and the influential nineteenth-century proposals by Survival Analysis, Gompertz [19] and Makeham [37], who modeled the Overview hazard function as bcx and a + bcx , respectively. Motivated by the controversy over smallpox inoc- ulation, D. Bernoulli [5] laid the foundation of the Survival analysis is the study of the distribution of theory of competing risks; see [44] for a histori- life times, that is, the times from an initiating event cal account. The calculation of expected number of (birth, start of treatment, employment in a given job) deaths (how many deaths would there have been in to some terminal event (death, relapse, disability pen- a study population if a given standard set of death sion). A distinguishing feature of survival data is the rates applied) also dates back to the eighteenth cen- inevitable presence of incomplete observations, par- tury; see [29] and the article on Historical Controls ticularly when the terminal event for some individuals in Survival Analysis. is not observed; instead, it is only known that this Among the important methodological advances in event is at least later than a given point in time: right the nineteenth century was, in addition to the para- censoring (see Censored Data). metric survival analysis models mentioned above, the The aims of this entry are to provide a brief graphical simultaneous handling of calendar time and historical sketch of the long development of survival age in the Lexis Diagram [35, cf. 30]. analysis and to survey what we have found to be Two very important themes of modern survival central issues in the current methodology of survival analysis may be traced to early twentieth century analysis. Necessarily, this entry is rich in cross- actuarial mathematics: references to other entries that treat specific subjects Multistate modeling in the particular case of dis- in more detail. However, we have not attempted to ability insurance [41] and nonparametric estimation include cross-references to all specific entries within in continuous time of the survival function in the survival analysis. competing risk problem under delayed entry and right censoring [13]. At this time, survival analysis was not an inte- History grated component of theoretical statistics. A charac- teristic scepticism about “the value of life-tables in The Prehistory of Survival Analysis in statistical research” was voiced by Greenwood [20] Demography and Actuarial Science in the Journal of the Royal Statistical Society,and Westergaard’s [50] guest appearance in Biometrika Survival analysis is one of the oldest statistical dis- on “Modern problems in vital statistics” had no ciplines with roots in demography and actuarial reference to sampling variability. This despite the science in the seventeenth century; see [49, Chap- fact that these two authors were actually statistical ter 2]; [51] for general accounts of the history of vital pioneers in survival analysis: Westergaard [48] by statistics and [22] for specific accounts of the work deriving what we would call the standard error of before 1750. the standardized mortality ratio (rederived by Yule The basic life-table methodology in modern ter- [52]; see [29]) (see Standardization Methods); and minology amounts to the estimation of a survival Greenwood [21] with his famous expression for “the function (one minus distribution function) from life ‘errors of sampling’ of the survivorship tables”, (see times with delayed entry (or left truncation;see below). below) and right censoring. This was known before 1700, and explicit parametric models at least since the linear approximation of de Moivre [39] (see e.g. The “Actuarial” life table and the Kaplan–Meier [22, p. 517]), later examples being due to Lambert Estimator [33, p. 483]: In the mid-twentieth century, these well-established


 x 2 x demographic and actuarial techniques were presented 1 − − 0.6176 exp − 96 31.682 to the medical–statistical community in influential
 x surveys such as those by Berkson and Gage [4] − exp − (1) 2.43114 and Cutler and Ederer [13]. In this approach, time 2 Survival Analysis, Overview is grouped into discrete units (e.g. one-year inter- a cure model assuming eternal life with probability vals), and the chain of survival frequencies from c and lognormally distributed survival times other- one interval to the next are multiplied together to wise. The exponential distribution was assumed by form an estimate of the survival probability across Littell [36], when he compared the “actuarial” and several time periods. The difficulty is in the devel- the maximum likelihood approach to the “T -year sur- opment of the necessary approximations due to the vival rate”, by Armitage [3] in his comparative study discrete grouping of the intrinsically continuous time of two-sample tests for clinical trials with staggered and the possibly somewhat oblique observation fields entry, and by Feigl and Zelen [16] in their model in cohort studies and more complicated demographic for (uncensored) lifetimes whose expectations