifetime X with probability density function f( x; 0, z) and survivor function S( x; 6, z), where 0 is a parameter vector. Often we may write 0 = ((5, <(>), where P is a vector of parameters of particular interest and <}> is a nuisance parameter vector. We assume that a set of data of the form (ti, 5i, z,), i = l,...,n are collected. Notice that the tj's are the times on study; namely, ti=min(xi Ci), where Ci is some censoring time. Let Si's be event indicator variables: Si = 1 if the ith subject failed (ti=xO; Si = 0 if the i* subject is censored (ti=Ci). As before, z\ is a p-vector of covariates. If we observe a subject who failed at time t, then the contribution to the likelihood is f(t; 0 , z), the density function at t. The contribution from a subject censored at t is S(t; 0, z), the probability of survival beyond t Thus the full likelihood based on the data (ti, Sj, Zi ), 1 = 1,....,n, IS
L( 0) = n f(ti; 0, Zi)5i S(ti; 0, Zi)1-5* . (1.6) i=l
The log likelihood is
L (0 ) = X ln f(li J e> z*) + X ln s (*i 5 0 * Z1> ’ (1-7> i e u i e c where the two summations are taken over uncensored and censored subjects respectively. We now consider the Weibull regression model, and substitute (1.3) into (1.7). Thus, we have the log likelihood function as n L (X, p, p ) = d ln A. + d ln p + (p -1) I ln ti + X P Zi - X. I tip exp(p zi) (1.8) u u i=l where d is the total number of deaths. 10
The first and second derivatives of (1.8) are
(1.9) OK X 1=1
a L (X, p, p )= d + J lnti. x £ tiPlntiexp(pZi) (1.10) 3 p p u i=l
^ L ^ "" ■ 2 zik - ^ I tip zik exp(P ^i) ...... k=l.....»P (1.11) 3 Pic u i=l
d*L q, pt p) d_ (1.12) ax2 ~ ’x2
a2L(X, pt p £tiP(lnti)2exp(pZi) (1.13) 3 p2 p2 i=l
(1.14) 92 \a x a p = - i=i It|Plntiexp(pZi)
(1.15) 82ta x f a ^ pK ’P i=i •£tf^=P(P«0...... k=1...... p
a 2L p ’ P-i = - X I tjp In ti Zik exp(p Zi) k=l ,p (1.16) a p a pk i=i
a L (X, p, P ) __ ^ £ tiPzjkzuexp(pZi) k,l =1.....,p (1.17) a pk a pi i=i We can obtain the maximum likelihood estimator p, (?) by setting equations (1.9), (1.10), and (1.11) equal zero and solving by the Newton-Raphson iterative procedure. Since our primary concern is the regression coefficients (3, we consider the form of hypothesis tests for Ho: (3 - Po» where Po is a specified vector. Here, we denote the parameter vector (p, p, X) as 6, and partition it as (P, ( I L _ \ a p t rui(6^ u( e ) = d L •••••* d p u 2o) d L \ b x j f d 2L a 2l a2l ^ 9 Pk 3 Pi a pk a x a pk a p d2L 32L d2L « e) = - f lll(0) Il2(0) d X 9 pi dx2 a x a p " t 121(9) 122(0) J a2 L a 2l a2l ^ a p a pi a p a x a p2 j IH(0) 112(0) A 1-H 0 ) ■ ( 121(0) 122(0) J 12 Three large sample test statistics derived from the likelihood function and the maximum likelihood estimator are widely used in application. These are the score statistic, R, based on the efficient score vector, Wald's statistic, W, based on the asymptotic distribution of the m.l.e.'s, and the likelihood ratio statistic, A, which is based on the likelihood function. These statistics are defined as R = Ui(0)' iUtfOUiO) (1.18) W = (P-pO)I1l(0)-1(P-Po)’ (1.19) A = -21og H i 2 (1.20) -L(§). where § = ($,$) is the m.l.e. under the full model and 0 = (Po. 2.3. Log Normal and Log Logistic Regression Models As noted in Section 2.1, the Weibull regression model is a special case of the accelerated failure time family. In this section, we investigate two other linear models: the log logistic and the log normal regression models. We investigate the linear model for Y=logX: logX = Y = n + p z + aE, where p. is a location parameter, a is a scale parameter, and E is a random variable of zero mean with a distribution not depending on z. 14 It is very natural to consider that E follows a standard normal distribution; thus, we have an ordinary regression problem if there are no censored data and the distribution of lifetime T given covariates z is log normal (Lawrence, 1988). However, with censoring, estimation for the log normal regression model becomes difficult since there is no explicit expression for the survivor function S(x). Another candidate distribution for E is the logistic with p.d.f. f(e) = exp(e) / [l+exp(e)]2 since the shape of a logistic distribution is quite similar to that of a normal distribution (see Figures 2.1 and 2.2). This family has an advantage over the normal in that it has the simple explicit forms for survivor, density, and hazard functions. Therefore, by specifying a standard logistic distribution for E, the time T given the covariates z is distributed as log logistic distribution (Lawrence, 1988). Note that one can specify any convenient distribution for E to obtain possible distribution for T. For example, the Weibull regression model is given by f(e) = exp(e - exp(e)), the extreme value distribution; and the generalized gamma regression model (Lawless, 1982) is given by a log gamma error distribution with p.d.f. kk-1/2 _ _ exp( Vke - k exp(e/Vk)) 0 ( 2 tc)1 /2 We have mentioned that the estimation for the log normal model becomes difficult especially with heavy censoring. Lawless (1982) proposed an EM algorithm like procedure, which is a generalization of the method of Sampford and Taylor (1959) to obtain the maximum likelihood estimates for the log normal model. This method applies the weighted least squares estimation procedure and gives different weights to death times 15 and censoring times. Lawless remarked that this EM algorithm like procedure is easily programmed and appears to give fewer convergence problems than the Newton-Raphson iteration. Section 2.2 considers the Weibull regression model as an illustration of several inference procedures. Similarly, the same inference procedures (likelihood ratio, score and Wald tests) can be applied to test the effect of the regression coefficients (3 and to obtain confidence regions for those accelerated life models. 2.4. Discussion We have mentioned several parametric regression models and three inference procedures for them. It is very important to know which model to choose for our inferences are to be based on the appropriate model. The parametric approach has stronger distributional assumption and these inference procedures may not be robust to model misspecification. Solomon (1984, 1986), Struthers and Kalbfleisch (1986) indicated that estimators of regression parameters based on the proportional hazards model will reflect the relative importance of the covariable when the underlying distribution is the accelerated failure time model; and analogous conclusion is drawn for the converse situation. That is, when the accelerated failure time model is assumed when the 'true' model is a proportional hazards model. Nevertheless, the effect of departures from assumptions on inferences is difficult to calculate. In Chapter n, we have conducted Monte Carlo experiments to evaluate this effect in a hypothesis testing framework. To solve the problem of model choice, it is natural to consider more general parametric models such as the generalized gamma regression model in (1.21), where it is an extreme 16 distribution for k=l, and a normal distribution for k=°o. However, this method may be computationally complicated or need a larger size of data set for inferences. The second approach is to perform graphic methods, such as histograms, Q-Q plots on probability paper and hazard plotting, for model diagnosis. The graphic methods often provides good information about how well a given model fits the data. Another method is to consider models that are nonparametric, or only partially parametric. The next section will describe the semiparametric approach for the proportional hazards and log linear models. 3. Semi Parametric Approach This section reviews the semiparametric approach to the proportional hazards (1.2) and accelerated failure time models (1.4). We have introduced the parametric setting for these two models by specifying both the baseline distribution and the relationship between the covariates and the lifetime in Section 2. Here we relax the assumption on the baseline distribution so that ho(x) is an unspecified function. In section 3.1, we discuss the construction of a likelihood based on Cox's proportional hazards model as well as the large sample properties of the estimators derived from the likelihood. For the accelerated failure time models we introduce, in section 3.2, linear rank tests which are the appropriate inference procedures based on rank statistic, and least square estimators where the standard linear regression analysis is extended to right censored data. 17 3.1. Cox Proportional Hazards Model Cox (1972) proposed a semiparametric inference procedure for the proportional hazards model He specified the model as h( x; z) = ho(x) exp(p z), (1.22) where the baseline hazard rate ho(x) is an unknown nonnegative function. He derives a partial likelihood for P as follows. Let (tj, Si, Z[) denote the observed data set. Let t(i)< t(2)<... (1.23) £exp(p zi) * l«=R(t(i)) which is the product of the conditional probabilities over the death times and can be treated as an ordinary likelihood. 18 There has been much discussion in the literature about what type of likelihood L(f3) is. Kalbfleisch and Prentice (1973) showed that (1.23) is a marginal likelihood of ranks under no censoring when there are no time dependent covariates. That is, P(t(l) = ^ exp(P Zfl)) i=l £ ex p (p zl) leR(t(i)) They asserted that the rank statistic is marginally sufficient for the estimation of P so that ordinary asymptotic results can be derived from the marginal likelihood. However, the marginal likelihood model can not handle general independent censoring and time dependent covariates since the more general model does not possess the property of group invariance. Cox (1975) gave a definition of partial likelihood to generalize the ideas of conditional and marginal likelihood, and showed that (1.23) is one example of partial likelihood. Let t(i) denote the i* ordered observed death in the proportional hazards model. Consider transforming tQ into die sequence (Y i ,S i ,Y2,S2 ,Yk,Sk), where Yi specifies the censoring in [t(j.i), t(i)) and the information that a failure occurs at time t(i); and Si specifies the particular individual with covariates Z(j) that failed at t(i). 19 Thus, the full likelihood can be divided as P(Yi ,Yk; Si, Sk I P) k = .I^P(Yi,Si I Yi ,Yn; Si,....,Sn; P) k k ~ II P(Yj I Yi,....,Yi.i; P) n P(Si 1 Yi,....,Yi; p). i=l i=l Cox called the second product the partial likelihood. He claimed that the partial likelihood contains most of the information about p so that the first product can be ignored. Johanson (1983), using the counting process theory, showed that Cox's partial likelihood for the regression parameters P is simply a conventional likelihood maximized with respect to die nuisance parameter ho(t). That is, the partial likelihood can be viewed as a profile likelihood. We set out this idea very informally. Assume that there is no censoring and consider the full likelihood on the observed data as n L(h, P) = I I { ho(t(i))exp(p Z(i>)} exp{-Ho(t(i))exp(p z q )} (1.24) i=l This is maximized with respect to ho(t) for fixed P when 1 if t=t(i) X exp(P zi) £>(t)=< leR(to)) (L25) 0 if t is not a death time Note that the non-parametric maximum likelihood estimate of the integrated baseline hazard function, Ho(t), is a step function with jumps at those observed death time points. That is, the Nelson- Aalen estimator 20 rio(t)= I — = -J------(1.26) 2/?xp(P zi) leR(t(i)) Substitution of (1.25) in (1.24) yields a profile likelihood for f) which is proportional to (1.23). Anderson and Gill (1982), and Gill (1984) reformulated Cox's regression model as a model for the random intensity of a multivariate counting process. Furthermore, using modem martingale and stochastic integral theory, the asymptotic properties of the estimators can be derived in a completely natural way. We briefly introduces Anderson and Gill's approach to Cox proportional hazards model Let N denote a multivariate counting process N = (Ni(t): 0 £ t < <*>; i=l,....,n} with an intensity process A = (Ai(t): 0 £ t < oo; i=l,....,n} which is defined by Ai(t)dt = P{ Ni jumps in a time interval [t, t+dt) I Ft.}, where Ft. denotes the history up to time t. In Cox's model, we define Ni(t) = I{Xi 5St,Xi Therefore, (1.23) can be rewritten as “ ( Yj(t)exp(ft Zi) \jNj(t) (1-27) 1 “ ^Yj(t)exp(p Zj) Vj=l / 21 Note that if the covariates z are time dependent, we can replace zj by Zi(t) and the results still hold. Anderson and Gill showed that the Cox partial likelihood can be treated as an ordinary likelihood so that the inference procedure discussed in section 2.2 can be applied to (1.23). 3.2. Accelerated Failure Time Model Linear Rank Tests As stated in Section 2, the accelerated failure time model can be expressed as logX = \i + P z + ctE (1.28) where E has a density f(£). We have studied the large sample tests for parametric models such as Weibull, and log normal regression models where the error density is specified. Peto and Peto (1972), and Kalbfleisch and Prentice (1980) have developed linear rank tests which do not require a strong assumption on the error distribution, and showed the log rank test and the generalized Wilcoxon test as examples of such rank tests. We briefly sketch their approach to the accelerated failure time model. We consider model (1.28) and observe a set of data (q, Zi, SO i=l,...,n, where 8i=l if ti is a death time, 8i=0 o.w., and Zi is a vector of covariates for the i1*1 individual. With no censoring present, let y(i) Let R=(Ri,....,Rn)' be the rank vector of Y. Without loss of generality, we assume |i=0, o*l, and consider the hypothesis Ho: (3 = 0. Thus, the probability of the rank vector r can be calculated as P(rW ; , *«>>dy® A locally most powerful rank statistic for testing Ho: P = 0 against Hi: p * 0 is given by the score statistic based on (1.30) v= d log P(r) dp M = IciZ(i) It can be shown that q = n! J- * ‘ / f'd -° l f(y(n- n f(y(j))dy(j>" y(i)<- Letting the error distribution be the extreme value distribution f(e)=exp(e-e£), then the locally most powerful rank statistic is of the form (1.29) with scores Peto and Peto named this statistic the log rank test. Similarly a logistic error density f(e)=ee/(l+ee)2 gives scores 2| °i = (n+T) " lf so that the optimum test statistic is the Wilcoxon test 23 Let Ep denote expectation over the permutation distribution. Then, the expectation and the covariance matrix of the rank statistic is given by Ep(v) = 0, V = Ep(vv') = (n-1)*1 X c? Z'Z respectively, where Z is the n x p matrix of regression vectors centered about their means. It can be shown that under mild restrictions on the regression vectors, v is asymptotically normal. Thus, under Ho: (3 = 0, v'y -lv 2 is asymptotically X ^ y where it has been assumed that V is nonsingular. Similarly, when type II censoring is present, we suppose that y(i)<... P where S(u) = f(x)dx. Using the score statistic based on (1.31) d log P(rc) d p p=o then the rank statistic is of the form k r m X fciz(i) + S* CiZijl (1*32) i=l V J=1 ) 24 where Cj and Q are the scores for the uncensored and censored data, respectively. The scores can be shown explicidy to be and where nj is the number of items at risk just prior to the death time y(j). Note that for the log rank test the scores are (1.33) Then the test statistic (1.32) with scores (1.33) can be written as 2 21 - 2(i) l€R(t(i)) which is the negative of the score statistic from the Cox partial likelihood (1.23). The rank tests are invariant under monotone increasing transformations on the response variable. Thus, one can consider the proportional hazards model as arising from a model where the monotone transformation of lifetime Y= logX has an extreme value error distribution. For this reason, the log rank test for Ho: P = 0 is exactly the score test based on the partial likelihood (1.23). In the comparison of the rank tests with parametric tests, Kalbfleisch and Prentice (1980) stated that although the rank tests are derived with some assumption on the error 25 distribution for which optimum parametric procedures exist, they are generally less sensitive to misspecification of error distribution or to extreme observations in the sample than the parametric procedures. However, with censored data and small sample size the efficiency of the rank tests could be questionable. In addition, the exact significance level based on the rank statistic involves the calculation of probability over the permutation distribution which can be very complicated. Least Squares Estimators As noted in Section 3.1, Cox (1972) developed the partial likelihood which involves only the parameters of interest for the proportional hazards model, and showed that the estimation of regression coefficients could be obtained by treating the partial likelihood as an ordinary likelihood. As a competitor, Miller (1976), and Buckley and James (1979) proposed methods of estimating parameters in the accelerated failure time model which allows the residual distribution to be unspecified. Here we discuss their methods of estimation and the properties of the estimators. For simplicity, we consider the case of p=l in (1.28) so logXi = + Pzi + 6i i=l .....,n where the ej's are independently and identically distributed with unspecified distribution function F, mean zero and finite variance. As before, suppose we observe a set of data (yi=logti, zi, Si), i=l,...,n. In the absence of censoring, the estimation of (p.,(3) could be accomplished by the usual least squares procedure. With censoring present, Miller considered the estimates which minimize 26 n fc> u2 d £(u) = X wi(p) (yi - \i - pzO2, (1.34) J i=l where uj = yi - |i - pzj, £ is the Kaplan-Meier (KM) estimator based on (ui,5j) i=l,...,n, and the weights Wi(p) are the jumps of the KM estimator. Differentiating (1.34) with respect to |i, we have ft = I wi(P)yi - p X wi(P)zi (1.35) i=l i=l Substitution of (1.35) into (1.34) yields a function of P alone: Q(P) = Iwi(P)(yi-(i-Pzi)2 which can be minimized by a search procedure. However, since the function Q(P) is not continuous, the computation could be very difficult, especially in multiple regression models. Miller suggested the following modified procedure. 1. Let P °, the least squares estimate based on death observations, be the initial estimate. 2. Using this, calculate u? = yi - 0°zi, , and wi($°), i=l,...,n. Define the new estimate J wj (0°) yi(zi - zd) J Wj (§°) (zi - zd) where i(6°) -♦ 3. Repeat the process until convergence. Unfortunately, convergence does not always occur. The sequence of estimates of P may oscillate between two values in the end. In addition to the convergence problem, the 27 consistency and die asymptotic variance of Miller's estimators depend on the assumption that the censoring distribution is of the form Gz(c) = Go(c-pz), for some distribution function Go. Buckley and James gave a method different from that of Miller in that the normal equations rather than the sum of squares of residuals are modified. The following procedure is thus suggested. With censoring present, we define the pseudo random variables Y* = YiSi + E(logXi I logXi > YO (1-80, i=l >n. * * It then can be shown that E(Yj) = p. + pzi. Substitution of yj for yi gives the normal equations I (y*-li-pzi) = 0 (1.36) i=l I (zi - z) (y*-PzO = 0 (1.37) * * Since we can not observe when i01 item is censored, we estimate yj as AZ,A wk(S) Uk = < for5i= 0 a * < yi for Si = 1 where 0 is the initial estimate of P, ui = yi - $zj, £ is the KM estimator based on (ui,80 i=l,...,n , and wj(0) i=l,...,n are the jumps of £. 28 Substituting (1.38) into (1.36) and (1.37), then we obtain the new estimates |i and p. One iterates this procedure until convergence. It should be noted that this procedure is a nonparametric EM algorithm. Again, the sequence of estimates of (3 does not always converge. However, Buckley and James claimed that if the estimates oscillate, then the difference between their two values is smaller than that for the Miller's estimate. Another advantage of their method over Miller's is the relaxation of the assumption on censoring distribution. Nevertheless, the properties of their estimators are still unknown. CHAPTER II A Monte Carlo Study of Univariate Regression Models for Survival Data 1. Introduction We have studied, in CHAPTER I, several parametric regression models (the exponential, Weibull, log logistic, log normal, and normal) and Cox's proportional hazards model as well as their inference procedures. Although the large sample properties of the test statistics of covariate effects for these models have been studied, their performance in the small samples typically encountered in practice has not been thoroughly investigated. In addition, the behavior of these statistics under model misspecification is not clear. Johnson et al. (1982) examined the small sample performances of the maximum partial likelihood estimators of the Cox model under type II censoring and the misspecification of the covariate model. Lagakos and Schoenfeld (1984), and Lagakos(1988) investigated the properties of proportional-hazards score tests under misspecification of the functional form of the regression portion of Cox model. Solomon (1984), and Struthers and Kalbfleisch (1986) studied the properties of an estimator based on a proportional hazards model when the true model is accelerated failure time. However, the size and the power of the likelihood ratio, score, and Wald test statistics involving independent censoring and the 29 30 missassumption of the proportional hazards have not been evaluated. Nor have the small- sample study of size and power of the statistic tests for the parametric regression models been examined when the model is incorrect Therefore, in this chapter, we conducted Monte Carlo experiments to compare the finite sample properties of the parametric approach to inference on covariate effects with those of Cox's semiparametric approach in the presence of model misspecification and an independent censoring mechanism. Our interest in this research has been focused on the following questions: (i) How sensitive is the test of Ho: P=0 in the parametric analyses to model misspecification? (ii) How robust is the test of Ho: P=0 of Cox's proportional hazards model when the underlying distribution does not have proportional hazards? (iii) What differences are there in the power for continuous versus discrete covariates? (iv) Which test statistic, the likelihood ratio (LR), score (SC), or Wald (WA) statistics is more efficient? (v) What are the effects of censoring? In section 2, we describe the methods and simulation procedures to assess covariate effects. Section 3 includes and tabulates the Monte Carlo results with respect to size, power, types of covariates, and test statistics. Section 4 contains the conclusion and the discussion of this simulation study. 31 2. Methods 2.1 Definition of the Statistics and the Assumed Models In the simulation study, we consider a single-covariate regression problem. We assume that the dichotomous covariate, z, equal 1 for treatments and -1 for controls. Also, we examine the case where z is uniformly distributed on -1 to +1. We investigate five fully parametric regression models (the exponential, Weibull, log logistic, log normal, and normal) and Cox's semiparametric regression model because these models are widely applied in practice without model checking. These models are the "assumed" models in the Monte Carlo experiments. We briefly review these models. (1) Cox proportional hazards model h ( x; z) = ho(x) eP z , x £ 0 , where ho(.) is any arbitrary nonnegative function. (2) Exponential regression model h ( x; z) = X eP z , X > 0 , x ^ 0 , where the baseline hazard function remains constant. (3) Weibull regression model h ( x; z) = X p tP*1 eP z , X , p > 0 , x ^ 0 . The baseline hazard rate increases with time when p > 1 and decreases when p < 1. (4) Log logistic regression model Y = log X = p. + Pz + oE , 32 where E follows a standard logistic distribution. Figure 2.1 shows log logistic hazard functions for |1=0, P=0, and various values of o. Notice that the hazard function is monotone decreasing for o £ 1 and that it increases initially to a maximum, then approaches to zero as time approaches infinity for a < 1. (5) Log normal regression model Y = log X = p. + Pz + cE , where E has a standard normal distribution. Figure 2.2 shows log normal hazard functions for p=0, P=0, and various values of o. For all log normal distributions, the hazard function is hump-shaped. That is, h(x) has h(0)=0, increases to a maximum, then approaches 0 monotonically as x— However, for a £ 0.2, h(x) increases over most of the distribution. For a ^ 0.8, h(x) decreases over most of the distribution. (6) Normal regression model X = (X + Pz + aE , where E has a standard normal distribution, and the hazard function is strictly increasing. For this reason, the normal distribution may be appropriate for the life of products with wear-out types of failure. For Cox model, we use bisection method to solve the score equation of p and perform the likelihood ratio (LR), score (SC), and Wald (WA) tests of Ho: P = 0 based on the partial likelihood (1.23). For the exponential model where there exists closed form solution for the nuisance parameter, we use bisection method to search the root of p. We applied the Marquart's iteration (David and Moeschberger, 1978) to solve the score systems of the Weibull and log logistic models. We performed the EM algorithm like procedure (Lawless, 1982) to search the maximum likelihood estimates for the log normal and normal models. 33 0.5- Figure 2.1 Log logistic hazard function solid line for 0=1, dashed line for 0=2, dotted line for o=.5 h(x) 0.84 Figure 2.2 Log normal hazard function solid line for 0=1, dashed line for 0=2, dotted line for o=.5 34 These algorithms are stopped and considered to have converged to a stationary point when A log L < 0.01 or maximum ( A 9j) < 0.001 , where A log L is the increase in log L over the last iteration and A 0i is the absolute value of the change in the ith parameter over the last step. After obtaining the maximum likelihood estimates, we calculate the LR, SC and WA tests of Ho: (5=0 for each model. 