Multivariate Survival Analysis

Previously we have assumed that either (Xi, δi) or (Xi, δi,Zi), i = 1, ..., n, are i.i.d.. This may not always be the case.

Multivariate survival data can arise in practice in diﬀerence ways:

• Clustered survival data. This happens when failure times (often of the same type, eg. death) from the same cluster are dependent, such as in – multicenter clinical trials: cluster = clinical center – familial data: cluster = family – matched pairs: cluster = pair • Recurrent event data. This happens when an individual may experience events of the same type more than once, eg, infections. It is actually a special case of clustered data, where the failure times are ordered.

What are the failure time random variables in each case?

1 Work has been done (and still being done) to estimate non- parametrically the joint survival distribution, without any predictors, eg. bivariate survival function. This turns out to be much harder than the one dimensional case (KM), mainly due to censoring.

In practice, we are often interested in relating certain covariates to the survival time (regression setting), while taking into consideration the dependence among the survival times.

Here we will introduce extensions of the Cox model to handle correlated survival data.

There are two types of approaches:

• the conditional approach (using random eﬀects) • the marginal (WLW) approach

2 The random eﬀects approach

The dependence among failure times may be viewed as induced by unmeasured factors that are common to subjects within a cluster but may vary from cluster to cluster.

For example,

• EST 1582 is a phase III multi-institutional lung cancer trial to compare two treatments, CAV vs. CAV-HEM. It was shown (Gray 1995) that there is significant institutional variation in the treatment effects (see plot); • for familial data, common genetic or environmental factors might have induce the dependence of survival times among the family members; • in recurrent events, multiple events of the same individual are likely correlated due to some factor unique to that individual but different from individual to individual.

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· · · · Estimated treatment effects (log HR) · · · · · · 16 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 · 18

0 5 10 15 20 25 30 Institution

Figure 1: Estimated institutional treatment eﬀects (E1582). Institutions are sorted in increasing order of number of patients.

One may model these cluster-to-cluster variation as fixed effects. For example, in the E1582 data, if Z = 1 for CAV- HEM, and Z = 0 for CAV, instead of using a single β for the treatment difference, we may use β1 for the treatment effect in institution 1, β2 for the treatment effect in institution 2, ... there are 31 institutions (579 observations). How do we write down such a fixed effects Cox model?

Obviously the above fixed effects model will introduce many parameters. In the above figure, that is NOT how we obtained the estimated effects.

4 Alternatively one may try to use a stratiﬁed Cox model. But this will still have problems with small strata. In fact, the number of patients per institution varied from 1 to 56.

One way to solve this problem is to turn the many (31) fixed treatment effects into random effects, i.e. put a (prior) distribution on them (in a Bayesian sense). This effectively shrinks the magnitude of the estimated effects towards zero.

Shrinkage estimators are known to be biased, but nonetheless have smaller MSE (Morris, 1983).

5 The univariate frailty model

The simplest random eﬀects model for clustered survival data is the univariate frailty model

0 λij(t) = λ0(t) exp{β Zij + bi}. Here i indicates the cluster, and j the individual from cluster i (i = 1, ..., n, j = 1, ...ni), and bi is the ‘random cluster eﬀect’. This is sometimes called a hierarchical model.

The assumption of the random effects model is that b1, ..., bn are i.i.d. random variables, from a distribution known up to 2 a finite number of parameters. For example, bi ∼ N(0, σ ). This can substantially reduce the number of parameters compared to a fixed effects model.

Such modeling is most suitable when there are many clusters and each of relative small size.

In general bi induces correlation among the individuals in the same cluster i.

6 We can equivalently write the above model as

0 λij(t) = λ0(t) exp{β Zij}ωi. so that bi = log ωi; ωi is often called the frailty term. Early use of frailty models was mainly to count for ‘un- observed heterogeneity’, or unmeasured covariates, for i.i.d. data (ni = 1).

In the past, the commonly used distribution for the ωi’s is the gamma distribution with mean of one and unknown variance. This is mainly due to mathematical convenience (conjugate prior).

The mean of bi (or ωi) needs to be ﬁxed to avoid problems of identiﬁability (why?).

The univariate frailty model can be ﬁtted using coxph() with option frailty. The distributions allowed for ωi are gamma and log-normal.

7 In terms of the data generating mechanism, we can think of the clusters as being sampled from a population.

bi here is sometimes called the random intercept; it’s a random eﬀect on the baseline hazard.

This model counts for the correlation among the failure times within a cluster, but not the covariate eﬀects (like in E1582) that vary from cluster to cluster.