were situations. The penetrating study by Kaplan and allowed to depend linearly on covariates, generalized Meier [28] (see Kaplan–Meier Estimator), the fas- to censored data by Zippin and Armitage [53]. cinating genesis of which was chronicled by Bres- Cox [11] revolutionized survival analysis by his low [8], in principle, eliminated the need for these semiparametric regression model for the hazard, approximations in the common situation in medical depending arbitrarily (“nonparametrically”) on time statistics where all survival and censoring times are and parametrically on covariates (see Cox Regres- known precisely. Kaplan and Meier’s tool (which sion Model). For details on the genesis of Cox’s they traced back to Bohmer¨ [7]) was to shrink the paper, see [42, 43]. observation intervals to include at most one observa- tion per interval. Though overlooked by many later Multistate Models authors, Kaplan and Meier also formalized the age- old handling of delayed entry (actually also covered Traditional actuarial and demographical ways of by Bohmer)¨ through the necessary adjustment for modeling several life events simultaneously may be the risk set, the set of individuals alive and under formalized within the probabilistic area of finite-state observation at a particular value of the relevant time Markov processes in continuous time. An impor- variable. tant and influential documentation of this was by Among the variations on the actuarial model, we Fix and Neyman [18], who studied recovery, relapse, will mention two. and death (and censoring) in what is now commonly Harris et al. [23] anticipated much recent work in, termed an illness–death model allowing for compet- for example, AIDS survival studies in their general- ing risks (see Fix–Neyman Process). Chiang [9], for ization of the usual life-table estimator to the situation example, in his 1968 monograph, extensively docu- in which the death and censoring times are known mented the relevant stochastic models (see Stochastic only in large, irregular intervals (see Grouped Sur- Processes), and Sverdrup [46], in an important paper, vival Times). gave a systematic statistical study. These models have Ederer et al. [(14)] developed a “relative survival constant transition intensities, although subdivision rate... as the ratio of the observed survival rate in a of time into intervals allows grouped-time method- group of patients to the survival rate expected in a ology of the actuarial life-table type, as carefully group similar to the patients ...” thereby connecting documented by Hoem [24]. to the long tradition of comparing observed with expected; see, for example, [29] and the article on Survival Analysis Concepts Historical Controls in Survival Analysis. The ideal basic independent nonnegative random = Parametric Survival Models variables Xi ,i 1,...,n are not always observed directly. For some individuals i, the available piece Parametric survival models were well-established in of information is a right-censoring time Ui ,that actuarial science and demography, but have never is, a period elapsed in which the event of interest dominated medical uses of survival analysis. How- has not occurred (e.g. a patient has survived until ever, in the 1950s and 1960s important contributions Ui ). Thus, a generic survival data sample includes   to the statistical theory of survival analysis were ((Xi ,Di), i = 1,...,n) where Xi is the smaller of based on simple parametric models. One example is Xi and Ui and Di is the indicator, I(Xi ≤ Ui ),of the maximum likelihood approach by Boag [6] to not being censored. Survival Analysis, Overview 3

Mathematically, the distribution of Xi may be dying. Under these assumptions, S(t) is estimated by described by the survival function the Kaplan–Meier estimator [28]. This is given by   = Di Si (t) Pr(Xi >t). (2) S(t) = 1 − ,(6) Y(X ) ≤ i If the hazard function Xi t

 ≤ + | where Y(t) = I(Xi ≥ t) is the number of individ- = Pr(Xi t ∆t Xi >t) αi (t) lim (3) uals at risk just before time t. The Kaplan–Meier ∆t→0 ∆t estimator is a nonparametric maximum likelihood  exists, then estimator and, in large samples, S(t) is approximately normally distributed with mean S(t) and a variance Si (t) = exp(−Ai (t)), (4) that may be estimated by Greenwood’s formula:

 Di where  σ 2(t) = [S(t) ]2 .(7) t   − =  Y(Xi )[Y(Xi ) 1] Ai (t) αi (u) du(5) Xi ≤t 0 From this result, pointwise confidence intervals for is the integrated hazard over [0,t). If, more generally, S(t) are easily constructed and, since one can also the distribution of the X has discrete components, i show weak√ convergence of the entire Kaplan–Meier then Si (t) is given by the product-integral of the curve { (n)[S(t) − S(t)]; 0 ≤ t ≤ τ},τ ≤∞ to a cumulative hazard measure. Owing to the dynamical mean zero Gaussian process (see Brownian Motion nature of survival data, a characterization of the dis- and Diffusion Processes), simultaneous confidence tribution via the hazard function is often convenient. bands for S(t) on [0,τ] can also be set up. (Note that αi (t)∆t when ∆t > 0issmall is approx- As an alternative to estimating the survival distri- imately the conditional probability of i “dying” just bution function S(t),thecumulative hazard function after time t given “survival” till time t.) Also, αi (t) is A(t) =−log S(t) may be studied. Thus, A(t) may the basic quantity in the counting process approach be estimated by the Nelson–Aalen Estimator to survival analysis (see e.g. [2], and the article on Survival Distributions and Their Characteristics).  = Di A(t)  .(8) Y(Xi ) Xi ≤t