2.2 Underlying Distributions In the simulation study, we have generated failure time data from several survival regression distributions, that is, the underlying distributions. The distributions used are the five parametric regression models in previous section and (a) Log Laplace distribution Y = log X = p. + pz + oE , where E has a Laplace distribution with p.d.f given by f(e) = 1/2 exp(-l e |) , - °° < e < °o. The hazard function for |i=0 and 0=1 is shown in Figure 2.3. (b) Gamma and Log gamma distribution Y = logX = |i + Pz + oE , where a > o is a scale parameter, and E has a log gamma distribution with p.d.f f(e) = 1/T(k ) exp(KE- e e) , k > 0 , - <» < e < «>. When k =1, f(e) is the standard extreme value distribution. When 0 < k < 1, the hazard rate decreases monotonically; and when k > 1, the hazard rate increases 35 monotonically as time increases. For o=l, X has a gamma distribution with scale parameter exp(p+|3z) and shape parameter k . h(x) 2.5 0.9 0.8 0.7 0.6 0.5 Figure 2.3 Log Laplace hazard function (c) Gompertz distribution (Elandt-Johnson and Johnson, 1980) h ( x; z) = exp(A. + y x +|3z) ,A.,y>0,x£0. The Gompertz regression model is a member of the proportional hazards family. (d) Smith-Bain distribution (Smith and Bain, 1975) h( x; z) = (y / a?) xY_1 exp[(x/a)7] eP z , y »a > 0, x > 0. The Smith-Bain regression model is another member of the proportional hazards family. Here, a is a scale parameter. The baseline hazard function is strictly increasing if Y^ 1; it allows for a bathtub shaped form if y < 1. Also the hazard function is exponentially increasing for large x. 36 2.3 Simulation Procedure We have written FORTRAN subroutines (see Appendix A) on Cray Y-MP machine to estimate the single-covariate effect and to perform the likelihood ratio, score, and Wald tests of Ho: P=0 for the assumed models [models (l)-(6)]. We wish to estimate size and power of the LR, SC, and WA statistics for models (l)-(6), sample sizes of 40 and 100, 0%, 20% and 40% of censoring, and various types of covariates. Equal numbers of individuals are allocated to each group with covariates +1 for treatments and -1 for controls. To study the size of the tests, we set P = 0. For each size study, the estimate is based on 3000 simulated samples. To study the power, we choose alternatives p = 0.22, 0.35, 0.55 and 0.70 so that the relative risk equals 1.5, 2, 3 and 4, respectively. Each power estimate is based on 1000 replicates. We suppose each individual has a lifetime X and a censoring time L. Let X and L be independent continuous random variables. We assume that X follows one of the underlying distributions, and L has an exponential distribution with parameter Xp. For a given censoring percentage P and failure time distribution with p.d.f., fx(x), we solved iteratively for Xp in P = 100 Ex[ Pr( L < x I X=x) ] = 100 j°[l-e-**]fx(x)dx Table 2.1 gives the numerical value of X based on different failure time distributions and 20 and 40 percent of censoring. The failure and the censoring times were generated from the uniform 0-1 congruendal generator function RANF available on Cray computers. For 37 each subject we generate two independent variates X and Lp. Then min (X, Lp) represents the observed time of that subject corresponding to an expected percentage censorship, P. In the Monte Carlo studies, we use RNNOA routine in IMSL, where an acceptance and rejection technique (Kinderman and Ramage, 1976) is used, to generate pseudorandom numbers from a standard normal distribution. RNGAM routine is called to generate variates from Gamma (k) in which squared and halved normal deviates are used for k=0.5 and a ten-region rejection procedure developed by Schmeiser and Lai (1980) is used for k=2. We apply the general inverse-transform method to simulate the Weibull, log logistic, log Laplace, Gompertz and Smith-Bain random variables from the RANF U(0,1) random number generator on the Cray Y-MP computer. We briefly describe the algorithms as follows. 1. Weibull The Weibull survival function is easily inverted to obtain S*l(u) = [-lnu/(XePz)] l/P. Thus, the Weibull pseudorandom numbers can be generated by setting X = [-lnU/(XeP*)] l/P,whereU~U(0,1). 2. Log logistic The inverse of the standard logistic survival function is denoted by S-1(u) = ln[u-1- 1]. Set X = exp { p. + 0z + a In [U-1 -1]} to generate log logistic variates. 3. Log Laplace The inverse of the Laplace distribution function is given by F*f(u) = In 2u if 0 £ u £ 1/2 = -ln[2(l-u)] if 1/2 < u ^ 1 To obtain log Laplace random numbers, perform the following algorithm. a. Generate U ~ U(0,1) b. If U £ 1/2, set X = exp { |i + Pz + o [ln2U] }. Otherwise, set X = exp { |x + Pz - a In [2(1-U)] }. 4. Gompertz The survival function of Gompertz random variable is of the form S(x) = exp [ - 1/y exp(X+Pz) (e^ -1)]. The Gompertz random variable is simulated by setting X = 1/yln [1 - yexp(Upz) In U]. 5. Smith-Bain The survival function of Smith-Bain random variate follows S(x) = exp{ (1 - exp[(t/a)Y]) e Pz}. Thus to simulate a Smith-Bain random variable we can simply set X = a{ln[l-lnU /ePz] . 39 Table 2.1 The Value of X under the Independent Exponential Censoring for Some Failure Time Distributions Failure Time Distribution 20% censoring 40% censoring Exponential 0.25 0.67 Weibull (X=2, p=.5) 0.42 1.93 Weibull (X=l, p= 4) 0.24 0.58 Log Logistic (|l=0, CT=1) 0.10 0.36 Log Logistic (p=0, a=2) 0.03 0.24 Log Logistic (p.=0, a=.5) 0.17 0.44 Log Laplace (p=0,0=1) 0.13 0.41 Log Laplace (|i=0, 0=2) 0.06 0.32 Log Normal (p=0, o=l) 0.16 0.43 Log Normal (p=0, 0=2) 0.08 0.32 Log Normal (p=0, o=.5) 0.20 0.48 Gamma (p=0, k =.5) 0.56 1.78 Gamma (|i=0, k =2) 0.12 0.29 Gompertz (a=l, 7=1) 0.85 2.17 Gompertz (a=.5, y=.5) 0.50 1.28 Smith-Bain (a=l, 7=2) 0.32 0.75 Smith-Bain (a=.5, y=.5) 1.01 3.05 40 3. Monte Carlo Results 3.1 Studies on Size We first investigate the hypothesis-testing properties by means of a 'size' comparison. With the assumption of asymptotic normality and a nominal 5% significance level, the 'size' in the Monte Carlo simulation is defined as the fraction of replications for which the test statistic (LR, SC, WA) is greater than 3.841. Here, due to space consideration, we present only partial simulation results (see TABLE 2.2). We compare the precision of estimates for the six models by counting the number of times that the observed sizes of the likelihood ratio tests fall into the symmetric 95% probability interval {0.05 ± 1.96 V0.05*0.95/5000 } = (0.043, 0.057). The Cox model maintains the nominal significance level fairly well. In only 13 out of 102 independent experiments, the observed sizes fall above the 95% probability interval, and there is no observation falling below the interval in our experiments. When the true value of (3 equals zero, all the underlying distributions are simply the survival distributions without covariates. In this case, the Cox model is 'nonparametric'. That is, no matter what the underlying distribution is the proportional-hazards assumption holds. However, the partial likelihood ratio test (we also investigate the score and Wald tests in sec 3.3) tends to overestimate the nominal size with small sample (N=40). Also, for fixed N, the observed sizes for 40% censoring are less than those for 0% censoring, and the sizes at 20% censoring are less than those at 0% in most cases. The exponential model often does poorly in maintaining the desired level of significance. Without censoring, the observed test sizes are much lower than the nominal 5% level when the underlying distribution has an increasing hazard rate and much higher than the 5% level when the underlying distribution has a decreasing hazard rate. In the cases of overestimation, the censorship will decrease the 'power' of the test so that the observed sizes appear closer to the prescribed level in the presence of censoring. As a matter of fact, the exponential model only holds the nominal level when the underlying distribution is exponential. For the Weibull model, we noticed that the nominal level of the test is slightly overestimated when the underlying distributions are Weibull and the sample size is small. The observed test sizes of the Weibull model maintain nearly the desired a = .05 when the underlying distribution is Gompertz or Gamma (k =0.5) for N=40. Nevertheless, the Weibull model tends to underestimate the a level for N=100 in these cases. For the log logistic, log normal, log Laplace and Gamma (k =2) underlying distributions, the estimates are larger than the nominal level of the test Again, the observed sizes become closer to the desired a level with heavy censoring due to the censoring effect When the underlying distribution is Smith-Bain, the observed sizes appears lower than the nominal level. In 48 and 29 out of 102 independent samples, the observed sizes of the log logistic and log normal models do not fall into the 95% probability interval. These two models maintain the nominal level better than the Weibull model and worse than the Cox model. Again, there is a tendency that the censoring effect will decrease the observed sizes of tests. 42 Table 2.2 Observed Sizes of the Likelihood Ratio Tests of Single-Covariate Effects with Nominal size a = .05 Underlying Assumed 0% Censoring 20% Censoring 46% Censoring Distribution Model n=40 n=100 n=40 n=100 n=40 n=100 Cox .054 .049 .049 .059 .047 .633 Exp .052 .048 .049 .055 .043 .055 Exponential Wei .057 .052 .055 .058 .054 .057 LLog .050 .059 .054 .062 .049 .064 LNorm .050 .054 .057 .056 .055 .056 Norm .056 .050 .050 .055 .055 .059 Cox .052 .058 .057 .053 .656 .653 Exp .369 .370 .291 .273 .196 .184 Weibull Wei .061 .056 .061 .055 .057 .056 A, = 2, p = .5 LLog .055 .094 .054 .086 .048 .081 LNorm .056 .053 .054 .053 .057 .055 Norm .042 .046 .054 .060 .053 .067 Cox .056 .053 .054 .054 .050 .057 Exp .000 .000 .000 .000 .000 .001 Weibull Wei .061 .057 .061 .054 .059 .059 A =1, p = 4 LLog .055 .055 .055 .052 .054 .056 LNorm .052 .054 .056 .053 .058 .059 Norm .055 .055 .060 .054 .061 .059 Cox .053 .65$ .059 .049 .052 .649 Exp .568 .665 .243 .250 .135 .132 Log Logistic Wei .158 .165 .097 .087 .077 .073 H=0, 0 = 1 LLog .062 .057 .063 .057 .059 .059 LNorm .059 .060 .060 .055 .059 .054 Norm .020 .021 .057 .053 .061 .053 Cox .035 .054 .056 .049 .042 .049 Exp .833 .883 .408 .390 .224 .229 Log Logistic Wei .151 .156 .075 .071 .061 .058 |i=0, o = 2 LLog .057 .054 .056 .050 .055 .053 LNorm .057 .056 .057 .049 .054 .053 Norm .006 .004 .055 .044 .055 .050 43 Table 2.2 (continued) Underlying Assumed 6% Censoring 20% Censoring 40% Censoring Distribution Model n=40 n=100 n=40 n=100 n=40 n=100 Cox .054 .054 .054 .050 .035 ■ 4531 ■ Exp .127 .163 .047 .052 .033 .032 Log Logistic Wei .158 .155 .122 .116 .102 .094 p.=0, o =.5 LLog .055 .055 .057 .063 .062 .066 LNorm .055 .051 .064 .068 .064 .060 Norm .040 .038 .055 .050 .059 .058 Cox .056 .058 .057 .049 .049 .053 Exp .103 .132 .071 .067 .042 .050 Log Normal Wei .119 .132 .108 .103 .089 .089 |i=0, o = l LLog .062 .067 .066 .063 .060 .060 LNorm .058 .056 .063 .056 .057 .050 Norm .047 .053 .056 .057 .051 .052 Cox .65$ .054 .056 .047 .04$ .044 Exp .526 .593 .288 .278 .160 .168 Log Normal Wei .118 .118 .096 .080 .074 .067 p.=0, o = 2 LLog .063 .066 .065 .056 .063 .058 LNorm .056 .053 .060 .050 .056 .051 Norm .025 .027 .057 .054 .058 .047 Cox .062 .05 i .057 .050 .052 .051 Exp .000 .000 .000 .000 .003 .001 Log Normal Wei .130 .123 .108 .118 .106 .113 |i=0, o =.5 LLog .063 .090 .066 .091 .061 .078 LNorm .057 .047 .063 .062 .060 .055 Norm .056 .050 .056 .055 .060 .055 Cox .053 .052 .054 .053 .055 .050 Exp .489 .596 .101 .119 .028 .031 Log Laplace Wei .186 .208 .152 .160 .099 .107 p.=0, o = l LLog .049 .045 .043 .046 .045 .044 LNorm .057 .053 .074 .071 .063 .055 Norm .021 .020 .050 .048 .059 .056 Cox .054 .053“ .054 .054 .051 .053 Exp .805 .883 .336 .378 .116 .119 Log Laplace Wei .178 .200 .108 .110 .064 .068 li=0, o = 2 LLog .040 .045 .044 .046 .045 .045 LNorm .051 .053 .054 .056 .058 .052 Norm .006 .008 .045 .048 .056 .055 44 Table 2.2 (continued) Underlying Assumed Wo Censoring 20% Censoring 40% Censoring Distribution Model n=40 n=100 n=40 n=100 n=40 n=100 Cox .059 .054 .055 .048 .052 .646 Exp .178 .171 .148 .133 .126 .121 Gamma Wei .046 .036 .049 .039 .053 .040 p=0, k = .5 LLog .051 .077 .052 .069 .049 .059 LNorm .055 .050 .060 .053 .055 .048 Norm .054 .052 .091 .079 .099 .102 Cox .054 .057 .054 .651 .050 .652 Exp .005 .008 .011 .010 .014 .012 Gamma Wei .076 .076 .080 .067 .066 .066 (1=0, K= 2 LLog .059 .061 .061 .051 .056 .052 LNorm .057 .060 .061 .051 .053 .050 Norm .055 .058 .061 .052 .053 .058 Cox .058 .059 .051 .051 .054 .648 Exp .018 .018 .024 .024 .037 .030 Gompertz Wei .047 .044 .048 .042 .052 .041 a = 1, y= 1 LLog .058 .141 .054 .165 .052 .106 LNorm .062 .052 .056 .048 .056 .045 Norm .060 .057 .065 .072 .071 .070 Cox .056 .040 .05 £ .051 .056 .048 Exp .020 .017 .031 .026 .037 .034 Gompertz Wei .048 .040 .051 .041 .053 .046 a = .5, y = .5 LLog .053 .102 .054 .083 .051 .077 LNorm .052 .049 .061 .045 .059 .050 Norm .055 .049 .071 .067 .080 .075 Cox .053 .0 6 l .065 .054 .05$ .054 Exp .000 .000 .000 .000 .001 .001 Smith-Bain Wei .036 .033 .043 .033 .045 .036 Y= 1, a = 2 LLog .055 .266 .051 .167 .050 .093 LNorm .055 .049 .058 .054 .063 .054 Norm .058 .053 .064 .052 .065 .058 Cox .056 .051 .048 .647 .056 .650 Exp .125 .119 .123 .121 .117 .112 Smith-Bain Wei .039 .027 .039 .038 .051 .044 Y = .5, a = .5 LLog .056 .053 .043 .093 .051 .079 LNorm .055 .051 .052 .051 .058 .054 Norm .055 .051 .083 .088 .099 .102 45 The behavior of normal model strongly depends on the censorship percentage and the underlying distributions. When the underlying distribution is Weibull, 7 observed sizes out of 18 independent trials do not fall into the 95% confidence interval. For the non monotone hazard underlying distributions such as log logistic, log normal and log Laplace, the normal model performs poorly in maintaining the desired a level. However, the observed sizes are closer to the a level when censoring is present For the Gamma, Gompertz and Smith-Bain underlying distributions, the normal model maintains the prescribed a level well without censoring but fails to hold it when censoring is present This implies that the normal assumption is not appropriate for survival data. 3.2 Studies on Power We examine the powers of the tests for the assumed models against the alternatives (p = .22, .35, .55 and .70) by computing the fraction of replications for which the test statistic is larger than 3.841. Thus, each test has nearly nominal size a = .05 when the model is correct Table 2.3 displays the empirical powers for P=.35 and .55 of the six likelihood ratio tests of dichotomous-covariate effects under various underlying distributions with N=40 and 0% and 40% censoring, respectively. The Cox partial likelihood ratio test has comparable power which is slightly smaller than that of Weibull model when data are generated from a Weibull distribution. The Cox model possesses competitive power even when the proportional hazards assumption does not hold showing that the Cox procedure is robust For data generated from log logistic, log normal and log Laplace distributions, the empirical powers of the Cox model are a bit less than those of log logistic and log normal models. It should be noted that the observed test sizes 46 of the parametric models are slightly higher than the prescribed a level even when the model is correctly specified. Hence, the powers of the parametric tests are somewhat inflated due to the small sample size. In comparison with the power of the Cox partial likelihood ratio test, the Weibull model is fairly powerful when data follow the Weibull, Gamma, Gompertz and Smith-Bain distributions. It has a bit larger power than Cox model when the underlying distributions are Weibull or Gamma, and smaller power when the underlying distributions are Gompertz or Smith-Bain. As stated before, the observed sizes of the Weibull model are a little higher than the desired a level when the underlying distributions are Weibull and Gamma (k =2). Thus the higher power of the Weibull model is somehow due to the overestimated test size. For the log logistic, log normal, and log Laplace underlying distributions, the power has been inflated since the observed size is much higher than the nominal level a = .05. The log logistic and log normal models have less power than those of the Weibull and Cox models when data are not from log logistic or log normal although they maintain the nominal level relatively well. Their powers are not much higher than the observed powers of the Cox model even when the data follow the log logistic and log normal distributions. The normal model maintains the desired nominal level reasonably well and has competitive power when the hazard rate of the underlying distribution is monotone increasing. However, it performs poorly in the cases of log logistic, log normal and log Laplace underlying distributions. Again, this indicates that the ordinary normal regression analysis is not always appropriate for the survival data. Similar results hold for 40% censoring. For fixed model and data distribution, the powers of the tests with censoring are consistently smaller than those without censoring. Moreover, heavy censoring could affect the tests by resulting in noticeable loss of power. Table 2.3 Empirical Powers of the Likelihood Ratio Test ( a=.05, and N=40 ) of Dichotomous-Covariate Effects for 0% and 40% Censoring Underlying Assumed (3 = 0.00 P = 0.35 P = 0.55 Distribution Model 0% 40% 0% 40% 0% 40% Cox .054 .047 .570 .363 .886 .711 Exp .052 .043 .606 .385 .929 .772 Exponential Wei .057 .054 .609 .382 .912 .763 LLog .050 .049 .466 .324 .802 .684 LNorm .050 .055 .425 .302 .762 .618 Norm .056 .055 .573 .362 .893 .687 Cox .052 .050 .548 .306 .905 .252 Exp .369 .196 .883 .483 .990 .762 Weibull Wei .061 .057 .590 .421 .933 .791 X = 2, p = .5 LLog .055 .048 .458 .368 .810 .739 LNorm .056 .057 .429 .373 .777 .734 Norm .042 .053 .395 .318 .708 .641 tox .056 .050 .558 .341 .808 .636 Exp .000 .000 .000 .000 .000 .000 Weibull Wei .061 .059 .606 .379 .918 .683 X =1, p = 4 LLog .055 .054 .476 .313 .808 .609 LNorm .052 .058 .449 .291 .770 .569 Norm .055 .061 .510 .316 .854 .603 Table 2.3 (continued) Underlying Assumed p = 0.00 p = 0.35 P = 0.55 Distribution Model 0% 40% 0% 40% 0% 40% Cox .054 .055 .615 .526 .917 Exp .127 .033 .591 .478 .885 .704 Log Logistic Wei .158 .102 .660 .538 .901 .842 (i=0, a = .5 LLog .055 .062 .741 .599 .979 .919 LNorm .055 .064 .699 .581 .964 .904 Norm .040 .059 .497 .506 .778 .812 Cox .056 .049 .552 .427 \905r .756 Exp .103 .042 .609 .416 .923 .745 Log Normal Wei .119 .089 .639 .473 .931 .757 li=0, a = 1 LLog .062 .060 .650 .510 .947 .835 LNorm .058 .057 .653 .509 .957 .839 Norm .047 .051 .473 .414 .820 .710 Cox .055 .043 .232 .158 .466 .3o6~ Exp .526 .160 .649 .233 .758 .290 Log Normal Wei .118 .074 .314 .182 .514 .298 (1=0, 0 = 2 LLog .063 .063 .251 .192 .474 .364 LNorm .056 .056 .242 .193 .480 .359 Norm .025 .058 .090 .146 .188 .245 Cox .062 .052 .083 .914 1.000 .996 Exp .000 .003 .747 .740 .997 .995 Log Normal Wei .130 .106 .991 .925 1.000 1.000 (1=0, a = .5 LLog .063 .061 .993 .958 1.000 1.000 LNorm .057 .060 .994 .959 1.000 1.000 Norm .056 .060 .990 .941 1.000 1.000 Cox .053 .055 .352 .218 .643 .522“ Exp .489 .028 .649 .479 .759 .661 Log Laplace Wei .186 .099 .435 .380 .633 .568 |i=0, o = l LLog .049 .045 .445 .363 .783 .656 LNorm .057 .063 .393 .266 .693 .561 Norm .021 .059 .130 .146 .265 .261 Cox .054 .051 .117 .099 .ITS" .152“ Exp .805 .116 .789 .288 .825 .487 Log Laplace Wei .178 .064 .235 .166 .362 .219 (1=0, o = 2 LLog .040 .045 .139 .098 .293 .155 LNorm .051 .058 .142 .139 .265 .188 Norm .006 .056 .015 .107 .030 .180 49 Table 2.3 (continued) Underlying Assumed 0 = 0.00 0 = 0.35 0 = 0.55 Distribution Model 0% 40% 0% 40% 0% 40% Cox .059 .052 .435 .170 .720 .301 Exp .178 .126 .849 .257 .955 .398 Gamma Wei .046 .053 .454 .194 .735 .393 |i=0, K = .5 LLog .051 .049 .261 .166 .509 .340 LNorm .055 .055 .255 .175 .464 .350 Norm .054 .099 .413 .196 .693 ,316 Cox .054 .050 .624 .657 .905 .961 Exp .005 .014 .593 .587 .967 .958 Gamma Wei .076 .066 .873 .723 .998 .972 p = 0 , k = 2 LLog .059 .056 .788 .651 .982 .954 LNorm .057 .053 .774 .633 .972 .940 Norm .055 .053 .834 .667 .994 .945 Cox .058 .054 .536 .353 .902 .004 Exp .018 .037 .392 .276 .836 .611 Gompertz Wei .047 .052 .521 .361 .899 .671 a= 1, y = 1 LLog .058 .052 .444 .327 .774 .608 LNorm .062 .056 .406 .291 .725 .548 Norm .060 .071 .538 .404 .909 .666 Cox .056 .050 .574 .370 .890 .667 Exp .020 .037 .442 .302 .850 .633 Gompertz Wei .048 .053 .550 .373 .900 .691 a = .5, y = -5 LLog .053 .051 .451 .351 .788 .613 LNorm .052 .059 .414 .307 .741 .552 Norm .055 .080 .591 .402 .901 .662 Cox .055 .6 W .505 .309 .907 .644 Exp .000 .001 .002 .005 .040 .007 Smith-Bain Wei .036 .045 .506 .294 .885 .608 Y= 1, oc = 2 LLog .055 .050 .412 .265 .771 .558 LNorm .055 .063 .379 .246 .700 .519 Norm .058 .065 .505 .303 .850 .613 Cox .050 .050 .550 .331 .805 .037 Exp .125 .117 .711 .621 .965 .899 Smith-Bain Wei .039 .051 .478 .354 .879 .666 Y = .5, a = .5 LLog .056 .051 .410 .310 .777 .618 LNorm .055 .058 .389 .283 .725 .576 Norm .055 .099 .547 .433 .901 .737 50 3.3 Studies on Type of Covariates and Test Statistics In Table 2.4, we have compared the powers for (3=.35 and .55 of testing the discrete- covariate (values -1 or +1) and continuous-covariate (uniform [-1,1]) effects for a sample size of 40. When the true value of (3 equals 0, the observed sizes of the tests for the continuous-covariate effect lead to the same conclusion as the tests for die discrete-covariate effect (see sec 3.1). It should be noted that the pattern of the powers of detecting the continuous-covariate effect is similar to that for discrete covariates. However, we lose substantial power in detecting the uniformly distributed covariate effect The loss of power is even more considerable in the presence of censoring. We also calculated the powers of the Wald and score statistics for each case (see Table 2.5). The empirical test sizes and powers of the score and Wald statistics are very close to those of the likelihood ratio test except that the powers of the score test of the Cox model Gog rank test) are slightly inflated. The observed sizes for the score test are larger than the nominal level so that it appears to have more power than the other tests. Actually, for small sample size (N=40), the observed test sizes of the LR, SC and WA statistics of the Cox model fall above the symmetric 95% probability interval (.043, .057) in 47%, 100% and 40% of 15 independent observations, respectively; and there is no observation falling below the interval. Thus, all three tests based on the Cox partial likelihood are prone to overestimate the nominal level for small-sample observations. 51 Table 2.4 Empirical Powers of the LR Test ( P = .35 P = .35 p = .55 P = .55 Underlying Assumed 0% 40% 0% 40% Distribution Model D C D C D C D C Cox .57 .19 .36 .13 .89 .38 .71 .25 Exp .61 .20 .39 .13 .93 .43 .77 .27 Exponential Wei .61 .22 .38 .14 .91 .42 .76 .28 LLog .47 .18 .32 .11 .80 .32 .68 .23 LNorm .43 .17 .30 .10 .76 .30 .62 .21 Norm .57 .18 .36 .15 .89 .37 .69 .29 Cox .66 .19 .46 .14 .91 .38 .75 .35 Exp .88 .55 .48 .32. .99 .78 .76 .47 Weibull Wei .59 .21 .42 .16 .93 .41 .79 .34 X = 2, p = .5 LLog .46 .19 .37 .16 .81 .31 .74 .28 LNorm .43 .16 .37 .15 .78 .28 .73 .28 Norm .40 .14 .32 .13 .71 .27 .64 .25 Cox .57 .1 8 .34 .15 .90 .39 .64 .25 Exp .00 .00 .00 .00 .00 .00 .00 .00 Weibull Wei .61 .21 .38 .16 .92 .44 .68 .26 X=l,p=4 LLog .48 .16 .31 .13 .81 .32 .61 .20 LNorm .45 .15 .29 .13 .77 .29 .57 .19 Norm .51 .17 .32 .15 .85 .34 .60 .22 Cox .22 .09 .22 .10 .43 .14 .41 .14 Exp .66 .53 .25 .18 .75 .53 .37 .20 Log Logistic Wei .32 .18 .23 .12 .53 .23 .40 .16 a = 1 LLog .26 .11 .25 .11 .55 .19 .45 .17 LNorm .26 .11 .26 .11 .52 .18 .45 .16 Norm .08 .07 .19 .09 .16 .11 .33 .12 Cox .0$ .67 .1 6 .66 .17 .68 . 1 6 .09 Exp .86 .77 .23 .25 .83 .79 .25 .29 Log Logistic Wei .20 .17 .11 .07 .29 .18 .17 .10 0 = 2 LLog .11 .07 .11 .06 .19 .09 .17 .09 LNorm .11 .08 .11 .06 .18 .08 .17 .09 Norm .01 .04 .09 .07 .02 .06 .11 .07 52 Table 2.4 (Continued) P = .35 P = .35 P = .55 .55 Underlying Assumed P- 0% 40% 0% 40 % Distribution Model D C D C D C D C Cox .56 .16 .43 .15 .91 .36 .16 .28 Exp .61 .21 .42 .14 .92 .42 .75 .26 Log Normal Wei .64 .25 .47 .21 .93 .46 .76 .33 o = 1 LLog .65 .22 .51 .19 .95 .44 .84 .33 LNorm .65 .21 .51 .18 .96 .46 .84 .33 Norm .47 .18 .41 .16 .82 .36 .71 .27 Cox .23 .08 .16 .08 .41 .15 .30 .13 Exp .65 .50 .23 .20 .76 .55 .29 .23 Log Normal Wei .31 .15 .18 .10 .51 .21 .30 .15 0 = 2 LLog .25 .09 .19 .09 .47 .17 .36 .14 LNorm .24 .08 .19 .09 .48 .16 .36 .14 Norm .09 .07 .15 .09 .19 .11 .25 .10 Cox .35 .14 .22 .12 .54" .24 .52 .20 Exp .65 .47 .48 .16 .76 .50 .66 .33 Log Laplace Wei .44 .25 .38 .19 .63 .31 .57 .25 0 = 1 LLog .45 .15 .36 .14 .78 .29 .67 .19 LNorm .39 .15 .27 .13 .69 .26 .56 .16 Norm .13 .08 .15 .07 .27 .14 .26 .10 Cox .12 .09 .10 .08 .26 .11 .16 .10 Exp .79 .75 .29 .18 .83 .77 .49 .27 Log Laplace Wei .24 .20 .17 .13 .36 .23 .22 .19 o = 2 LLog .14 .07 .10 .10 .29 .10 .16 .14 LNorm .14 .08 .14 .11 .27 .11 .19 .10 Norm .02 .05 .11 .08 .03 .07 .18 .10 Cox .44 .12 .18 .09 .72 .24 .36 .12 Exp .85 .28 .26 .15 .96 .48 .40 .21 Gamma Wei .45 .09 .19 .09 .74 .22 .39 .14 shape = .5 LLog .26 .09 .17 .07 .51 .18 .34 .12 LNorm .26 .09 .18 .09 .46 .17 .35 .12 Norm .41 .13 .20 .10 .69 .25 .32 .12 53 Table 2.4 (Continued) P = .35 II .35 P = .55 Underlying Assumed P = .55 0% © % 0% 40% Distribution Model D C C D C D C Cox .82 .34 .66 .25 1.00 .67 .96 .48 Exp .59 .12 .59 .13 .97 .39 .96 .35 fiamma Wei .87 .38 .72 .29 1.00 .74 .97 .53 shape = 2 LLog .79 .30 .65 .23 .98 .63 .95 .47 LNonn .77 .28 .63 .22 .97 .60 .94 .45 Norm .83 .35 .67 .24 .99 .70 .95 .47 Cox .54 .20 .35 .13 .90 .39 .66 .22 Exp .39 .11 .28 .08 .84 .26 .61 .17 Gompertz Wei .52 .17 .36 .13 .90 .38 .67 .22 a = 1, Y= 1 LLog .44 .17 .33 .10 .77 .30 .61 .21 LNorm .41 .15 .29 .10 .73 .28 .55 .20 Norm .54 .19 .40 .15 .91 .39 .67 .29 Cox .61 .IS .37 .14 .90 .37 .61 .25 Exp .44 .10 .30 .10 .85 .24 .63 .20 Gompertz Wei .55 .17 .37 .15 .90 .37 .69 .26 a = .5, y = .5 LLog .45 .16 .35 .13 .79 .30 .61 .22 LNonn .41 .17 .31 .13 .74 .26 .55 .21 Norm .59 .18 .40 .18 .90 .38 .66 .30 Cox .57 .18 .31 .13 .91 .38 .64 .11 Exp .00 .00 .01 .00 .04 .00 .01 .00 Smith-Bain Wei .51 .15 .29 .11 .89 .33 .61 .23 Y= l , a = 2 LLog .41 .15 .27 .11 .77 .27 .56 .23 LNorm .38 .14 .25 .10 .70 .26 .52 .21 Norm .51 .17 .30 .13 .85 .34 .61 .27 Cox .5 6 .20 .38 .13 .90 .38 .69 .16 Exp .71 .34 .62 .26 .97 .57 .90 .47 Smith-Bain Wei .48 .16 .35 .12 .88 .33 .67 .25 Y = .5, a = .5 LLog .41 .16 .31 .11 .78 .30 .62 .21 LNorm .39 .15 .28 .11 .73 .29 .58 .20 Norm .55 .19 .43 .19 .90 .37 .74 .34 Table 2.5 Empirical Powers of the LR, Score, and Wald Tests (a =.05, N=40) of Single-Covariate Effects for 0% Censoring Underlying Assumed 3 = 0.00 3 = 0.35 3 = 0.55 Distribution Model LR SC WA LR SC WA LR SC WA Cox .03 .06 .03 .57 .59 .57 .89 .89 .88 Exp .05 .06 .05 .61 .62 .61 .93 .93 .93 Exponential Wei .06 .06 .06 .61 .62 .61 .91 .92 .91 LLog .05 .07 .05 .47 .51 .48 .80 .83 .81 LNorm .05 .07 .06 .43 .48 .45 .76 .80 .78 Norm .06 .07 .06 .57 .63 .59 .89 .91 .90 Cox .03 .06 .03 .53 .55 .54 .91 .91 .90 Exp .37 .38 .38 .88 .89 .88 .99 .99 .99 Weibull Wei .06 .07 .06 .59 .61 .60 .93 .94 .93 X = 2, p = .5 LLog .06 .07 .06 .46 .51 .47 .81 .84 .83 LNorm .06 .08 .06 .43 .48 .46 .78 .80 .79 Norm .04 .06 .05 .40 .48 .43 .71 .78 .73 Cox .06 .06 .06 .57 .58 .5tf .90 .90 .90 Exp .00 .00 .00 .00 .00 .00 .00 .00 .00 Weibull Wei .06 .07 .06 .61 .63 .61 .92 .92 .92 X =1, p = 4 LLog .06 .07 .06 .48 .53 .49 .81 .84 .82 LNorm .05 .07 .06 .45 .50 .47 .77 .81 .78 Norm .06 .07 .06 .51 .56 .53 .85 .88 .86 Cox .03 . 0 6 .03 .22 .23 .22 .43 .44 .44 Exp .57 .58 .57 .66 .67 .66 .75 .75 .75 Log Logistic Wei .16 .17 .16 .32 .33 .33 .53 .54 .53 0 = 1 LLog .06 .08 .07 .26 .30 .28 .55 .60 .57 LNorm .06 .08 .06 .26 .28 .27 .52 .57 .54 Norm .02 .03 .03 .08 .11 .09 .16 .21 .18 Cox .06 .06 .05 .00 .09 .09 .17 .17 .17 Exp .83 .84 .83 .86 .86 .86 .83 .84 .83 Log Logistic Wei .15 .16 .15 .20 .21 .20 .29 .29 .29 a = 2 LLog .06 .07 .06 .11 .13 .12 .19 .22 .20 LNorm .06 .08 .06 .11 .14 .12 .18 .22 .19 Norm .01 .01 .01 .01 .02 .01 .02 .03 .02 55 Table 2.5 (Continued) Underlying Assumed 8 = 0.00 B = 0.35 B = 0.55 Distribution Model LR SC WA LR SC WA LR SC WA Cox .06 .06 .06 .56 .5? .56 .71Ck 1 .91 .91 Exp .10 .11 .11 .61 .62 .61 .92 .93 .92 Log Normal Wei .12 .13 .12 .64 .65 .64 .93 .93 .93 0 = 1 LLog .06 .08 .07 .65 .69 .67 .95 .94 .95 LNorm .06 .08 .07 .65 .70 .67 .96 .94 .96 Norm .05 .07 .05 .47 .53 .49 .82 .87 .83 Cox .05 .06 .05 .23 .24 .23 .41 .41 .41 Exp .53 .54 .53 .65 .66 .65 .76 .77 .76 Log Normal Wei .12 .13 .12 .31 .33 .32 .51 .53 .52 ii LLog .06 .08 .07 .25 .29 .27 .47 .51 .48 Q to LNorm .06 .07 .06 .24 .29 .26 .48 .52 .50 Norm .03 .04 .03 .09 .13 .10 .19 .24 .21 Cox .05 .06 .05 .35 .32 .36 .64 .66 .65 Exp .49 .50 .49 .65 .66 .65 .76 .77 .76 Log Laplace Wei .19 .21 .11 .44 .45 .40 .63 .64 .62 o = 1 LLog .05 .07 .05 .45 .51 .47 .78 .82 .80 LNorm .06 .08 .06 .39 .45 .41 .69 .72 .71 Norm .02 .03 .02 .13 .17 .14 .27 .31 .27 Cox .05 .06 .05 .12 .13 .12 .26 .27 .26 Exp .81 .81 .81 .79 .79 .79 .83 .83 .83 Log Laplace Wei .18 .21 .10 .24 .26 .16 .36 .39 .30 o = 2 LLog .04 .05 .04 .14 .17 .15 .29 .34 .31 LNorm .05 .07 .06 .14 .16 .15 .27 .32 .29 Norm .01 .01 .01 .02 .03 .02 .03 .05 .03 Cox .06 .06 .06 .44 .44 .43 .22 .23 .22 Exp .18 .19 .18 .85 .86 .86 .96 .96 .96 Gamma Wei .05 .05 .05 .45 .47 .46 .74 .75 .74 shape = .5 LLog .05 .07 .06 .26 .31 .28 .51 .56 .52 LNorm .06 .07 .06 .26 .30 .27 .46 .51 .48 Norm .05 .08 .06 .41 .49 .44 .69 .78 .72 56 Table 2.5 (Continued) Underlying Assumed B = 0.00 0 = 0.35 0 = 0.55 Distribution Model LR SC WA LR SC WA LR SC WA Cox .05 .06 .05 .82 .83 .82 1.00 1.00" o o Exp .01 .01 .01 .59 .62 .60 .97 .97 .97 Gamma Wei .08 .08 .08 .87 .88 .88 1.00 1.00 1.00 shape =2 LLog .06 .08 .07 .79 .82 .80 .98 .91 .98 LNorm .06 .08 .06 .77 .81 .79 .97 .88 .98 Norm .06 .08 .06 .83 .87 .85 .99 .99 1.00 Cox .06 .06 .06 .54 .55 .54 .40 .41 .40 Exp .02 .02 .02 .39 .41 .39 .84 .85 .84 Gompertz Wei .05 .05 .05 .52 .54 .53 .90 .91 .90 a = 1, y= 1 LLog .06 .07 .06 .44 .49 .45 .77 .81 .79 LNorm .06 .08 .07 .41 .45 .43 .73 .76 .74 Norm .06 .08 .07 .54 .60 .56 .91 .93 .92 Cox .06 .06 .06 .51 .58 .57 .40 .40 .40 Exp .02 .02 .02 .44 .47 .45 .85 .87 .86 Gompertz Wei .05 .05 .05 .55 .57 .56 .90 .91 .90 a = .5, y = .5 LLog .05 .07 .06 .45 .49 .47 .79 .81 .80 LNorm .05 .07 .06 .41 .46 .43 .74 .78 .76 Norm .06 .07 .06 .59 .64 .61 .90 .93 .92 Cox .05 .06 .05 .57 .5 1 .56 .41 .41 .41 Exp .00 .00 .00 .00 .00 .00 .04 .05 .04 Smith-Bain Wei .04 .04 .04 .51 .53 .51 .89 .89 .89 Y= 1, a = 2 LLog .06 .07 .06 .41 .46 .42 .77 .81 .79 LNorm .06 .08 .06 .38 .43 .39 .70 .75 .72 Norm .06 .08 .06 .51 .56 .53 .85 .88 .87 Cox .06 .06 .06 .55 .56 .54 .40 .40 .40 Exp .13 .13 .13 .71 .72 .71 .97 .97 .97 Smith-Bain Wei .04 .04 .04 .48 .49 .48 .88 .88 .88 Y = .5, a = .5 LLog .06 .07 .06 .41 .46 .43 .78 .81 .79 LNorm .06 .07 .06 .39 .43 .40 .73 .75 .74 Norm .06 .08 .06 .55 .60 .56 .90 .93 .91 57 In table 2.5, there are 63%, 84% and 71% of 90 non independent experiments where the observed sizes of the LR, SC and WA tests fall above the interval, respectively. In 16%, 13% and 14% of the 90 observations, the empirical sizes of the LR, SC and WA tests fall below the symmetric 95% probability interval, respectively. In summary, these test statistics are inclined to overestimate the true size when the sample size is small. However, of the three statistics, the score test has a greater propensity to inflate the size and power. 4. Summary and Discussion In this chapter, we have studied the finite sample robustness properties of Cox's semi- parametric approach and five standard parametric approaches with respect to model misspecification. We have also studied the small sample performances of the likelihood ratio, the score and the Wald tests based on these models. In addition, we have examined the effects of an independent censoring mechanism on these inference procedures. The Cox proportional hazards regression model in a hypothesis-testing framework appears to be fairly robust with respect to the proportional hazards assumption. That is, the Cox model will reflect the relative importance of the covariate effect when the underlying distribution is not proportional hazards. However, this does not imply we should not search for good parametric models in various circumstances. For instance, if the hazard rate is strictly increasing (e.g. Gompertz), then the normal model is more competitive than the Cox model. For monotone hazard functions, the power of testing single-covariate effects in the Weibull model is comparable to that from the Cox model. For hump-shaped hazard functions, the log logistic and log normal models have slightly higher empirical 58 power than the other models. Indeed, the appropriate parametric models can always improve the precision and power of inference. In comparing the small-sample performances of the likelihood, score and Wald tests, the likelihood ratio and Wald tests closely agree with each other. The score test has a greater propensity to inflate the size and power than the other two tests. However, the observed sizes of all three tests are apt to exceed the nominal significance level when the sample size is small. We have also investigated the effect of an independent exponential censorship on the power. For fixed model and data distribution, the powers of tests with censoring are consistently smaller than those without censoring. In fact, the loss of power can be substantial with high rate of censoring. Johnson et al. (1982) studied the small-sample performance of the maximum partial likelihood estimator in a two-covariate model. They showed that multiple covariate analyses could result in obscure conclusions. Censoring has a more complicated effect which is confounded by the second covariate. Lagakos and Schoenfeld (1984) and Lagakos (1988) have investigated the effects of another type of model misspecification, missassumption of the functional form of the covariates, for the proportional hazards model on partial likelihood score test They have found that the omissions of a quantitative covariate and a small or moderate interaction between treatment and a covariate will cause a slight loss in power. However, in the cases of the non proportionality of the treatment hazard functions ho and hi ,and in many other situations, they claime that the inefficiency can be considerable. Lin and Wei (1989) have suggested a consistent variance-covariance estimator for the maximum partial likelihood estimator when Cox model is misspecified. Based on that 59 consistent estimator, they have proposed a robust statistical inference and demonstrated that their procedure retains the nominal level better than the conventional inference procedures in the empirical studies. Of course, other problems arise concerning the finite-sample performance of these models. The effects of other censoring mechanisms and of dependencies between some subgroups of the population are of particular interest for future work. CHAPTER HI Multivariate Regression Models Induced by Random Effects 1. Introduction In Chapters I and n, we have studied the regression analysis of survival data when there is a single, possibly, censored event time. However, many clinical and epidemiologic studies are concerned with multiple survival times of individuals who are naturally or artificially grouped. For example, in familial research of disease incidence, the times to the event of interest among related individuals such as parents and offspring, siblings, or husbands and wives may be associated because they have shared common unobservable risk factors, either genetic or environmental. In epidemiology, multiple events within the same subject may also tend to be associated. Often we are interested in the relationship between the survival times and certain covariates, such as treatment or patient characteristics. It is very unrealistic to assume that the survival times of individuals in such subgroups are independent Furthermore, the assumption of independence could result in erroneous conclusions about the effects of covariates on lifetimes. Thus, it is necessary to consider a multivariate lifetime model for this situation. Vaupel et al. (1979) introduced the term 'frailty' to account for unobserved heterogeneity between individuals. There has been much recent work addressing the problem of the 60 61 frailty models since their paper. Clayton (1978) and Oakes (1982, 1986) studied the semiparametric inference for association in a bivariate survival model induced by the gamma frailty. Lee and Klein (1988) also examined the dependence properties of the frailty model. Hougaaid (1984, 1986a,b, 1991) has discussed the impact of using other frailty distributions, such as the inverse Gaussian and positive stable distributions. Oakes (1989) has developed a cross-ratio function to characterize the degree of association for bivariate frailty models. In section 2, we review these frailty models and study their dependence properties. Our interest has focused on the problem of estimating the effects of covariates on survival times after an adjustment for the unobservable frailty. We have shown that the Cox proportional hazards model is fairly powerful in assessing covariate effects regardless of underlying distributions in our previous Monte Carlo study. A natural approach to the multivariate lifetime regression analysis is to incorporate an unobservable random effect into the Cox-type proportional hazards model. Clayton and Cuzick (1985) have generalized the proportional hazards model to a multivariate regression analysis. That is h( x I zy, w i) = ho(x) exp(p zy) Wi, i = 1 G, j = 1 m where ho(.) is an unknown baseline hazard function, zy are the fixed covariates for the j 1*1 individual in the i 1*1 group, and Wi is a random effect in that group. There is a simple interpretation for this model. The individuals of the same group share an unobservable random variable, W, which may be some genetic and/or environmental characteristics. Conditional on that random effect, W, the survival times are independent. We can construct the joint distribution of the lifetimes of the subjects within the same group by assuming that random effect W follows a specified distribution. Hence, the association of 62 the survival times arc induced by the common random variable (which is called a frailty in some literature) shared among the family members. In section 3, we review the papers of Clayton and Cuzick (1985), Self and Prentice (1986), and Klein (1991) where three different estimation procedures for the model in which W follows a gamma distribution are developed. Finally, section 4 contains a discussion about the properties of those estimators and models. 2. Frailty Models For clarity, we consider bivariate frailty models without covariate in this section. Suppose we observe the times to occurrence of two distinct events, Xi and X 2 . We assume that a positive random effect, W, acts multiplicatively on the hazard rate of each event time. Thus, the conditional cumulative hazard function for each event time is given by Hi(x I w) = Hoi(x) w , i = 1,2. Also, conditional on the random variate, the two event times are statistically independent Let Lap(s) denote the Laplace transform of the random effect That is, Lap(s) = Ew[exp(-sW)] . Thus, the unconditional joint survival function of Xi and X 2 can be obtained from S(xi, X2) = Ew[ exp(- W{Hoi(xi)+Ho 2(x 2 )})] = Lap[ Hqi(xi) + Hq 2(x 2 ) ]. 63 Lee and Klein (1988) have shown that the joint probability density function is totally positive of order 2, TP2 (Barlow and Proschan, 1982). That is, * * * * * * f(xi, X2) f(xj, X 2 ) - f(xi, X2) f(xj, X 2) ^ 0 for all xi < Xj, X 2 £ x2 . This implies that Xi and X 2 are positively associated regardless of the frailty distribution. Let h(xi I X 2 = X2 ) and h(xi I X 2 > X2 ) denote the hazard rates of the conditional distribution of Xi, given X 2 = X2 and given X 2 > X2 , respectively. Lee and Klein (1988) also show that h(xi IX 2 = X2) > h(xi IX 2 > X2), for all xi and X 2 . Oakes (1989) introduces a cross-ratio function, 0(x), defined for x = (xi, X 2) by M v x ______f(xil X2)/S(xil X2) ^ x ; " f(x 11 X2 > x2)/S(xil X2 > x2) ’ This function can be interpreted as the ratio of h(xi I X 2 = X2 ) to h(xi I X 2 > X2 ). Accordingly, this quantity is always greater than one for frailty models. Oakes notes that bivariate distributions generated by a frailty are a subclass of the Archimedean distributions (Genest and MacKay, 1986). He shows that for Archimedean distributions 0(x) depends on x only through the joint survival function, S(x). It turns out that the function Lap(.) is uniquely determined by S(.) up to a scale factor, through the inverse function of Lap(.) which can be derived from 0(.). Oakes uses Kendall's coefficient of concordance (x), which is defined by 2Pr[ (Xi-X*)(X2-xJ) > 0 ] - 1 where (Xi, X2) and (Xj, X*2) are independent copies of (Xi, X 2 ), to measure the global dependence for the frailty models. He claims that the function 0(.) can be related to a conditional version of x , defined by {0(x)-l }/{0(x)+l}, which can be used to measure the local dependence. Several frailty models have been studied in the literature. We briefly review the properties of these models in the sequel. 64 Gamma Frailty Model The gamma distribution has been extensively proposed for the random effect in the literature (Clayton, 1978; Oakes, 1982,1986; Aalen, 1987, 1988; Klein, 1991; etc.) for its mathematical convenience. Let W follow a gamma distribution with probability density function ,(«)- " Wa' 1)exp(;w/a) .a a o (3.1) 5 r(l/a)ai/« The joint survival function for Xi and X 2 is thus S(xi, X2) - [ 1 + a (Hoi(xi)+Ho 2 (x2)} ]'1/a, (3.2) Here a is an association parameter. It can be shown that a -* 0 corresponds to the independence of Xi and X 2; a ®o corresponds to the model with maximum positive association, that is, S(xi, X 2) = min {exp[-Hoi(xi)], exp[-Ho2(x2)]} • Kendall's x and the cross-ratio function 0(x) for this model are a/(a+2) and (a+1), respectively. Klein and Costigan et al. (1991) define a conditional version of Kendall's tau by Xx = 2Pr[ (Xi-X*) (X 2 -X2 ) >0 I Xi>xi, X*>xi, X2>x2, -1 , where (Xi, X 2) and (Xj, X 2 ) are independent copies of (Xi, X 2), and x denotes (xi, X 2). They calculate the value of xx, which is a/(a+2) for all x, for the gamma model. The constant values of conditional Kendall's tau and cross ratio imply that the strength of the association between the subjects does not change over time for the gamma fiailty model. 65 Power Variance Family for Frailty Hougaard (1991) considers a general class of distributions, positive stable distribution, for the random effect, W. The density and the Laplace transform of the positive stable distribution, P(y,o,a), are given by g(w; y,a,a) = - exp(-aw+ — a*) (rcw)-1 £ U (. £ w_Y)k sin(yk7c), (3.3) y k=l K- Y where w>0, ye (0,1), and Lap(s) = exp£- ^ {(a + s)* - a Y}j . (3.4) Notice that here a is a scale parameter. We can always choose a = a 1_Y so that E(W) = 1. It has been shown that, for fixed y and a, P(y,o,a) is a natural exponential family with power variance function. Considering the bivariate frailty model, in the case of y =1, W is degenerate, where lifetimes are independent.; For y =0, W follows a gamma distribution so that the joint survival function is of the form of (3.2). When y = 1/2, W has an inverse Gaussian distribution with density given by g(w ; a) = (a/x)1^2 exp(2a) w -3/2 exp(-aw-a/w) . (3.5) The inverse Gaussian distribution has been used by Hougaard (1984) to model heterogeneity in univariate survival models, by Hougaard (1991) to model association in the lifetimes of twins and by Whitmore and Lee (1991) to model batch effects in reliability. 66 Here, we reparameterize (3.5) by setting T)=l/a. Thus, the p.d.f. and the Laplace transform of the inverse Gaussian distribution are given by g(w ; tj) = exp(2/n) w exp^if-W-T]-1 w-1) (3.6) and Lap(s) = exp{ 2ri—1 - 2 (n-2 + t^ s)1/2 } , respectively. Accordingly, the joint survival function for Xi and X 2 is S(xi, X 2 ) = exp{ 2ri-l - 2[ T]-2 +11-1 (Hoi(x1)+H02(x 2))]1/2 } (3.7) Again, x\ is the association parameter. It can be shown that tj->0 corresponds to the independence case. Oakes (1989) has computed the value of Kendall's x and the cross- ratio function for this model. Expressed in our notation they are t = .5 - 2/n + (8/n2) exp(4/r|) j e*y/y dy (3.8) 4/q and 0(x) = 1 + T] / [2-t] In S(x)] We have calculated the conditional value of Kendall's x which is OO Tx = [S(x)]-2 { (1.5-2c/n)exp[4(l-c)/n] + 8c2/n2 e4/n j e*y/y dy } , (3.9) 4c/q where c = 1+T](xi+X2) . Notice that both the values of the cross-ratio function and conditional Kendall's x are decreasing in x. This implies that the strength of association under the inverse Gaussian model lessens over time. Another interesting property of the 67 inverse Gaussian model is the maximum value of x can only be 0.5. Thus, the inverse Gaussian model is not appropriate for survival data with a high degree of association. Hougaard (1986a,b) has investigated the standard stable distribution, P(y,y,0), where CT=y and a=0 are chosen, with density and Laplace transform given by 00 n k v + 11 g(w; y) = (Jtw)-1 £ A (-W-Y)k sin(yk7t) , 0< y z 1 (3.10) k=l and Lap(s) = exp(- sT) • (3.11) Notice that for this distribution, the tails are sufficient thick that the mean is infinite. The joint survival function for Xi and X 2 is S( xi, X2 ) = exp{ -[ Hoi(xi)+Ho2(x2)]7 } (3.12) Clearly, y = 1 gives the independence case. Here, the value of Kendall's x is 1-y. Oakes (1989) found the formula for the cross-ratio function which is defined as 6(x) = 1+ (l-y)/[-y log S(x)] . Klein and Costigan et al. (1991) found the formula for the conditional version of Kendall's x which is OO Xx = (1-Y) {1 - 2W(- log S(x))Y exp[2(- log S(x))] j y-^e-y dy } , 2(-logS(x)) Again, both 0(x) and xx are decreasing functions of time x. Actually, Hougaard (1991) has shown that for the power variance family of distributions, only the gamma model has 68 the property that the association is constant over time. For the remaining members of this class the strength of the association is a decreasing function of time. Hougaard (1986b, 1987,1991) has emphasized that if covariates are included in the model assuming conditional proportional hazards, then any frailty distribution with a finite mean has the theoretical drawback (see Elbers & Ridder, 1982) that the dependence can be identified based on the marginal distribution alone. Thus, only for the positive stable model with infinite mean, it is impossible to identify all parameters from knowledge of the marginals alone. However, if no covariates are observable, the dependence cannot be identified from the marginals for the inverse Gaussian and gamma models. 3. Semiparametric Estimation for Conditional Proportional Hazards Models Our primary concern is to estimate the effects of covariates on survival times after an adjustment for the unobserved frailty. In this section, we apply three semiparametric schemes proposed by Clayton and Cuzick (1983), Self and Prentice (1986) and Klein (1991) for the gamma frailty model. We include covariates in the model assuming conditional proportional hazards; that is h( x I zy, w i) = ho(x) exp(p z,j) W{, i = 1,...,G, j = l,...,ni (3.13) where ho(.) is an unknown baseline hazard function, zjj are the fixed covariates of the jth individual in the i 1*1 group, and Wi's are independent and identical gamma random variables with densities g(w; a) [see (3.1)]. 69 Given a fully parametric specification of the baseline hazard rates ho, one can draw inference about (a,(3) based on the ordinary likelihood theory. The likelihood ratio, score or Wald procedures stated in Chapter I may be used to obtain large sample tests. However, when the baseline hazard functions are unknown, we have to use some 'semi-parametric' estimation procedure. Clayton and Cuzick (1985) have suggested an estimation procedure based on the generalized rank statistics; Self and Prentice (1986) have developed analysis from a counting process point of view; and Klein (1990) has proposed an analysis using the EM algorithm on a profile likelihood construct. Their approaches are outlined in the following sections. 3.1 Method of Generalized Ranks Considering model (3.13), we assume a set of data (t;j,§ij) (i=l,...,G, j=l,...,ni) have been observed. As usual, 5y is an indicator of censoring (8y=l, if ty is a death time; 8ij=0, otherwise). Let R be a set of rank vectors based on (tij,5ij), i=l,...,G. Similar to the argument for the development of linear rank statistics (Prentice, 1978), the likelihood based on the generalized rank vector R is given by L(R; a, |3) “ L ^ n ri exp[-Ho(tij)exp(pZij)wi]{ho(tij)exp(Pzij)wi}5ijg(wi;a)dtijdwi (3.14) i=lj=l 70 where g(w; a) is the density of the gamma random variable. Notice that the integration over R yields the accumulated probability of any possible underlying rank vectors based on (ty» $ij). For convenience, let 6ij = exp(|3 zjj) and consider the monotone transformation on the observed data: yy = Ho(tij). Thus, we can rewrite (3.14) as f f f ^ _ i, II II exp(-yij0ijwi) (0ijWi)6ij g(wi;a) dyy dwi (3.15) i=lj=l Then the score vector based on (3.15) is given by ii° g L = ° E p> |o8 ;(»■;“ ) | R 1 (3.16) 9 a i=l L 9 a J =X E[Zij(5ij-WiyijQij) IR ] (3.17) a p i=l Note that 31oS gf r g > = i { w - >ogw -1 + loga + Since E[ wi IR] = E[ E(wi I y, R)], we can substitute (3.18) into (3.16) and (3.17) to yield a score vector in terms of yij alone. That is 1 m 2i2S k= x f « ------+ “Jj |i5« + log(l+a X yijOij) I R 3 a a2 i=l ^ 1 + a X yijOij J“ J=1 i G ni (3.19) a 2 1=1 j=i 1 + a X 5ij (3.20) ’»j nii r — * ieu IR l + a l yij©ij j=i Because it is impossible to compute E(yij IR), Clayton and Cuzick has suggested a series of approximations. These yield Ski yij = E[yij i R] s I (3.21) ykj^yij £ wi0ij leR(tkj) where wj = E[ w] I y, R ] and R(tkj) is the risk set just prior to time tty To solve the system of equations (3.19) and (3.20), the following procedure is suggested. 1. Set zero as an initial estimate for a , which is the case of independence. Compute 0 from (3.20) as in a Cox proportional hazards model. 2. Iterate between the following approximations: 72 S (3.22) ykj^yy Z W1®U leR(tkj) A “ » 1 + a £ Sij wS------^ ------(3.23) i + £ X yiAj j=l 3. Substituting 0 and yy into (3.19), then one can obtain new estimate of a, say a. 4. Repeat this process until convergence. As we can see, the search for the maximum likelihood estimates is computationally difficult Clayton and Cuzick also obtained die information matrix for the likelihood based on generalized ranks. However, it is even more complex to calculate the observed information matrix. Furthermore, the properties of (&$) are still unknown. There is a need for further work both on the properties of this method and upon possible improvements. 3.2 Partial Likelihood Based on Counting Process Formulation Self and Prentice (1986) reformulate model (3.13) using a counting process setup. Similar to the notation introduced in Chapter I, we consider a multivariate counting process N = { Ny(t): 0 £ t £ «»; i = 1,...,G, j = l,...^ii} with an intensity process 73 A = { Ay(t): 0 £ t £ <» ; i = 1,...,G, j = l,...,ni} defined by Ay(t) dt = Yy(t) ho(t) exp{p Zij(t)}E[ Wi I Ft.] dt (3.24) where Ft- denotes the past just prior to time L Yij(t) denotes the censoring process (Yij(t)=l, if the j* individual of the i 1*1 group is under observation just before time t; Yij(t)=0, otherwise.). The covariate process Zy(t) is the concomitant information on subject (ij) prior to time t, and Wi's are independent and identical random block effects. Applying the innovation theorem (Aalen, 1978) and assuming that the covariate and censoring processes are independent of Wi, we can show that E[WIF,.]= Ew[ W ^ 1«p(-H,WW)] EwtW^expt-Hi.ttW}] where Ni.(t)= 2 Ny(t-), and H|.(t) = 2 f*- Yy(s)exp{P Zy(s)}dHo(s). J=1 J=1 Supposing that W follows a gamma distribution as shown in (3.1), then (3.25) has a simple form given by E[ W I Ft. ] =V(t; Thus, the intensity process (3.24) can be rewritten as Yij(t) ho(t) exp{p Zy(t)}Vij(t; a,P,Ho) (3.27) Based on (3.27), we can formulate a Cox-type partial likelihood as 74 L(a,p,Ho)= n n pNij(t) (3.28) t s o M I Yki(t) exp{p Zki(t)} Vki(t;a,P,Ho) (k,l) The score vector can be obtained by taking the first derivatives of the log likelihood. That is, 8 logL d a ^8 logVji(t) , v m (k,l) n £7______kl(t) “ p f P Zkl(t) 1 7 7 ^ 8 a dNij(t) (3.29) =Jo~(S) 9 a ' I Y ki(t)exp{pZki(t)} Vki(t) V (k,l) 8 logL 8 p f \ I Yki(t)exp{pZki(t)}[Zki(t)Vki(t)+^^] [Zki(t). ? log ij(t)] - « ------dNij(t) = |o“ (5 ) 8 P ZYki(t)exp{pZki(t)}Vki(t) \ On,I) (3.30) However, these system equations depend on the baseline cumulative hazard function Ho. Therefore, for the estimation procedure, we have to consider some nonparametric estimate of Hq. Self and Prentice suggest the Nelson-Aalen estimator of Hq given by TdNij(t) (*d) (3.31) S Yki(t) expfp'ZkiW) Vki(t;a,p,Ho) (k,l) Hence, one can derive the maximum partial likelihood estimates of (a,P) by iteration among equations (3.29) (3.30) and (3.31). 75 There is no conceptual difficulty in extending this method to a more general frailty model, in which Wi's are drawn from some family of distributions (e.g. the stable distribution), or the baseline intensity is stratified. However, the computation of the estimates could be more complicated in these cases. In addition, the variance and the properties of the maximum partial likelihood estimates of (a, (3) are still not clear. 