Nonparametric Estimation and Testing The relation between the estimators S(t) and A(t) is given by the product-integral from which it fol- The simplest situation encountered in survival anal- lows that their large-sample properties are equivalent. ysis is the nonparametric estimation of a survival Though the Kaplan–Meier estimator has the advan- distribution function based on a right-censored sam- tage that a survival probability is easier to interpret   ple of observation times (X1,...,Xn), where the true than a cumulative hazard function, the Nelson–Aalen survival times Xi ,i = 1,...,n, are assumed to be estimator is easier to generalize to multistate situ- independent and identically distributed with common ations beyond the survival data context. We shall survival distribution function S(t), whereas as few return to this below. To give a nonparametric esti- assumptions as possible are usually made about the mate of the hazard function α(t) itself requires some right-censoring times Ui except for the assumption smoothing techniques to be applied (see Smoothing of independent censoring (see Censored Data). The Hazard Rates). concept of independent censoring has the interpreta- Right censoring is not the only kind of data- tion that the fact that an individual, i, is alive and incompleteness to be dealt with in survival analysis; uncensored at time t, say, should not provide more in particular, left truncation (or delayed entry)where information on the survival time for that individual individuals may not all be followed from time 0 but than Xi >t, that is, the right-censoring mechanism maybe from a later entry time Vi conditionally on should not remove individuals from the study who having survived until Vi , occurs frequently in, for are at a particularly high or a particularly low risk of example, epidemiological applications. Dealing with 4 Survival Analysis, Overview left truncation only requires a redefinition of the risk to H0 (see Linear Rank Tests in Survival Analy-  set from the set {i: Xi ≥ t} of individuals still alive sis). An important such test statistic is the logrank  and uncensored at time t to the set {i: Vi 0). For this test, of individuals with entry time Vi hazards alternatives, Zh given by (9) reduces to = − the size of the risk set at time t both (6), (7), and (8) Zh Oh Eh with Oh the total number of observed   are applicable though one should be aware of the fact failures in group h and Eh = Di Yh(Xi )/Y (Xi ) an that estimates of S(t) and A(t) may be ill-determined “expected” number of failures in group h. For the for small values of t due to the left truncation (see two-sample case (k = 2), one may of course use the Truncated Survival Times). square root of X2 as an asymptotically normal test When the survival time distributions in a number, statistic for the null hypothesis. For the case where k, of homogeneous groups have been estimated the k groups are ordered, and where a score xh (with nonparametrically, it is often of interest to test x1 ≤ ...≤ xk) is attached to group h,atest for trend 2 =  2  =  the hypothesis H0 of identical hazards in all is given by T (x Z) /x Σx with x (x1,...,xk) groups. Thus, on the basis of censored survival and it is asymptotically chi-squared with 1 df. data ((Xhi ,Dhi), i = 1,...,nh) for group h, h = The above linear rank tests have low power  against certain important classes of alternatives such 1,...,k, the Nelson–Aalen estimates Ah(t) have as “crossing hazards”. Just as for uncensored data, been computed, and based on the combined sample =  = this has motivated the development of test statis- of size n h nh with data ((Xi ,Di), i 1,...,n), an estimate of the common cumulative hazard tics of the Kolmogorov–Smirnov and Cramer–von` Mises types, based on maximal deviation or inte- function A(t) under H0 may be obtained by a Nelson–Aalen estimator A(t) . As a general statistic grated squared deviation between estimated hazards, cumulative hazards or survival functions. for testing H0, one may then use a k-vector of sums of weighted differences between the increments of   Ah(t) and A(t): Parametric Inference n   ˆ  Zh = Kh(Xi )[∆Ah(Xi ) − ∆A(Xi )].