3.3 EM Algorithm on a Profile Likelihood Klein (1991) has followed the suggestion of Gill (1985) in the discussion of Clayton and Cuzick’s paper. He develops an estimation procedure based on the profile likelihood construction (Johansen, 1983). In this approach, the EM algorithm (Dempster et al., 1977) is used to avoid the computational difficulty in the unobserved Wi's. His analysis is summarized as follows. Given the observed data set (ty.Sij) (i=l,...,G, j=l,...,nO and the unobserved frailty Wi's, the log likelihood based on model (3.13) is given by L [ a, P, Hq I w, (tij.Sij)] = Li( p, Hq) + 1«2( a), where G ni Li(p,Ho) = X X 5ijt logho(tij) + P Zjj] - wjHo(tij)exp(P Zij) (3.32) i=lj=l and G G L2 (a) = -G [logr(a-i) + a-Uoga] + X (or1 + Di -1) logwj - or1 I wj (3.33) 76 ni where Di' = X Sy. We use the E-step of the EM algorithm to estimate the unobserved Wi's j=l A 1 + a Xi Sij Wi = E[Wi I (tij,8ij)] = ------j J (3.34) 1 + a Xj Ho(tij)exp({* zy) Notice that this estimate is exacdy the same as that of Clayton and Cuzick [ see (3.18)]. We then apply the M-step of the algorithm; that is, maximize Li(P,Ho) with respect to Ho for fixed p. Hence, we have a nonparametric estimate of Ho given by Ao(o= x — a (3-35) tij^t 2 Wk exp(p Zki) tkl^tij Substitution of fto into jLi(P,Ho) yields a profile likelihood given by 2 I Sij[ PZij - log{ X Wkexp(Pzjd) } ] (3.36) i=1J=1 tki^tij Thus, the estimate of (a,P) can be derived by solving the score equation based on (3.36).and (3.33). The observable full log likelihood is given by T(a,p) = X 2 Sij [p zij + logh^Ktij)] + i=l j=l X Diloga + logr(l/a+Di) - (1/a+DO log[l+ a X I$)(tij)exp(p zy)] (3.37) i=l j=l Thus, standard errors for estimates can be obtained from the observed information matrix based on this likelihood. Detailed variance formulas for the estimates of (a,P) can be 77 found in Klein (1991). Likelihood ratio tests of the hypothesis Ho : oc=0 of no association can also be derived from the comparison of this likelihood and the likelihood under independence. Again, the properties of the estimator (&,$) based on this profile likelihood are difficult to justify . Neilsen et al. (1991) has implemented this EM algorithm to maximum likelihood estimation for the gamma frailty model and conducted Monte Carlo experiments to study the performances of the estimates. Their simulation results do lead positive evidence that the maximum profile likelihood estimates obtain reasonable precision in estimating the association parameter for large sample size. 4. Discussion We have studied three semiparametric estimation procedures for estimation of the frailty parameter and the risk coefficients. The estimation derived from profile likelihood (Klein, 1991) seems to be more natural and less complicated than those of the generalized ranks and counting process methods. Wang (1991) shows that the EM algorithm on the profile likelihood technique yields identical estimates to that of the generalized ranks method (Clayton and Cuzick, 1985). However, the convergence speed of the EM algorithm is relatively slow. It would be nice if an algorithm can be developed to improve the convergence speed. In addition, there is a need for further work on a large sample theory of the estimates. Hougaard (1986b) suggests a two stage estimation procedure for the frailty models without covariates. In the first stage one estimates the integrated hazards of the marginal 78 distribution by the Nelson-Aalen estimator. In the second stage one restricts the marginal distributions to the estimated ones and finds the maximum likelihood estimate for the dependence parameter. An advantage of this procedure is that the same estimation technique can be applied to a variety of frailty distributions. Although this technique is simpler, it can not handle the estimation of the covariate effects and its statistical properties are still unknown. Recently, Clayton (1991) applies the Gibbs sampling method to a Bayesian inference in the gamma frailty model As stated before, we have several candidates for the random effects. Hougaard et al. (1991) indicate that both positive stable and the gamma distributions are on the boundary of the parameter set for the power variance function family. As a matter of fact, they are two extreme models. The thick right tail of the stable distribution reflects strong early dependence. The fact that its cross-ratio function and conditional Kendall's x decreases in time implies the association washes out over time as the more frail pairs die. For the gamma model, the large left tail leads to high association at old ages. Furthermore, the constant values of cross-ratio and conditional Kendall's x implies that the strength of association under the gamma model is not affected by truncation. The inverse Gaussian model can be considered to have properties between the gamma and positive stable models. In next chapter, we will study the parametric and semiparametric estimation for the inverse Gaussian frailty model. CHAPTER IV Parametric and Semiparametric Estimation for the Inverse Gaussian Frailty Model 1. Introduction We have considered, in Chapter m , the incorporation of a random effect into the Cox's proportional hazards model, that is, a frailty model. This random effect can explain the association between multiple event times, such as the times to occurrence of cardiovascular activities or coronary heart disease, within the same study subject Also, the random effect can account for the dependence of survival times where individuals within certain subgroups, such as families, households or litter mates, share a common unobservable trait There are several candidate distributions for the random effect Among them, the gamma model for the random effect has been extensively studied for its mathematical tractability. Hougaard (1986a, b) has considered the positive stable frailty model. We have indicated in Chapter in that the gamma and positive stable frailty models are two extreme cases. On one hand, the large left tail of the gamma distribution reflects high association at old ages; while on the other hand, the thick right tail of the stable distribution leads to strong early dependence. Another important distribution for the random effect is the inverse Gaussian. 79 80 It shares the mathematical conveniences as the gamma distribution. However, the inverse Gaussian model has properties between the gamma and positive stable models. In Klein and Moschberger et al. (1991b), we have applied the inverse Gaussian and gamma frailty models to the Framingham Heart Study. In our numerical results, the inverse Gaussian and gamma models lead to similar conclusions. It is crucial to investigate the impact of the choice of the frailty distributions on the inferences being made. Figures 4.1-4.4 compare the probability density functions of the inverse Gaussian and gamma variates with parameter values which yield the same values of Kendall's x. As T] and a get closer to 0 , corresponding to no association, the shapes of the two densities become similar. In this chapter, we presents the methodology to perform parametric and semiparametric estimation for the inverse Gaussian frailty model. In section 2 the semiparametric estimation using EM algorithm based on a profile likelihood (Klein, 1991) for two special situations are considered. In section 3, we introduce the generalized inverse Gaussian distribution (Jorgensen, 1982) and investigate some properties of the modified Bessel function of the third kind. In section 4, we calculate the information matrix for the estimates derived in section 2. In section 3, we implement the parametric estimation for the inverse Gaussian model, assuming a Weibull regression form. As an illustration, in section 6 , the procedures developed are applied to a large scale longitudinal risk assessment study, the Framingham Heart Study. 81 g(w) 1.2 ■ 0.6 • Figure 4.1 Inverse Gaussian Probability Function with T| = 0.25,0.50 and 0.75 corresponding to Kendall's x = 0.05,0.10 and 0.13 g ( v ) 1 . 2 - 0 . 8- 0 . 6- 0.4- 0 . 2 - Figure 4.2 Gamma Probability Function with a = 0.11,0.21 and 0.29 corresponding to Kendall's x = 0.05,0.10 and 0.13 82 g (v ) 1 . 2- 0 . 8- 0 . 6- 0.4- 0.2 Figure 4.3 Inverse Gaussian Probability Function with Tj = 1 ,2 and 3 corresponding to Kendall’s x = 0.15,0.27 and 0.30 0 . 6- 0 . 2- Figure 4.4 Gamma Probability Function with a = 0.36,0.73 and 0.85 corresponding to Kendall’s x = 0.15,0.27 and 0.30 83 2. Models and Semiparametric Estimation Models We consider the most general model where, conditional on the random effect Wi, the hazard function of a study subject has the form of hk(x I zijk, Wi) = h 0k(x) exp{PkZijk} Wj where Zijk are the covariates for the jth subject of the kth stratum in the ith group. Here hokC) is an unknown baseline hazard function for the kth stratum, and Wi is the random effect in the ith group. We now assume that the Wi's are iid one parameter inverse Gaussian variates with density given by g(w; T)) = (ri7t)- 1/2 exp( 2 Ai)w-3 /2 expf-Ti'lw-iylw"!), ti > 0. (4.2) Note that E(W)=1, and Var(W)=Ti. We have shown in Chapter HI that the unconditional joint survival function for the bivaiiate frailty model is S(xi, x2) = exp{ 2TT 1 - 2[ t t 2 + t t 1 (H0 i(xi)+H 02(x 2 ) ) ] 1/ 2 } , (4.3) where t\ is the association parameter and t| —> 0 corresponds to the independence case. oo The value of Kendall's x can be calculated as .5-2/q+(8/n2) exp(4Al) Notice 4 that it increases as T) increases, and its range is ( 0 ,0.5). 84 We can apply the frailty model in two distinct situations. For model I, we observe the times to occurrence of two events, Xi and X 2 . These events can be considered as the times to the first detection of two chronic diseases within an individual. Let Zk and (3k be the covariates and the risk coefficients for the kth event time, respectively. By this formulation, we allow the covariates either to be specific to a given event or to be common for both Xi and X 2 . Therefore, we have a special case of model (4.1) hk( x I zjk, Wj) = hokOO exp(pkZik) W* , k= 1 ,2 , i = 1,...,G (4.4) where hok(.) is an unknown baseline hazard function for disease k and Wi is the common frailty for the two diseases of the ith individual For model K, we consider the survival times of related individuals such as husbands and wives, parents and offspring, or siblings who share a common unobservable risk factor, either genetic or environmental. The conditional hazard function for the jth individual in the ith subgroup, given the random effect Wi, is defined as h ( x I Zjj, W j) * ho(x) exp(P zy) Wj , i = 1,...,G; j = l,...,nj (4.5) where zy is the vector of observable covariates for the jth individual in the ith group and ho(.) is an unknown baseline hazard function for all subjects. SemipaEametric Estimation via the.EM Algorithm Klein (1991) develops an estimation procedure using an EM algorithm (Dempster, et. al, 1972) on a profile likelihood (Johansen, 1983) for the gamma frailty model. To 85 estimate the parameters T| and (3, we apply this semiparametric technique to the inverse Gaussian frailty model. For model I, conditional on a set of data (tn, 5ii, to, 812 ) (i=l,...,G) and the unobserved frailty wi's, the log likelihood is given by L Dn.Pk.Hok 1 (tik.8ik.Zik.Wi)] = L j[pk4i(& 1 (tik,Sik,Zik.Wi)] +L2 [T| I wjj , where G 2 L j [Pk, Hok] = £ 2 8 jk [In hok(tik) + Pkzik] _wi Hok(tik) expCPfcZyJ , (4.6) i=l k=l and G G L 2 [Til ~ G {-1/2 lnri + 2/q) - 1/q I (wi + w fl) + I (Di -3/2) In Wi . (4.7) j=l i=l Notice that Di is the number of events for the ith individual. Similarly, for model n, given the observed data set (tij.Sij) (i=l,...,G, j=l,...,ni) and the unobserved frailty wi's, the log likelihood is defined as L [T|,P,Ho I (tij, 8 ij,Zij,Wi)] = L j [p,Ho I (tij,5ij,Zij,Wi)] +£»2 [’H lwi] « where L y [p, Ho] = I I Sij [In ho(tij) + pZij] -Wi Ho(tij) exp(pzij) (4.8) i=l j=l It can be shown that, given the observable data, the Wi’s are conditionally independent generalized inverse Gaussian variables as defined by Jorgensen (1982), say N^Aj.Bi.iv1) (see section 3), where 86 2 2 Ai = £ 8ik - 1/2, and Bi = 1T 1 + 2 Hok(tik) exp(PicZik) . for model I, te l tel or Ai= £ Sij -1/2, and Bi = Tp1 + 2 Ho(tij) exp(Pzjj) , for model II. M H Applying the E-step of the EM algorithm to estimate the unobserved wi's, we have that wi = E[Wi I data] = CnBi ) ' 1^2 RAjfV^iT^Bi) , and v r h = E[W fl I data] = (tiBj) 1/2 R ^p^rr^) , where Rx(-) = Kx+lO/KxC) and Kx is the modified Bessel function of the third kind with index X. M-step: Maximizing Z.2 with respect to T|, it follows that f i = § 2 ( w i + w V 4 (4.9) tel Maximizing L ilPfeHok] with respect to Hok for fixed pk and maximizing L ^[P, Ho] with A s p e c t to Ho for fixed P, we have the nonparametric estimates (Nelson-Aalen estimates) of % or Ho given by ftok(t) = 2 ------^ ------,k=U , for model I (4.10) tik^t 2 wm exp(pkzmk) *mk2tik and ft0(t) = 2 ------t —^------, for model H (4.11) tjjSt 2 wkexp(pzki) tkl^tij respectively. Substitutions of Abk and Ao into L j [Pk.Hok] and L ^[P,Ho] yield profile likelihoods given by £3 T G A 3 ( P k) = S Sik [ Pkzik - ln{ 2 wm exp(pkZmk)} ] k=l,2 , for model I (4.12) 1-1 tmk^tik and £ 3 ( P ) = .2 .2 Sij [ pzy - ln{ 2 Wk exp(Pzti) } ] , for model II (4.13) 1=1 i=1 tjd^tij respectively. Thus, the estimate of pk or P can be derived by solving the score equations based on (4.12) and (4.13) for model I and n, respectively. We summarize the estimation procedure for model n as follows: Step 1: Set i)=0, that is, wi=l. Use Newton-Raphson iterative method to compute initial estimates of P and Ho from (4.13) and (4.11), respectively. Step 2: Calculate wi and w ^i based on the most current value of T), P, and Ho. Step 3: Update the estimates of T|, P. Ho using (4.9), (4.13), and (4.1 1), respectively. Step 4 : Iterate between steps 2 and 3 until convergence. This estimation procedure can also be applied for model I. We have written two FORTRAN programs (see Appendixes B and C) implemented in the CRAY Y-MP computer to execute these estimation procedures for models I and n, respectively. 88 It can be shown that the observable full log likelihoods are G 2 £(n>Pk) = X X Sik [pkzik + In hok(tik)] + G [ln2-ln(jc)/2-ln(T])/2 +2/n] i=l k=l + £ {-Aj/2 [In Bi + In ti] + In [Ka W ^ ' ^ ) ] } for model I (4.14) i=l LOl.P) = £ £ Sij [pzij + In ho(tij)] + G [ln 2 -ln( 7t)/2 -ln(Ti ) / 2 +2 /r|] fc=l j=l + X {-Aj/2 [In Bi + In T]] + In [KA.(V4TT 1Bi ) ] } for model n (4.15) i=l where 2 2 Ai = £ Sik * 1/2, and Bi = T)-1 + £ Hok(tik) exp(pkzik) , for model I, k=l k=l or "i ni Ai = £ Sij -1/2, and Bi = tt1 + £ Ho(ty) exp(Pzij) , for model n, j=l j=l and Kx,(.) is the modified Bessel function of the third kind with index X. Thus, standard errors for estimates can be obtained from the observed information matrix (see section 4) based on the unaugmented likelihoods. Likelihood ratio tests of the hypothesis Ho : T1 = 0 of no association can also be derived from the comparison of this likelihood and the likelihood under independence. 89 3. Generalized Inverse Gaussian Distribution We now consider a three-parameter family of distributions, Generalized Inverse Gaussian distribution (Jorgensen, 1982), whose probability density function is denoted by N-(X.,T|,<|)) and given by (tM) ^ 2 w^ ' 1 expC-riw-^w'1), w>0, (4.16) where Kx is the modified Bessel function of the third kind and with index X. Notice that the domain of variation for the parameters in (4.16) is given by X 6 R, (n,<|>) e ©x, where f {(T|, Special cases of (4.16) are the gamma distribution ( (1965) and J 0 rgensen (1982). 90 The modified Bessel function of the third kind and with index X e R is denoted by OO Kx ( 00) = 1/2 J x *--1 expf-lAZcofx+x*1)] dx , g> > 0 . (4.17) The Bessel functions Kx, X e R, satisfy the following recurrence formula Kx(a>)=K.x(o>) (4.18) Kx+i (co) = 2X/(0 Kx (£0) + Kx-i (co) (4.19) f Kx-i (co) + Kx+i (co) = - 2 K^ (co) (4.20) where Ki/ 2 (co) = K.i/ 2 (to) = V ^ c o exp(-co) and K (to) = 9 . A. 3 to It follows that one can use a finite series to compute the Bessel functions of half order. Kn+i/ 2(co) = V*/2©‘ 1/2 c® (1 + .2 ) for X = n+i/ 2 andn = 0,1,... (4.21) We define the function Rx by Rx(co)=Kx+i(co)/Kx(co) (4.22) The following relations are easily derived from (4.18), (4.19) and (4.20). R-X (co) = Rx (co) - 2 X/co (4.23) » • R x (to) = 2X/co2 (4.24) R^(co) = = -1/2[1 + Kx+ 2 (co)/Kx(g>)] + [Rx(co)]2 - A/co Rx(co) (4.25) 91 4. Calculation of the Observed Information Matrix In this section, we calculate the observed information matrix of the estimates (tj, $) for model n. Similar formulas with minor modifications can be used to obtain the observed information matrix of the estimates for model L The second derivatives of the observable full log likelihood (4.15) are 32L(T1, P) ^ = 4GT1-3 + | T1 - 2 + ^ (-2 B-JV 3 + B : V ) d • [ *AiW4Bm-l) - ~ 7= i = - 1 - Qi [ R a X V ^ 1) + d ^ - 7 ] 1 (4.26) * 1 A‘ 4BiTr J where Ai = S S ij-1/2 , Bj = T 1 + I Hoftj) exp(pzij), Qi = - B.1/2Tr2/3 - B;lyV 5/2 j=l j=l and ^ = 3/2 b }%-5/2 + 3 B ? \ m -1/2 Bj3/2T|-9/2. d»l Notice that is a function of p. Thus, the derivatives of lio(tij) are as follows. 92 , v 8ij S WkZklyexp(pzki) _ 3 ho(tii) _ tki^ijj hOyOij) 8 Py [ 2 Wk cxp(pZki) ]2 tkl^tij 2 Wk zkiYZkia exp(pZki) i / , , a 2 ho(tii) z m ______[ I wk exp(pzki) ] 2 2 [ I wk ZkiY exp(pZki)] [ 2 wm zmna exp(pzmn)] tkl^tii______tmn^tij ______[ 2 exp(pzki) ]3 tkl^tij Notice that H(>y(t) = X h()Y(*ij) Ho(xy(t) — X hOaY^ij) • tij^t tij^t The derivatives with respect to P’s follow f t(a’BP)- I, f r l - I 1 (nBi ) - 2 -^ [ RAiCV^T-1) - - f ^ = 1 9 tj 3 PY i=l 3Py| 2 d®i ^4B,-r|-l - Qi OlBO-l/2 [ R^(V4BiTi-l) + - ^ - ] I (4.27) where fir - 2 expCpztj) [HoyCtij) + ZijYHo(tij)] and = -1/2 b ; 1/2t t 3/2 + 1/2 b:3/V 5/2 • opY j=l oBi 93 32L(ll, ft) 2 [ 2.1 g.. r hQav(tii)ho(Iii) - hov(tii)hon(tii) ^ 3 P « 3 ftr - i 1 { j?l ‘J 1 bfo) 1 Ai r --1 32B, -2 a Bi 3 Bi 2 ' 3 ft„ 3 ft, > 3 ft0 3 ft, 3 Bi 3 Bj r 1 where d2Bj 2j — S exp(pZij) [ HoayOij) + ZijYHoa(tij) + 2ijaH()y(tij) + ZijotzijY^O(tij) ] • 3 Pa 3 Py j = 1 5. Parametric Estimation Based on a Weibull Regression Model We now present a parametric approach to the inverse Gaussian frailty model assuming a baseline Weibull hazard function. This underlying Weibull assumption allows us to use the maximum likelihood inference procedure, which is less computationally demanding than the EM-algorithm estimation scheme. Since the asymptotic theory of the maximum likelihood estimates is well established, we can compare the inferences based on the maximum likelihood with the numerical results derived from the maximum protile likelihood. 94 We have studied the properties of the Weibull distribution in Chapters I and II. The Weibull model can be viewed as a parametric proportional hazards model. The Weibull hazard function is flexible enough to account for an increasing, decreasing or constant hazard rate. It has been used, among other applications, to investigate the risk factors for many cancer data sets (Byar, 1982). In addition, the Weibull distribution often provides a good fit for the age of incidence of chronic diseases (Peto and Lee, 1973). We reformulate model II [see (4.5)] into a fully parametric model assuming an underlying Weibull hazard function as follows: h (x I zy, W i) = a x a _ 1 exp(P zy) Wi i=l,...G; j=l,....,ni . (4.29) Here, we let zyi be identically 1 to allow for the estimation of the baseline Weibull scale parameter. The remaining zyt's are prognostic factors. As in the semiparametric formulation, the Wfs are iid inverse Gaussian variates with one parameter 'n, and a is the shape parameter for the baseline Weibull distribution. It can be easily seen that the joint survival for the nj individuals in the ith group is Pr[ Xjj>X|j, j=l,...,ni I Zjj, j=l,...,nj ] = exp( 2TT1 - 2n*1[ 1 + T] Z Xija exp(pzy) ] W } (4.30) j=l Notice that the marginal hazard function for this model is of the form a xa_1 exp(pz) [1+T] xa exp(pz)]*1/2 r\ > 0, a > 0. The marginal hazard rate will be strictly decreasing if a is less than or equal to 1 and be 2 (a-l) . . hump-shaped increasing to a maximum at x = [ ——— ------]1/a if a is greater than one. T[(2-a)exp(pz) 95 This apparent lack of aging for older individuals is due to individuals with a large value of the frailty tending to experience the event of interest early. Throughout the discussion in this work, allowance has been made for possible right censoring. We now consider a more general analysis accounting for left truncated failure time data. Suppose we observe a set of data (tij.Sij.Xij) (i=l,...,G; j=l,...,nj) in which the tij's are times on study and Sy's are censoring indicators as before. Here Ty is the left truncation time, possibly zero, for the jth individual in the ith group. Since we only observe values that are greater than the truncation times, the contribution of a left truncated failure time to the likelihood is the conditional distribution, given on the survival time is greater than the truncation time. Based on model (4.29) the likelihood can be derived from f TT exp[-1 g exp(pzij)wi] [a t y’1 exp(pzij)wi]5ij g(wi; r}) dwi L«x,|5,tD = J [ < > i=l / II exp[- x g exp(pzij)wi] g(wi; t j ) dwi w j= l J Integration over the inverse Gaussian variates Wi, one can obtain a log likelihood as follows: G ni L (a,p,ri) = X I Sjj [Pz,j + lna + (a-1) In tij] + G {ln2-ln(ji)/2} i=l j=l (4.31) 96 ni ni /y * n» of where Ai = ^Sij - 1 /2 , Bi = Tr1 + ]5^t “ exp(Pzij) , Bj s i r 1 + £^x ® exp(Pzij) and K\Q is the modified Bessel function of the third kind with index X. The first and second derivatives of (4.31) are as follows. 9 a i=l I j=l L 9 a - (Bitl)-l/2|^i[RAW4Tl-lBi ) r . (4-32) da 1 9 a where i d . n; dB i: ni - — 1 = i t ? In tij e x p (p Z y )— = £ x ® In xy exp(pzy) and Rx(.) = Kx+i(.)/Kx(.). 9 a j=l y 9 a j=l y 9 L (a,p,ri) G r ni A k .-I 3 Bi 9 B; O m J - w f g l , (4.33) where 3 g- ni ^ ni TT 2W exp(pzij) and — = Z x ® zijY exp(Pzij). d Py j = i y d Py j = i y 97 Qi [ RAi( V ^ B i ) - ] + Q* 1 , (4.34) V4TT1Bi J where Qi = - b J 'V 2' 3 - b : 1/2t t 5/2 and Q* = - B 'J 'V 2' 3 - B*: 1/2i r 5/2 I j Sf - ay a-2 - ( Af ) [ Bi1 b:2 ( U i , 2 ] 3 a 2 i=i j j=l L 9 a 2 1 9 a •BitR^cV^Bi)- 3 — 1 where Ei = (Bm)-1/2| - y -1/2 B:2/3T]-1/2( | ^ i )2 , ^ = I t “ (lntij ) 2 expCpzy) , 9 a 2 1 9 a 9 a 2 j=l 5 32 B‘ 3B* 3 2 b" R* - Os7Tl)'1/2'^—5“- *^2 B*I2/3n_1/2<-r— 9 and - ~ ^ r = I * ? 0”tij) 2 exp(Piij) . 1 1 9 a 2 9 a 9 a z i=l y 98 3 2 l« x ,M ) S J Ak - 1 92 Bj -2 9 Bj 9 Bj 9 a 9 pY " i=l| 2 1 9 a 9 pY ' 1 9 a 9 pY - Hi [ RAW4TTlBi) - - p ^ = J + [ RA.(V4Tl-1 Bi)+ -^ -] + H* (4.36) 9 a 9 pY A' 4‘n-1Bi 1 where Hi = (BiTD-1/2 -1/2 B j^rr , 9 a 9 PY 1 9 a 9 PY 9 2 B- ni J - T 7 = . I t a 0“ tip 2ijyexP(Pzij) . o a o pY j=i y 92 B* 9 B* 9 B* H* = (B*T|)l/2-— T~r~-1/2 B*:2/V 1/2 9 a 9 PY 1 9a 9 Py * 32 b ; md F T T p ^ = # r ? (1" Tii) Zi» “ * N > • 92 L (a,P,T|) 1 9 a 9 t\ 1=1 1 z 9 a •\4ri"1Bi 99 where 8B* Fi = 1/2[ b:2/V 5/2 - B:1/2Tl-3/2 ] and F* = 1/2— - [ B*:2^ - 5'2 - B*[ 1/2n-3/2 ]. d Py d py d2 L (a.ft.Ti) GJ Aj -1 92 Bj p -2 9 Bj 9 Bj a p«p 3 pY ~i=i| 2 1 ap - Pi [ RAi(V 4T|”1B i) - ■ V41T1 Bi’ + (B in ) -1 IS [ R*(V^ Ti' >+^ ! + p ; } (438> where Pi. ...82 Bl — 1 a , 1 1 3 p » a pT 1 3 M Pt 02 B- ni = . 1 1 “ (lntij) zijYZijYCxp(pZij) , a p9 a Py j=i 9 a2 b* a b* a b* P* = (B*rj)’1/2 ------1/2 B*:2^ ' 1'2 a p9 a Py 1 a p9 a Py * 9 2 B- n. 1 i y and — — — = I x “ (InTjj) ZijY^ijyexpCpZij) . p + ( B m ) - l / 2 Q i [ R ^ c V ^ B i ) + ] + u * 1 (4.39) 9 pY A‘ 4ri-lBi j where 9 B* Ui = 1/2 [ Br273!)’572. b : 1/2ti-3/2 ] and U* = 1/2— — [ B*:2/V 5'2 . b*:1/2tt 3/2 1 • 9 Py 9 Py a 2L(0t,p,Ti) = G | (Aj+1) 2 + A± (^ B '.S ’3 + B:2ti-4) 9 n i=l ( 2 2 1 1 3Q* ^ [RA (Vii^T-i) - - p ^ r ] - Q? [R a /V ^ t1) + 7^ - 7] + - r f- ” (4.40) 1 V ^ ir 1 4Bm ^ where — = 3/2 B.1/2Tr5/2 + 3 B '^ r y W - 1/2 b:3/V 9/2 9n 1 9Q* and -----= 3/2 B*.172^ - 5/2 + 3 B* :172^ - 7/2 - 1/2 B*T 37V 9/2 a n 1 1 To search for the maximum likelihood estimates of (a,p,n), we implement a numerical procedure of maximizing profile likelihoods suggested by Klein and Costigan et al. (1991). First, we use Marquart's algorithm to obtain the initial estimates for a and P under the 101 independent Weibull model. Second, we search a range of values for the dependence parameter, t\. Then, for fixed value of T], we maximize the profile likelihood to obtain new estimates of a and p. Finally, we use the update estimates of (oc,p,T|) as initial guesses for solving the score equations (4.32)-(4.34) by Marquart's method. We have written two FORTRAN programs (see Appendixes D and E) to perform this numerical procedures for models I and II, respectively. Our experience has found this search does not require extensive computer time. 6 . Examples From Framingham Heart Study The Framingham Heart Study began in 1948 as a long-term prospective study of chronic heart diseases. This study has contributed substantially to the understanding of the effect of various risk factors on heart diseases. In this study, 2,336 men and 2,873 women were followed with biannual examinations for more than 30 years. The sampling procedures, clinical protocol, response rates, follow-up techniques, and criteria for cardiovascular endpoints have been described in detail (see Dawber, 1980). The examinations recorded several endpoints such as age at death for various causes, times to the detection of coronaiy heart disease (CHD), cerebrovascular accident (CVA) and cancer. In addition, some important risk factors, such as gender, age, systolic blood pressure, serum cholesterol, body mass index, smoking behavior and glucose intolerance were recorded during the 30 year follow-up period. We consider a data set of individuals in the Framingham Heart Study who had no prior history of hypertension or diabetics and had not experienced a CHD or CVA event before 102 their forty-fifth birthday examination. Since examinations were conducted every two years, we included in our sample individuals who had an exam at age 44 or 45 and were disease-free at that exam. A consequence of this approach is that individuals will be in different cohorts, that is; some will be in the study several years prior to inclusion into the data set. To control for this cohort effect, we have included a covariate, waiting time (T- WATT) in the study until the individual reaches 45 years of age, along with the traditional risk factors of interest, namely; sex, cholesterol level (CHOL), body mass index (BMI), smoking (SMK), systolic blood pressure (SBP), and interactions with sex. All covariate values were taken from the biannual exam at which an individual was entered into the sample. Individuals were followed for a maximum of 30 years from entry into the study and were excluded from the sample if their covariate values on inclusion were missing. Table 4.1 presents some summary statistics for the covariates and the event of interests. In our analysis, all continuous covariates were centered at their mean. The response variable is time from entry into the study until the event of interest is experienced. Example I We first apply the inverse Gaussian frailty model in the situation where two endpoints, first evidence of coronary heart disease (CHD) and cerebrovascular accident (CVA), are investigated within the same individual. That is, Model I. Tables 4.2 and 4.3 present the results of semiparametric and parametric analyses under the independence and frailty models, respectively. It should be noted that the bivariate survival times in our analysis are measured from the birth until the CHD or the CVA events occurred or censoring. The individuals recruited in our sample must survive until age 45 disease free. To adjust the truncation effect, we consider the conditional distribution, given on an individual does not experience CHD or CVA until age 45, in the parametric analysis. 103 Table 4.1 Summary Statistics for Covariates and Events of Interest Continuous Covariates COVARIATES Mean Std. Dev. CHOL 230.86 mg/dL 41.394 BMI 24.83 kg/m 2 3.603 SBP 122.03 mm.Hg. 12.398 T-WATT 7.24 years 3.928 Discrete Covariates Freq.(percent) Female (1) Male(0) SEX 920 (58.6%) 651 (41.4%) Ereq.