(9) The nonparametric methods outlined in the previ- i=1 ous section have become the standard approach to the analysis of simple homogeneous survival data  = Here, ∆Ah(t) 0ift is not among the observed without covariate information. However, parametric survival times in the hth sample and Kh(t) is 0 survival time distributions are sometimes used for = whenever Yh(t) 0, in fact all weight functions used inference, and we shall here give a brief review. = in practice have the form Kh(t) Yh(t)K(t). With Assume again that the true survival times X1,...,Xn this structure for the weight function, the covariance are independent and identically distributed with sur- between Zh and Zj given by (9) is estimated by vival distribution function S(t; θ) and hazard func- tion α(t; θ) but that only a right-censored sample   (X ,D ), i = 1,...,n, is observed. Under indepen- n Y (X ) Y (X ) i i = 2  h i − j i dent censoring, the likelihood function for the param- σhj K (Xi )  δhj  Di , i=1 Y(Xi ) Y(Xi ) eter θ is (10) n  Di  L(θ) = (α(Xi ; θ)) S(Xi ; θ). (11) and, letting Z be the k-vector (Z ,...,Z ) and 1 k i=1 Σ the k by k matrix (σhj ,h,j = 1,...,k) the test statistic X2 = ZΣ −Z is asymptotically chi-squared The function (11) may be analyzed using standard distributed under H0 with k − 1 degrees of freedom large-sample theory. Thus, standard tests, that − if all nh tend to infinity at the same rate. Here, Σ is is, Wald-, score-, and likelihood ratio tests are a generalized inverse for Σ (see Matrix Algebra). used as inferential tools (see Chi-square Tests). Special choices for K(t) correspond to test statis- Two frequently used parametric survival models tics with different properties for particular alternatives are the Weibull distribution with hazard function Survival Analysis, Overview 5

ρ−1  αρ(αt ) , and the piecewise exponential distribution where β = (β1,...,βp) is a vector of unknown with α(t, θ) = αj for t ∈ Ij with Ij = [tj−1,tj ), 0 = regression coefficients and α0(t),thebaseline hazard, t0 Zij = 1 compared = As a special case of the nonparametric tests discussed to those with Zij 0 all other covariates being the above, a one-sample situation may be studied. This same for the two individuals. Similar interpretations may be relevant if one wants to compare the observed hold for parameters corresponding to covariates tak- survival in the sample with the expected survival ing more than two values. based on a standard life table. Thus, assume that a The model is semiparametric in the sense that the hazard function α∗(t) is given and that the hypothesis relative risk part is modeled parametrically while the ∗ baseline hazard is left unspecified. This semiparamet- H0 : α = α is to be tested. One test statistic for H0 is the one-sample logrank test (O − E∗)/(E∗)1/2 where ric nature of the model led to a number of inference ∗ problems, which was discussed in the literature in E , the “expected” number of deaths is given by ∗ ∗  ∗ ∗ the years following the publication of Cox’s article E = [A (Xi ) − A (Vi )] (with A the cumulative hazard corresponding to α∗). In this case, θ = O/E∗, in 1972. However, these problems were all resolved the standardized mortality ratio, is the maximum like- and estimation proceeds as follows. The regression lihood estimate for the parameter θ in the model coefficients β are estimated by maximizing the Cox α(t) = θα∗(t). Thus, the standardized mortality ratio partial likelihood   arises from a multiplicative model involving the D ∗ n  i known population hazard α (t). Another classical exp(β Zi ) L(β) =  ,(13) tool for comparing with expected survival, the so- ∈ exp(β Zj ) i=1 j Ri called expected survival function, arises from an addi-    tive or excess hazard model (see Excess Mortality; where Ri ={j: Xj ≥ Xi },therisk set at time Xi , Expected Number of Deaths; Historical Controls is the set of individuals still alive and uncensored in Survival Analysis). at that time. Furthermore, the cumulative baseline hazard A0(t) is estimated by the Breslow estima- tor The Cox Regression Model D  = i A0(t)  ,(14) In many applications of survival analysis, the interest exp(β Zj ) Xi ≤t ∈ focuses on how covariates may affect the outcome; j Ri in clinical trials, adjustment of treatment effects for which is the Nelson–Aalen estimator one would effects of other explanatory variables may be crucial use if β were known and equal to the maximum if the randomized groups are unbalanced with respect partial likelihood estimate β. The estimators based to important prognostic factors, and in epidemiolog- on (13) and (14) also have a nonparametric maxi- ical cohort studies, reliable effects of exposure may  be obtained only if some adjustment is made for con- mum likelihood interpretation. In large samples, β is founding variables. In these situations, a regression approximately normally distributed with the proper model is useful and the most important model for mean and with a covariance, which is estimated by survival data is the Cox [11] proportional hazards the information matrix based on (13). This means regression model. In its simplest form, it states the that approximate confidence intervals for the relative hazard function for an individual, i, with covariates risk parameters of interest can be calculated and that  the usual large-sample test statistics based on (13) Zi = (Zi1,...,Zip) to be are available. Also, the asymptotic distribution of the  αi (t; Zi ) = α0(t) exp(β Zi ), (12) Breslow estimator is normal; however, this estimate 6 Survival Analysis, Overview is most often used as a tool for estimating survival Other Regression Models for Survival probabilities for individuals with given covariates, Data Z0. Such an estimate may be obtained by the prod-    uct integral S(t; Z0) of exp(β Z0)A0(t).Thejoint Though the semiparametric Cox model is the regres- asymptotic distribution of β and the Breslow esti- sion model for survival data that is applied most mator then yields an approximate normal distribution frequently, other regression models, for example,  parametric regression models also play important for S(t; Z0) in large samples. A number of useful extensions of this simple Cox roles in practice. Examples include models with a model are available. Thus, in some cases, the covari- multiplicative structure, that is, models like (12) but = ates are time-dependent, for example, a covariate with a parametric specification, α0(t) α0(t; θ), of might indicate whether or not a given event had the baseline hazard, and accelerated failure-time occurred by time t, or a time-dependent covariate models. might consist of repeated recordings of some mea- A multiplicative model with important epidemio- logical applications is the Poisson regression model surement likely to affect the prognosis. In such cases, with a piecewise constant baseline hazard. In large the regression coefficients β are estimated replacing   data sets with categorical covariates, this model has exp(β Z ) in (13) by exp[β Z (X )]. j j i the advantage that a sufficiency reduction to the num- Also, a simple extension of the Breslow estima- ber of failures and the amount of person-time at risk tor (14) applies in this case. However, the survival in each cell defined by the covariates and the division function can, in general, no longer be estimated in of time into intervals is possible. This is in contrast to a simple way because of the extra randomness aris- the Cox regression model (12) where each individual ing from the covariates, which is not modeled in the data record is needed to compute (13). The substantial Cox model. This has the consequence that the esti- computing time required to maximize (13) in large mates are more difficult to interpret when the model samples has also led to modifications of this estima- contains time-dependent covariates. To estimate the tion procedure. Thus, in nested case–control studies survival function in such cases, a joint model for the the risk set Ri in the Cox partial likelihood is replaced hazard and the time-dependent covariate is needed  by a random sample Ri of Ri (see Case–Control (see Joint Modeling of Longitudinal and Event Study, Nested). Time Data). In the accelerated failure-time model, the focus Another extension of (12) is the stratified Cox is not on the hazard function but on the survival model where individuals are grouped into a num- time itself much like in classical linear models. Thus, ber, k of strata each of which has a separate baseline  this model is given by log Xi = α + β Zi + εi ,where hazard (see Stratification). This model has impor- the error terms are assumed to be independent and tant applications for checking the assumptions of identically distributed with expectation 0. Examples (12). The model assumption of proportional haz- include normally distributed (εi ,i = 1,...,n),and ards may also be tested in a number of ways, error terms with a logistic or an extreme value the simplest possibility being to add interaction distribution, the latter giving rise to a regression terms of the form Zij f(t) between Zij and time model with Weibull distributed life times. where f(t) is some specified function. Also, vari- Finally, we shall mention some nonparametric ous forms of residuals as for normal linear models hazard models. In Aalen’s additive model, αi (t) =  maybeusedformodel checking in (12) (see Good- β0(t) + β(t) Zi (t) (see Aalen’s Additive Regression ness of Fit in Survival Analysis; Residuals for Model), the regression functions β0(t), . . . , βp(t) are Survival Analysis). In (12), it is finally assumed left completely unspecified and estimated nonpara- that a quantitative covariate affects the hazard log- metrically much like the Nelson–Aalen estimator linearly. This assumption may also be checked in discussed above. This model provides an attractive several ways and alternative models with other rel- alternative to the other regression models discussed  ative risk functions r(β Zi ) may be used. Special in this section. There also exist more general and care is needed when covariates are measured with flexible models containing both this model and the error (see Measurement Error in Survival Analy- Cox regression model as special cases (see Addi- sis). tive–Multiplicative Intensity Models). Survival Analysis, Overview 7