(percent) Yes (1) No (0) SMK 987 (62.8%) 584 (37.2%) Events of Interest CVA CHD EVENT EVENT YES NO Total YES 17 34 51 NO 233 1287 1520 Total 250 1321 1571 Several points should be mentioned here. First, both of the semiparametric and parametric analyses indicate significant association between the CHD and CVA event times within the same study subject However, the likelihood ratio and Wald tests of dependence in the parametric analysis do not agree closely [p-value=.0182 by (one-sided) Wald test p- value=.0006 by likelihood ratio test]. It appears that the variance estimate of the dependence parameter in the parametric analysis is so large that the Wald test becomes more conservative than the likelihood ratio test In the semiparametric analysis, the Wald and likelihood ratio tests are more consistent with each other [p-value=.0008 by (one-sided) Wald test p-value=.0018 by likelihood ratio test]. The reason for this phenomenon is still not very clear. It could be simply due to the increased number of parameters in the parametric setting. Another possible reason is the Weibull baseline hazard function may be not an appropriate assumption. There is a need for an investigation of the asymptotic behavior of the frailty parameter (n). We have shown that the Kendall's t for the inverse Gaussian model is given by 90 f(H) = .5-2M+8M2 exp(4/n) J e‘u/u du. *h\ and its range is (0,0.5). The standard error, SE($) = f *(^)SE(f\), is found by applying the delta method. Here, oo f '(n) = 2tv 2 + 8 iy 3 - (16t|-3 + llry 4) exp(4/n) f e-u/u du. 4Ai However, the behavior of $ is unclear since the properties of ^ are still unknown. A second point should be noted is that the absolute magnitudes of the estimates are smaller under the assumption of independence in both the parametric and semiparametric 105 settings. Among them, the magnitudes of the estimates and their variances are larger in the frailty model assuming a conditional Weibull hazard rate. Notice that these values should be interpreted carefully. We will explain the assessment of risk factors later. Among the potential risk factors, cholesterol, smoking, and systolic blood pressure are consistently significant in predicting CHD. Systolic blood pressure is significant (p-value = .049) for CVA event only in the parametric frailty analysis. This is the case that the independence assumption of multiple event times within the same subject can lead to an erroneous conclusion about the importance of a particular risk factor. Finally, we examine the truncation effect on the strength of association. The conditional value of Kendall's x for the inverse Gaussian frailty model has been shown as oo Tx = [S(x)]-2 {(1.5 - 2c/n) exp[4(l-c)M] + 8 c2/n 2 e4*! j e*y/y dy ) 4c/n where x=(xi, X 2 ) and c = V l+TUxl+x 2 ) • Notice that this function is decreasing in the truncation time x. Thus, the strength of dependence between the CHD and CVA event times will be washed out due to the effects of delayed entry. 106 Table 4.2 Semiparametric Estimates of Parameters Based on Model I For Time to First Evidence of Coronary Heart Disease Cox Model under Inverse Gaussian Independence Frailty Model Effect P S fi' P-value P ------§£ P-value SEX •6.367 .253 .148 -0.415 ‘ "M .149 Ch OL 0.566 .196 .004 0.667 .237 .005 BMI 0.409 .265 .124 0.480 .317 .130 SMK 0.501 .2 2 1 .024 0.609 .256 .018 SBP 1.559 .520 .003 1.904 .603 .0 0 2 T-WATT 0 .0 0 1 .019 .955 -0 .0 0 1 .0 2 1 .968 Sex-Chol -0.367 .309 .236 -0.460 .356 .197 Sex-Bmi -0.143 .363 .694 -0 . 2 1 0 .424 .621 Sex-Smk -0.336 .301 .264 -0.397 .340 .243 Cerebrovascular Accident Cox Model under Inverse Gaussian Independence ______Frailty Model ------gg- Effect P P-value P ' ■§£" P-value "SEX 0.354 .6^4 .605" 0.37o .702 ■ m CHOL 0.468 .464 .313 0.531 .484 .273 BMI 0.939 .622 .131 1.013 .661 .125 1.048 .618 .090 1.133 .633 .074 SBP 2.082 1.128 .065 2.261 1.163 .052 T-Waj T 0.024 .041 .561 0.025 .042 .566 Sex-Chol -0.558 .678 .411 -0.611 .698 .382 Sex-Bmi -0.605 .793 .445 -0.627 .843 .456 Sex-Smk -0.619 .748 .408 -0.676 .768 .379 n 37223 1.0l7 .OOoS X 0.2750 .0343 Log Likelihood -2332.88 -2328.03 107 Table 4.3 Parametric Estimates of Parameters Based on Model I For Time to First Evidence of Coronary Heart Disease Weibull Model under Inverse Gaussian Independence ______Frailty Model Effect P .....SE P-value 3 SE P-value Shape (a) 7.111 6.605 8.316 6.752 Intercept -32.042 2.563 -36.809 3.141 $EX -0.370 0.254 .145 -0.422 0.295 .152 CHOL 0.572 0.196 .004 0.705 0.250 .005 BMI 0.418 0.266 .117 0.515 0.330 .119 Sm k 0.500 0 .2 2 1 .024 0.633 0.267 .018 SBP 1.560 0.520 .003 1.977 0.631 .0 0 2 T-watt 0.013 0.018 .465 0 .0 1 0 0 .0 2 1 .630 Sex-Chol -0.368 0.310 .235 -0.492 0.368 .182 Sex-Bmi -0.155 0.364 .671 -0.242 0.439 .581 Sex-Smk -0.335 0.301 .265 -0.413 0.350 .238 Cerebrovascular Accident Weibull Model under Inverse Gaussian Independence Frailty Model Effect SE P-value 3 SE P-value 3 Shape(a) 6.496” 1.321 6.945 1.343 Intercept -31.362 5.629 -33.250 5.709 ...... SEX 0.338 0.683 .621 0.359 0.706 .612 CHOL 0.463 0.463 .317 0.555 0.490 .258 BMI 0.933 0.621 .133 1.028 0.670 .125 SMK 1.040 0.617 .092 1.157 0.639 .070 SBP 2.080 1.131 .066 2.317 1.182 .049 T-WATT 0.026 0.040 .510 0.026 0.041 .529 Sex-Chol -0.547 0.677 .419 -0.625 0.705 .375 Sex-Bmi -0.606 0.792 .444 -0.640 0.853 .453 Sex-Smk -0.601 0.748 .422 -0.681 0.775 .379 3.601 i.76S .6l82 X 0 .2 8 0 6 0.0511 Log Likelihood -1716.16 -1710.25 108 Example II The second example focuses on studying a single endpoint, the first evidence of coronary heart disease, where there are dependencies between study subjects, siblings. Table 4.4 summarizes the information on sibling group size and the number of CHD events experienced in groups for our study sample. Again, the time scale used for our analysis is from birth until the CHD event time occurred or censored. The survival times of our study subjects are truncated at age 45. Tables 4.5 and 4.6 present the semiparametric and the parametric estimates for the independence model and frailty model adjusted for sibling effects. Here, the likelihood ratio (p-value=.0234) and one-sided Wald tests (p-value=.0935) of association do not agree with each other in the parametric analysis. They even draw totally different conclusions about the association parameter. In the semiparametric analysis, the p-values of the one-sided Wald statistic (.0172) and the likelihood ratio tests (.0600) are not consistent, either. There are several possible reasons for this conflict. First, different models and different asymptotic tests can result in various conclusions since these p-values are on the border line. Second, the asymptotic tests may not be valid due to the heavy censoring in our example. Third, we are testing a parameter on the boundary of the parameter space. The legitimacy of the likelihood ratio test should be considered carefully. Finally, this implies the inverse Gaussian model can not explain the sibling data well. However, further study on the properties of the association estimator is necessary. 109 T a b le 4.4 Summary Information on Sibling Group Size and number of CHD Events in Groups for Each Size Group for 45 year old Disease-free Individuals Size of Number Number Number Number Sibling with no with 1 with 2 with 3 Total group Events Event Events Events 1 1 0 6 6 2 0 1 0 0 llb l 2 82 19 1 0 1 1 1 3 5 5 3 1 14 4 2 0 1 0 3 Total 1159 235 6 1 T a b le 4.5 Semiparametric Estimates of Parameters For Time to First Evidence of Coronary Heart Disease Adjusted For Sibling Effects Cox Model under Inverse Gaussian Independence ______Frailty Model Effect sfi P-value .... 5E P-value P P SE* -0.367 .253 .141 -0.421 .2 1 6 .1 2 2 cho L 0.566 .196 .004 0.615 .226 .006 BMI 0.409 .265 .124 0.447 .304 .142 SM* 0.501 .2 2 1 .024 0.571 .246 .0 2 0 SBP 1.559 .520 .003 1.771 .579 .0 0 2 T-WAff 0 .0 0 1 .019 .955 -0.000 . 0 2 0 .988 Sex-Chol -0.367 .309 .236 -0.403 .340 .237 Sex-Bmi -0.143 .363 .694 -0.191 .406 .637 Sex-Smk -0.336 .301 .264 -0.382 .327 .243 2.038 .9£f .6172 ' V X 0.2247 .65 lS Log -1934.00 Likelihood -1935.77 110 Table 4.6 Parametric Estimates of Parameters For Time to First Evidence of Coronary Heart Disease Adjusted For Sibling Effects Weibull Model under Inverse Gaussian Independence Frailty Model Effect P ' - SE-...... P-value P Se P-value Shape(a) 7.211 0.665 8.243 0.362 Intercept -32.042 2.563 -36.250 3.574 SEX -0.370 0.254 .145 -0.457 0.292 .117 CHOL 0.572 0.196 .004 0.645 0.243 .008 BMI 0.418 0.266 .117 0.490 0.327 .134 SMK 0.500 0.221 .024 0.603 0.263 .022 SBP 1.560 0.520 .003 1.867 0.625 .003 T-WATT 0.013 0.018 .465 0.010 0.021 .626 Sex-Chol -0.368 0.310 .235 -0.422 0.357 .237 Sex-Bmi -0.155 0.364 .671 -0.236 0.430 .583 Sex-Smk -0.335 0.301 .265 -0.403 0.343 .239 3.076 2.331 .0935 i\ X 0.2699 6.6827 Log -1352.34 Likelihood -1354.91 A traditional method of assessing the effects of risk factors is to examine the relative risk of experiencing the event of interest for an individual with covariate vector zi as compared to an individual with covariate vector Z 2 . This quantity is estimated by the ratio of the respective hazard rates. Both the Cox and Weibull models, under the independence assumption, have proportional hazards. That is, the ratio of the hazard rates, given by exp{p(zi-Z 2 )}, is free of the survival time. In a frailty model we defme the relative risk, R(t), by the ratio of the unconditional marginal hazard rates of individuals with covariates zi and Z2 , respectively. For the inverse Gaussian frailty model the relative risk is Ill R(,).^(P(z,.z2)l[j^ lM « E (M ]1z2 , 1+T1 Ho(t) ex p (p zi) where fto(t) is the Nelson-Aalen estimator for the semiparametric analysis and fto(t) = t & for the parametric (Weibull baseline hazard) analysis. Thus, R(t) depends on the time on study, t, and all components of the covariate vector. Figures 4.S and 4.6 illustrate the relative risk of coronary heart disease, R(t), for a smoker as compared to a nonsmoker for males and females based on the three parametric models (Weibull model under independence, Models I and II). In this figure cholesterol, body mass index, blood pressure and waiting time covariates were set at their mean value. Notice that the relative risk is decreasing in age for the frailty models. For both genders, # the relative risk of smoking is higher in the frailty model I (considering association between CHD and CVA event times) than in model II (adjusted for sibling effect). This is for the reason that the degree of association, rj = 3.691 and x = .2896, in the model I is slightly stronger than that (rj = 3.076 and x - .2699) in model n. For males between 45 and 65 years old, the relative risk of smoking's effect on time to CHD is higher in the frailty models than in the independence case (1.65). For most female in our sample, the relative risk of smoking's effect is higher in the frailty models than in the independence case (1.18). This implies that had one ignored any possible association and assume an independent model that the relative risk of smoking's effect on time to CHD would be underestimated. The relative risk results of the semiparametric analysis is similar to those of the parametric analysis since there is no significant difference in the estimates of the risk coefficients. 112 Relative Risk 1.85 ■ 1 . 8-' 1.75" 1.7" 1.65 »- age 50. 60. 65.55. 1.55- Figure 4.5 Relative Risk of Smoking for Males solid line for Model I, dashed line for Model II, dotted line for Weibull model under independence Relative Risk 1.24" 1.23- 1.22 1 . 21 -- age 0 5.55. 60.50. 1.19- Figure 4.6 Relative Risk of Smoking for Females solid line for Model I, dashed line for Model n, dotted line for Weibull model under independence CHAPTER V Discussion of Future Work We have demonstrated that the Cox proportional hazards regression analysis is fairly robust with respect to model misspecification in univariate problems. The power of Cox model is competitive for assessing treatment differences or risk factors even when the proportional hazards assumption does not hold. Due to the success of Cox's semiparametric estimation procedure, we have incorporated an inverse Gaussian random effect in to the proportional hazards regression analysis. Through this random effect, the dependence of multiple event times within the same subject or survival times of related individuals may be explained. In addition, the assessment of risk factors is obtained when the association of survival times is considered. However, several problems have arisen in our research. One of the most difficult problems facing the applied statistician is the choice of an inference procedure from the many that might be applicable to the problem at hand. In Klein and Moschberger et al. (1991), we have applied the inverse Gaussian and gamma frailty models to the Framingham Heart Study. In our numerical results, the inverse Gaussian and gamma models lead to similar conclusions. It is crucial for further comparison between these two models to investigate the impact of the choice of the frailty 113 114 distributions on the inferences being made. It is also important to develop some diagnostic checks or tests of goodness-of-fit for the validity of the assumed frailty distribution. A second problem is that there are limitations to the inverse Gaussian frailty model as well as to the frailty approach. We have noted that the range of Kendall's x is (0, .5) for the inverse Gaussian model. That is, this model can not account for negative or strong positive association between failure times. One should be cautious about the biological implication when applying these models. A third question is that we have shown in our examples that the test of association parameter in the semiparametric analysis does not closely agree with those in the parametric analysis. There is a need for an investigation of the asymptotic behavior of the frailty parameter. Finally, another important class of multivariate failure time models arises because of competing causes of death or recurrences of certain event on each study subject. One may argue that the occurrence of a failure has a direct effect on the hazard rate for the occurrence of subsequent failures. In this case, the frailty model does not provide good biological interpretation. Kalbfleisch and Prentice (1980) have provided a framework based on the hazard function formulation for the analysis of competing risk problems. A second approach is to consider some stochastic process models. These are especially helpful when failure times arise in a natural order. A traditional example is the Marshall-Olkin model (1967) which may be interpreted in the Markov process framework of event history analysis. Klein et al. (1989) have generalized the Marshall-Olkin model to semiparametric regression analysis and applied it to the Danish data on occurrence of metastases from breast cancer. APPENDIX A FORTRAN PROGRAM FOR THE MONTE CARLO STUDY ★ ★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★■A****** *m a i n p r o g r a m ***************************** ★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★Hr********* IMPLICIT REAL*8 (A-H,L,0-Z) CHARACTER* 8 0 DIST PARAMETER (N= 10 0,TRUB=0.3 5D 0,NSMP LE=10 0 0,ELAMB=0.42, $ DIST='WEIBULL WITH LAMBDA=2.0, ALPHA=0.5, BETA=0.35 $ AND 20% CENSORING') COMMON T(100),IC(100),X(100) DIMENSION ST(100),CT(100) CSB-0.D0 CSB2-0.D0 CSB4-0.D0 NCOX-O NCLR1=0 NCLR5=0 NCLR10-0 NCSC1-0 NCSC5=0 NCSC10=0 NCWA1-0 NCWA5-0 NCWA10-0 ESB=0.DO ESB2=0.D0 ESB4=0.D0 NEXP=0 NELR1=0 NELR5=0 NELR10=0 NESC1=0 NESC5=0 NESC10=0 NEWA1-0 NEWA5-0 NEWA10-0 WSB-0.D0 WSB2-0.D0 WSB4-0.D0 115 116 NWEI=0 NWLR1=0 NWLR5=0 NWLR10-0 NWSC1-0 NWSC5-0 NWSC10-0 NWWA1-0 NWWA5-0 NWWA10*0 GSB=O.DO GSB2=O.DO GSB4=0.D0 NLLG=0 NGLR1=0 NGLR5=0 NGLR10=0 NGSC1=0 NGSC5=0 NGSC10=0 NGWA1=0 NGWA5=0 NGWA10=0 LSB=0.DO LSB2-0.DO LSB4-0.D0 NLNM=0 NLLR1«=0 NLLR5-0 NLLR10-0 NLSC1=0 NLSC5-0 NLSC10-0 NLWA1-0 NLWA5-0 NLWA10-0 ZSB-0.DO ZSB2-0.D0 ZSB4=0.DO NZ=0 NZLR1=0 NZLR5=0 NZLR10*=0 NZSC1-0 NZSC5-0 NZSC10-0 NZWA1-0 NZWA5*=0 NZWA10-0 NK-0 NKK-0 r it it it i c i tit it it it it h i t 1t h 1t t i t i h t i h ic h t i t i t i t i t i 1t ic ic h t i t i t i ic t i t i t i t i t i t i t f * 1r t i t i t i * Q 0 X(I)=2*RANF()-1. 10 PROMRGESO NLSS O SXMDL * MODELS SIX FOR ANALYSIS REGRESSION PERFORM C CONTINUE 2 ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ a** 1 CONTINUE 11 o o NL5NL1,NSC NS5NS 10,NLWA1,NLWA5,NLWA10) ,NLSC5,NLSC C1 NLS NLLR5,NLLR10, $ N R NL1,CSC1,NCSC5,NCSC10,NCWA1,NCWA5,NCWA10) 1 C S ,NCLR10,NC LR5 NC $ DO 10 1=1, 1=1, 10N DO ALRANSET(457329) T E S N A R CALL O J=1,NSMPLE 1 DO O 1 1=1,N 11 DO CALL RANSET(ISEED1) CALL O 1=1,N 2 DO RANSET(ISEED2) CALL ISEED2-38748344 ISEED1-73824983 CALL NORM (N,NORM 1,CALL 0 ., 0 . , 1., ALPHA, BET5, SIGMA, LR, SC, WA, NWORK) CALL COX(N,BETA,BETA1,NWORK,LR,SC,WA) CALL ELSE CALL WEXP(N,BETA,BETA2,LR,SC,WA,NWORK) CALL ENDIF F NOK E. ) THEN 1) .EQ. (NWORK IF K=0 CALL RANGET(ISEED1) CALL I)(AO(AF)/2.*EXP(TRUB*X(I)))**2 2 )=(-ALOG(RANF())/ (I T S ENDIF ELSE F NOK E. ) THEN 1) .EQ. (NWORK IF F NOK E. ) THEN 1) .EQ. (NWORK IF IUAEDT * DATA SIMULATE TI = -ALOG(RANF())/(ELAMB*EXP(TRUB*X(I))) = CT(I) ELSE ALRANGET(ISEED2) K=K+IC(I) T E G N A R CALL END IF F S I .E C I ) THEN ) CT (I) .LE. ( ST (I) IF RN *' OCNEGNEFRCXMDL J= ' = ,J ,J MODEL COX FOR CONVERGENCE NO *,' PRINT RN *' OCNEGNE O ONRA OE, J=',J MODEL, LOGNORMAL FOR CONVERGENCE NO *,' PRINT NCOX=NCOX+NWORK CALL COUNT(TRUB, BETA1,CSB, CSB2,CSB4,LR,SC,WA,NCLR1, BETA1,CSB, COUNT(TRUB, CALL NLNM-NLNM+NWORK CALL COUNT(TRUB,BET5,LSB,LSB2,LSB4,LR,SC,WA,NLLR1, CALL (BET5/SIGMA) BETA— 1./SIGMA THETA- LAMBDA=DEXP(-ALPHA/SIGMA) T (I) = CT CT T = (I)(I) ST T = (I)(I) CI = 1 = IC(I) CI = 0 = IC(I) 117 118 NEXP=NEXP+NWORK CALL COUNT(TRUB,BETA2,ESB,ESB2,ESB4,LR,SC,WA,NELR1, $ NELR5,NELR10,NESC1,NESC5,NESC10,NEWA1,NEWA5,NEWA10) ELSE PRINT ' NO CONVERGENCE FOR EXPONETIAL MODEL, J=',J ENDIF CALL WEI(N,THETA,BETA,BETA3,WALPHA,LR,SC,WA,NWORK) IF (NWORK .EQ. 1) THEN NWEI=NWEI+NWORK CALL COUNT(TRUB,BETA3,WSB,WSB2,WSB4,LR,SC,WA,NWLR1, $ NWLR5,NWLR10,NWSC1,NWSC5,NWSC10,NWWA1,NWWA5, NWWA10) ELSE PRINT ' NO CONVERGENCE FOR WEIBULL MODEL, J=',J ENDIF CALL NORM(N,2,0.,0.,1.,ALP,BETA6,SIGMAN,LR,SC,WA,NWORK) IF (NWORK .EQ. 1) THEN NZ-NZ+NWORK CALL COUNT(TRUB,BETA6,ZSB,ZSB2,ZSB4,LR,SC,WA,NZLR1, $ NZLR5,NZLR10,NZSC1,NZSC5,NZSC10,NZWA1,NZWA5, NZWA10) ELSE PRINT *,' NO CONVERGENCE FOR NORMAL MODEL, J=',J ENDIF CALL LOGIST(N,ALP,BET5,SIGMA,BET4,A,SCL,LR,SC,WA,NWORK) IF (NWORK .EQ. 1) THEN NLLG-NLLG+NWORK CALL COUNT(TRUB,BET4,GSB,GSB2,GSB4,LR,SC,WA,NGLR1, $ NGLR5,NGLR10,NGSC1,NGSC5,NGSC10,NGWA1,NGWA5, NGWA10) ELSE PRINT *,'NO CONVERGENCE FOR LLOGISTIC MODEL, J=',J ENDIF NK-NK+K NKK-NKK+K* * 2 1 CONTINUE C COMPUTE CENSORING PERCENTAGE, BIASE, MSE * AVECEN-FLOAT(NK)/FLOAT(NSMPLE) DEVCEN-SQRT((FLOAT(NKK)-NSMPLE*AVECEN**2)/FLOAT(NSMPLE)) BIAS1-CSB/DBLE(NCOX) BIAS2=ESB/DBLE(NEXP) BIAS3=WSB/DBLE(NWEI) BIAS1=CSB/DBLE(NCOX) BIAS2-ESB/DBLE(NEXP) BIAS3=WSB/DBLE(NWEI) BIAS4=GSB/DBLE(NLLG) BIAS5«=LSB/DBLE (NLNM) BIAS6*=ZSB/DBLE(NZ) RMSE1-CSB2/DBLE(NCOX) RMSE2-ESB2/DBLE(NEXP) RMSE3-WSB2/DBLE(NWEI) RMSE4-GSB2/DBLE(NLLG) RMSE5~LSB2/DBLE(NLNM) RMSE6=ZSB2/DBLE(NZ) SEB1=DSQRT((CSB2-DBLE(NCOX)*BIAS1**2)/DBLE(NCOX*(NCOX-1 SEB2-DSQRT((ESB2-DBLE(NEXP)*BIAS2**2)/DBLE(NEXP*(NEXP-1 SEB3=DSQRT((WSB2-DBLE(NWEI)*BIAS3**2)/DBLE(NWEI*(NWEI-1 SEB4-DSQRT((GSB2-DBLE(NLLG)*BIAS4**2)/DBLE(NLLG*(NLLG-1 SEB5=DSQRT((LSB2-DBLE(NLNM)*BIAS5**2)/DBLE(NLNM*(NLNM-1 SEB6“DSQRT((ZSB2-DBLE(NZ)*BIAS6**2)/DBLE(NZ*(NZ-1))) SEM1=DSQRT((CSB4-DBLE(NCOX)*RMSE1**2)/DBLE(NCOX*(NCOX-1 SEM2-DSQRT((ESB4-DBLE(NEXP)*RMSE2**2)/DBLE(NEXP*(NEXP-1 SEM3-DSQRT((WSB4-DBLE(NWEI)*RMSE3**2)/DBLE(NWEI*(NWEI-1 SEM4-DSQRT((GSB4-DBLE(NLLG)*RMSE4**2)/DBLE(NLLG*(NLLG-1 SEM5=DSQRT((LSB4-DBLE(NLNM)*RMSE5**2)/DBLE(NLNM*(NLNM-1 SEM6=DSQRT((ZSB4-DBLE(NZ)*RMSE6**2)/DBLE(NZ*(NZ-1))) CALL POWER(NCOX,NCLR1,PCLR1,SEPCL1) CALL POWER(NCOX,NCLR5, PCLR5,SEPCL5) CALL POWER(NCOX,NCLR10,PCLRO,SEPCLO) CALL POWER(NCOX,NCSC1,PCSC1,SEPCS1) CALL POWER(NCOX,NCSC5,PCSC5,SEPCS5) CALL POWER(NCOX,NCSC10,PCSCO,SEPCSO) CALL POWER(NCOX,NCWA1, PCWA1,SEPCW1) CALL POWER(NCOX,NCWA5, PCWA5,SEPCW5) CALL POWER (NCOX, NCWA10 , PCWAO, SEPCWO) CALL POWER(NEXP,NELR1,PELR1,SEPEL1) CALL POWER(NEXP,NELR5,PELR5,SEPEL5) CALL POWER(NEXP,NELR10,PELRO,SEPELO) CALL POWER(NEXP,NESC1,PESC1,SEPES1) CALL POWER(NEXP,NESC5,PESC5,SEPES5) CALL POWER(NEXP,NESC10,PESCO,SEPESO) CALL POWER(NEXP,NEWA1, PEWA1,SEPEW1) CALL POWER(NEXP,NEWA5,PEWA5,SEPEW5) CALL POWER(NEXP,NEWA10,PEWAO, SEPEWO) CALL POWER(NWEI,NWLR1,PWLR1,SEPWL1) CALL POWER(NWEI,NWLR5, PWLR5,SEPWL5) CALL POWER(NWEI,NWLR10,PWLRO,SEPWLO) CALL POWER(NWEI,NWSC1,PWSC1,SEPWS1) CALL POWER(NWEI,NWSC5 , PWSC5,SEPWS5) CALL POWER(NWEI,NWSC10,PWSCO,SEPWSO) CALL POWER(NWEI,NWWA1,PWWA1,SEPWW1) CALL POWER(NWEI,NWWA5, PWWA5,SEPWW5) CALL POWER(NWEI,NWWA10,PWWAO, SEPWWO) CALL POWER(NLLG,NGLR1, PGLR1,SEPGL1) CALL POWER(NLLG,NGLR5,PGLR5,SEPGL5) CALL POWER(NLLG,NGLR10,PGLRO, SEPGLO) CALL POWER(NLLG,NGSC1,PGSC1,SEPGS1) CALL POWER(NLLG,NGSC5,PGSC5,SEPGS5) CALL POWER(NLLG,NGSC10,PGSCO, SEPGSO) CALL POWER(NLLG,NGWA1, PGWA1,SEPGW1) CALL POWER(NLLG,NGWA5,PGWA5,SEPGW5) CALL POWER(NLLG,NGWA10, PGWAO, SEPGWO) CALL POWER(NLNM,NLLR1, PLLR1,SEPLL1) CALL POWER(NLNM,NLLR5,PLLR5,SEPLL5) CALL POWER (NLNM, NLLR10, PLLRO, SEPLLO) 120 CALL POWER(NLNM,NLSC1,PLSC1,SEPLS1) CALL POWER (NLNM,NLSC5,PLSC5,SEPLS5) CALL POWER(NLNM,NLSC10,PLSC0,SEPLSO) CALL POWER(NLNM,NLWA1,PLWA1,SEPLW1) CALL POWER(NLNM, NLWA5,PLWA5,SEPLW5) CALL POWER (NLNM,NLWA10,PLWAO,SEPLWO) CALL POWER(NZ,NZLR1,PZLR1,SEPZL1) CALL POWER(NZ,NZLR5,PZLR5,SEPZL5) CALL POWER(NZ,NZLR10,PZLR0, SEPZLO) CALL P0WER(NZ,NZSC1,PZSC1,SEPZS1) CALL POWER(NZ,NZSC5,PZSC5,SEPZS5) CALL POWER(NZ,NZSC10,PZSC0,SEPZSO) CALL POWER(NZ,NZWA1,PZWA1,SEPZW1) CALL POWER(NZ,NZWA5,PZWA5,SEPZW5) CALL POWER(NZ,NZWAl0,PZWAO,SEPZWO) C OUTPUT FORMAT * WRITE (6,12) DIST PRINT 14 WRITE (6,15) BIAS1,SEB1,RMSE1,SEMI WRITE (6,16) BIAS2,SEB2,RMSE2,SEM2 WRITE (6,17) BIAS3,SEB3,RMSE3,SEM3 WRITE (6,18) BIAS4,SEB4,RMSE4,SEM4 WRITE (6,19) BIAS5,SEB5,RMSE5,SEM5 WRITE (6,20) BIAS6,SEB6,RMSE6,SEM6 PRINT 30 WR I T E (6,31) PCLR1,SEPCL1,PCSC1,SEPCS1,PCWA1,SEPCW1 WRITE(6,32) PELR1,SEPEL1,PESC1,SEPES1,PEWA1,SEPEW1 WRITE(6,33) PWLR1,SEPWL1,PWSC1,SEPWS1,PWWAl, SEPWW1 W R I T E (6,34) PGLR1,SEPGL1,PGSC1,SEPGS1,PGWA1, SEPGW1 WRITE(6,35) PLLR1,SEPLL1,PLSC1,SEPLS1,PLWA1, SEPLW1 WRITE(6,36) PZLR1,SEPZL1,PZSC1,SEPZS1,PZWA1, SEPZW1 PRINT 37 WRITE(6,31) PCLR5,SEPCL5,PCSC5,SEPCS5,PCWA5, SEPCW5 WRITE(6,32) PELR5,SEPEL5,PESC5,SEPES 5,PEWA5,SEPEW5 WRITE(6,33) PWLR5,SEPWL5,PWSC5,SEPWS 5,PWWA5,SEPWW5 WRITE(6,34) PGLR5,SEPGL5,PGSC5,SEPGS5,PGWA5,SEPGW5 WRITE(6,35) PLLR5,SEPLL5,PLSC5,SEPLS5,PLWA5, SEPLW5 W R I T E (6,36) PZLR5,SEPZL5,PZSC5,SEPZS5,PZWA5,SEPZW5 PRINT 38 WRITE(6,31) PCLRO,SEPCLO,PCSCO,SEPCSO,PCWAO, SEPCWO WRITE(6,32) PELRO,SEPELO,PESCO,SEPES0,PEWAO,SEPEWO W R I T E (6,33) PWLRO,SEPWLO,PWSCO,SEPWSO,PWWAO,SEPWWO W R I T E (6,34) PGLRO,SEPGLO,PGSCO,SEPGSO,PGWAO,SEPGWO W R I T E (6,35) PLLRO,SEPLLO,PLSCO,SEPLSO,PLWAO,SEPLWO WRITE(6,36) PZLRO,SEPZLO,PZSCO,SEPZSO,PZWAO, SEPZWO WRITE (6,13) NCOX,NEXP,NWEI,NLLG,NLNM,NZ,NSMPLE 12 FORMAT(//,5X,'DATA ARE GENERATED FROM',T30,A80) 13 FORMAT(//,5X,'NCOX-',14,4X,'NEXP-',14,4X,'NWEI-',14,4X,'NLLG-', $ 14,4X,'NLNM-',14,4X,'NZ=*,I4,4X,'NSMPLE-',14) 14 FORMAT(//,T9,'MODEL',T26,'BIAS',T49,'S.E.(BIAS)',T73, * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ************* * * * * * * * * MODEL * * * * * * * REGRESSION * cox * * * * * * * * **************** ******************************************************** o o o o .3,5X, 3X, 7 F CENSORING-', OF 'AVERAGE , FORMAT(T6 370 0 FORMAT(T6,'NORMAL',T19,D20.10,T42,D20.10,T65,D20.10,T88,D20.10) 20 6 OMTT,EPNTA'T9D01, 1, 5D01,88, 8 65,D20.10,T8 .10,T 0 2 , D , 2 4 T19,D20.10,T42,D20.10,T65,D20.10,T88,D20.10) FORMAT C 121 c------c C USING BISECTION METHOD TO FIND MLE C C THE ROOT SHOULD LIE BETWEEN Z1 AND Z2. C C------C PDIF-0.5D0 IERR-0 331 Zl- BSTART - PDIF Z2« BSTART + PDIF CALL LIKCOX(N,Zl,LIKE,SI,S2) F1=S1 CALL LIKCOX(N,Z2,LIKE,SI, S2) F2-S1 IF (F1*F2 .GT. 0 .DO) THEN PDIF=PDIF*2. IERR-IERR+1 IF(IERR.LT.9) GO TO 331 WRITE(6,*) 'ERROR FOR COX, Ul=',F, 'U2=',FMID NWORK=0 RETURN ELSE NWORK-1 ENDIF JMAX—10+IERR DO 332 J—1,JMAX ZMID— (Z1+Z2)/2.D0 CALL LIKCOX(N,ZMID,LIKE,SI,S2) FMID—SI IF (F2*FMID .GT. 0.D0) THEN Z2-ZMID F2-FMID ELSE Zl— ZMID FI—FMID ENDIF 332 CONTINUE C------C C COMPUTING THE LIKELIHOOD RATIO, THE WALD, C C AND THE SCORE TEST STATISTICS C C------C BETA-ZMID B-ZMID CALL LIKCOX(N,B,S,SI,S2) CALL LIKCOX(N,0.DO,L,LI,L2) LR— (-2.DO)* (L-S) SCORE- (Ll**2)/ (-L2) WALD- B**2 * (-S2) RETURN END o o o o C ------C THIS SUBROUTINE CALCULATES THE VALUES OF LOGLIKELIHOOD C SCORE, AND OBSERVED INFOMATION FUNCTIONS C------* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *********** * * * * * * MODEL * * * REGRESSION * * * * * EXPONENTIAL * * * ************* 3 Z-BTR - PDIF - BSTART Zl- 231 o o o o 0 CONTINUE 50 c c 0 -, N J-l, 50 O D OMNT(0) I(0) X(100) IC(100), T (100), COMMON END RETURN OMNT10, C10, X(100) IC(100), T(100), COMMON 52-0.D0 .DO 0 51— S-O.DO PDIF-0.5D0 WEXP(N,BSTART,BETA,LR,SCORE,WALD,NWORK) SUBROUTINE SXX-O.DO SX-O.DO SUM-0.DO FI -SB1(N,Zl,LAMBDA,S,SB2,SL2, SLB) -SB1(N,Zl,LAMBDA,S,SB2,SL2, FI SB1(N, — Z2,LAMBDA,S,SB2,SL2,SLB) F2 O 32 J-l,JMAX 332 DO URUIE LIKCOX(N,B,SUM,SI,S2) SUBROUTINE ELSE ENDIF (A-H,L,O-Z) REAL IMPLICIT MLCT EL AH L O-Z) L, (A-H, REAL IMPLICIT 2 SAT PDIF + BSTART Z2- IERR-0 MX 10+IERR JMAX— F F*2 G. D) THEN .DO) 0 .GT. (F1*F2 IF SN IETO MTO OFN ME C MLE TO FIND METHOD USING BISECTION H OT HUDLEBTEN lAD C . 2 Zl Z AND LIE SHOULD BETWEEN ROOT THE —D P(X(J)*B) XP DE Y— X- X+ Y*X(J)**2 SXX+ SXX- (J) X * Y SX+ SX- S+Y S- U-U X()B -DLOG(S) (J)*B X + -SX**2)/S**2 SUM-SUM (SXX*S S2- SX/S S2= - 50 TO X(J) GO ) S1+ 0 Sl= .EQ. ( IC(J) IF RITE(,) ERRFREXP','Ul=',F,'U2-',FMID P X E FOR NWORK-0 'ERROR (6,*) E T I WR NWORK-1 . 2 * F I PDIF—PD RETURN MD SB1(N,ZMID,LAMBDA,S,SB2,SL2,SLB) - FMID FIR.T9 G O 231 TO GO IF(IERR.LT.9) IERR-IERR+1 MD (Z1+Z2)/2.D0 ZMID— F F*MD G. .0 THEN 0.D0) .GT. (F2*FMID IF 123 Z2-ZMID F2-FMID ELSE Zl-ZMID Fl-FMXD ENDIF 332 CONTINUE c------c C COMPUTING THE LIKELIHOOD RATIO, THE WALD, C C AND THE SCORE TEST STATISTICS C c ------C BETA=ZMID B=ZMID UBETA=SB1(N,B,LAMBDA,S,SB2,SL2,SLB) UBETA0=SB1(N,0.DO,LAMBO,L,LB2,LL2,LLB) LR= (-2 .DO) * (L-S) c------C C COMPUTING THE INVERSE OF OBSERVED C C INFOMATION MATRIX C C SCORE - SCORE STATISTIC C C WALD - WALD STATISTIC C C------C DET=LL2*LB2-LLB**2 FINV=1.DO/DET*(-LL2) SCORE*(UBETA0**2)*FINV DET“SL2*SB2-SLB**2 OINV=l.DO/DET*(-SL2) WALD*(B**2)/OINV RETURN END C------C C THIS SUBROUTINE CALCULATES THE VALUES OF LOGLIKELIHOOD C C SCORE, AND OBSERVED INFOMATION FUNCTIONS C C------C DOUBLE PRECISION FUNCTION SB1(N,B,LAMBDA, S, SB2,SL2,SLB) IMPLICIT REAL (A-H,L,O-Z) COMMON T(100), IC(IOO), X(100) K-0 SUM1-0.D0 SUM2=0 .DO SUM3=0.D0 SUMX-O.DO SUMBX-0 .DO DO 70 J*1,N Y=DEXP(X(J)*B) SUM1-SUM1+T(J)*Y SUM2-SUM2+T(J)*X(J)*Y SUM3-SUM3+T (J) * (X (J) **2) *Y IF ( IC(J) .EQ. 0) GO TO 70 K-K+l SUMBX-SUMBX+B*X(J) SUMX-SUMX+X(J) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * **************** * * * * * MODEL * * * * REGRESSION * * * WEIBULL * * * * * * * * * * **************** 0 Zl(1) l PDIFL Z “ 807 o o o o o 0 CONTINUE 70 UO() 2, 2, 22 FISH0(,)FISH(3,3) H S I (3,3),F 0 H S (2,2) I O F ,F N I (2),X D I M (2),Z 0 B (2), O U $ ,LNRA+)X 22, 2,ISHF(3, (2,2) F Q H S A 3) (2),FI (2,2),Q ,FINV(3,3),OINV(3,3) F Q H (2,2),U » F (2,2),Q $ F N Q ,$ L(NTRIAL+1),XI ------LAMBDA=DBLE(K)/SUM1 S= K*DLOG(LAMBDA) + SUMBX - DBLE(K) - SUMBX + SB1=SUMX-DBLE(K)*SUM2/SUM1 K*DLOG(LAMBDA) S= END RETURN L—SUM2 (K)SLB— ) (SUM1**2/DBLE (DBLE(K)*SUM3/SUMl) SL2— SB2— IESO 2, 22, 2, 22, 2, 2, (2), 2 (2),Z l (2),Z R O (2,2),C (2),A (2,2),U F N I (2),X B DIMENSION OMNT10, CIO, X(100) IC(IOO), T(100), COMMON PARAMETER(NTRIAL=100,TOLB=0.001D0,TOLF=0.01D0) SUBROUTINE WEI (N,ASTART,BSTART,BETA,ALPHA, LR, SCORE, WALD, NWORK) (N,ASTART,BSTART,BETA,ALPHA, WEI SUBROUTINE NERR=0 PDIFL-ASTART/10.DO (A-H,L,O-Z) REAL IMPLICIT PDIFU=ASTART*2.DO I =U1(N,Zl) FI =U1(N,Z2) F2 2() .DO 0 Z2 (2)“ PDIFU (1)- 2 Z (2)=0.DO l Z F F*2 G. D) THEN .DO) 0 .GT. (F1*F2 IF ENDIF DO 332 J-l, J-l, JMAX 332 DO ELSE MX(LGDB(l1-Z2())DO(.0D)) (2.0D0)+1 G O L D )/ (1)))-DLOG(0.001D0) 2 Z JMAX-(DLOG(DABS(Zl(1)- SN IETO EHD OFN TE L NE H C THE MLE UNDER THE TO FIND METHOD BISECTION USING ETITDMDL C MODEL RESTRICTED H RO SOL LEBTENZ() N 2l . C . AND Z2(l) Zl(l) LIE BETWEEN SHOULD ROOT THE NERR-NERR+1 PDIFU-PDIFU*1.5D0 PDIFL-PDIFL/20.DO 6* 'E ER 3',F,FMID ERR 'WEI NWORK=0 (6,*) E T I R W RETURN FNR.T1) OT 807 TO GO IF(NERR.LT.15) NWORK-1 FMID-U1(N,ZMID) D I M Z (1) — (Zl (1) 2 Z + (1) ) /2 .DO ZMID(2)-0.DO F F*MD G. .0 THEN 0.D0) .GT. (F2*FMID IF 1—ZMID(1) D I M Z (1)— 2 Z C C 125 6 D 2 1=1,2 22 DO 567 3 CONTINUE 332 3 CONTINUE CONTINUE 23 22 o o o o 4 CONTINUE 24 C c 9 RF -1.DO 1 - = ERRF 79 3 CONTINUE 73 4 CONTINUE 74 SN AQATSPOEUE OFN H LS C TO FIND MLES THE PROCEDURE MARQUART1S USING O 1 J=1,NTRIAL 71 DO BLEND=2.0D0 B (2)=BSTART B (1) (1)=ZMID NERRS=0 CALL LIKF1(N, B, LIKF1(N, LAMBDA, S,U, CALL XINF, FISH) O 3 1=1,2 23 DO M=l,2 22 DO (U(2))) BS DMAX1(ERRF,DABS(U(1)),DA ERRF= J= -S (J)= L O 4 1=1,2 24 DO XINF(2,2) * (1,1) (2,2)**2 F Q N I X (2,2)= * Q H Q (1,1)**2 Q (1,1)= Q H Q 22=QHQ(,) DET / (1,1) Q H Q (2,2)= /DET A (2,2) Q ) H Q (2,2)-(QHQ(1,2) **2 (1,1)= Q H Q A (1,1)* Q H Q = T DE (1,2) Q H Q (2,1)= Q H Q (1,1)*Q(2,2)*XINF(1,2) Q (1,2)= Q H Q 80 TO GO TOLF) .LE. (ERRF IF 12=-QHQ(,) /DET (1,2)A(2,1)=A(1,2) Q H Q - (1,2)= A 22=Q(2,2) Q (2,2)= Q A Q O 4 1=1,2 74 DO 1=1,2 73 DO (1,1)*Q(2,2)*A(1,2) Q (1,2)= Q A Q 11=Q(,)* * (1,1) A * (1,1)**2 Q (1,1)= Q A Q ERRB=-1.DO (1,2) Q A Q (2,1)= Q A Q F ER .E TL) O O 80 TO GO TOLB) .LE. (ERRB IF ENDIF ELSE I=QAQ( 1*U() (1,2)*U(2) Q A Q (1)+ U ,1)* (I Q A Q (I)= R O C F1=FMID F2=FMID 1=ZMID(1) D I M Z (1)= l Z IM=0.DO 0 Q (I,M)= II“ D/ DSQRT(DABS(XINF(1,1))) .DO/ 1 Q (I,I)“ 11=QHQ(,) BLEND + (1,1) Q H Q (1,1)= Q H Q I=B( +COR(I) R O C )+ (I B (I)= B DMAX1(ERRB,DABS(COR(I))) = ERRB NE TEFL MDL C MODEL THE FULL UNDER **2 Q(2,2) * 126 ICOUNT-O 89 IF (B(l) .LE. 0.001D0) THEN DO 88 1-1,2 COR(I)-COR(I)/2.D0 B (I)—B (I)-COR(I) 88 CONTINUE ICOUNT—ICOUNT+1 IF (ICOUNT .LE. 10) GOTO 89 WRITE(6, *) 'WEI ERR 4, B(l)=' ,B(1) NWORK— 0 RETURN ENDIF CALL LIKF1 (N,B, LAM, S,U, XINF, FISH) IF (-S .LE. L (J) ) GOTO 231 DO 555 1=1,2 B (I)= B (I)-C O R (I) 555 CONTINUE NERRS-NERRS+1 BLEND—BLEND * 2 IF (NERRS .LE. 20) GOTO 567 WRITE(6,*) 'ERROR IN WEI(FULL)',L(J),-S,BLEND NWORK-0 RETURN 231 BLEND-BLEND/2. 71 CONTINUE W R I T E (6,*) 'WEI ERR 2 ,', 'ERRB-',ERRB,'COR(1) = ',C O R (1), $ 'CO R (2)-' , CO R (2) NWORK-0 RETURN 80 NWORK-1 BETA—B (2) ALPHA—B (1) CALL LIKF1(N,B,LAMBF,SF,UF,XINFF,FISHF) C ------C C COMPUTING THE LIKELIHOOD RATIO, THE WALD, C C AND THE SCORE TEST STATISTICS C C ------C B O (1)- Z M I D (1) B O (2)—0 .DO CALL LIKF1(N,BO,LAMBO,SO,U0,XINFO,FISHO) LR— (-2 .DO) * (SO-SF) CALL INVER(FISHO,OINV) SCORE-(U0(2)**2)*OINV(3,3) CALL INVER(FISHF,FINV) WALD— (B(2)**2)/FINV(3,3) RETURN END C ------C C THIS SUBROUTINE CALCULATES THE VALUES OF LOGLIKELIHOOD C C SCORE, AND OBSERVED INFOMATION FUNCTIONS C C------C SUBROUTINE LIKF1(N,B,LAMBDA,S,U,XINF,FISH) 128 IMPLICIT REAL (A-H, L, O-Z) COMMON T (100), IC(100), X (100) DIMENSION B (2),XINF(2,2), U (2),FISH(3,3) K=0 SUM1=0.D0 SUM2-0.D0 SUM3=0.D0 SUM4=0.DO SUM5-0.D0 SUM6-0.DO SUMX=0.DO SUMLT=0.D0 DO 100 J=1,N Y-DEXP (X(J)*B(2)) LT=DLOG(T(J)) Z=0.D0 IF (LT*B(1) .GT. -20.DO) Z=DEXP(LT*B(1)) SUM1=SUM1+Z*Y SUM2=SUM2+Z*LT*Y SUM3=SUM3+Z*X(J)*Y SUM4—SUM4+Z*LT*X(J)*Y SUM5«SUM5+Z*LT**2*Y SUM6-SUM6+Z*X(J)**2*Y IF ( IC(J) .EQ. 0) GO T O 100 K-K+l SUMLT-SUMLT+LT SUMX-SUMX+X(J) 100 CONTINUE LAMBDA-DBLE(K)/SUM1 S-DBLE(K)*DLOG(LAMBDA)+DBLE (K)*DLOG (B (1)) + (B(1)-1)*SUMLT $ +B (2) *SUMX-DBLE (K) U(1)-DBLE(K)/B(1)+SUMLT-DBLE(K)*SUM2/SUM1 U (2)—SUMX-DBLE(K)*SUM3/SUM1 XINF(1,1)-DBLE(K)/B(l)**2+DBLE(K)* (SUM5/SUM1-(SUM2/SUM1)**2) XINF (1,2) -DBLE (K) * (SUM4/SUM1-(SUM2 /SUM1) * (SU M 3 /SUM1) ) XINF (2, 2) -DBLE (K) * (SUM6/SUM1- (SUM3/SUM1) **2) XINF(2,1)-XINF(1,2) FISH(1,1)-SUM1**2/DBLE(K) FISH(2,2)—DBLE(K)/B(1)**2+DBLE(K)*SUM5/SUMl FISH(3,3)-DBLE(K)*SUM6/SUM1 FISH(1,2)—SUM2 FISH(1,3)-SUM3 FISH(2,3)-DBLE(K)*SUM4/SUM1 FISH(2,1)—FISH(1,2) FISH(3,1)—FISH(1,3) FISH (3,2)—FISH(2,3) RETURN END C ------U O O C THIS SUBROUTINE COMPUTES THE FIRST ELEMENT OF SCORE VECTOR U(l) C ------DOUBLE PRECISION FUNCTION U1(N,B) IMPLICIT REAL (A-H, L, O-Z) COMMON T(100),IC(100),X(100) DIMENSION B (2) K=0 SUM1=0.DO SUM2-O.DO SUMLT-O.DO DO 100 J=1,N Y=DEXP (X (J)*B (2)) LT-DLOG(T(J) ) Z-0.D0 IF(LT*B(1) .GT. -20.0D0) Z-DEXP(LT*B(1)) SUM1=SUM1+Z*Y SUM2=SUM2+Z*LT*Y IF ( IC(J) .EQ. 0) GO TO 100 K=K+1 SUMLT=SUMLT+LT 100 CONTINUE Ul-DBLE (K) /B (1) +SUMLT-DBLE (K) *SUM2/SUM1 RETURN END ************************************************************** ************** loglogistic regression model **************** ************************************************************** SUBROUTINE LOGIST (N, ASTART, BSTART, CSTART, BETA, ALPHA, SIGMA $ ,LR,SCORE,WALD,NWORK) IMPLICIT REAL (A-H, L,O-Z) PARAMETER (NTRIAL-8 0, TOLB-O . 001D0 , TOLF=0 . 01 D 0 ) COMMON T (100), IC(100), X(100) DIMENSION B<3),U(3),FISH(3,3),F I N V (3, 3),C O R (3),FI SH O(3,3) $ ,L(NTRIAL+1),U0(3),FISHF(3,3),UF(3),B0(3),OINV(3,3) , S (3,3), $ SHS(3,3),TEMP (3,3),SHSINV(3,3) C------c C FIND MLE UNDER THE RESTRICTED MODEL C C------C CALL LOGLOG (N, ASTART, CSTART, A1 , R1 , NWORK) IF (NWORK .EQ. 0) RETURN B0 (1) *1 .D0/R1 B 0 (2)=-DLOG (A1 )* B0(1) B0 (3)=0.DO C------c C USING MARQUARDT'S COMPRMISE PROCEDURE TO FIND MLES C C UNDER THE FULL MODEL CONTROLL WITH NTRIAL, C C------C 120 B (1)= B 0 (1) B (2)= B 0 (2) B( 3) -BSTART B L E N D - 5 .ODO DO 101 J-l,NTRIAL NERRS-0 CALL LIKF2 (N,B, LIKE,U,FISH) 130 L (J) =-LIKE ERRF=-1.DO ERRF-DMAX1(ERRF,DABS(U(l)),DABS(U(2)),DABS(U(3))) IF (ERRF .LE. TOLF) GO TO 110 3823 DO 1 1=1,3 D O 1 M=l,3 S (I,M)= 0 .DO 1 CONTINUE DO 2 1=1,3 S(I,I)=1.DO/DSQRT(DABS(FISH(I, I) )) 2 CONTINUE CALL MULT(S,FISH,TEMP) CALL MULT(TEMP,S,SHS) DO 3 1=1,3 SHS (I,I)=SHS(I,I)+BLEND 3 CONTINUE CALL INVER(SHS,SHSINV) CALL MULT(S,SHSINV,TEMP) CALL MULT(TEMP,S,FINV) DO 103 1=1,3 COR(I)= 0 .DO 103 CONTINUE DO 104 1=1,3 DO 105 M=l,3 COR(I)=COR(I)-FINV(I,M)*U(M) 105 CONTINUE 104 CONTINUE ERRB=-1.D0 DO 106 1=1,3 ERRB-DMAX1( ERRB,DABS(COR(I) ) ) B (I)= B (I)-COR(I) 106 CONTINUE IF (ERRB .LE. TOLB) GO TO 110 ICOUNT=0 109 IF (B(1) .LE. 0.0001D0) THEN DO 1080 1=1,3 COR(I)“COR(I)/2.D0 B (I)= B (I) +C O R (I) 1080 CONTINUE ICOUNT=ICOUNT+l IF (ICOUNT .LE. 10) GOTO 109 W R I T E (6, *) 'LLOGIST ERR 5, B(1) = ',B(1) NWORK=0 RETURN ENDIF CALL LKLEI(N,B,LIKE) IF(—LIKE.LE.L (J)) GO TO 675 DO 789 1=1,3 789 B (I )= B (I)+ C O R (I ) NERRS-NERRS+1 BLEND=BLEND*2 IF(NERRS.LE.20) GO TO 3823 131 W R I T E (6,*) 'ERROR IN MARQUART',LIKE,L(J),BLEND NWORK=0 RETURN 675 BLEND—BLEND/2. 101 CONTINUE PRINT *, ' LLOGIST ERR 6, ERRB=',ERRB NWORK-0 RETURN 110 NWORK—1 BETA—B (3) ALPHA—B (2) SIGMA—B (1) C ------C C COMPUTING THE LIKELIHOOD RATIO, THE WALD, C C AND THE SCORE TEST STATISTICS C C------C CALL LIKF2(N,B,LIKEF,UF,FISHF) CALL LIKF2{N,BO,LIKE0,U0,FISHO) LR— (-2.DO)* (LIKEO-LIKEF) CALL INVER (FISHO,OINV) SCORE-(U0(3)**2)*OINV(3,3) CALL INVER(FISHF,FINV) WALD— (B(3)**2)/FINV(3,3) RETURN END C ------C C THIS SUBROUTINE CALCULATES THE VALUES OF LOGLIKELIHOOD C C SCORE, AND OBSERVED INFOMATION FUNCTIONS C C ------C SUBROUTINE LIKF2(N, B, LIKE,U,FISH) IMPLICIT REAL (A-H, L,O-Z) COMMON T(100), IC(100), X(100) DIMENSION B(3),U(3),FISH(3,3) K— 0 SUMDZ-0.DO SUMD1-0.D0 SUMT1-0.D0 SUMDH-0.D0 SUMTH-0.D0 SUMDX-0.D0 SUMD2-0.DO SUMT2-0.D0 SUMD3-0.D0 SUMT3-0.DO SUMDF-0.DO SUMTF-0.DO SUMD4-0.DO SUMT4-0.DO SUMD5-O.DO SUMT5-0.D0 SUMD6-0.D0 SUMT6-0.D0 132 SUMD7=0.D0 SUMT7-0.D0 SUMD8=0.DO SUMT8=0.D0 DO 121 J=1,N Y=DLOG(T(J) ) Z=(Y-B(2)-B(3)*X(J))/B(l) C C THE FOLLOWING PROCEDURE IS TO AVOID OVERFLOWING PROBLEM C H=0.DO F=0.DO LZ=0.DO IF ( DABS(Z) .LT. 170.DO) THEN H-DEXP(Z)/ (1.D0+DEXP(Z)) F=H/(1.DO+DEXP(Z)) LZ=DLOG(l.DO+DEXP(Z) ) ELSE IF( Z .GE. 170.DO) THEN H-1.D0 F=0.DO LZ-Z ENDIF SUMT1=SUMT1+LZ SUMTH-SUMTH+H SUMT2-SUMT2+H*X(J) SUMT3«SUMT3+H*Z SUMTF=SUMTF+F SUMT4=SUMT4+F*X(J)**2 SUMT5-SUMT5+F*Z**2 SUMT6=SUMT6+F*X(J) SUMT7=SUMT7+F*Z SUMT8=SUMT8+F*Z*X(J) IF ( IC(J) .EQ. 0) GO TO 121 K-K+l SUMDZ-SUMDZ+Z SUMDX=SUMDX+X(J) SUMD1-SUMD1+LZ SUMDH-SUMDH+H SUMD2-SUMD2+H*X(J) SUMD3-SUMD3+H*Z SUMDF=SUMDF+F SUMD4=SUMD4+F*X(J)**2 SUMD5=SUMD5+F*Z**2 SUMD6-SUMD6+F*X(J) SUMD7=SUMD7+F*Z SUMD8=SUMD8+F*Z*X $ 2.D0*SUMT3+SUMD5+SUMT5) FISH(2,2)— (1.DO/B(1)**2)*(SUMDF+SUMTF) FISH(3,3)-(1.DO/B(1)**2)*(SUMD4+SUMT4) FISH(l,2)=(l.DO/B(l)**2)*(-DBLE(K)+SUMDH+SUMTH+SUMD7+SUMT7) FISH(1,3)=(1.DO/B(1)**2)*(-SUMDX+SUMD2+SUMT2+SUMD8+SUMT8) FISH(2,3)-(1.DO/B(1)**2)*(SUMD6+SUMT6) FISH(2,1)—FISH(1,2) FISH(3,1)—FISH(1,3) FISH (3,2)=FISH(2,3) RETURN END C ------C C COMPUTE THE LOG LIKELIHOOD OF THE LOG LOGISTIC MODEL C C ------C SUBROUTINE LKLEI(N,B,LIKE) IMPLICIT REAL (A-H,L,O-Z) COMMON T(100) , IC(IOO), X(100) DIMENSION B (3),U(3),FISH(3,3) K=0 SUMDZ-O.DO SUMD1— 0 .DO SUMT1-0.DO DO 121 J-1,N Y«DLOG(T(J)) Z— (Y-B(2)- B (3)* X (J) )/ B (1) c C THE FOLLOWING PROCEDURE IS TO AVOID OVERFLOWING PROBLEM C H-O.DO F-O.DO LZ-O.DO IF ( DABS(Z) .LT. 170.DO) THEN H-DEXP(Z)/(1.DO+DEXP(Z)) F-H/(l.DO+DEXP(Z)) LZ-DLOG(1.DO+DEXP(Z)) ELSE I F ( Z .GE. 170.DO) THEN H— 1 .DO F — 0 .DO LZ-Z ENDIF SUMT1—SUMT1+LZ IF ( IC(J) .EQ. 0) GO TO 121 K-K+l SUMDZ-SUMDZ+Z SUMD1—SUMD1+LZ 121 CONTINUE LIKE-(-DBLE(K))*DLOG(B(1))+SUMDZ-SUMD1-SUMT1 RETURN END SUBROUTINE LOGLOG(N,ALPHA,SIGMA,A,R,NWORK) IMPLICIT REAL(A-H,L,O-Z) COMMON T(100) , I (100) 134 DIMENSION L (50) R=l.DO/SIGMA A=DEXP (-ALPHA/SIGMA) SLOG-0.DO D “ 0.DO LIKO-O.DO DO 1 J=1,N L (J)-DLOG(T (J) ) Y=1.0D0+A*T(J)**R IF (I(J).E Q .0) GO TO 2 D=D+1.DO SLOG“SLOG+L (J) LIKO=LIKO-2.ODO*DLOG(Y) GO TO 1 2 LIKO-LIKO—DLOG (Y) 1 CONTINUE LIKO=D*DLOG(A)+D*DLOG(R)+ (R-l.0D0)*SLOG+LIKO DO 99 IT=1, 50 DA=0.DO DR-O.DO DAA=0.DO DRA-O.DO DRR—0.DO DO 20 J“l,N X = T (J)**R Y“1.0D0+A*X IF(I(J).E Q .0) GO TO 30 DA=DA-2.0D0*X/Y DR=DR-2.0D0*A*X*L(J)/Y DAA-DAA+2.0D0* (X/Y)**2 DRA-DRA-2.0D0*X*L(J) /Y**2 DRR-DRR-2.ODO*A*X*L(J)**2/Y**2 GO TO 20 30 DA-DA-X/Y DR=DR-A*X*L (J) /Y DAA-DAA+ (X/Y) **2 DRA=DRA-X*L (J) /Y**2 DRR=DRR-A*X*L (J) **2/Y**2 20 CONTINUE DA=DA+D/A DR-D/R+DR+SLOG DAA=DAA-D/A**2 DRR=DRR-D/R**2 DET=DAA*DRR-DRA* * 2 EA= (DA*DRR-DR*DRA)/DET ER= (DR*DAA-DA*DRA) /DET 71 IF(A.GT.EA) GO TO 70 EA-EA/2. GO TO 71 70 IF(R.GT.ER) GO TO 72 ER-ER/2. GO TO 70 2 A-A-EA 72 c c O O O O * * * NWORK=l 88 VD ui * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * **************** * * * * * * * * * * * model * * * * * * * * regression * * * * * * * lognormal * * * * * * * * * * * * * * *************** * * * * * * CONTINUE 9 3 CONTINUE 136 3 CONTINUE 132 3 CONTINUE 133 SN TE TRTV POEUEO APODAD ALR C OF TAYLOR PROCEDURE AND SAMPFORD ITERATIVE THE USING ,03,UF()FISHF(,), (3,3) O H (3,3) S F I H ,F S I (3),F F U ,U0(3), ,NWORK) D L A LR, SCORE,W $ $ $ ,F I N V (3,3),OINV(3,3 ) ,W(100) ,W(100) ,R(100) ) (3,3),OINV(3,3 V N I ,F $ O J=1,N 4 DO OT 4 TO GO Y=1.0D0+A*T(J)**R CONTINUE LIKE-LIKE-2.ODO*DLOG(Y) LIKE=0 R=R-ER DIMENSION B (3),B O (3),B O L D (3),BNEW(3) ,COR<3) (3),BNEW(3) D L O (3),B O (3),B B DIMENSION EA,ER LOGLOG', IN NWORK=0 'NOCONV , * PRINT IEDDO()DDO() (R-l.0D0)*SLOG+LIKE LIKE=D*DLOG(A)+D*DLOG(R)+ LIKE-LIKE-DLOG(Y) O 3 1=1,N 136 DO LIKO=LIKE IF (I (J) . 5 . TO Q E GO 0) OMN (0) I(0) X(100) IC(100), T(100), COMMON 01D0) 0,TOLB-O.0 (NTRIAL=3 PARAMETER END RETURN RETURN K=0 (3)-BSTART B (2)-ALPHAO B SIGMAO (1)“ B 88 TO GO .LT. .001) (DABS((LIKE-LIKO)/LIKO) IF O 3 J-l,NTRIAL 131 DO SIGMA, (N,MODEL,ALPHAO,BSTART,SIGMAO,ALPHA,BETA, NORM SUBROUTINE MLCTRA (A-H,L,O-Z) REAL IMPLICIT O 3 1=1,3 132 DO O 3 1=1,3 133 DO AL TRAN(N,MODEL,B,W,R) CALL AL ROOT(N,K,W,R,B) CALL RB 1.DO ERRB— F I() E. ) K=K+1 1) .EQ. (IC(I) IF I— I) (I (I)—B D L O B I— I) (I (I)—B W E N B O IDTEMESUDRTE ULMDL C FULL MODEL THE UNDER MLE'S FIND THE TO 135 T ID H L' UDRTE ETITDMDL C C MODEL RESTRICTED TAYLOR THE SAMPFORD AND OF UNDER MLE'S THE PROCEDURE FIND TO ITERATIVE THE USING C C C CNRLE WT TIL AD C C INCREMENTS VARIABLE AND SUMMED = NTRIAL, TOLB WITH C CONTROLLED C C o o o ------c 4 NWORK-1 140 CONTINUE 131 CONTINUE 134 4 CONTINUE 142 4 CONTINUE 143 4 CONTINUE 144 4 CONTINUE 145 4 CONTINUE 146 4 CONTINUE 141 LP - (2) A-B PH AL NWORK-0 CALL LIKF3 (N,K,MODEL,B,LIKEF,UF,FISHF) LIKF3 CALL (3) BETA—B O 4 1=1,NTRIAL 141 DO RETURN SIGMA-B(l) OPTN H IEIODRTO TEWL, C WALD, THE RATIO, LIKELIHOOD THE COMPUTING O 4 J=l,2 142 DO O 4 J 1,N J= 144 DO J=1,N 143 DO AL TRAN(N,MODEL,BO,W,R) CALL .DO 0 (3)— O B BO (2) —B (2) (1) (1) O B B = BO (2) — SUMW/DBLE BO (N)(2) SUMW/DBLE — U 0.D0 = SUM SUMW-0.DO DO 146 J=l, J=l, 2 146 DO J=l,2 145 DO UR 0.DO 0 = SUMR RB 1 .DO ERRB— (SUM/ DSQRT (DBLE BO ) = (K)+SUMR) (1) F ER .T TL) OT 150 TO GO TOLB) .LT. (ERRB IF O 3 1=1,3 134 DO F ER .T TL) OT 140 TO GO TOLB) .LT. (ERRB IF (J)=B0(J) D L O B UW SUMW+W(J) SUMW— (J)“BNEW(J)-BOLD(J) R O C (J)BNEW =B0 (J) U SM(( (2)) O B - **2 ) SUM+(W(J = SUM SUMR = SUMR+R (J) SUMR+R = SUMR ERRB—DMAX1 (ERRB, DABS (COR (J)ERRB—DMAX1 ) ) F I() E. ) OT 144 TO GO 1) .EQ. (IC(J) IF I=NWI)BL( ) )-BOLD(I (I)=BNEW(I R O C ERRB=DMAX1(ERRB,DABS(COR(I))) C c 136 137 C AND THE SCORE TEST STATISTICS C C ------C 150 CALL LIKF3(N,K,MODEL,BO,LIKE0,U0,FISHO) LR=(-2.D0)* (LIKEO-LIKEF) CALL INVER (FISHO,OINV) SCORE-(UO(3)**2)*OINV(3,3) CALL INVER (FISHF, FINV) WALD-(B (3) **2)/FINV (3, 3) RETURN END C ------C C THIS SUBROUTINE TRANSFORMS THE LIFETIME DATA TO C C THE STANDARDIZED NORMAL C C P(Z) = THE STANDARD NORMAL P.D.F C C Q(Z) = THE STANDARD NORMAL SURVIVOR FUNCTION C C V(Z) = P(Z)/Q(Z) C C R(Z) = V(Z) (V(Z)-Z) C C------C SUBROUTINE TRAN (N, MODEL, B, W, R) IMPLICIT REAL (A-H,L,O-Z) COMMON T(100),IC(100),X(100) DIMENSION B(3),R(N),W(N) PARAMETER (PI - 3.14159265358979323846264338327950288D+00) DO 151 J-1,N IF (MODEL .EQ. 1) THEN Y - DLOG (T (J) ) ELSE Y-T (J) ENDIF Z - (Y—B (2)- B (3)*X(J))/ B (1) IF (IC(J) .EQ. 1) THEN W (J) - Y R(J) = 0.D0 ELSE C C AVOID FLOATING EXCEPTIONS C IF ((Z .LT. 4.) .AND. (Z .GT. -4.)) THEN P - 1 .DO/DSQRT(2.D0*PI)*DEXP(-(Z**2) /2.DO) Q = l.DO-ANORDF(Z) V - P/Q ELSE IF (Z .GE. 4.) THEN V = 1 . ELSE V— 0 . ENDIF R (J) = V*(V-Z) W(J) = B (2)+ B (3)*X(J)+B(1)*V ENDIF 151 CONTINUE RETURN END C ------c C THIS SUBROUTINE CALCULATES THE SOLUTION TO THE C C NORMAL EQUATIONS C C ------C SUBROUTINE ROOT (N,K,W,R,B) IMPLICIT REAL (A-H,L,O-Z) COMMON T(100),IC(100),X(100) DIMENSION W(N),R(N),B (3) SUMX-0.D0 SUMXX-0.DO SUMW=0.DO SUMWX-0.D0 DO 161 J=1,N SUMX=SUMX+X(J) SUMXX=SUMXX+X(J)**2 SUMW=SUMW+W(J) SUMWX=SUMWX+W(J)*X(J) 161 CONTINUE DET = DBLE(N)*SUMXX - SUMX**2 B (2)= 1.D0/DET*(SUMXX*SUMW-SUMX*SUMWX) B (3)= 1 .DO/DET*(-SUMX*SUMW+ DBLE(N) * SUMWX) SUM = 0 .DO SUMR - 0 .DO DO 162 J- 1,N SUM - SUM+(W(J)-B(2)-B(3)*X(J))**2 IF (IC(J) .EQ. 1) GO TO 162 SUMR - SUMR+R(J) 162 CONTINUE B (1) - DSQRT(SUM/(DBLE(K)+SUMR)) RETURN END C ------C C THIS SUBROUTINE CALCULATES THE VALUES OF LOGLIKELIHOOD C C SCORE, AND OBSERVED INFOMATION FUNCTIONS C C------C SUBROUTINE LIKF3(N,K,MODEL,B,LIKE,U,FISH) IMPLICIT REAL (A-H,L,O-Z) COMMON T(100), IC(100), X(100) DIMENSION B(3),U(3),FISH(3,3) PARAMETER (SUMXX+SUMX2R) * * 2 * SUMZ=SUMZ+Z SUMZZ=SUMZZ+Z**2 SUMXX=SUMXX+X(J)**2 SUMX2R«SUMX2R+X(J)**2*R SUMXZR«SUMXZR+X(J)*Z*RSOMZ2R=SUMZ2R+Z**2*R SUMXZ=SUMXZ+X(J)*Z SUMZR-SUMZR+Z*R V SUMZV=SUMZV+Z * SUMLQ-SUMLQ+DLOG(Q)SUMR-SUMR+R SUMV-SUMV+V SUMX=SUMX+X(J) Q*1.DO-ANORDF(Z) R«V*(V-Z) SUMXR-SUMXR+X(J)*R SUMXV-SUMXV+X(J)*V Q*.000001 P-l.DO/DSQRT(2*PI)*DEXP(-(Z**2)/2.DO)V-P/Q V-l. v-o. Q-l. Y=DLOG(T(J)) IF IF ((Z .LT. 4.) .AND. (Z .GT. -4.)) THEN ENDIF Y=T(J) ELSE IF (Z .GE. 4.) THEN ELSE ENDIF IF IC(J) ( .EQ. 1) THEN Z=(Y-B(2)-/B(l) B) (3)*X(J) ELSE IF (MODEL .EQ. 1) THEN ELSE IF END FISH(2,2)-1.DO/B(1)**2*(DBLE(K)+SUMR) FISH(3,3)-1.D0/B(1) LIKE=-(DBLE(K)*DLOG(B(l)))-(0.5D0*SUMZZ)+SUMLQ SUMZ2R-0.D0 SUMX2R-0.D0 SUMXZR-O.DO FISH(1,1)-1.D0/B(1)**2*(-DBLE(K)+3.D0*SUMZZ+2.D0*SUMZV U (1)«1.DO/B(1)*(-DBLE(K)+SUMZZ+SUMZV)U U (2)=1.DO/B(1)*(SUMZ+SUMV)U(3)=1.D0/B(1)*(SUMXZ+SUMXV) DO 171 J=1,N $ $ +SUMZ2R) 171 CONTINUE AVOID FLOATING EXCEPTIONS o o o o C o o NSC1,NSC5,NSC10,NWA1,NWA5,NWA10) $ CHI10=2.70554351806640625DO) CHI5=3.84143829345703125DO, $ $ FISH (1,FISH 2) -1 .DO/B (1) (2,1)=FISH(1, H S I F FISH (2, 3) =1 .DO/B FISH (1,(1) 3)(SUMX+SUMXR) **2* =1 .DO/B (1) AAEE (CHI1=6.63481235504150391DO, PARAMETER END RETURN (3, 2) FISH =FISH(2,3) (3,1)=FISH(1,3) H S I F SUBROUTINE COUNT COUNT (TRUB, BETA, SB, SB2, SB4, SUBROUTINE LR, SC, WA, NLR1, NLR5 , NLR10, LE F(R G. H1) THEN CHI10) .GE. IF (LR ELSE B = SB2+(BETA-TRUB)**2 = SB2 SB+(BETA-TRUB) = SB (A-H,LfO-Z) REAL IMPLICIT ENDIF THEN CHI5) .GE. IF(LR ELSE B = SB4+(BETA-TRUB)**4 = SB4 LE FS .E CI) THEN CHI5) .GE. IF(SC ELSE LE FS .E CI0 THEN CHI10) .GE. IF(SC ELSE F L .E CI) THEN CHI1) .GE. (LR IF ENDIF F S .E CI) THEN CHI1) .GE. (SC IF LE F(A G. H1) THEN CHI10) .GE. IF (WA ELSE THEN CHI5) .GE. IF(WA ELSE ENDIF F W .E CI) THEN CHI1) .GE. (WA IF RETURN END NLR10-NLR10+1 NLR5-NLR5+1 NLR1=NLR1+1 NSC5=NSC5+1 NSC1-NSC1+1 NLR10-NLR10+1 NLR10-NLR10+1 NLR5=NLR5+1 NSC10=NSC10+1 NSC10=NSC10+1 NSC5-NSC5+1 NSC10=NSC10+1 NWA1=NWA1+1 NWA10=NWA10+1 NWA5=NWA5+1 NWA10=NWA10+1 NWA5=NWA5+1 NWA10=NWA10+1 2 ) * * * * 2 2 * * (2 .DO*SUMXZ+SUMXV+SUMXZR) (2 .DO*SUMZ+SUMV+SUMZR) 140 CONTINUE 1 o o o o o o c C c c HS URUIECMUE TE NES FA 33MTI C MATRIX 3*3 A OF INVERSE THE COMPUTES SUBROUTINE THIS HS URUIE EFRS ARXMLILCTO. C MULTIPLICATION. MATRIX PERFORMS SUBROUTINE THIS A(,) 32*(,) (3,3)*A(2,1)*A(1,2) A (3,2)*A(1,1)- A *A<3,1)- 2) (2,3)* A 2)-A(l,3)*A(2, (3,1)*A(2,3)*A(1, A $ $ D E T = A (1/1)* (1/1)* A = A T (2, E D (3,3)A 2) V (3, N * I 3),A DIMENSION A (3, 3)+A(2,1)*A(3,2)*A(1,3)+ IESO 33 ,(,) C(3,3) 1=1,3 ,B(3,3), 1 (3,3)DO A DIMENSION )*A(3,1)) (l,2 (3,2)-A AINV(3,3)=(1.DO/DET)*(A(1,1)*A(2,2)-A(1,2)*A(2,1)) (l,1)*A (A AINV<3,2)«(-1.DO/DET)* AINV(3,1)=(1.DO/DET)*(A(2,1)*A(3,2)-A(2,2)*A(3,1)) (A(1,1)*A(2,3)-A(1,3)*A(2,1)) AINV(2,3) = (-l.DO/DET)* AINV(2 , (1,3)*A(2,2)) 2) = AINV(2,1)(1.DO/DET)*(A(1,1)*A(3,3)-A(1,3)*A(3,1)) = (-1.DO/DET)*(A(2,1)*A(3,3)-A(2,3)*A(3,1)) (l,2)*A(2,3)-A (A AINV(1,3)=(1.DO/DET)* (A(1,2)*A(3,3)-A(1,3)*A(3,2)) (-1.DO/DET)* (2,3)*A(3,2)) AINV(1,2)= AINV(1,1)=(1.DO/DET)*(A(2,2)*A(3,3)-A END RETURN DBLE(NRJCT)/DBLE(NTOTAL) = PRJCT DO 1 J=l, J=l, 1 3 DO A, 8 * B,C REAL END RETURN ( PRJCT*(1.D0-PRJCT)/DBLE(NTOTAL)) T R Q S D - SEP ) P E S POWER(NTOTAL,NRJCT,PRJCT, SUBROUTINE END INVER(A,AINV) SUBROUTINE RETURN MULT(A,B,C) SUBROUTINE (A-H,L,0-Z) REAL IMPLICIT (A-H,L,0-Z) REAL IMPLICIT C(I, J)=A(I,1) *B C(I, (1,J)=A(I,1) J) +A (I, 2) *B (2, J) +A (I, 3) *B (3, J) 141 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Q q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Q CONTINUE 3 0 CONTINUE 10 ZT(J,K)=0 T Z 4 6 ZT(J,K)=ZT(J,K)+Z(I,IZ(J,K) ) ZT(J,K)=ZT(J,K)+Z(I,IZ(J,K) 6 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *I4AIN ******************** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * I-* * * * * * N> * * * * * * * * * * * * * * * * * * * * * * * * * o * * o o o o o o o o OBSERVED TIME AND CENSORED IND. FOR DISEASE 2, T(I,2) AND T(I,2) IND(I,2)* AND T(I,1) * IND(I,1)* INDIVIDUAL 2, ITH THE FOR 1, COVARIATE KTH THE Z(I,K), DISEASE FOR IND. COVARIATES DISEASE FOR CENSORED IND. AND AND CENSORED TIME AND OBSERVED TIME OBSERVED NU UBR O CVRAE FRDSAE 1 N 2 N() N Z2 * NZ(2) AND NZ(1) 2, AND 1 DISEASES FOR COVARIATES OF NUMBERS INPUT DIMENSION H (2500,2),W(2500),I T (2500, (2500, T 2) (10,2) I (2),B (2500,2),W(2500),I I H (10,2),A T (2),Z DIMENSION A (10,2),V(23,23) DIMENSION O B DIMENSION OMN 20,)ID20,)Z(509, 1,) (2),NTOTAL Z N (10,2), Z (2500,9),I (2500,2),IND(2500,2),Z T COMMON MLCT REAL(A-H,L,O-Z) IMPLICIT Pr am of par rc Esi ton f Moe I odel M r fo n atio stim E tric e m ra a ip m e S f o m ra g ro P N A R T R O F R E A D (3,*) T (1,1),IND(I,1),T(I,2),IND(1,2) , (Z(I,K),K=l,NTZ) (1,1),IND(I,1),T(I,2),IND(1,2) T (3,*) D A E R UBR O IDVDAS NOA * * NTZ 2, AND 1 DISEASES FOR COVARIATES TOTAL NTOTAL OF INDIVIDUALS, OF NUMBERS NUMBERS OAITS NIAO ZIK * IZ(I,K) INDICATOR COVARIATES O 1=1,NZ(K) 2 DO K-1,2 1 DO 3* NZ()NZ(2) Z (1),N Z N (3,*) D A E R O 1=1,NTOTAL 3 DO CONTINUE CONTINUE IZ(I,K) (3,*) D A E R 3* NTOTAL,NTZ (3,*) D A E R W(I)=1.0 O 1=1,NTOTAL 5 DO J=1,NZ(K) 4 DO O 0 K=l,2 80 DO O J=l,NZ(K) 6 DO FIDIK.Q0 G O 5 TO GO IF(IND(I,K).EQ.0) O 0 J=l,10 10 DO 1=1,2 10 DO OIJ = 0. = BO(I,J) m a r g o r p A P P E N D IX B B IX D N E P P A 142 **************************** 143 5 CONTINUE 80 CONTINUE C------C C COMPUTE THE RANK OF OBSERVED TIMES C C ------C CALL SORT(IT) LIKETI=0 C ------C C PERFORM THE COX ANALYSIS UNDER INDEPENDENCE C C------C DO 8 K=l, 2 FLAG=1 CALL COX (BO, ZT, IT, W, K,FLAG, H) 8 CONTINUE THE=-1. LIKEO=FLIKE(THE,BO,H,IT,ZT) WRITE(6,*) 'INITAL LIKELIHOOD®',LIKEO C------C C EM ALGORITHM C C------C THE® 3.22 DO 99 ITER=1,1000 DO 50 K=l,2 DO 50 J-1,NZ(K) 50 BI(J,K)®BO(J,K) THEN=THE CALL FINDTHE(THEN,BI,H,W) DO 52 K-1,2 FLAG=-1 CALL COX(BI, ZT,IT,W,K, FLAG,H) 52 CONTINUE LIKEN=FLIKE(THEN,BI,H,IT,ZT) WRITE(6,*) ITER,' OLD®', LIKEO,' NEW®',LIKEN DLIKE®LIKEN-LIKEO LIKEO-LIKEN W R I T E (6,*) 'THE: OLD, N E W ',T HE,THEN IF(DLIKE.LT.0.) GO TO 777 IF(DLIKE.LT..00001) GO TO 98 EMAX=ABS(THE-THEN) 777 THE=THEN DO 54 K=l,2 DO 54 J=l,NZ(K) EMAX=AMAX1(ABS(BO(J,K)-BI(J,K)),EMAX) BO (J, K) =BI (J, K) 54 CONTINUE IF(EMAX.LT..0001) GO TO 98 99 CONTINUE WRITE(6,*) 'WARNING MAX ITT EXCEEDED' C ------C C COMPUTE COVARIANCE MATRIX C C------C 98 CALL VAR(V, H, THEN, BI, IT,W) 144 WRITE(6,*) 'DISEASE 1* DO 13 J=1,NZ(1) SE=SQRT(V(J,J)) ZQ-BI (J, 1) /SE P=2.*(1.O-ANORDF(ABS(ZQ))) WRITE(6,101) J,BI(J,1),SE,ZQ,P 101 FORMAT(' B(',I2,') = • , F12.6,2X,'SE=',F12.6,'Z = ',F10.4, 2X,'P=' , F 7 .4) 13 CONTINUE W R I T E (6,*) 'DISEASE 2' DO 14 J = 1 , N Z (2) SE-SQRT(V(J+NZ(1),J+NZ(1))) ZQ=BI(J,2)/SE P=2.*(l.0-ANORDF(ABS(ZQ))) WRITE(6,101) J,BI(J, 2),SE,ZQ,P 14 CONTINUE NN=NZ (1) +NZ (2) +1 SE =SQRT(V(NN, NN) ) ZQ=THEN/SE P=(1.0-ANORDF(ABS(ZQ))) W R I T E (6,103) THEN,SE,ZQ,P 103 FORMAT(' THETA=',F10.4,2X,'SE=', F12.4,2X,'Z=',F10.4, 2X, 'P=' , F 8 .4) WRITE(6,*) 'FINAL LIKELIHOOD*', LIKEN 788 STOP END C THIS SUBROUTINE PERFORMS COX PROFILE LIKELIHOOD ANALYSIS C £***************************************************************£ SUBROUTINE COX(B, ZT, IT,W, K,FLAG,H) IMPLICIT REAL(A-H,L,O-Z) COMMON T (2500,2),IND(2500,2),Z(2500,9),IZ(10,2),NZ(2),N DIMENSION B(10,2) ,ZT(10,2),F(11) ,F 2 (11,11) , FI (11,11),E (11) DIMENSION H T (2500),I T (2500,2),S U M 1 (10),S U M 2 (10,10) DIMENSION W(2500),H (2500,2) NPAR=NZ(K) DO 98 ITTER=1,50 SUM=0 DO 960 J=1,NZ(K) F(J)=ZT(J,K) SUM1(J)=0 DO 960 1=1,NZ(K) F2(J,I)=0. S U M 2 (J,I)=0. 960 CONTINUE DO 61 1=1,N II=IT(I,K) SB=0 . DO 62 J=l,NZ(K) 62 SB=SB+B(J,K)*Z(II,IZ(J,K)) Y=W(II)*EXP(SB) SUM-SUM+Y DO 63 J=l,NZ(K) SUM1(J)=SUM1(J)+Y*Z(II,IZ(J,K)) 145 DO 63 J1-J,NZ(K) SUM2(J,Jl)-SUM2(J,Jl)+Y*Z(II,IZ(J,K))*Z(II,IZ(Jl,K)) 63 SUM2 (Jl, J) -SUM2 (J, Jl) DO 64 J-1,NZ(K) DD-FL OAT (IN D (II,K) ) F (J)=F(J)-DD*SUM1(J)/SUM DO 64 J1=J,NZ(K) F2(J,J1)=F2(J,J1)-DD*(SUM2(J,Jl)/SUM-SUMl(J)*SUM1(Jl)/SUM**2) F2(J1,J)-F2(J,Jl) 64 CONTINUE 61 CONTINUE CALL LINRG(NPAR,F2,11,FI,11) EMAX=-100 FMAX=-100 DO 5 J=1,NPAR E (J) =0 DO 6 JJ=1,NPAR 6 E (J) =E (J) +F (JJ) *FI (J, JJ) EMAX-AMAX1(ABS(E(J)),EMAX) 5 FMAX—AMAX1(ABS(F(J)),FMAX) DO 7 J=l,NZ(K) B(J,K)=B(J,K)-E(J) 7 CONTINUE IF(FMAX.LE..01) GO TO 70 IF(EMAX.LE..001) GO TO 70 98 CONTINUE WRITE(6,*) 'NO CONVERGENCE IN COX ROUTINE' RETURN 70 CONTINUE IF(FLAG.LT.0.) GO TO 20 W R I T E (6, 100) K 100 FORMAT(' ESTIMATES FOR DISEASE',12,' ONLY') DO 10 J»1,NZ(K) SE-SQRT(-FI(J,J)) ZQ—B (J,K)/SE P-2.*(l.0-ANORDF(ABS(ZQ))) W R I T E (6,101) J , B (J,K),SE,ZQ,P 101 F O R M A T (' B(',I2,•)-',F 1 2 .6,2X, 'SE=',F12.6,'Z=',F10.4,2X, 'P=',F7.4) 10 CONTINUE 20 SUM-0. SUMH-0 D O 50 1=1,N SB-0 DO 30 J— 1,NZ(K) 30 SB—SB+B(J,K)*Z(IT(I,K),IZ(J,K)) SUM—SUM+W(IT(I,K))*EXP(SB) HT (IT (I, K) ) —FLOAT (IND (IT (I, K) , K) ) /SUM 50 CONTINUE SUMH-0 DO 60 I— 1,N II—IT(N-I+l,K) SUMH—SUMH+HT(II) H (II, K) =SUMH 60 CONTINUE RETURN END Q***★★*★*★★★*★*★*★★★★★★★★*★*****★*★*★★★★★★**★★★*★***0 C THIS SUBROUTINE COMPUTES THE RANKS OF OBSERVED TIMES C C FOR DISEASES 1 AND 2, RESPECTIVELY. C SUBROUTINE SORT(IT) IMPLICIT REAL (A-H,L,O-Z) COMMON T (2500,2),IND(2500,2),Z(2500,9),IZ(10,2),NZ(2),NTOTAL DIMENSION IT (2500,2) DO 99 K = 1 ,2 DO 1 J=l,NTOTAL 1 I T (J, K)=J NT=NTOTAL-l DO 7 J=l,NT DO 7 I=J+1,NTOTAL IF(T(IT(J,K),K)-T(IT (I,K),K)) 2,3,7 2 ITEMP=IT(J,K) IT(J,K)=IT(I,K) IT (I, K) =ITEMP GO TO 7 3 IF(IND(IT (J,K),K) .EQ.l) GO TO 2 7 CONTINUE 99 CONTINUE RETURN END C THIS SUBROUTINE FINDS THE ROOT FOR THE ASSOCIATION C C PARAMETER FOR THE INVERSE GAUSSIAN FRAITY MODEL C SUBROUTINE FINDTHE(THE, B, H, W) IMPLICIT REAL(A-H,L,0-Z) COMMON T (2500,2),IND(2500,2),Z(2500,9),IZ(10,2),NZ(2),N DIMENSION B(10,2),H (2500,2),W(2500),D(2500),S (2500) SUM-0 DO 1 J— 1,N S(J)-0 D (J) =0 DO 2 K=l,2 SB-0 DO 3 I— 1, NZ(K) 3 SB-SB+B(I,K)*Z(J, IZ(I,K)) D (J) —D (J) +FLOAT (IND (J, K) ) S (J) =S (J) +EXP (SB) *H (J, K) 2 CONTINUE RIND—D (J)—0 .5 A-l./THE+S(J) C-SQRT(4.*A/THE) CALL RBESS (RIND,C, RB) W (J)-1./SQRT(THE*A) *RB 147 WINV-SQRT(THE*A)* (RB-2.*RIND/C) SUM=SUM+W(J) +WINV * 1 CONTINUE THE=2./N*SUM-4. DO 4 J=1,N RIND=D(J)-0.5 A=1./THE+S(J) C=SQRT(4.*A/THE) CALL RBESS(RIND, C,RB) W (J) -1. /SQRT (THE*A) *RB 4 CONTINUE RETURN END C THIS SUBROUTINE COMPUTES THE VALUE OF FULL LOG LIKELIHOOD C REAL FUNCTION FLIKE(THE,B,H, IT,ZT) IMPLICIT REAL(A-H,L,O-Z) PARAMETER (PI=3.141592653589793238) COMMON T (2500,2),IND(2500,2),Z (2500,9),IZ (10,2),NZ (2),N DIMENSION H (2500,2),ZT (10,2),B (10,2),I T (2500,2) W R I T E (6,*) 'IN FLIKE T H E = 1,THE FLIKE-0. DO 1 K=l,2 DO 2 J=1,NZ(K) 2 FLIKE=FLIKE+B(J, K ) * Z T (J,K) II-IT(N,K) IF(IND(II,K).EQ.0) GO TO 20 FLIKE=FLIKE+ALOG(H(II,K)) 20 CONTINUE DO 3 1=2,N II=IT(N— I+1,K) IF(IND(II,K).EQ.O) GO TO 3 JJ=IT(N-I+2,K) FLIKE=FLIKE+ALOG(H(II,K)-H(JJ,K)) 3 CONTINUE I CONTINUE IF(THE.LT.0.) GO TO 90 FLIKE=FLIKE+FLOAT(N)* (ALOG(2.)-0.5*ALOG(PI) ) & + 2. *FLOAT(N)/THE-FLOAT(N)/4.*ALOG(THE) DO 10 1=1,N D=0 S=0 DO 11 K = 1 ,2 SB=0 DO 12 J=1,NZ(K) 12 SB= SB+B(J,K)*Z (I , I Z (J,K) ) D-D+FLOAT(IND(I, K)) S=S+H (I, K) *EXP (SB) II CONTINUE RIND=D-0.5 A = 1 ./THE+S 148 C=SQRT(4.*A/THE) CALL BESS3(RIND,C,RK) FLIKE=FLIKE-D/2.*ALOG(THE)-RIND/2.*ALOG(A)+ALOG(RK) 10 CONTINUE RETURN 90 DO 15 1=1,N DO 15 K=l,2 SB=0 DO 16 J=l,NZ(K) 16 SB=SB+B(J,K)*Z(I,IZ(J,K)) 15 FLIKE=FLIKE-H(I,K)*EXP(SB) RETURN END C THIS SUBROUTINE CALCULATES THE COVARIANCE MATRIX C SUBROUTINE VAR(VI,H,THE,B,IT,W) IMPLICIT REAL(A-H,L,O-Z) COMMON T (2500,2),IND (2500,2),Z (2500, 9),IZ(10,2),NZ(2),N DIMENSION G (2500, 2),G B (2500,2),G B B (2500),H B (2500) DIMENSION VI (23,23),V(23,23),H (2500,2),B (10, 2) DIMENSION A(2500),C(2500),Q(2500),IT(2500,2),BZ(2500,2) DIMENSION RB (2500),DERB(2500),SH(2500),W(2500) ,D(2500) N T = N Z (1)+ N Z (2)+1 DO 1 J=l,NT DO 1 K=l,NT 1 V (J, K) =0 V (NT,NT)=4.