Multistate Models One important extension of the two-state model for survival data is the competing risks model with Models for survival data may be considered a special one transient alive state 0 and a number, k, of absorb- case of a multistate model, namely, a model with a ing states corresponding to death from cause h, h = transient state alive (0) and an absorbing state dead 1,...,k. In this model, the basic parameters are the (1) and where the hazard rate is the force of transi- cause-specific hazard functions αh(t), h = 1,...,k, tion from state 0 to state 1. Multistate models may and the observations for individual i will consist  conveniently be studied in the mathematical frame- of (Xi ,Dhi), h = 1,...,k,whereDhi = 1 if individ- work of counting processes with a notation that actu- ual i is observed to die from cause h,andDhi = 0 ally simplifies the notation of the previous sections otherwise. On the basis of these data, k counting pro-  and, furthermore, unifies the description of survival cesses for each i can be defined by Nhi (t) = I(Xi ≤ data and that of more general models like the com- t,Dhi = 1) and letting Nh = Nh1 +···+Nhn,the peting risks model and the illness–death model to integrated cause-specific hazard Ah(t) is estimated be discussed below. We first introduce the counting by the Nelson–Aalen estimator replacing N by Nh in processes relevant for the study of censored sur- (17). A useful synthesis of the cause-specific hazards = vival data [1] Define, for i 1,...,n, the stochastic is provided by the transition probabilities P0h(0,t)of processes being dead from cause h by time t. This is frequently called the cumulative incidence function for cause h =  ≤ = Ni (t) I(Xi t,Di 1)(15) and is given by  t and = =  ≥ P0h(s, t) S(u)αh(u) du, (18) Yi (t) I(Xi t). (16) s

Then (15) is a counting process counting 1 at time X and hence it may be estimated by (18) by inserting i the Kaplan–Meier estimate for S(u) and the Nel- if individual i is observed to die; otherwise N (t) = 0 i son–Aalen estimate for the integrated cause-specific throughout. The process (16) indicates whether i is hazard. In fact, this Aalen–Johansen estimator of still at risk just before time t. Models for the survival the matrix of transition probabilities is exactly the data are then introduced via the intensity process, product-integral of the cause-specific hazards. λ (t) = α (t)Y (t) for N (t),whereα (t), as before, i i i i i Another important multistate model is the ill- denotes the hazard function for the distribution of ness–death or disability model with two transient X . Letting N = N +···+N and Y = Y +···+ i 1 n 1 states, say healthy (0) and diseased (1) and one Y the Nelson–Aalen estimator (8) is given by the n absorbing state dead (2). If transitions both from 0 stochastic integral to1andfrom1to0arepossible,thediseaseis  t recurrent, otherwise it is chronic. On the basis of  J(u) A(t) = dN(u), (17) such observed transitions between the three states, it Y(u) 0 is possible to define counting processes for individual where J(t) = I(Y(t) >0). In this simple multistate i as Nhj i (t) = number of observed h → j transitions model, the transition probability P00(0,t),thatis, in the time interval [0,t] for individual i and, further- the conditional probability of being in state 0 by more, we may let Yhi (t) = I (i is in state h at time time t given state 0 at time 0 is simply the survival t−). With these definitions, we may set up and ana- probability S(t), which, as described above, may be lyze models for the transition intensities αhj i (t) from estimated using the Kaplan–Meier estimator, which state h to state j including nonparametric compar- is the product-integral of (17). In fact, all the mod- isons and Cox-type regression models. Furthermore, els and methods for survival data discussed above, transition probabilities Phj (s, t) may be estimated by which are based on the hazard function have imme- product-integration of the intensities. diate generalizations to models based on counting processes. Thus, both the nonparametric tests and the Other Kinds of Incomplete Observation Cox regression model may be applied for counting process (multistate) models (see Counting Process A salient feature of survival data is right censoring, Methods in Survival Analysis). which has been referred to throughout in the present 8 Survival Analysis, Overview overview. However, several other kinds of incom- Concluding Remarks plete observation are important in survival analysis. Often, particularly when the time variable of inter- Survival analysis is a well-established discipline in est is age, individuals enter study after time 0. This statistical theory as well as in biostatistics. Most is called delayed entry and may be handled by left books on biostatistics contain chapters on the topic truncation (conditioning) or left filtering (“viewing and most software packages include procedures for the observations through a filter”). There are also handling the basic survival techniques (see Survival situations when only events (such as AIDS cases) Analysis, Software). Several books have appeared, that occur before a certain time are included (right among them the documentation of the actuarial and truncation)(see Truncated Survival Times). The demographical know-how by Elandt–Johnson and phenomenon of left censoring, though theoretically Johnson [15]; the research monograph by Kalbfleisch possible, is more rarely relevant in survival analysis. and Prentice [27], the first edition of which for a When the event times are only known to lie in an decade maintained its position as main reference on interval, one may use the grouped time approach of the central theory; the comprehensive text by Law- classical life tables (see Grouped Survival Times; less [34] covering also parametric models, and the Life Table), or (if the intervals are not synchronous) concise text by Cox and Oakes [12], two central con- techniques for interval censoring may be relevant. tributors to the recent theory. The counting process A common framework (coarsening at random) approach is covered by Fleming and Harrington [17] was recently suggested for several of the above types and by Andersen et al. [2]; see also [25]. Later, books of incomplete observation. intended primarily for the biostatistical user have appeared. These include [10, 31, 32, 38, 40]. Also, books dealing with special topics, like implementa- tion in the S-Plus software [47], multivariate survival Multivariate Survival Analysis data [26], and the linear regression model [45] have appeared. For multivariate survival, the innocently looking problem of generalizing the Kaplan–Meier estimator References to several dimensions has proved surprisingly intri- cate. A major challenge (in two dimensions) is how [1] Aalen, O.O. (1978). Nonparametric inference for a to efficiently use singly censored observations, where family of counting processes, Annals. of Statistics 6, one component is observed and the other is right 701–726. censored. [2] Andersen, P.K., Borgan, Ø., Gill, R.D. & Keiding, N. For regression analysis of multivariate survival (1993). Statistical Models Based on Counting Processes. Springer, New York. times, two major approaches have been taken. One is [3] Armitage, P. (1959). The comparison of survival curves, to model the marginal distributions and use estimation Journal of the Royal Statistical Society A 122, 279–300. techniques based on generalized estimating equa- [4] Berkson, J. & Gage, R.P. (1950). Calculation of survival tions leaving the association structure unspecified rates for cancer, Proceedings of the Staff Meetings of the (see Marginal Models for Multivariate Survival Mayo Clinic 25, 270–286. Data.) The other is to specify random effects models [5] Bernoulli, D. (1766). Essai d’une nouvelle analyse for survival data based on conditional independence de la mortalite´ causee´ par la petite verole,´ et des avantages de l’inoculation pour la prevenir,´ M´emoires (see Frailty.) An interesting combination between de Math´ematique et de Physique de l’Acad´emie Royale these two methods is provided by copula models des Sciences, Paris Annee´ MDCCLX, pp. 1–45 of in which the marginal distributions are combined Memoires.´ via a so-called copula function thereby obtaining an [6] Boag, J.W. (1949). Maximum likelihood estimates of the explicit model for the association structure. proportion of patients cured by cancer therapy, Journal For the special case of repeated events, both of the Royal Statistical Society B 11, 15–53. [7] Bohmer,¨ P.E. (1912). Theorie der unabhangigen¨ Wahr- the marginal approach and the conditional (frailty) scheinlichkeiten, Rapports, M´emoires et Proc´es – ver- approach have been used successfully (see Repeated baux du 7 e Congr`es International d’Actuaires, Amster- Events). dam 2, 327–343. Survival Analysis, Overview 9

[8] Breslow, N.E. (1991). Introduction to Kaplan and [28] Kaplan, E.L. & Meier, P. (1958). Non-parametric esti- Meier (1958). Nonparametric estimation from incom- mation from incomplete observations, Journal of the plete observations, in Breakthroughs in Statistics II, American Statistical Association 53, 457–481, 562–563. S. Kotz & N.L. Johnson, eds. Springer, New York, [29] Keiding, N. (1987). The method of expected number 311–318. of deaths 1786–1886–1986, International Statistical [9] Chiang, C.L. (1968). Introduction to Stochastic Pro- Review 55, 1–20. cesses in Biostatistics. Wiley, New York. [30] Keiding, N. (1990). Statistical inference in the Lexis [10] Collett, D. (2003). Modelling Survival Data in Medical diagram, Philosophical Transactions of the Royal Society Research, 2nd Ed., Chapman and Hall, London. London A 332, 487–509. [11] Cox, D.R. (1972). Regression models and life-tables [31] Klein, J.P. & Moeschberger, M.L. (2003). Survival (with discussion), Journal of the Royal Statistical Society Analysis. Techniques for Censored and Truncated Data, (B) 34, 187–220. 