*FLOAT(N)/THE**3 + FLOAT(N)/4./THE**2 DO 2 1=1,N 2 SH(I) =0 DO 3 1=1,N DO 4 K=l,2 BZ (I, K)=0 DO 5 J=1,NZ(K) 5 BZ(I,K)=BZ(I,K)+B(J,K)*Z(I,IZ(J,K)) 4 SH(I)=SH(I)+EXP(BZ(I, K))*H(I, K) D (I)=FLOAT(IND(1,1)+IND(1,2)) RIND=D(I)-0.5 A (I)= 1./THE + SH(I) C (I )= S Q R T (4.*A (I)/THE) Q(I)=-1./SQRT(A(I))/THE**2.5 - SQRT(A(I) )/THE**1.5 DQT= 3./SQRT(A(I))/THE**3•5 - 0.5/A(I)**1.5/THE**4.5 + & 1.5*SQRT(A(I))/THE**2.5 CALL RBESS(RIND,C(I),RB(I)) CALL DRBESS(RIND,C(I) ,DERB(I) ) RB (I) =RB(I) -RIND/C(I) DERB(I)=DERB(I)+RIND/C(I)**2 V (NT,NT)=V(NT,NT)+ D(I)/2./THE**2 + RIND/2*(-2./A(I)/THE**3 & +1./A(I)**2/THE**4)- DQT*RB(I) - Q(I)**2*DERB(I) 3 CONTINUE DO 20 K=l, 2 II=IT(N,K) 149 G(II,K)=H(II, K) DO 20 1=2,N G(IT(N-I+l,K),K)“H(IT(N-I+l,K),K)-H(IT(N-I+2,K),K) 20 CONTINUE DO 30 J = l , N Z (1) SUM=0 DO 31 1=1,N II=IT(I,1) SUM=SUM+W(II)*Z(II,IZ(J,1))*EXP(BZ(II,1)) 31 GB(II,1)=-G(11,1)**2*SUM DO 32 Jl=J,NZ(1) SUM=0 SUM1=0 DO 33 1=1,N II=IT(I ,1) SUM=SUM+W (II) *Z(II,IZ (Jl, 1) ) *EXP (BZ (II, 1) ) SUM1=SUM1+W(II)*Z(II,IZ(J, 1) ) *Z(II,IZ(Jl,1))*EXP(BZ(11,1)) GB(II,2)=-G(11,1)**2*SUM GBB(II) =-G(II,1)**2*SUM1-2*GB(II,1)*SUM*G(II,1) 33 CONTINUE DO 35 1=1,N II=IT(I,1) IF(IND(II,1) .EQ.O) GO TO 35 V(J,J1)=V(J,J1)+GBB(II)/G(II,1)-GB(II, 1)*GB(II,2)/G(II,1)**2 35 CONTINUE SUM1-0 SUM2-0 SUM12-0 DO 36 1=1,N II—IT(N-I+l,1) SUM1=SUM1+GB(11,1) SUM2-SUM2+GB (11,2) SUM12=SUM12 +GBB (II) X=SUM12+SUM1*Z(II,IZ(Jl,1))+SUM2*Z(II,IZ(J,1)) X=EXP(BZ(II,1)) * (X+H(II,1)*Z(II,IZ(J,1))*Z(II,IZ(Jl,1))) DB1=EXP(BZ(11,1)) * (SUM1+Z(II,IZ(J,1))*H(11,1)) HB(II)=DB1 DB2=EXP(BZ(II,1)) * (SUM2+Z(II, IZ(J1,1) )*H(II,1)) P=l./SQRT(THE)*(1./SQRT(A(II))*X - 0.5*DB1*DB2/A(II)**1.5) V(J, J1)=V(J, J1)-(D(II)/2.-0.25)* (X/A(II)-DB1*DB2/A(II)**2) & -(1./THE/A(II))*DB1*DB2*DERB(II) -P*RB(II) 36 CONTINUE 32 V(J1, J)=V(J, Jl) D O 37 1=1,N II=IT(N-I+l,1) DQA=0.5/A(II)**1.5/THE**2.5 -0.5/SQRT(A(II))/THE**1.5 V (J,NT)=V(J,NT)+HB(II)*(-(D(II)/2.-0.25)/A(II)**2/THE**2 & - Q(II)*DERB(II) /SQRT(A(II) ) / SQRT(THE) - DQA*RB(II)) 37 CONTINUE V (NT, J) =V (J, NT) SUM=0 150 DO 40 J 1 = 1 , N Z (2) SUM-0 SUM2=0 DO 41 1=1, N II = I T (1,2) SUM=SUM+W(II)*Z(II,IZ(Jl, 2) ) *EXP (BZ (11,2)) GB(II,2)=-G(II,2)**2*SUM 41 CONTINUE SUM2=0 DO 43 1=1,N II=IT(N-I+l,2) SUM2=SUM2+GB(11,2) X=EXP(BZ(11,2))*(SUM2+Z(II,IZ(J1,2) ) *H(II,2) ) V ( J , N Z (1)+J1)=V(J,NZ(1)+J1)+X*HB(II) * &((D(II)/2.-0.25)/A(II)**2 + 0.5/SQRT(THE)/A (II)**1.5 *RB(II) & - 1./A(II)/THE*DERB(II)) 43 CONTINUE 40 V ( N Z (1)+J1,J ) = V (J, NZ (1) +J1) 30 CONTINUE DO 130 J = 1 , N Z (2) SUM=0 DO 131 1=1,N II=IT(I, 2) SUM=SUM+W(II)*Z(II,IZ(J,2))*EXP (BZ(II, 2) ) 131 GB(II,1)=-G(II,2)**2*SUM DO 132 J1 = J , N Z (2) SUM-0 SUM1— 0 DO 133 1=1,N II=IT(1,2) SUM— SUM+W (II) *Z(II,IZ (Jl, 2)) *EXP (BZ (II, 2)) SUM1—SUM1+W(II)*Z(II,IZ(J,2))*Z(II,IZ(Jl,2)) *EXP (BZ(11,2)) GB(II,2)=-G(II,2)**2*SUM GBB(II)=-G(11,2)**2*SUM1-2*GB(II, 1) *SUM*G(II,2) 133 CONTINUE DO 135 1=1,N II— I T (1,2) IF(IND(II,2 ) .EQ.0) GO TO 135 XZ=GBB(II)/G(II,2)-GB(11,1)*GB(II,2)/G(II,2)**2 V(J+NZ(1),J1+NZ(1))=V(J+NZ(1),Jl+NZ(1))+XZ 135 CONTINUE SUM1=0 SUM2-0 SUM12-0 DO 136 1=1,N II=IT(N-I+l,2) SUM1— SUM1+GB (11,1) SUM2=SUM2+GB(11,2) SUM12-SUM12+GBB(II) X=SUM12+SUM1*Z(II,IZ(Jl,2))+SUM2*Z(II,IZ (J,2) ) X=EXP(BZ(II,2)) * (X+H(II,2)*Z(II,IZ(J,2))*Z(II,IZ(J1,2))) DB1—EXP(BZ(11,2)) * (SUM1+Z(II,IZ(J, 2))*H (II,2) ) 151 HB (II) =DB1 DB2=EXP(BZ (11,2) ) * (SUM2+Z(II,IZ(Jl,2) ) *H (11,2) ) P=l./SQRT (THE)*(I./SQRT(A(II))*X - 0.5*DB1*DB2/A(II)**1.5) V (J+NZ (1) , Jl+NZ (1)) =V( J+NZ (1) , J1+NZ(1) )- & (D(II)/2.-0.25)*(X/A(II)—DB1*DB2 / A (11)**2) & -(l./THE/A(II))*DB1*DB2*DERB(II) -P*RB(II) 136 CONTINUE V (Jl+NZ (1) , J+NZ (1))=V( J+NZ (1) , Jl+NZ (1) ) 132 CONTINUE DO 137 1=1,N II=IT(N-I+l,2) DQA=0 . 5/A(II)**1.5/THE**2.5 -0.5/SQRT(A(II))/THE**1.5 V(J+NZ(1),NT) =V(J+NZ(1), NT)+HB(II)*(-(D(II)/2.-0.25)/A(II)**2 & /THE**2 - Q(II)*DERB(II)/SQRT(A(II))/SQRT(THE) - DQA*RB(II)) 137 CONTINUE V(NT, J+NZ (1) )=V(J+NZ (1) ,NT) 130 CONTINUE DO 50 J = 1 , NT DO 50 K=1,NT 50 V(J,K)*-V(J,K) CALL LINRG(NT,V,23, VI, 23) RETURN END C THE FOLLOWING ROUTINES COMPUTE THE BESSEL FUNCTION, BESS3 C C THE RATIO OF TWO BESSEL FUNCTIONS, RBESS C C THE DERIVATIVE OF RBESS, DRBESS C SUBROUTINE BESS3 (RIND, W, RK) PARAMETER(PI<=3.141592653589793238) SUMW-0.0 IF (ABS (RIND) .EQ. 0.5) S U M W = 0 .0 IF(RIND .EQ. 1.5) SUMW=1./W IF (RIND .EQ. 2.5) SUMW=3 ./W+3./W**2 IF(RIND .EQ. 3.5) SUMW=6./W+15./W**2+15./W**3 RK=SQRT(PI/2 . ) * (l.+SUMW) *EXP (-W)*W**(-0.5) RETURN END C SUBROUTINE RBESS (RIND, W, RB) IF (RIND .GT. 0.) THEN CALL BESS3 (RIND,W,RK) CALL BESS3 (RIND+1., W, RK 1 ) RB= RK1/RK ELSE RI=-RIND CALL BESS3 (RI, W,RK) CALL BESS3(RI+1.,W,RKl) RB=RKl/RK-2.*RI/W ENDIF RETURN END SUBROUTINE DRBESS(RIND,W,DRB) CALL BESS3(RIND, W,RK) CALL BESS3(RIND+2,W,RK2) CALL RBESS(RIND,W,RB) DRB=RB**2-RIND/W*RB-0.5-0.5*RK2/RK RETURN END APPENDIX C FORTRAN Program of Semiparametric Estimation for Model II 0********************************************************Q c* ****************m a i n PROGRAM***************************0 0********************************************************0 IMPLICIT REAL (A-H,L,O-Z) COMMON N,NZ,I(5000),Z(5000,25),W(5000) DIMENSION V (26,26),BOLD(25),H (5000),IG(5000),B (25),ST(25) DIMENSION D (5000),SM(5000),EBZ(5000),HP(5000),F(26,26) C ------C INPUT NUMBERS OF STUDY SUBJECTS, N, COVARIATES, NZ, AND GROUPS, NG C OBSERVED TIMES, TI C CENSERING INDICATOR, I(J) C GROUP INDICATOR, IG(J) C AND COVARIATES Z(J,K) FOR THE KTH COVARIATE OF THE JTH INDIVIDUAL C NOTE THAT THE DATA HAVE TO BE SORTED FROM THE LARGEST EVENT TIME C TO THE SMALLEST EVENT TIME IN ADVANCE C ------o o o o o o o o o R E A D (3,*) N,NZ,NG W R I T E (6,100) N,NZ,NG 100 FORMAT(IX,'SAMPLE SIZE-',15,' NUM. BETAS-',15,' NUM. GROUPS-',15) DO 1 J =1,N READ(3,*) TI,I(J),IG(J), (Z(J,K),K=1,NZ) W(J)»1.0 1 CONTINUE DO 2 J=l,25 BOLD(J)=0.0 2 CONTINUE DO 3 J=l,NZ 3 S T (J)=0 DO 4 J=l,N DO 4 K=l,NZ 4 ST (K) —ST (K) +Z (J, K) *FLOAT (I (J) ) DO 80 J=1,NZ BOLD(J)—B (J) 80 CONTINUE THE-1.7 NITT— 1000 C ------C C PERFORM THE COX ANALYSIS UNDER INDEPENDENCE C C ------C 153 154 FLAG=1 CALL COX(B,H,HP,ST,FLAG) LIKEO=PIND (B, H, HP, IG,NG) C ------C C EM ALGORITHM C C ------C DO 99 IT— 1,NITT THEO=THE CALL FINTHE(THE,B,H,NG,IG) THEN=THE WRITE(6,*) IT,'THE: OLD, NEW', THEO, THEN FLAG=-1 CALL COX(B,H,HP,ST,FLAG) LIKEN=PLIKE(THE,B,H,HP,IG,NG) W R I T E (6,*) IT,' OLD=',LIKEO,* NEW=',LIKEN DLIKE=LIKEN-LIKEO LIKEO=LIKEN IF (DLIKE .LT. 0.) GOTO 800 IF (DLIKE .LT. 0.00001) GOTO 1000 800 EMAX=-10 0 DO 71 J=1,NZ E=ABS(B(J)-BOLD(J)) EMAX=AMAX1(E,EMAX) BOLD(J)=B(J) 71 CONTINUE E-ABS(THEO-THEN) EMAX—AMAX1(E,EMAX) IF (EMAX .LT. .0001) GOTO 1000 99 CONTINUE WRITE(6,*) 'WARNING MAX IT EXCEEDED' C ------C C COMPUTE COVARIANCE MATRIX C C ------C 1000 CALL VAR(V,H,HP,THE,B,NG,IG) SE-SQRT(V(NZ+1,NZ+1)) TEST-THE/SE PVAL-1.-ANORDF(ABS(TEST) ) WRITE(6,771) THE,SE, PVAL 771 FORMAT(' ESTIMATE OF THE-',F12.7,2X,'SE=',F16.8,2X, 'PVAL=', F7.4) DO 9876 J—1,NZ SE-SQRT(V(J,J)) PVAL—2.*(1.-ANORDF(ABS(B(J)/SE))) WRITE(6,8761) J,B(J),SE,PVAL 8761 FORMAT(' B(',I2,')— ',F13.8,' SE-',F13.8,2X,'PVALUE-',F7.5) 9876 CONTINUE DO 35 J=1,NZ+1 DO 35 K—1,NZ+1 W R I T E (6,108) J , K , V (J,K) 108 FORMAT('ESTIMATED COVARIANCE V (',12,’,',12,')=•,F20.10) 35 CONTINUE STOP END 155 £*★**★★****★★**★******★**★★**★*****★★★★★*★******★**★*★★**★★★£ C THIS SUBROUTINE PERFORMS COX PROFILE LIKELIHOOD ANALYSIS C SUBROUTINE COX(B,H,HL, S,FLAG) COMMON N,NZ,I(5000),Z (5000,25),W(5000) DIMENSION S (25),H (5000),F (25),Tl(25),T2 (25,25),F2(25,25),E (25) DIMENSION H L (5000),FI(25,25),B(25) DO 90 IT=1,50 T-0 DO 3 K-1,NZ F(K)-S(K) Tl(K)=0 DO 3 KK=1,NZ T2(K,KK)=0 3 F2(K,KK)=0 DO 5 J=l,N BZ=0 DO 6 K=l,NZ 6 BZ=BZ+Z(J,K)*B(K) EWZ-W(J)*EXP(BZ) DO 7 K-1,NZ Tl(K)=T1(K)+Z(J,K)*EWZ DO 7 KK=1,NZ 7 T2(K,KK)-T2(K,KK)+Z(J,K)*Z(J,KK)*EWZ T-T+EWZ DO 8 K=1,NZ F (K) “F (K) -FLOAT (I (J) ) *T1 (K) /T DO 8 KK=1,NZ 8 F2(K,KK)=F2(K,KK)-FLOAT(I(J))*(T*T2(K,KK)-Tl(K)*T1(KK))/T**2 5 CONTINUE CALL LINRG(NZ,F2,25,FI,25) EMAX— 100 FMAX— 100. DO 10 J=1,NZ E (J) =0 DO 11 K»=1,NZ E (J) -E (J) +FI (J, K) *F (K) 11 CONTINUE EMAX*=AMAX1 (EMAX,ABS (E (J) ) ) FMAX=AMAX1(FMAX,ABS(F(J))) B(J)=B(J)-E(J) 10 CONTINUE IF(FMAX .LT. .01) GOTO 88 IF(EMAX.LT..001) GO TO 88 90 CONTINUE WR IT E(6,*) 'NO CONVERGENCE IN COX STEP',EMAX RETURN 88 CONTINUE IF (FLAG .LT. 0.) GOTO 99 DO 31 J«1,NZ SE=SQRT(-FI(J, J) ) ZQ=B(J)/SE 156 P - 2.*(1.0-ANORDF(ABS (ZQ))) WRITE(6/101) J,B(J),SE,ZQ,P 101 FORMAT(' B(',I2,')-',F12.6,2X,'SE= \ F10.6,2X,'Z=',F10.4, & 2X,'P=',F7.4) 31 CONTINUE 99 T-0 DO 20 J=l,N BZ=0 DO 21 K=1,NZ BZ-BZ+Z(J,K)*B(K) 21 CONTINUE T=T+W(J)*EXP(BZ) 20 HL(J)*FLOAT(I(J))/T SUM=0 DO 22 J=1,N K=N-J+l SUM=SUM+HL(K) H(K)=SUM 22 CONTINUE RETURN END Q ★★ ★ ★* * ★ ★* ★ ★★ *★ i t ie * ★ ★ ★ ★ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0 C THIS SUBROUTINE FINDS THE ROOT FOR THE ASSOCIATION C C PARAMETER FOR THE INVERSE GAUSSIAN FRAITY MODEL C £********************************************************£ SUBROUT INE FINTHE (THE ,B,H ,NG,IG) COMMON N,NZ,I (5000),Z(5000,25),W(5000) DIMENSION IG(5000),B(25),D(5000) , S(5000),H (5000) DO 1 J~1,NG S(J)«0. D(J)-0. 1 CONTINUE DO 2 J»1,N D (IG (J) ) -D (IG (J) ) +FLOAT (I (J)) BZ-0 DO 3 K*1,NZ 3 BZ-BZ+B(K) *Z (J,K) S(IG(J) )=S(IG(J) )+H(J) *EXP (BZ) 2 CONTINUE SUM=0. DO 5 K«*1,NG RIND=D(K)-0 . 5 A-l./THE+S(K) C=SQRT (4. *A/THE) CALL RBESS(RIND,C,RB) WK*1./SQRT(THE*A)*RB WINV=SQRT(THE*A)* (RB-2.*RIND/C) SUM=SUM+WK+WINV 5 CONTINUE THE=2./NG*SUM-4. DO 6 J*1,N RIND*D(IG(J))-0.5 157 A=l./THE+S(IG(J)) C=SQRT(4.*A/THE) CALL RBESS (RIND,C,RB) W (J)=1./SQRT(THE*A)*RB 6 CONTINUE RETURN END C THIS SUBROUTINE COMUPTES COVARIANCE MATRIX C SUBROUTINE VAR(VI,H,HP,THE,B,NG, IG) COMMON N,NZ, I (5000),Z (5000,25),W(5000) DIMENSION H (5000),HP(5000),EBZ(5000),IG(5000),B(25) DIMENSION D(5000),V (26,26),VI(26,26),GB(5000,2),GBB(5000) DIMENSION SMI(5000),SM2(5000),SM12(5000),SH(5000),A (5000) DIMENSION C (5000),Q(5000),RB(5000),DERB(5000) NT=NZ+1 DO 1 J=l,NT DO 1 K=l, NT 1 V (J, K) =0 V (NT,NT)=4.*FLOAT(NG)/THE**3 + FLOAT(NG)/4./THE**2 DO 2 J-1,NG S H (J)=0 D (J) =0 2 CONTINUE DO 3 J«1,N K=IG(J) D (K)—D(K)+FLOAT(I (J)) BZ-0. DO 5 M-1,NZ 5 BZ-BZ+B(M) *Z(J,M) EBZ (J)-EXP (BZ) SH (K)-SH(K)+EBZ(J)*H(J) 3 CONTINUE DO 4 K—1,NG RIND-D(K)-0.5 A(K)-1./THE + SH(K) C (K)-SQRT(4.*A(K)/THE) Q (K) — 1. /SQRT (A (K) ) /THE**2 . 5 - SQRT (A (K) )/THE**1. 5 DQT= 3./SQRT(A(K))/THE**3.5 - 0.5/A(K)**1.5/THE**4.5 + & 1.5*SQRT(A(K))/THE**2.5 CALL RBESS (RIND,C(K),RB(K)) CALL DRBESS (RIND,C(K),DERB(K)) RB (K)-RB(K)-RIND/C(K) DERB(K) -DERB(K)+RIND/C(K)**2 V(NT, NT)-V(NT,NT)+ D(K)/2./THE**2 + RIND/2*(-2./A(K)/THE**3 & +1./A(K)**2/THE**4)- DQT*RB(K) - Q(K)**2*DERB(K) 4 CONTINUE DO 30 J—1,NZ SUM-0 DO 31 II— 1,N SUM-SUM+W(II) *Z (II, J) *EBZ (II) 158 31 GB(II,1)=-HP(II)**2*SUM DO 32 J1-J,NZ SUM=0 SUM1=0 DO 33 11=1,N SUM=SUM+W(II)*Z(II,Jl)*EBZ(II) SUM1=SUM1+W(II)*Z(II,J)*Z(II, Jl) *EBZ(II) GB (11,2) =-HP (II) **2*SUM GBB(II)— HP (II) **2*SUM1-2*GB (II, 1) *SUM*HP(II) 33 CONTINUE DO 35 11=1,N IF (I (II) .EQ. 0) GO TO 35 V(J,J1)=V(J, J1)+GBB(II)/HP(II)-GB(II,1)*GB(11,2)/HP(II)**2 35 CONTINUE SUM1=0 SUM2=0 SUM12=0 DO 34 K=l,NG SMI (K) =0 SM2(K)=0 SM12(K)=0 34 CONTINUE DO 36 M=1,N K-N-M+l KG=IG(K) SUM1=SUM1+GB(K,1) SUM2=SUM2 +GB(K,2) SUM12-SUM12+GBB (K) SMI (KG) -SM1 (KG) + (Z (K, J) *H (K) +SUM1) *EBZ (K) SM2(KG)=SM2(KG) + (Z(K,Jl)*H(K)+SUM2)*EBZ(K) SM12(KG)“SM12(KG) + (Z(K,J)*Z(K,J1) *H(K)+Z(K, J)*SUM2 & +Z(K,Jl)*SUM1+SUM12)*EBZ(K) 36 CONTINUE DO 10 11=1,N G P=l./SQRT(THE)* (1./SQRT(A(II))*SM12(II)-0.5*SM1(II) & *SM2(II) /A(II)**1.5) V(J, J1)=V(J,Jl)-(D(II)/2.-0.25)*(SM12(II)/A(II)-SMI(II)*SM2(II) & /A(II)**2)- P*RB(II) & -(1./THE/A(II) )*SM1(II)*SM2(II)*DERB(II) 10 CONTINUE V(J1,J)=V(J,Jl) 32 CONTINUE DO 11 11=1,NG DQA=0.5/A(II)**1.5/THE**2.5 -0.5/SQRT(A(II) )/THE**1.5 V (J,NT)=V(J,NT)+SM1 (II)*(-(D(II)/2.-0.25)/A(II) **2/THE**2 & - Q(II)*DERB (II)/SQRT(A(II))/SQRT(THE) - DQA*RB(II)) 11 CONTINUE V (NT, J) =V (J, NT) 30 CONTINUE DO 50 J=1,NT DO 50 K=1,NT 50 V (J, K) — V (J, K) 159 CALL LINRG(NT,V,2 6,VI,2 6) RETURN END C********************************************************* C THIS SUBROUTINE COMPUTES THE VLUAE OF FULL LOG LIKELIHOOD C FUNCTION PLIKE(THE,B,H,HP,IG,NG) COMMON N, NZ , I (5000),Z (5000, 25) ,W(5000) DIMENSION B (25),D(5000),EBZ(5000),SM(5000),H (5000) , HP (5000) DIMENSION I G (5000) PARAMETER (PI=3.141592653589793238) P L IKE=0. DO 10 J=l,NG D (J) =0 S M (J)=0 10 CONTINUE DO 12 J=1,N K = I G (J) D(K)=D(K)+FLOAT(I (J) ) BZ=0 DO 11 M=l,NZ 11 BZ=BZ+Z(J,M)*B(M) EBZ(J)=EXP(BZ) SM(K)=SM(K)+H(J)*EBZ(J) 12 CONTINUE DO 15 J“1,N IF ( I(J) .EQ. 0) GOTO 15 PLIKE=PLIKE+ALOG(HP(J))+ALOG(EBZ(J)) 15 CONTINUE PLIKE-PLIKE+FLOAT(NG)* (ALOG(2.)-0.5*ALOG(PI)) & +2.*FLOAT (NG)/THE-FLOAT(NG)/4.*ALOG(THE) DO 16 K“1,NG RIND=D(K)-0.5 A-l./THE + SM(K) C“SQRT(4.*A/THE) CALL BESS3(RIND,C,RK) PLIKE-PLIKE-D(K)/2.*ALOG(THE)-RIND/2.*ALOG(A)+ALOG(RK) 16 CONTINUE RETURN END C THIS SUBROUTINE COMPUTES THE LOG LIKELIHOOD UNDER INDEPENDENCE C FUNCTION PIND(B,H,HP,IG,NG) COMMON N,NZ,1(5000),Z(5000,25) ,W(5000) DIMENSION B (25),D(5000),EBZ(5000),SM(5000),H(5000),HP (5000) DIMENSION IG(5000) PIND-0. DO 12 J“1,N BZ-0 DO 11 M “ 1,NZ 11 BZ=BZ+Z(J,M)*B(M) 160 EBZ(J)=EXP(BZ) 12 CONTINUE DO 15 J-1,N PIND=PIND-H(J)*EBZ(J) IF(I(J).EQ.0) GO TO 15 PIND-PIND +ALOG(HP(J))+ALOG(EBZ(J)) 15 CONTINUE WRITE(6,*) ' LIKELIHOOD UNDER INDEPENDENCEPIND RETURN END C THE FOLLOWING ROUTINES COMPUTE THE BESSEL FUNCTION, BESS3 C C THE RATIO OF TWO BESSEL FUNCTIONS, RBESS C C THE DERIVATIVE OF RBESS, DRBESS C SUBROUTINE BESS3(RIND,W,RK) PARAMETER (PI=3.141592653589793238) SUMW=0. IF (ABS(RIND) .EQ. 0.5) THEN S UMW=0.0 ELSE IF(RIND .EQ. 1.5) THEN SUMW=1./W ELSE IF(RIND .EQ. 2.5) THEN SUMW=3./W+3./W**2 ELSE IF(RIND .EQ. 3.5) THEN SUMW=6 . /W+15 . /W**2+15 . /W**3 ELSE N=INT(RIND-0.5) DO 10 1=1,N A-FACT(N+I) B-FACT(N-I) C = F A C T (I) SUMW=SUMW+A/B/C/(2.*W)**I 10 CONTINUE ENDIF RK=SQRT (PI/2 .) * (1 .+SUMW) /EXP (W) /SQRT(W) RETURN END C REAL FUNCTION FACT(N) FACT=1. IF (N .EQ. 0) THEN FA CT =1. ELSE DO 10 1=1,N FACT=FACT*FLOAT(I) 10 CONTINUE ENDIF RETURN END C C o o o ELSE URUIE RBESS(RIND,W,RB) SUBROUTINE END RETURN ENDIF F RN .T 0) THEN 0.) .GT. (RIND IF DRB=RB**2—RIND/W*RB-0.5-0 . 5*RK2/RK DRB=RB**2—RIND/W*RB-0.5-0 W,RB) RBESS(RIND, CALL BESS3(RIND+2,W,RK2) CALL BESS3(RIND,W,RK) CALL SUBROUTINE DRBESS (RIND, DRBESS W,DRB) SUBROUTINE END RETURN AL BESS3(RIND+1.,W,RK1) CALL BESS3(RIND,W,RK) CALL RB=RK1/RK RB=RKl/RK-2.*RI/W BESS3(RI+1.,W,RK1) CALL W,RK) BESS3(RI, CALL RI=—RIND 161 APPENDIX D FORTRAN Program of Parametric Estimation for Model I 0*******************************************************Q q * ******** *********** *J4AIN PROGRAM**********************0 ******************************************************* 0 IMPLICIT REAL (A-H, L, O-Z) COMMON T (2500,2), IN D (2500, 2) , TA U (2500) , Z(2500, 9) , IZ (10,2) & ,NZ (2), NTOTAL DIMENSION BO(10,2), V(23,23) , TL N (2),DEAD(2) DIMENSION A(2),Z T (10,2),AI(2),BI(10,2) 0******************************************************************** C INPUT NUMBERS OF COVARIATES FOR DISEASES 1 AND 2, NZ(1) AND NZ(2) * C COVARIATES INDICATOR IZ(I,K) * C NUMBERS OF INDIVIDUALS, NTOTAL * C NUMBERS OF TOTAL COVARIATES FOR DISEASES 1 AND 2, NTZ * C OBSERVED TIME AND CENSORED IND. FOR DISEASE 1, T(I,1) AND IND (1,1)* C OBSERVED TIME AND CENSORED IND. FOR DISEASE 2, T(I,2) AND IND (1,2)* C LEFT TRUNCATED TIME, TAU (I) * C AND COVARIATES Z(I,K), THE KTH COVARIATE FOR THE ITH INDIVIDUAL * 0******************************************************************** IZ <1,1)-1 I Z (1,2)=1 READ (3,*) N Z (1),N Z (2) DO 1 K-1,2 NZ (K) “NZ (K) +1 DO 2 I=2,NZ(K) READ ( 3, *) IZ (I, K) 2 IZ (I,K)-IZ(I,K)+1 1 CONTINUE READ (3,*) NTOTAL,NTZ NTZ=NTZ+1 DO 3 1=1, NTOTAL READ (3, *) T (1, 1),IND(1,1),T(I,2),IND(1,2),TAU(I), (Z(I,K),K=2,NTZ) Z 162 163 IF (IND (I, K) .EQ.O) GO TO 5 TLN(K)=TLN(K)+ALOG(T(I,K)) DEAD(K)—DEAD(K)+1 DO 6 J=1,NZ(K) 6 ZT(J,K)=ZT(J,K)+Z(I,IZ(J,K)) 5 CONTINUE 80 CONTINUE A(l)-1 A(2)=l LIKETI-0 C ------C C PERFORM WEIBULL REGRESSION ANALYSIS UNDER INDEPENDENCE C C------C DO 8 K=l,2 CALL INDEP(A(K),BO,TLN(K),ZT,DEAD(K),K,LIKE) 8 LIKETI=LIKETI+LIKE WRITE(6,*) 'TOTAL LIKELIHOOD UNDER INDEPENDENCE*',LIKETI LIKEO=LIKETI THEI=0 C------C C PROFILE SEARCH FOR ESTIMATES OF A, B AND THE C C------C DO 10 IQ-2,8 THE-5*FLOAT(IQ)/100 CALL PROFILE(A,BO,THE,TLN,ZT,DEAD,LIKE) WRITE(6,*) 'THETA-',THE,'LIKE-',LIKE LIKEN—LIKE IF(LIKEN.LT.LIKEO) GO TO 10 THEI—THE DO 11 K-1,2 AI(K)-A(K) DO 11 J—1,NZ(K) 11 BI(J,K)—B O (J,K) LIKEO—LIKEN 10 CONTINUE 777 IF(THEI.EQ.O) GO TO 788 CALL FULL(AI,BI,THEI,V,TLN, ZT, DEAD,LIKEM) SE-SQRT(V(1,1)) WRITE(6,*) 'DISEASE 1' WRITE(6,100) AI(1),SE 100 F O R M A T (' A — ',F 1 6 .8,2X,'SE— ',F16.8) DO 13 J— 1,N Z (1) SE-SQRT(V(J+1,J+l)) ZQ—BI(J,1)/SE P - 2 .*(1.0-ANORDF(DABS(ZQ) ) ) WRITE(6,101) J,BI(J,1),SE,ZQ, P 101 FORMAT (' B (',12,') =',F12.6,2X, 'SE=',F12.6,'Z=',F10.4,2X, ’P=',F7.4) 13 CONTINUE WRITE(6,*) 'DISEASE 2' N N —N Z (1)+2 SE-SQRT(V(NN,NN)) W R I T E (6,100) A I (2),SE 164 DO 14 J«1,NZ(2) SE-SQRT(V(J+NN,J+NN)) ZQ=BI(J,2)/SE P=2.*(1.O-ANORDF(DABS(ZQ))) WRITE(6,101) J,BI(J,2),SE,ZQr P 14 CONTINUE N N = N Z (1)+ N Z (2)+3 SE—SQRT (V(NN,NN)) ZQ=THEI/SE P= (1.O-ANORDF(DABS(ZQ))) W R I T E (6,103) THEI,SE,ZQ,P 103 FORMAT(' THETA=',F10.4,2X,'SE=',F10.4, 2X,1Z=',F10.4,2X,’P=',F8.4) WRITE(6,*) 'FINAL LIKELIHOOD=',LIKEM 7 88 STOP END C THIS SUBROUTINE PERFROMS WEIBULL REGRESSION ANALYSIS C C UNDER INDEPENDENCE ASSUMPTION C SUBROUTINE INDEP(A, BD, TLN, ZT, DEAD, K, LIKE) IMPLICIT REAL(A-H,L,O-Z) COMMON T(2500,2),IND(2500,2),TAU(2500) ,Z(2500,9) , IZ (10,2),NZ(2),N DIMENSION BD(10,2),ZT(10,2),B(10),F(11) ,F 2 (11,11) ,F I (11,11),E(11) DIMENSION B N (10),S C (11,11) BL END=2.0 NPAR=NZ(K)+1 DO 1 J=1,NZ(K) 1 B(J)=0. LIKEO=DEAD*ALOG(A)+ (A-l.)*TLN DO 820 J=1,NZ(K) 820 LIKEO=LIKEO+B(J)*ZT(J,K) DO 850 1=1,N SB=0 DO 830 J=1,NZ(K) 830 SB=SB+B(J)*Z(I,IZ(J,K) ) ET=T(I,K)**A*EXP(SB) ETAU=TAU(I) **A*EXP(SB) 850 LIKEO-LIKEO-ET+ETAU DO 98 IT=1,50 F (1)=DEAD/A+TLN F2(1,1)=-DEAD/A**2 DO 2 J=2,NPAR F (J)=ZT(J—1,K) DO 2 JJ=1,NPAR F2 (J, JJ) =0 . F 2 (JJ,J)=0. 2 CONTINUE DO 80 1=1,N SB=0. DO 3 J=1,NZ(K) 3 SB=SB+B(J)*Z(I,IZ(J,K)) ET=EXP(SB+A*ALOG(T(I,K))) 165 ETAU=EXP(SB+A*ALOG(TAU(I)) ) F (1)=F(1)-ET*ALOG(T(I,K))+ETAU*ALOG(TAU(I)) F2(1,1)=F2(1,1)-ET*(ALOG(T(I,K)))**2+ETAU*(ALOG(TAU(I)))**2 DO 4 J=1,NZ(K) F (J+l)=F(J+l)-ET*Z(I,IZ(J,K) )+ETAU*Z(I, IZ(J,K)) F 2 (1,J +l )= F2(1,J+l)-ET*ALOG(T(I,K))*Z(I,IZ(J,K)) F2(1,J+1)=F2(1,J+l)+ETAU*ALOG(TAU(I))*Z(I, IZ(J,K)) F2(J+1,1)=F2(1,J+l) DO 4 JJ=J,NZ(K) F 2 (J+l,JJ+1)=F2(J+l,JJ+1) + (ETAU-ET)*Z(I, IZ (J,K))*Z(I,IZ(JJ,K) ) F2 (JJ+1, J+l) =F2 (J+l, JJ+1) 4 CONTINUE 80 CONTINUE 591 CONTINUE DO 33 J=1,NPAR DO 33 1=1,NPAR 33 SC(J,I)=-F2(J,I)/SQRT(ABS(F2(J,J)*F2(I, I) )) DO 34 J=l,NPAR SC(J,J)=SC(J,J)+BLEND 34 CONTINUE CALL LINRG(NPAR,SC,11,FI,11) IBLEND=0 EMAX=-100 FMAX— 100 DO 5 J-1,NPAR E(J)=0 DO 6 JJ-1,NPAR 6 E (J)=E(J)+F(JJ)*FI(J,JJ)/SQRT(F2(J,J)*F2(JJ,JJ)) EMAX-AMAX1(ABS(E(J)) , EMAX) 5 FMAX=AMAX1(ABS(F(J)),FMAX) AN«A+E(1) IF(AN.LT.0.) GO TO 811 DO 7 J«1,NZ(K) BN(J)=B(J)+E(J+1) 7 CONTINUE LIKEN=DEAD*ALOG(AN)+ (AN-1.)*TLN DO 520 J=1,NZ(K) 520 LIKEN=LIKEN+BN(J)*ZT(J,K) DO 550 1=1,N SB=0 DO 530 J=1,NZ(K) 530 SB=SB+BN(J)*Z(I,IZ(J, K)) ET=EXP(SB+AN*ALOG(T(I,K))) ETAU=EXP(AN*ALOG(TAU(I))+SB) 550 LIKEN=LIKEN-ET+ETAU IF(LIKEN.GT.LIKEO) GO TO 711 811 BLEND=BLEND * 3 IBLEND=IBLEND+1 WRITE(6,*) ' IN INDEP CORRECTION',BLEND IF(IBLEND.LT.10) GO TO 591 LIKEO=LIKEN 711 A=AN 166 DO 743 J=1,NZ(K) 743 B(J)«BN(J) EMAX=ABS((LIKEO-LIKEN)) IF(EMAX.LE..001) GO TO 70 LIKEO-LIKEN BLEND=BLEND/3. IF(FMAX.LE..01) GO TO 70 IF(EMAX.LE..001) GO TO 70 98 CONTINUE WRITE(6,*) 'NO CONVERGENCE BASED ON INDEPENDENCE' RETURN 70 CONTINUE CALL LINRG(NPAR,F2,11,FI,11) SE=SQRT(-FI (1,1)) WRITE(6,100) K 100 FORMAT(' ESTIMATES FOR DISEASE',12,' ONLY') WRITE(6,*) 'A-',A,' SE=',SE DO 10 J=1,NZ(K) SE=SQRT(-FI(J+l,J+l)) ZQ=B(J)/SE P=2.*(1.O-ANORDF(DABS(ZQ))) WRITE(6,101) J,B(J),SE,ZQ,P 101 F O R M A T (' B (',I2,')*',F 1 2 .6,2X, 'SE=',F12.6, 'Z = ',F 1 0 .4,2X, 'P=',F7.4) 10 BD(J,K)-B(J) LI KE=DE AD * ALOG (A) + (A-l.) *TLN DO 20 J=l,NZ(K) 20 LIKE-LIKE+B(J)*ZT(J,K) DO 50 1-1, N SB=0 DO 30 J«1,NZ(K) 30 SB“SB+B (J) *Z (I, IZ (J, K) ) ET-T(I,K)**A*EXP(SB) ETAU=TAU(I)**A*EXP(SB) 50 LIKE=LIKE-ET+ETAU WRITE(6,*) 'LOG LIKELIHOOD(INDEPENDENCE) = ',LIKE RETURN END £*★**★★★*************★**★*★★*★★★*★****★***★***★**★**★***★£ C THIS SUBROUTINE SEARCH FOR THE ROOTS OF A AND B C C FOR FIXED VALUE OF THE C SUBROUTINE PROFILE(A,B,THE,TLN,ZT,DEAD,LIKE) IMPLICIT REAL(A-H,L, O-Z) COMMON T(2500,2),IND(2500,2),TAU(2500),Z(2500,9),IZ(10,2),NZ(2),N DIMENSION B (10,2),B N (10,2),U(22),D U (22, 22) ,TLN(2),DEAD(2) DIMENSION A( 2 ) ,Z T (10,2),S Z (2),S C (22,22) , AN (2),S C I (22, 22 ) ,E(22) PARAMETER (PI=3.141592653589793238) BLEND=2 NPAR-2+NZ(1)+NZ(2) M**NZ (1) +2 LIKEO=FLOAT(N)* (ALOG(2.)-0.5*ALOG(PI)) & -FLOAT(N)/4.*ALOG(THE) 167 DO 1 K-1,2 DO 2 J-1,NZ(K) 2 LIKEO«LIKEO+B(J,K)*ZT(J,K) 1 LIKEO-LIKEO+DEAD (K) *ALOG (A (K)) + (A EA— 0.5/H**l.5/SQRT(THE)* (HI*ALOG(T(I,1)))**2 + & 1./SQRT(H*THE)*Hl*ALOG(T(I,1))**2 ETAU— 0.5/G**l.5/SQRT(THE)* (Gl*ALOG(TAU(I) ) ) **2 + & 1./SQRT(G*THE)*Gl*ALOG(TAU(I))**2 U (1)-U(1) +G1*ALOG(TAU(I))/SQRT(THE*G) - & Hl*ALOG(T(I,1))*(RIND/2./H+ RB/SQRT(THE*H)) D U (1,1)=DU(1,1)+ETAU-RIND/2.*(HI*ALOG(T(I,1))**2/H & - (Hl*ALOG(T(I,1)))**2/H**2) - EA*RB - & DERB*(Hl*ALOG(T(I,1)))**2/(H*THE) EA— 0.5/H**l. 5/SQRT (THE)* (H2*ALOG(T(I,2)))**2 + & 1./SQRT(H*THE)*H2*ALOG(T (1, 2) ) **2 ETAU— 0.5/G**l.5/SQRT(THE)* (G2*ALOG(TAU(I)))**2 + & 1./SQRT(G*THE)*G2*ALOG(TAU(I))**2 U(M)=U(M)+G2*ALOG(TAU(I))/SQRT(THE*G) - & H2*ALOG(T(1,2))*(RIND/2./H+ RB/SQRT(THE*H) ) DU(M,M)“DU(M, M)+ETAU-RIND/2.*(H2*ALOG(T(I,2))**2/H & - (H2*ALOG(T(I,2)))**2/H**2) - EA*RB - & DERB*(H2*ALOG(T(I,2)))**2/(H*THE) EA“-0.5/H**l.5/SQRT (THE)*Hl*ALOG(T(I,1))*H2*ALOG(T(I,2)) ETAU— 0.5/G**l.5/SQRT(THE)*Gl*ALOG(TAU(I))*G2*ALOG(TAU(I)) DU (1, M) -DU (1, M) +ET AU+RIND / 2 . *H2 * ALOG (T (I, 2) ) * & Hl*ALOG(T(I,1))/H**2 - EA*RB - & DERB*Hl*ALOG(T (1,1))*H2*ALOG(T(I,2 ) ) / (H*THE) DU(M,1)—DU(1,M) DO 20 J—1,N Z (1) EA— 0.5/H**l.5/SQRT(THE)*Hl*ALOG(T(I,1))*H1*Z(I,IZ(J,1)) & +1./SQRT(H*THE)*Hl*ALOG(T(1,1))*Z(I,IZ(J, 1)) ETAU— 0.5/G**l.5/SQRT(THE)*Gl*ALOG(TAU(I))*G1*Z(I,IZ(J,1)) & +1./SQRT(G*THE)*Gl*ALOG(TAU(I) ) *Z(I,IZ(J, 1) ) U(J+l)—U(J+l)+G1*Z(I,IZ(J,1))/SQRT(THE*G) - & H1*Z(I, IZ(J,1))*(RIND/2./H+ RB/SQRT(THE*H)) D U (1,J+l)-DU(1,J+l)+ETAU-RIND/2.*(HI*ALOG(T(1,1) ) * & Z(I,IZ(J,1))/H-Hl*ALOG(T(I,1))*H1*Z(I,IZ(J, 1))/H**2) - & EA*RB - DERB*Hl*ALOG(T(1,1))*H1*Z(I,IZ(J,1))/(H*THE) EA— 0.5/H**l.5/SQRT(THE)*H2*ALOG(T(I,2))*H1*Z(I,IZ(J,1)) ETAU— 0.5/G**l.5/SQRT(THE)*G2*ALOG(TAU(I))*G1*Z(I, IZ(J,1)) DU(M,J+l)-DU(M,J+l)+ETAU+RIND/2.* & H2*ALOG (T (I, 2) ) *H1*Z (I, IZ (J, 1) ) /H**2 - & EA*RB - DERB*H2*ALOG(T(I,2))*H1*Z(I,IZ(J,1))/(H*THE) DU(J+l,1)=DU(1,J+l) DU (J+l, M) -DU 777 LIKE-LIKEO RETURN END C****************************************************** q C THIS SUBROUTINE SEARCH FOR MLE'S OF A, B AND THE C C USING MARQUART'S ITERATIVE PROCEDURE C SUBROUTINE FULL(A,B,THE,V,TLN,ZT,DEAD,LIKE) IMPLICIT REAL(A-H,L, O-Z) COMMON T (2500,2),IND(2500,2),TAU(2500),Z(2500,9),IZ(10,2),NZ(2),N DIMENSION B (10,2),BN(10,2),U(23),DU(23,23),TLN(2),DEAD(2), &V(23,23),A(2),ZT(10,2),SZ(2),SC(23,23),A N (2), SCI(23,23),E (23) PARAMETER (PI-3.141592653589793238) BLEND=2 N P =3 +NZ (1)+ N Z (2) M - N Z (1) +2 LI KEO=F LOAT(N)* (ALOG(2.)-0.5*ALOG(PI)) & -FLOAT(N)/4.*ALOG(THE) DO 1 K=l,2 DO 2 J=1,NZ(K) 2 LIKEO-LIKEO+B(J,K)*ZT(J,K) 1 LIKEO-LIKEO+DEAD(K)*ALOG(A (K)) + (A(K)-1.)*TLN(K) DO 3 1=1,N DO 4 K=l,2 SZ(K)=0 DO 4 J=l,NZ(K) 4 SZ(K)=SZ(K)+B(J,K)*Z(I,IZ(J,K) ) G=TAU(I)**A(1)*EXP(SZ(1))+TAU(I)**A(2)*EXP(SZ(2)) G- G + l ./THE H-T(I,1)**A(1)*EXP(SZ(1))+T(I,2)**A(2)*EXP(SZ(2)) H- H+ l./THE D—F L O A T (IN D (1,1)+ I N D (1,2)) C=SQRT(4.*H/THE) RIND-D-0.5 CALL BESS3(RIND,C,RK) CTA U-S QR T(4.*G/THE) LIKEO—LIKEO-D/2.*ALOG(THE)-RIND/2.*ALOG(H) +ALOG(RK) 3 LIKEO—LIKEO+CTAU DO 999 IT-1,50 IB-0 U (1)—D E A D (1)/A (1)+ T L N (1) DU (1,1)— DEAD (1) /A(l) **2 U(M)—D E A D (2)/ A (2)+ T L N (2) DU (M,M) — DEAD (2) /A<2) **2 DU(M,1)=0 DU(1,M)— 0 DU(NP,1) —0 DU (NP,M) —0 DU(1,NP)— 0 DU(M,NP)— 0 DO 5 J—1,NZ(1) U(J+l)—ZT(J,1) 5 DU(JJ, J+1)=0 DU(JJ, 5 SZ (K)=SZ(K)+ 7 o o DU ( oJJ» J+M) =0 6 5 1. / SQRT (H*THE) *Hl*ALOG(T 1.(I, (H*THE) SQRT 1)/ *Hl*ALOG(T ) **2 5 & -(Hl*ALOG(T(I, 1) ) -(Hl*ALOG(T(I, ) **2/H**2) & 6 1. / SQRT 1. (G*THE) (TAU )) SQRT (I / *Gl*ALOG **2 6 & Hl*ALOG(T(I, 1) )* (RIND/2./H+ RB/SQRT (THE*H) ) RB/SQRT 1) )* (RIND/2./H+ Hl*ALOG(T(I, & 1. / (G*THE)(TAU SQRT (I)*G2*AL0G ) & **2 1. / (H*THE)SQRT *H2 *ALOG (T ,&(1 2) ) (1,1)) )**2/ (Hl*ALOG(T (H*THE) DERB* **2 & & H2*ALOG (T(I,2) )* H2*ALOG (RIND/2 . (THE RB/SQRT *H)) /H+ & & DERB* (H2*ALOG - (T DERB* EA*RB (I,) - 2)) )) -(H2*ALOG(T(I,2) **2/H**2) & **2/ (H*THE) & DU(l,l)-DU(l,l)+ETAU-RIND/2.* (Hl*ALOG(T(I, 1) )(Hl*ALOG(T(I, **2/H DU(l,l)-DU(l,l)+ETAU-RIND/2.* UMM U D (M,M)+DU(M,M) ® DU DU (1,M) U D .(1,M)- (T *H2*ALOG +ETAU+RIND/2 ,(1 2) ) * U U (1) U (1)= +G1 U *AL0G A (T (I)) - (THE*G) /SQRT (THE) )* ) EA--0 (Hl*ALOG(T(1,1) .5/H**l.5/SQRT + **2 ETAU— 0.5/G**l.5/SQRT (THE) (I)* )(Gl*ALOG(TAU 0.5/G**l.5/SQRT ) + **2 ETAU— U(M)=U (M) U(M)=U U A (T (I)+G2*ALOG ) / - (THE*G)SQRT TU 0.5/G**l. ETAU— EA TU 0 .ETAU— 5 0 /G*. (THE)(T.5/SQRT EA— * 5/H**l *Hl*ALOG (1,1)1 )(I, 2))*H2*ALOG(T =-0 O JJ=1,NP 6 DO U( =ZT J=1,NZ(2) J+M) (J,2 6 ) DO O 0 1=1, 10 N DO (N) / (DEAD DU + (NP,NP)=FLOAT 2 (1) 4./THE* * (2)+DEAD -FLOAT “ (N))U(NP) /4 ./THE- *0.5/THE**2 (1)(DEAD (2)+DEAD )*0. 5/THE DU DU (J+M, JJ) “0 JJ=1,NP 5 DO DU DU (J+l* JJ)=0 D=FL0AT(IND(I,1)+IND(I,2) ) D=FL0AT(IND(I,1)+IND(I,2) J=1,NZ(K) 7 DO K=l, 2 7 DO =l+ + ; 2+l +G /THE G=Gl ) *EXP(SZ(1) G1=TAU(I)**A(1) (SZ*EXP (1) H1=T(I,1)**A(1) ) DERB-DERB+RIND / C DERB-DERB+RIND * * 2 C=SQRT(4.*H/THE) .5 0 - D - D N RI H»Hl+H2+l./THE G2*TAU(I) **A(2) (SZ*EXP G2*TAU(I) (2) ) (1,2) H2=T **A(2) (SZ*EXP (2) ) CALL DRBESS (RIND, DRBESS C, DERB) CALL b r l l a c SZ (K) =0 .5/H**l. 5/SQRT (THE) 5/SQRT .5/H**l. * (T(H2*ALOG (I, 2) ) ) + **2 - b b s s e b r - d n i r ( / c d n i r .5 b e 5 / (THE)SQRT *G1 * ALOG (TAU )(I ) *G2 * ALOG (TAU (I) ) TAU-RIND/2.* (H2*ALOG(T(I,2) ) **2/H (H2*ALOG(T(I,2) TAU-RIND/2.* /SQRT (THE)* (TAU (G2*ALOG /SQRT (I) ) ) + **2 (J,K)*Z(I,IZ(J,K)) , c , b r ) -EA*RB- 172 173 6 HI*ALOG(T(I,1) )/H**2 - EA*RB - & DERB*Hl*ALOG(T(I,1))*H2*ALOG(T(1,2))/(H*THE) DU(M,1)-DU(1,M) Q— SQRT(H)/THE**1.5 - 1./SQRT(H)/THE**2.5 QTAU— SQRT(G)/THE**1.5 - 1./SQRT (G)/THE**2 . 5 DQA— 0.5/SQRT (H) /THE**1.5 + 0 .5/H**l. 5/THE**2 .5 DQATAU— 0.5/SQRT (G)/THE**1.5 + 0.5/G**l.5/THE**2.5 DQT-1.5*SQRT(H)/THE**2.5 + 3./SQRT(H)/THE**3.5 6 - 0•5/H**l.5/THE**4.5 DQTTAU—1.5*SQRT(G)/THE**2.5 + 3./SQRT(G)/THE**3.5 & - 0.5/G**l.5/THE**4.5 U(NP)-U(NP)+ QTAU + RIND/2./H/THE**2 - RB*Q DU(NP,NP)-DU(NP,NP)+ DQTTAU + RIND/2.*(1./H**2/THE**4 - & 2./H/THE**3) -DQT*RB - Q**2*DERB DU (NP, 1) =DU (NP, 1) +Gl*ALOG (TAU (I) ) *DQATAU - Hl*ALOG (T (I, 1) ) & * (RIND/2./(H*THE)**2 + RB*DQA + Q*DERB/SQRT(THE*H)) DU(NP,M)=DU(NP,M)+G2*ALOG(TAU(I))*DQATAU - H2*ALOG(T(I,2)) & * (RIND/2./(H*THE)**2 + RB*DQA + Q*DERB/SQRT(THE*H)) DU (1, NP )=DU(NP,1) DU (M, NP) “DU (NP,M) DO 20 J=1,NZ(1) EA— 0.5/H**l.5/SQRT(THE)*Hl*ALOG(T(I,1))*Hl*Z(I,IZ(J,1)) & +1. /SQRT (H*THE) *Hl*ALOG (T (I, 1) ) *Z (I, IZ (J, 1) ) ETAU— 0 . 5/G**l .5/SQRT (THE) *Gl*ALOG(TAU (I) ) *G1*Z (I, IZ (J, 1) ) & +1./SQRT(G*THE)*Gl*ALOG(TAU(I))*Z(I,IZ (J,1)) U(J+l)—U(J+l)+G1*Z(I,IZ(J, 1))/SQRT(THE*G) - & H1*Z(I,IZ(J,1))*(RIND/2./H+ RB/SQRT(THE*H)) DU(1,J+l)-DU(1,J+l)+ETAU-RIND/2.*(HI*ALOG(T(I,1))* & Z(I,IZ(J,1))/H-H1*AL0G(T(I,1))*H1*Z(I,IZ(J,1))/H**2) - & EA*RB - DERB*Hl*ALOG(T(1,1))*H1*Z(I,IZ(J,1))/(H*THE) EA— 0.5/H**l.5/SQRT(THE)*H2*ALOG(T(I, 2))*H1*Z(I, IZ(J, 1) ) ETAU— 0 .5/G**l .5/SQRT (THE) *G2 * ALOG (TAU (I))*G1*Z(I,IZ(J,1)) DU(M,J+l)-DU(M,J+l)+ETAU+RIND/2.* & H2*ALOG(T(I, 2) )*H1*Z(I,IZ(J,1))/H**2 - & EA*RB - DERB*H2*ALOG(T(I,2))*H1*Z(I,IZ(J,1))/(H*THE) DU(J+l,1)—D U (1,J+l) DU (J+l, M)-DU (M, J+l) DU (NP, J+l) —DU (NP, J+l) +G1*Z (I, IZ (J, 1) ) *DQATAU- H1*Z (I, IZ (J, 1) ) & * (RIND/2./(THE*H)**2 + DQA*RB + Q*DERB/SQRT(THE*H)) DU(J+l,NP)=DU(NP,J+l) DO 21 J1=J,NZ(1) EA— 0 . 5/H**l. 5/SQRT (THE) *H1*Z (I, IZ (Jl, 1) ) *H1*Z (I, IZ (J, 1) ) & +1./SQRT(H*THE)*H1*Z(I, IZ(Jl,1))*Z(I,IZ(J,1)) ETAU— 0 . 5/G**l. 5/SQRT (THE) *G1*Z (I, IZ (Jl, 1) ) *G1*Z (I, IZ (J, 1) ) & +1. /SQRT (G*THE) *G1*Z (I, IZ (Jl, 1) ) *Z (I, IZ (J, 1) ) DU(Jl+1,J+l)-DU(Jl+1,J+l)+ETAU-RIND/2.*(H1*Z(I,IZ(Jl,1))* & Z(I,IZ(J,1))/H-H1*Z(I,IZ(J1,1))*H1*Z(I,IZ(J,1))/H**2) - & EA*RB - DERB*H1*Z(I,IZ(Jl,1))*H1*Z(I,IZ(J,1))/(H*THE) DU (J+l, Jl+1) -DU (Jl+1, J+l) 21 CONTINUE DO 22 J2-1,NZ(2) EA— 0.5/H**l.5/SQRT(THE)*H2*Z(I,IZ(J2, 2))*H1*Z(I, IZ(J, 1) ) 174 ETAU— 0.5/G**l.5/SQRT(THE)*G2*Z(I,IZ(J2,2))*G1*Z(I,IZ(J,1)) D U (J2+M,J+l)=DU(J2+M,J+l)+ETAU+RIND/2.* & H2*Z(I, IZ(J2,2))*H1*Z(I,IZ(J,1))/H**2 - & EA*RB - DERB*H2*Z(I,IZ(J2,2))*H1*Z(I, IZ(J,1))/(H*THE) DU(J+l,J2+M)-DU(J2+M,J+l) 22 CONTINUE 20 CONTINUE DO 30 J“l,NZ(2) EA— 0.5/H**l. 5/SQRT (THE) *H2*ALOG(T(I,2) ) *H2*Z (I, IZ (J,2) ) & +1./SQRT(H*THE)*H2*ALOG(T(I,2))*Z(I, IZ(J,2) ) ETAU— 0.5/G**l. 5/SQRT (THE) *G2*ALOG (TAU (I))*G2*Z(I,IZ(J,2)) & +1. / SQRT (G*THE) *G2*ALOG(TAU(I) ) *Z (I, IZ (J,2) ) U(J+M)=U(J+M)+G2*Z(I,IZ(J,2))/SQRT(THE*G) - & H2*Z(I,IZ(J,2))*(RIND/2./H+ RB/SQRT(THE*H)) DU (M, J+M) =DU (M, J+M) +ETAU-RIND/2 . * (H2*ALOG (T (I, 2) ) * & Z (I, IZ (J, 2) ) /H-H2*ALOG (T (I, 2) ) *H2*Z (I, IZ (J,2) ) /H**2) - & EA*RB - DERB*H2*ALOG(T(I,2))*H2*Z(I, IZ(J,2))/(H*THE) EA— 0.5/H**l. 5/SQRT (THE)*Hl*ALOG(T(I, 1) ) *H2*Z(I,IZ(J,2)) ETAU— 0.5/G**l. 5/SQRT (THE)*Gl*ALOG(TAU(I))*G2*Z(I,IZ (J,2)) D U (1,J+M)=DU(1,J+M)+ETAU+RIND/2.* & Hl*ALOG (T (I, 1) ) *H2*Z (I, IZ (J, 2) ) /H**2 - & EA*RB - DERB*Hl*ALOG(T(1,1))*H2*Z(I, IZ(J,2))/(H*THE) DU (J+M, 1) —DU (1, J+M) DU (J+M, M) “DU (M, J+M) DU (NP, J+M)“DU(NP,J+M)+G2*Z(I,IZ(J,2))*DQATAU- H2*Z(I,IZ(J,2)) & * (RIND/2./(THE*H)**2 + DQA*RB + Q*DERB/SQRT(THE*H)) DU(J+M,NP)-DU(NP,J+M) DO 31 J1— J , N Z (2) EA— 0.5/H**l.5/SQRT(THE)*H2*Z(I,IZ(Jl, 2) )*H2*Z(I, IZ (J,2)) & +1./SQRT(H*THE)*H2*Z(I,IZ(Jl,2))*Z(I,IZ(J,2)) ETAU— 0.5/G**l.5/SQRT(THE)*G2*Z(I, IZ(Jl, 2))*G2*Z(I,IZ(J,2)) & +1./SQRT(G*THE)*G2*Z(I,IZ(Jl,2))*Z(I,IZ (J, 2)) DU(Jl+M,J+M)“DU(Jl+M,J+M)+ETAU-RIND/2.*(H2*Z(I,IZ(J1,2))* & Z(I,IZ(J,2))/H-H2*Z(I,IZ(Jl,2))*H2*Z(I,IZ(J,2))/H**2) - & EA*RB - DERB*H2*Z(I,IZ(Jl,2))*H2*Z(I, IZ(J,2))/(THE*H) DU (J+M, Jl+M) “DU (Jl+M, J+M) 31 CONTINUE 30 CONTINUE 10 CONTINUE 70 DO 33 J“ 1,NP DO 33 K“1,NP 33 SC (J, K) — DU (J, K) /SQRT (ABS (DU (J, J) *DU (K, K) ) ) DO 34 K“1,NP 34 SC(K,K)“BLEND+SC(K,K) CALL LINRG(NP,SC,23,SCI,23) EMAX— 100 UMAX— 100 DO 35 J“l,NP E (J) —0 IF(ABS(U(J)) .GT.UMAX) UMAX-ABS(U(J) ) DO 36 K—l,NP 175 36 E (J) - E (J)+ (SCI(J,K)/SQRT(DU(J,J)*DU(K,K)))*U(K) 35 EMAX—AMAX1(EMAX,ABS(E(J))) A N (1)= A (1)+ E (1) DO 37 J=1,NZ(1) 37 BN (J, 1) —B (J, 1) +E (J+l) AN (2) =A (2) +E (M) DO 38 J=1,NZ(2) 38 BN (J, 2) =B (J, 2) +E (M+J) THEN—THE+E(NP) LIKEN=FLOAT(N)* (ALOG(2.)-0.5*ALOG(PI)) -FLOAT(N)/4.*ALOG(THEN) DO 41 K=l,2 DO 42 J=1,NZ(K) 42 LIKEN=LIKEN+BN(J, K)*ZT(J,K) 41 LIKEN-LIKEN+DEAD (K) *ALOG (AN (K) ) + (AN (K) -1.) *TLN (K) DO 43 1=1,N DO 44 K=l,2 SZ(K)=0 DO 44 J=1,NZ(K) 44 SZ (K) =SZ (K) +B N (J,K) *Z (I, IZ (J,K) ) G=TAU(I)**AN(1)*EXP(SZ(1))+TAU(I)**AN(2)*EXP(SZ(2)) G=G+1./THEN H=T(1,1)**AN(1)*EXP(SZ(1))+T(I,2)**AN(2)*EXP(SZ(2)) H-H+l./THEN D-FLOAT(IND(I,1)+ I N D (1,2)) C—SQRT(4.*H/THEN) RIND-D-0.5 CALL BESS3(RIND,C,RK) CTAU-SQRT(4.*G/THEN) LIKEN—LIKEN-D/2.*ALOG(THEN)-RIND/2.*ALOG(H)+ALOG(RK) 43 LIKEN—LIKEN+CTAU IF(LIKEN.GT.LIKEO) GO TO 55 BLEND—BLEND * 3 W R I T E (6,*) 'IN FULL CORECT IN BL E N D ',BLEND IB-IB+1 IF(IB.L E .7) GO TO 70 WRITE(6,*) 'ERROR IN FULL' RETURN 55 DO 60 K-1,2 A (K) =AN (K) DO 60 J=1,NZ(1) 60 B (J, K) —BN (J, K) THE-THEN IF(UMAX.LE..01) GO TO 777 IF(EMAX.LE..001) GO TO 777 EMAX-ABS((LIKEO-LIKEN) ) IF(EMAX.LE..001) GO TO 777 LIKEO-LIKEN BLEND—BLEND/3 999 CONTINUE WRITE(6,*) 'NO CONV IN FULL',EMAX 777 LIKE—LIKEO 176 DO 88 J-1,NP DO 88 K-1,NP DU(J,K)—DU(J,K) WRITE(6,*) J,K, DU(J,K) 88 CONTINUE CALL LINRG (NP,DU, 23, V, 23) WRITE(6, *) ' *******VAR*********' DO 333 J=1,NP DO 333 K=1,NP WRITE (6, *) J,K, V (J, K) 333 CONTINUE RETURN END C THE FOLLOWING ROUTINES COMPUTE THE BESSEL FUNCTION, BESS3 C C THE RATIO OF TWO BESSEL FUNCTIONS, RBESS C C THE DERIVATIVE OF RBESS, DRBESS C SUBROUTINE BESS3 (RIND, W, RK) PARAMETER(PI=3.141592653589793238) SUMW=0.0 IF (ABS(RIND) .EQ. 0.5) SUMW=0.0 IF(RIND .EQ. 1.5) SUMW-1./W IF (RIND .EQ. 2.5) SU M W = 3 ./W+3./W**2 IF(RIND .EQ. 3.5) SUMW=6./W+15./W**2+15./W**3 RK=SQRT(PI/2.)*(1.+SUMW)*EXP(-W)*W**(-0.5) RETURN END C C SUBROUTINE RBESS (RIND, W, RB) IF (RIND .GT. 0.) THEN CALL BESS3 (RIND,W,RK) CALL BESS3(RIND+1.,W,RK1) RB- RK1/RK ELSE RI—RIND CALL BESS3 (RI,W,RK) CALL BESS3(RI+1. , W,RK1) RB-RK1/RK-2.*RI/W END IF RETURN END C C SUBROUTINE DRBESS(RIND,W,DRB) CALL BESS3(RIND,W,RK) CALL BESS3(RIND+2,W,RK2) CALL RBESS(RIND,W,RB) DRB»RB**2-RIND/W*RB-0.5-0.5*RK2/RK RETURN END A P P E N D IX E FORTRAN Program of Parametric Estimation for Model II Q-k-kifk ****************************** ************************ c********★★**★**★*★***PROGRAM************************* C********************************************************** IMPLICIT REAL C ------o o o o o o o RE A D (3,*) N,NZ,NG WRITE(6,100) N,NZ,NG 100 FORMAT(IX,'SAMPLE SIZE-',15,' NUM. BETAS-',15,' NUM. GROUPS=',I5) NZ-NZ+1 DO 1 J — 1,N R E A D (3,*) T (J),T A U (J),I (J) ,I G (J) , (Z (J,K),K=2,NZ) Z (J, 1) —1. 1 CONTINUE A— 1.0 DO 2 J—1,NZ 2 BO(J)-0.0 DO 3 J—l,NZ 3 Z T (J)— 0 DO 4 J=1,N DO 4 K—1,NZ 4 ZT (K) —ZT (K) +Z (J, K) *FLOAT (I (J) ) TLN-0 DEAD-0 DO 5 J=1,NG 5 D (J) —0 DO 6 J— 1,N K-IG(J) D (K) -D (K) +FLOAT (I (J) ) DEAD-DEAD+FLOAT(I (J) ) TLN—TLN+FLOAT(I (J))*ALOG(T(J) ) 177 178 6 CONTINUE C------C C PERFORM WEIBULL REGRESSION ANALYSIS UNDER INDEPENDENCE C C------C CALL INDEP (A,BO, TLN, ZT,DEAD,LIKETI) WRITE(6,*) 'LIKELIHOOD UNDER INDEPENDENCE* LIKETI. LIKEO-LIKETI THEI-0.0 C------C C PROFILE SEARCH FOR ESTIMATES OF A, B AND THE C C------C DO 7 IQ=58,60 THE=5*FLOAT (IQ)/100 CALL PROFILE (A, BO, THE, TLN, ZT, DEAD, LIKE) WRITE (6,*) ' THETA— ', THE, 'LIKE*', LIKE LIKEN-LIKE IF (LIKEN .GT. LIKEO) THEN THEI=THE AI-A DO 8 J— 1,NZ 8 BI(J)=BO(J) LIKEO-LIKEN END IF 7 CONTINUE IF (THEI .EQ. 0.) G O TO 999 CALL FULL (AI,BI, THEI, V, TLN, ZT, DEAD, LIKEM) WRITE(6,*) 'FINAL LIKELIHOOD- *, LIKEM SE-SQRT(V(l,1)) WRITE(6, 77) AI,SE 77 FORMAT(' A-',F16.8,2X, ' SE«',F16.8) DO 9876 J— 1,NZ SE-SQRT (V (J+l, J+l)) PVAL-2 . * (1. -ANORDF (ABS (BI (J) /SE) ) ) WRITE(6, 8761) J,BI (J) ,SE,PVAL 8761 FORMAT(' B (•,12, • )-',F13.8,* SE— ',F13.8,2X,'PVALUE— ',F 7 .5) 987 6 CONTINUE SE-SQRT (V (NZ+2, NZ + 2 ) ) TEST—THEI/SE PVAL— 1. -ANORDF (ABS (TEST) ) WRITE (6, 771) THEI, SE, PVAL 771 FORMAT ( ' ESTIMATE O F T H E - ', F12 .7, 2X, ' SE= ’ , F16. 8, 2X,'PVAL= ', F7 . 4) DO 35 J=l,NZ+2 DO 35 K— l,NZ+2 WRITE(6,108) J,K,V(J,K) 108 FORMAT ('ESTIMATED COVARIANCE V (', 12, ', •, 12, ') =', F20 .10) 35 CONTINUE 999 STOP END C THIS SUBROUTINE PERFROMS WEIBULL REGRESSION ANALYSIS C C UNDER INDEPENDENCE ASSUMPTION C 179 SUBROUTINE INDEP(A,B,TLN,ZT,DEAD,LIKE) IMPLICIT REAL(A-H, L,O-Z) COMMON N,NZ,NG,T(5000),1(5000),IG(5000), TAU(5000),Z(5000,26) & , D (5000) DIMENSION B N (26), Z T (26),B(26),F (27) ,F 2 (27,27) , FI (27,27) , E (27) DIMENSION SC (27,27) BLEND=2.0 NPAR-NZ+1 LIKEO=DEAD*ALOG(A)+ (A-l.)*TLN DO 820 J=1,NZ 820 LIKEO=LIKEO+B(J) * Z T (J) DO 850 11=1,N SB=0 DO 830 J=1,NZ 830 SB=SB+B(J)*Z(II,J) ET=T(II)* *A*EXP(SB) ETAU=TAU(II)**A*EXP(SB) 850 LIKEO=LIKEO-ET+ETAU DO 98 IT=1,50 F (1)=DEAD/A+TLN F 2 (1,1)=-DEAD/A**2 DO 2 J=2,NPAR F (J)“Z T (J-l) DO 2 JJ=1,NPAR F2 (J, JJ) =0 . F2(JJ,J)=0. 2 CONTINUE DO 80 11=1,N SB = 0 . DO 3 J=l,NZ 3 SB=SB+B(J)*Z(II, J) ET-EXP(SB+A*ALOG(T(II))) ETAU=EXP(SB+A*ALOG(TAU(II) ) ) F (1)= F (1)-ET*ALOG(T(II))+ETAU*ALOG(TAU(II)) F2 (1,1)=F2(1,1)-ET*(ALOG(T(II)))**2+ETAU*(ALOG(TAU(II)))**2 DO 4 J=l,NZ F (J+l)=F(J+l)-ET*Z(II,J)+ETAU*Z(II, J) F2 (1,J+1)=F2(1,J+l)-ET*ALOG(T(II))*Z(II,J) F2 (1,J+1)=F2(1,J+l)+ETAU*ALOG(TAU(II))*Z(II,J) F2 (J+l, 1) =F2 (1, J+l) DO 4 JJ=J,NZ F2 (J+l, JJ+1) =F2 (J+l, JJ+1) + (ETAU-ET) *Z (II, J) *Z (II, JJ) F2 (JJ+1, J+l) =F2 (J+l, JJ+1) 4 CONTINUE 80 CONTINUE 591 CONTINUE DO 33 J=1,NPAR DO 33 K=l,NPAR 33 SC(J,K)=-F2(J,K)/SQRT(ABS(F2(J,J)*F2(K,K))) DO 34 J-l,NPAR SC(J,J)=SC(J,J)+BLEND 34 CONTINUE 180 CALL LINRG(NPAR,SC, 27,FI,27) IBLEND-0 EMAX— 100 FMAX— 100 DO 5 J-l,NPAR E(J)-0 DO 6 JJ-l,NPAR 6 E (J)-E(J)+F(JJ)*FI(J,JJ)/SQRT(F2(J,J)*F2(JJ,JJ)) EMAX—AMAX1(ABS(E(J)),EMAX) 5 FMAX-AMAX1(ABS(F(J)),FMAX) AN—A + E (1) IF(AN.LT.0.) GO TO 811 DO 7 J=1,NZ B N (J)=B(J)+E(J+l) 7 CONTINUE LIKEN—DEAD*ALOG(AN)+ (AN-1.)*TLN DO 520 J=1,NZ 520 LIKEN—LIKEN+BN(J)*ZT(J) DO 550 K=1,N SB=0 DO 530 J=1,NZ 530 SB-SB+BN(J)*Z(K, J) ET-EXP(SB+AN*ALOG(T(K))) ETAU-EXP(AN*ALOG(TAU(K))+SB) 550 LIKEN-LIKEN-ET+ETAU IF(LIKEN.GT.LIKEO) GO TO 711 811 BLEND—BLEND*3 IBLEND—IBLEND+1 WRITE(6,*) ' IN INDEP CORRECTION',BLEND IF(IBLEND.LT.10) GO TO 591 LIKEO-LIKEN 711 A-AN DO 743 J—1,NZ 743 B (J) —BN (J) EMAX-ABS((LIKEO-LIKEN)) IF(EMAX.LE..001) GO TO 70 LIKEO-LIKEN BLEND—BLEND/3. IF(FMAX.LE..01) GO TO 70 IF(EMAX.LE..001) GO TO 70 98 CONTINUE WRITE (6,*) 'NO CONVERGENCE BASED ON INDEPENDENCE' RETURN 70 CONTINUE CALL LINRG(NPAR,F2, 27,FI,27) SE-SQRT(-FI(1,1)) W R I T E (6,*) 'A-*,A,' SE— ',SE DO 10 J—1,NZ SE-SQRT(-FI(J+l,J+l) ) ZQ—B (J)/SE P—2.*(1.O-ANORDF(DABS(ZQ))) W R I T E (6,101) J , B (J) , SE,ZQ,P 181 101 FORMAT(' B(«,I2, ')=',F12.6,2X, «SE=’,F12.6, ' Z= ' , F10 . 4, 2X, 'P=',F7.4) 10 CONTINUE LIKE=DEAD*ALOG(A)+ (A-l.)*TLN DO 20 J=l,NZ 20 LIKE-LIKE+B(J)*ZT(J) DO 50 K-1,N SB=0 DO 30 J-l,NZ 30 SB=SB+B(J)*Z(K, J) ET-T(K)**A*EXP(SB) ETAU-TAU(K)**A*EXP(SB) 50 LIKE=LIKE-ET+ETAU WRITE(6,*) 'LOG LIKELIHOOD(INDEPENDENCE)=',LIKE RETURN END C THIS SUBROUTINE SEARCH FOR THE ROOTS OF A AND B C C FOR FIXED VALUE OF THE C SUBROUTINE PROFILE(A,B,THE,TLN,ZT,DEAD,LIKE) IMPLICIT REAL (A-H,L,O-Z) COMMON N,NZ,NG,T(5000),1(5000),IG(5000),TAU(5000),Z(5000,26) & , D (5000) DIMENSION B N (26) , ZT (26),B (26),U(27),D U (27,27),SCI(27,27) ,E (27) DIMENSION SC(27,27),SH(5000),S G (5000),C (5000),EBZ(5000) ,DHA(5000) DIMENSION DHAA(5000),DGA(5000),DGAA(5000),RB(5000),DERB(5000) & ,HM1(5000),HM2(5000),HM12(5000),GM1(5000),GM2(5000),GM12(5000) PARAMETER (PI-3.141592653589793238) BLEND-2 LIKEO-FLOAT(NG)* (ALOG(2.)-0.5*ALOG(PI)) & -FLOAT(NG)/4.*ALOG(THE) LIKEO—LIKEO+DEAD*ALOG(A)+ (A-l.)*TLN DO 11 J— 1,NG SH (J) —0 . SG (J)— 0 . 11 CONTINUE DO 12 J«1,N K-IG(J) BZ-0. DO 13 M=1,NZ 13 BZ—B Z + Z (J,M)*B(M) EBZ(J)-EXP(BZ) SH (K) —SH (K) +T (J) **A*EBZ (J) SG(K)-SG(K)+TAU(J)**A*EBZ(J) 12 CONTINUE DO 2 J— 1,N IF ( I(J) .EQ. 0) GOTO 2 LIKEO—LIKEO+ALOG(EBZ(J) ) 2 CONTINUE DO 3 K— 1,NG SH(K)—SH(K)+1./THE SG(K)—SG(K)+1./THE C (K) -SQRT(4.*SH (K) /THE) RIND=D (K) -0.5 CALL BESS3(RIND,C (K),RK) CTAU-SQRT(4.*SG EA— 0 .5/SH (K) **1. 5/SQRT (THE) *DHA (K) **2 + 1./SQRT(SH(K)*THE)*DHAA(K) ETAU— 0.5/SG(K)**l. 5/SQRT (THE) *DGA(K)**2 + 1./SQRT(SG(K)*THE)*DGAA(K) U(l)-U(l) +DGA(K) /SQRT (THE*SG(K) ) - DHA(K)* ( (D(K)/2.— 0.25)/SH(K) + RB(K)/SQRT(THE*SH(K))) 183 D U (1,1)=DU(1,1)+ETAU-(D(K)/2.-0.25)*(DHAA(K)/SH (K) - & DHA(K)**2/SH(K)**2) - EA*RB(K) - & DERB(K)*DHA(K)**2/(SH(K)*THE) 10 CONTINUE DO 20 J=1,NZ DO 25 K=1,NG HM1(K)=0. GM1 24 CONTINUE DU (J+l, Jl+1) -DU (Jl+1, J+l) 21 CONTINUE 20 CONTINUE C******************************************************************* 70 CONTINUE DO 33 J-l,NZ+1 DO 33 K=1,NZ+1 33 SC (J, K) — DU(J,K)/SQRT(DABS(DU(J,J)*DU(K,K) ) ) DO 34 K»1,NZ+1 34 SC(K,K)-BLEND+SC(K,K) CALL LINRG(NZ+1,SC,27,SCI,27) EMAX=-100 UMAX=-100 DO 35 J=l,NZ+1 E (J )=0 IF(ABS(U(J) ) .GT.UMAX) UMAX=ABS(U(J)) DO 36 K— 1,NZ+1 36 E (J)=E(J) + (SCI(J,K)/SQRT(DU(J,J)*DU(K,K)))*U(K) 35 EMAX-AMAX1 (EMAX, ABS (E (J) ) ) AN-A+E(1) DO 37 J-1,NZ 37 BN(J)-B(J)+E(J+1) LIKEN-FLOAT(NG)* (ALOG(2.)-0.5*ALOG(PI)) & -FLOAT(NG)/4.*ALOG(THE) LIKEN-LIKEN+DEAD*ALOG(AN)+ (AN-1.)*TLN DO 41 K-1,NG SH(K)“ 0. SG(K)-0. 41 CONTINUE DO 47 M«1,N K-IG (M) BZ-0. DO 42 J=1,NZ 42 BZ-BZ+Z(M,J)*BN(J) EBZ (M) “EXP(BZ) SH (K) -SH (K) +T (M) **AN*EBZ (M) SG(K)-SG(K)+TAU(M)**AN*EBZ(M) 47 CONTINUE DO 43 J=1,N IF(I(J) .EQ. 0) GOTO 43 LIKEN-LIKEN+ALOG(EBZ(J)) 43 CONTINUE DO 44 K=1,NG SH(K)—SH(K)+1./THE SG(K)—SG(K)+1./THE C (K) -SQRT (4 . *SH (K) /THE) RIND-D (K) -0 . 5 CALL BESS3(RIND,C(K),RK) CTAU-SQRT(4.*SG(K)/THE) LIKEN—LIKEN-D(K)/2.*ALOG(THE)-RIND/2.*ALOG(SH(K))+ALOG(RK) 44 LIKEN-LIKEN+CTAU 185 IF(LIKEN.GT.LIKEO) GO TO 55 BLEND=BLEND * 3 W R I T E (6,*) 'IN PROFILE CORECT IN BLEND',BLEND IB-IB+1 I F (IB .L E .7) GO TO 70 WRITE(6,*) 'ERROR IN PROFILE' RETURN 55 A=AN DO 60 J=1,NZ 60 B (J) —BN (J) IF(UMAX.LE..01) GO TO 777 IF(EMAX.LE..001) GO TO 777 EMAX-ABS((LIKEO-LIKEN)) IF(EMAX.LE..001) GO TO 777 LIKEO=LIKEN BLEND=BLEND/3 999 CONTINUE WRITE(6,*) 'NO CONV IN PROFILE',EMAX 777 LIKE=LIKEO RETURN END C THIS SUBROUTINE SEARCH FOR MLE'S OF A, B AND THE C C USING MARQUART'S ITERATIVE PROCEDURE C £******************************************************£ SUBROUTINE FULL(A,B,THE,V,TLN,ZT,DEAD,LIKE) IMPLICIT REAL(A-H,L,O-Z) COMMON N,NZ,NG,T(5000),1 (5000),IG(5000),TAU(5000),Z (5000,26) & , D (5000) DIMENSION B N (26),ZT(26),B (26),U(28),DU(28,28),SCI(28,28),E (28) DIMENSION SC(28,28),SH(5000),SG(5000) ,C(5000),EBZ(5000),DHA(5000) DIMENSION DHAA(5000),DGA(5000),DGAA(5000),R B (5000),D E R B (5000) & , HM1 (5000) ,H M 2 (5000),HM12 (5000) ,GM 1(5000),GM2 (5000),GM12 (5000) DIMENSION Q (5000),V(28,28),DQA(5000),DQATAU(5000) PARAMETER (PI-3.141592653589793238) NP=NZ+2 BLEND=2 LIKEO-FLOAT(NG)* (ALOG(2.)-0.5*ALOG(PI)) & -FLOAT(NG)/4.*ALOG(THE) LIKEO=LIKEO+DEAD*ALOG(A)+ (A-l.)*TLN DO 11 J=1,NG S H (J)=0. S G (J)=0. 11 CONTINUE DO 12 J=1,N K = I G (J) BZ=0 . DO 13 M=1,NZ 13 BZ=BZ+Z(J,M)*B(M) EBZ(J)-EXP(BZ) SH(K)-SH(K)+T(J)**A*EBZ(J) SG (K) -SG (K) +TAU (J) **A*EBZ (J) CONTINUE DO 2 J»1,N IF ( I(J) .EQ. 0) GOTO 2 LIKEO-LIKEO+ALOG (EBZ(J)) CONTINUE DO 3 K=1,NG SH (K) «=SH (K) +1. /THE SG(K)“SG(K)+1./THE C(K)-SQRT(4.*SH(K) /THE) RIND—D(K)—0.5 CALL BESS3(RIND,C (K) ,RK) CTAU-SQRT(4.*SG(K) /THE) LIKEO-LIKEO-D(K)/2.*ALOG(THE) -RIND/2.*ALOG(SH(K))+ALOG(RK) LIKEO—LIKEO+CTAU DO 999 IT—1,50 IB-0 U (1)-DEAD/A+TLN D U (1,1)— DEAD/A**2 DU(1,NP)=0. DU(NP,1)—0. DO 5 J—l,NZ U (J+l) —ZT (J) DO 5 JJ—1,NP DU(J+l,JJ)—0 DU(JJ,J+l)—0 U (NP) — FLOAT (NG) /4 . / THE-DEAD*0 . 5/THE DU 187 188 & HMl(K)*((D(K)/2.-0.25)/SH(K) + RB(K)/SQRT(THE*SH(K)) ) DU(1, J+l)=DU(1, J+l)+ETAU-(D(K)/2.-0.25)* (HM12(K)/SH(K) - & HM1(K)*DHA(K)/SH(K)**2) -EA*RB(K) - & DERB (K) *HM1 (K) *DHA (K) / (SH (K) *THE) DU(NP, J+l)=DU(NP, J+l)+GM1(K)*DQATAU(K)-HM1(K)*((D(K)/2.-0.25) & /(THE*SH(K))**2 +DQA(K)*RB(K) +Q(K)*DERB(K)/SQRT(THE*SH(K))) 27 CONTINUE DU (J+l, 1) =DU (1, J+l) DU(J+l,NP)=DU(NP,J+l) DO 21 J1=J,NZ DO 22 K=1,NG HM12 (K) =0. GM12 (K) =0. HM2(K)=0. GM2(K)=0. 22 CONTINUE DO 23 M=1,N K=IG (M) HM2(K)=HM2(K)+T(M)**A*EBZ(M)*Z(M,Jl) GM2(K)=GM2(K)+TAU(M)**A*EBZ(M)*Z(M,Jl) HM12 (K) =HM12 (K) +T (M) **A*EBZ (M) *Z (M, J) *Z (M, Jl) GM12 (K) =GM12 (K) +TAU (M) **A*EBZ (M) *Z (M, J) *Z (M, Jl) 23 CONTINUE DO 24 K*1,NG EA— 0 .5/SH (K) **1. 5/SQRT (THE) *HM1 (K) *HM2 (K) & +1./SQRT (SH(K)*THE)*HM12(K) ETAU— 0 . 5/SG(K) ** 1 . 5/SQRT (THE) *GM1 (K) *GM2 (K) & +1./SQRT (SG(K)*THE)*GM12(K) DU(Jl+1,J+l)“DU(Jl+1,J+l)+ETAU-(D(K)/2.-0.25)*(HM12(K)/SH(K) & -HM1(K)*HM2(K)/SH(K)**2) - EA*RB(K) - & DERB (K) *HM1 (K) *HM2 (K) / (SH (K) *THE) 24 CONTINUE DU(J+l,Jl+1)=DU(Jl+1,J+l) 21 CONTINUE 20 CONTINUE 70 CONTINUE DO 33 J“ 1,NP DO 33 K-1,NP 33 SC (J,K)=-DU(J, K)/SQRT(DABS(DU(J,J)*DU(K,K))) DO 34 K“ 1,NP 34 SC (K, K) “BLEND+SC (K, K) CALL LINRG(NP,SC,28,SCI,28) EMAX=-100 UMAX=-100 DO 35 J=l,NP E(J)“ 0 IF(ABS(U(J)) .GT.UMAX) UMAX=ABS(U(J) ) DO 36 K=1,NP 36 E (J)“E(J)+(SCI(J,K)/SQRT(DU(J,J)*DU(K,K)))*U(K) 35 EMAX“ AMAX1 (EMAX, ABS (E (J) ) ) AN=A+E(1) DO 37 J=1,NZ B N (J) = B (J) + E (J+l) THBN=THE+E(NP) LIKEN—FLOAT(NG)*(ALOG(2.)-0.5*ALOG(PI)) -FLOAT(NG)/4.*ALOG(THEN) LIKEN=LIKEN+DEAD*ALOG(AN)+ (AN-1.)*TLN D O 41 K-1,NG SH(K)=0. SG(K)=0. CONTINUE D O 47 M=1,N K=IG(M) B Z = 0 . D O 42 J=1,NZ BZ=BZ+Z(M,J)*BN(J) EBZ(M)=EXP(BZ) SH(K)=SH(K)+T(M)**AN*EBZ(M) SG(K)-SG DO 88 J=1,NP DO 88 K=1,NP DU(J,K)=-DU(J,K) WRITE(6,*) J,K, DU(J,K) 88 CONTINUE CALL LINRG(NP,DU,28,V,28) WRITE(6,*) ' *******VAR*********' DO 333 J=1,NP DO 333 K=1,NP W R I T E (6,*) J,K, V (J,K) 333 CONTINUE RETURN END C THE FOLLOWING ROUTINES COMPUTE THE BESSEL FUNCTION, BESS3 C C THE RATIO OF TWO BESSEL FUNCTIONS, RBESS C C THE DERIVATIVE OF RBESS, DRBESS C SUBROUTINE BESS3(RIND,W,RK) PARAMETER (PI=3.141592653589793238) SUMW-0. IF (ABS(RIND) .EQ. 0.5) THEN SUMW=0.0 ELSE IF(RIND .EQ. 1.5) THEN SUMW-1./W ELSE IF(RIND .EQ. 2.5) THEN SUMW=3./W+3./W**2 ELSE IF(RIND .EQ. 3.5) THEN SUMW-6./W+15./W**2+15. /W**3 ELSE N=INT(RIND-0.5) DO 10 1=1,N A=FACT(N+I) B=FACT(N-I) C=FACT(I) SUMW=SUMW+A/B/C/(2. *W) **I 10 CONTINUE ENDIF RK=SQRT(PI/2.)*(1.+SUMW)/EXP(W)/SQRT(W) RETURN END C C REAL FUNCTION FACT(N) FACT=1. IF (N .EQ. 0) THEN FACT=1. ELSE DO 10 1=1,N FACT=FACT*FLOAT(I) 10 CONTINUE ENDIF o o o o o ELSE END RETURN RETURN ENDIF DRB=RB**2-RIND/W*RB-0.5-0.5*RK2/RK RBESS (RIND, W,RB) CALL BESS3(RIND+2,W,RK2) CALL BESS3 (RIND, W, CALL RK) END SUBROUTINE RBESS (RIND,W,RB) RBESS SUBROUTINE F RN .T 0) THEN 0.) .GT. (RIND IF SUBROUTINE DRBESS (RIND,W, DRB) DRBESS SUBROUTINE END RETURN CALL BESS3(RIND+1., W, RK1) BESS3(RIND+1., CALL BESS3(RIND,W,RK) CALL CALL BESS3(RI+1., W,RKl) BESS3(RI+1., CALL BESS3(RI,W,RK) CALL RI=-RIND RB=RK1/RK BRIR— *RI/W . RB=RKI/RK—2 191 LIST OF REFERENCES Aalen, O. (1978). Nonparametric inference for a family of counting process. Annals of Statistics 6, 701-726. Aalen, O. (1987). Two examples of modelling heterogeneity in survival analysis. Scandinavian Journal of Statistics 6,701-726. Aalen, O. (1988). Heterogeneity in survival analysis. Statistics in Medicine 14,19-25. Abramowitz, M. and I. Stegun Eds. (1965). Handbook of Mathematical Functions. Dover, New York. Anderson, P.K. and R. D. Gill (1982). Cox's regression model for counting processes: a large sample study. Annals o f Statistics 10,1100-1120. Barlow, R. E. and F. Proschan (1982). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York. Buckley, J. and I. James (1979). Linear regression with censored data. Biometrika 66, 429-436. Byar, D. (1982). Analysis of survival data: Cox and Weibull models with covariates. Statistics in Medical Research (Mike and Stanley, Eds.). Wiley, New York, 365- 401. Clayton, D. G. (1978). A model for association in bivariate life tables and its application in epidemiologic studies of familial tendency in chronic disease incidence. Biometrika 65, 141-151. Clayton, D. and J. Cuzick (1985). Multivariate generalizations of the proportional hazards model (with discussion). Journal of the Royal Statistical Society A148, 82-117. Clayton, D. (1988). The analysis of event history data: a review of progress and outstanding problems. Statistics in Medicine 7,819-841 Clayton, D. (1991). A Monte Carlo method for Bayesian inference in frailty models. Biometrics. To appear. 192 193 Cox, D. R. (1972). Regression models and life tables (with discussion). Journal of the Royal Statistical Society B34,187-220. Cox, D. R. (1975). Partial likelihood. Biometrika 62, 269-276. Cox, D. R. and D. Oakes (1984). Analysis of Survival Data. Chapman and Hall, London. David, H. A. and M. L. Moeschberger (1978). The Theory of Competing Risks. Griffin No. 39, London. Dawber, T. (1980). The Framingham Study: The epidemiology of the atherosclerotic disease. Harvard University Press, Cambridge, Mass. Dempster, A. P., N. M. Laird and D. B. Rubin (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society B39,1-38. Elandt-Johnson R. C. and N. L. Johnson (1980). Survival Models and Data Analysis. Wiley, New York. Elbers, C. and G. Ridder (1982). True and spurious duration dependence: the identifiability of the proportional hazards model. Review of Economic Studies XLIX, 403-409. Feigl, P. and M. Zelen (1965). Estimation of exponential survival probabilities with concomitant information. Biometrics 21, 826-838. Genest, C. and J. MacKay (1986). The joy of copulas: Bivariate distributions with uniform marginals. The American Statistician 40,280-283. Gill, R. D. (1984). Understanding Cox’s regression Model: a Martingale approach. Journal o f the American Statistical Association 79,441-447. Gill, R. D. (1985). Contribution to discussion of paper by Clayton and Cuzick. Journal of the Royal Statistical Society B34,108-109. Hougaard, P. (1984). Life table methods for heterogeneous populations: distributions describing the heterogeneity. Biometrika 71,75-83. Hougaard, P. (1986a). Survival models for heterogeneous populations derived from stable distributions. Biometrika 73, 387-396. Hougaard, P. (1986b). A class of multivariate failure time distributions. Biometrika 73, 671-678. Hougaard, P. (1987). Modelling multivariate survival. Scandinavian Journal of Statistics 14, 291-304. 194 Hougaard, P. (1991). Genetic determination of life times elucidated by adult Danish twins bom 1881-1930. Journal o f the American Statistical Association, to appear. Hougaard, P., B. Harvard and N. Holm (1991). Assessment of dependence in the life times of twins. Invited paper NATO Advances Studies Workshop on Survival Analysis and Related Topics, Columbus, Ohio. Johanson, S. (1983). An extension o f Cox's Regression Model. International Statistical Review 51, 258-262. Johnson, M. E., H. D. Tolley, M. C. Bryson and A. S. Goldman (1982). Covariate analysis of survival data: A small-sample study of Cox’s model. Biometrics 38, 685-698. J0rgensen, B. (1982). Statistical Properties o f the Generalized Inverse Gaussian Distribution. Springer-Verlag, New York. Kalbfleisch, J. D. and R. L. Prentice (1973). Marginal likelihoods based on Cox's regression and life model. Biometrika 60, 267-278. Kalbfleisch, J. D. and R. L. Prentice (1980). The Statistical Analysis of Failure Time Data. Wiley, New York. Kay, R. (1977). Proportional h a z a r d regression models and the analysis of censored survival data- Applied Statistics. 26,227-237. Kinderman, A. J. and J. G. Ramage (1976). Computer generation of normal random variables. Journal of the American Statistical Association 71,893-896. Klein, J. P., N. Keiding and C. Kanflby (1989). Semiparametric Marshall-Olkin models applied to the occurrence of metastases at multiple sites after breast cancer. Biometrics 45, 1073-1086 Klein, J. P. (1991). Semiparametric estimation of random effects using the Cox model based on the EM algorithm. Biometrics, to appear. Klein, J. P., T. M. Costigan and M. L-. Moeschberger (1991). Assessment of risk factors and the strength of dependence when event times are associated within groups. Ohio State University Technical Report No.455. Klein, J. P., M. Moeschberger, Y. H. Li and S. T. Wang (1991). Estimating random effects in the Framingham heart study. Invited paper NATO Advances Studies Workshop on Survival Analysis and Related Topics, Columbus, Ohio. Lagakos, S. and D. Schoenfeld (1984). Properties of proportional hazards score tests under misspecified regression models. Biometrics 40, 1037-1048. 195 Lagakos, S. (1988) The loss in efficiency from misspecifying covariates in proportional hazards regression models. Biometrika 75,156-160. Lawless, J. F. (1982). Statistical models and methods for lifetime data. Wiley, New York. Lawrence, R. J. (1988). The lognormal as event-time distribution. Lognorm al Distributions. (Crow and Shimizu Eds.) Marcel Dekker, Inc., New York. Lee, S. and J. Klein (1988) Bivariate models with a random environmental factor. Indian Journal of Productivity, Reliability and Quality Control 13(2), 1-18. Lin, D. Y. and L. J. Wei (1989). The robust inference for the Cox proportional hazards model. Journal of the American Statistical Association 84,1074-1078. Marshall, A. and I. Olkin (1967). A multivariate exponential distribution. Journal o f the American Statistical Association 62,30-40. Miller, R. (1976). Least squares regression with censored data. Biometrika 63, 449- 464. Neilsen, G. G ., R. D. Gill, P. K. Andersen and T. I. A. S0rensen (1991). A counting process approach to maximum likelihood estimation in frailty models. Research Report 91/2, Statistical Research Unit, University of Copenhagen. Oakes, D. (1982). A model for association in bivariate survival data. Journal of the Royal Statistical Society B44, 414-422. Oakes, D. (1986). Semiparametric inference in a model for association in bivariate survival data. Biometrika 73, 353-361. Oakes, D. (1989). Bivariate survival models induced by frailties. Journal of the American Statistical Association 84,487-493. Peto, R. and P. Lee (1973). Weibull distributions for continuous carcinogenesis experiments. Biometrics 29,457-470. Peto, R. and J. Peto (1972). Asymptotically efficient rank invariant procedures (with discussion). Journal of the Royal Statistical Society A135,185-206. Prentice, R. L. (1973). Exponential survival with censoring and explanatory variables. Biometrika 60, 279-288. Prentice, R. L. and R. Shillington (1975) Regression analysis of Weibull data and the analysis of clinical trials. Utilitas Mathematica 8,257-276. Prentice, R. L. (1978). Linear rank tests with right censored data. Biometrika 65, 167- 179. 196 Sampford, M. and J. Taylor (1959). Censored observations in randomized block experiments. Journal of the Royal Statistical Society B21,214-237 Schmeiser, B. W. and R. Lai (1980). Squeeze methods for generating gamma variate.s, Journal o f the American Statistical Association 75,679-682. Self, S. G. and R. L. Prentice (1986). Incorporating random effects into multivariate relative risk regression models in Modern Statistical Methods in Chronic Disease Epidemiology (Moolgavkar and Prentice, Eds). Wiley, New York, 167-177, Smith, R. M. and L. J. Bain (1975). An exponential power life-testing distribution. Communications in Statistics 4(5), 469-481. Solomon, P. J. (1984). Effects of misspecification of regression models in the analysis of survival data. Biometrika 71, 291-298. Amendment (1986), 73,245. Struthers, C. A. and J. D. Kalbfleisch (1986). Misspecified proportional hazard models. Biometrika 73, 363-369. Wang, S. T. (1991). Estimation for the gamma and positive stable frailty models. Ph.D. Dissertation, Ohio State University. Whitmore, G. A. and M. L. T. Lee (1991). A multivariate survival distribution generated by an inverse Gaussian mixture of exponentials. Technometrics 33,39-50. Williams, J. S. (1978). Efficient analysis of Weibull survival data from experiments on heterogeneous patient populations. Biometrics 34,209-222. Vaupel, J. W., K. G. Mantom and E. Stallard (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16,439-454.