2nd Ed., Springer, New York. [12] Cox, D.R. & Oakes, D. (1984). Analysis of Survival [32] Kleinbaum, D.G. (1996). Survival Analysis. A Self- Data. Chapman and Hall, London. Learning Text. Springer, New York. [13] Cutler, S.J. & Ederer, F. (1958). Maximum utilization [33] Lambert, J.H. (1772). Beytr¨age zum Gebrauche der of the life table method in analyzing survival, Journal Mathematik und deren Anwendung, Vol. III, Verlage des of Chronic Diseases 8, 699–713. Buchlages der Realschule, Berlin. [14] Ederer, F., Axtell, L.M. & Cutler, S.J. (1961). The rel- ative survival rate: A statistical methodology, National [34] Lawless, J.F. (2002). Statistical Models and Methods for Cancer Institute Monographs 6, 101–121. Lifetime Data, 2nd Ed., Wiley, New York. [15] Elandt-Johnson, R.C. & Johnson, N.L. (1980). Survival [35] Lexis, W. (1875). Einleitung in die Theorie der Bev¨olker- Models and Data Analysis. Wiley, New York. ungsstatistik.Trubner,¨ Strassburg. [16] Feigl, P. & Zelen, M. (1965). Estimation of exponen- [36] Littell, A.S. (1952). Estimation of the T -year survival tial survival probabilities with concomitant information, rate from follow-up studies over a limited period of time, Biometrics 21, 826–838. Human Biology 24, 87–116. [17] Fleming, T.R. & Harrington, D.P. (1991). Counting [37] Makeham, W.M. (1860). On the law of mortality, and Processes and Survival Analysis. Wiley, New York. the construction of mortality tables, Journal of the [18] Fix, E. & Neyman, J. (1951). A simple stochastic model Institute of Actuaries 8, 301. of recovery, relapse, death and loss of patients, Human [38] Marubini, E. & Valsecchi, M.G. (1995). Analysing Biology 23, 205–241. Survival Data from Clinical Trials and Observational [19] Gompertz, B. (1825). On the nature of the function Studies. Wiley, Chichester. expressive of the law of human mortality, Philosophical [39] de Moivre, A. (1725). Annuities upon Lives: or, The Transactions of the Royal Society of London, Series A Valuation of Annuities uon any Number of Lives; as 115, 513–580. also, of Reversions. To which is added, An Appendix [20] Greenwood, M. (1922). Discussion on the value of concerning the Expectations of Life, and Probabilites of life-tables in statistical research, Journal of the Royal Survivorship. Fayram, Motte and Pearson, London. Statistical Society 85, 537–560. [40] Parmar, K.B. & Machin, D. (1995). Survival analysis. A [21] Greenwood, M. (1926). The natural duration of can- practical approach. Wiley, Chichester. cer, in Reports on Public Health and Medical Sub- [41] du Pasquier, L.G. (1913). Mathematische Theorie jects, Vol. 33 His Majesty’s Stationery Office, London, der Invaliditatsversicherung,¨ Mitteilungen der Vereini- pp. 1–26. gung der Schweizerische Versicherungs-Mathematiker 8, [22] Hald, A. (1990). A History of Probability and Statistics 1–153. and Their Applications before 1750. Wiley, New York. [42] Prentice, R.L. (1991). Introduction to Cox (1972) [23] Harris, T.E., Meier, P. & Tukey, J.W. (1950). The timing Regression models and life-tables, in Breakthroughs in of the distribution of events between observations, Statistics II, S. Kotz & N.L. Johnson eds. Springer, New Human Biology 22, 249–270. [24] Hoem, J.M. (1976). The statistical theory of demo- York, pp. 519–526. graphic rates. A review of current developments [43] Reid, N. (1994). A conversation with Sir David Cox, (with discussion), Scandinavian Journal of Statistics 3, Statistical Science 9, 439–455. 169–185. [44] Seal, H.L. (1977). Studies in the history of probability [25] Hosmer, D.W. & Lemeshow, S. (1999). Applied Survival and statistics, XXXV. Multiple decrements or competing Analysis. Regression Modeling of Time to Event Data. risks, Biometrika 64, 429–439. Wiley, New York. [45] Smith, P.J. (2002). Analysis of Failure Time Data. [26] Hougaard, P. (2000). Analysis of Multivariate Survival Chapman and Hall/CRC, London. Data. Springer, New York. [46] Sverdrup, E. (1965). Estimates and test procedures in [27] Kalbfleisch, J.D. & Prentice, R.L. (2002). The Statistical connection with stochastic models for deaths, recoveries Analysis of Failure Time Data, 2nd Ed., Wiley, New and transfers between different states of health, Skandi- York. navisk Aktuarietidskrift 48, 184–211. 10 Survival Analysis, Overview

[47] Therneau, T.M. & Grambsch, P.M. (2000). Modeling [52] Yule, G. (1934). On some points relating to vital statis- Survival Data: Extending the Cox Model. Springer, New tics, more especially statistics of occupational mortality York. (with discussion), Journal of the Royal Statistical Society [48] Westergaard, H. (1882). Die Lehre von der Mortalit¨at 97, 1–84. und Morbilit¨at. Fischer, Jena. [53] Zippin, C. & Armitage, P. (1966). Use of concomitant [49] Westergaard, H. (1901). Die Lehre von der Mortalit¨at variables and incomplete survival information with esti- und Morbilit¨at, 2. Aufl., Fischer, Jena. mation of an exponential survival parameter, Biometrics [50] Westergaard, H. (1925). Modern problems in vital statis- 22, 655–672. tics, Biometrika 17, 355–364. [51] Westergaard, H. (1932). Contributions to the History of PER KRAGH ANDERSEN &NIELS KEIDING Statistics. King, London.