8 General random eﬀects Cox model

A more general random eﬀects Cox model is

0 0 λij(t) = λ0(t) exp{β Zij + biWij},

Here Wij is a vector of covariates that have random effects, such as treatment in E1582 data; bi is the ‘covariate by cluster interaction’ (compare to the fixed effects model we might have written down earlier).

When Wij = 1, the above is the univariate frailty model.

Often the covariates in W are also included in Z, so that they have fixed effects (can be thought of as ‘main effects’), and we impose that E(bi) = 0.

The above model is also called the proportional hazards mixed-effects model (PHMM). It parallels the linear mixed- effects model (LMM) and generalized mixed-effects model (GLMM).

The bi’s are often assumed to be normal. Gamma distribution is no longer suitable, as it is not scale invariant.

This model was considered in Vaida and Xu (2000), and Ripatti and Palmgren (2000).

9 Cluster

log hazard Treatment -1.0 -0.5 0.0 0.5 1.0

0246810 time

Trt, Cluster 1

Trt, Cluster 2 log hazard -1.0 -0.5 0.0 0.5 1.0

0246810 time

Figure 2: Top: cluster eﬀects (βZ + bi0); bottom: cluster x treatment interaction (βZ + bi0 + bi1Z).

10 Inference

Inference under the random eﬀects Cox model is carried out using the full likelihood (as opposed to the partial likelihood).

• The assumptions are: data from diﬀerent clusters are independent; within the same cluster i, the dependence is induced by bi; in other words, conditional on bi, the individuals in cluster i are independent.

• Conditional on bi, we may write down the full likelihood for cluster i: ni Y δij Li = λij(Xij) Sij(Xij) j=1

ni 0 0 0 0 Y β Zij+b Wij δij β Zij+b Wij = {λ0(Xij)e i } exp{−Λ0(Xij)e i } j=1

• The whole likelihood condition on the bi’s is n Y Li. i=1

11 • The likelihood for observed data is then n Y Z L(β, λ0, σ) = Li · p(bi; σ)dbi. i=1 Here σ denotes the unknown parameter(s) in the distribution of the random eﬀects, and p(bi; σ) is the density of bi. The parameters to be estimated are (β, λ0, σ).

In order to solve for MLE, one needs to evaluate the above integrals. Often these integrals are not of closed forms. It- erative algorithms such as EM are often used.

An R package ‘phmm’ is available for the above NPMLE.

Table 1: Estimates (standard errors) from E1582 lung cancer trial

d = 0 d = 1 d = 2 β β Var(b) β Var(b) treatment -0.254 (0.085) -0.250 (0.104) 0.071 (0.069) -0.247 (0.119) 0.046 (0.184) bone 0.223 (0.093) 0.212 (0.095) - 0.230 (0.144) 0.129 (0.083) liver 0.429 (0.090) 0.423 (0.091) - 0.393 (0.094) - ps -0.602 (0.104) -0.641 (0.109) - -0.649 (0.131) - wt loss 0.200 (0.087) 0.218 (0.089) - 0.208 (0.092) -

12 Notes

In a previous ﬁgure of estimated treatment eﬀects by institution, those are the ‘estimated’ bi’s under the d = 1 model. They are obtained as empirical Bayes estimate following the MLE; they are the posterior means (can also be mode) of the bi’s given the observed data.

In addition,

• R package ‘phmm’ gives the NPMLE, as well as the empirical Bayes estimate of the random eﬀects, and various testing, AIC etc.; • Asymptotic properties for the NPMLE has been estab- lished, with also stable numerical properties (Gamst et al, 2009). • coxph() with ‘frailty’ option gives a penalized partial likelihood (PPL) estimator; • there is also an R package ‘coxme’ that gives the PPL under the general PHMM. • No theoretical properties for the PPL has been estab- lished (inconsistent in general). It has numerical problems sometimes, and the estimated variances (hence CI’s) can be inaccurate. • PPL is faster to compute.

13 (a)

· · · · ·· · ····· · · · ···· · · · · ·· ·· · ·· · ·· ·· · · · · · · · ·· · · ·· · ········ · ··· · · ············ ·· · ····· ··· · ··· · ·· ···· · · · ·· ·· ···· · · · · ·· · · ···· · · · ··· · · Random effect

1 event/patient 2 events/patient 3 4 -3 -2 -1 0 1 2 3 0 20 40 60 80 100 120 Patient

(b)

· ·· ·· ·· ·· · · · ·· · · · · ·· ·· · · · ·· · · · · ··· · · · ·· ··· ··· · ··· · · ·· ·· ·· ···· · ············ · ··· · ·· ········ · · · · · · ····· · ··· ·· · · · · · ·· · · · · · · ··· · · · · · · Random effect

1 event/patient 2 events/patient 3 4 -3 -2 -1 0 1 2 3 0 20 40 60 80 100 120 Patient

Figure 3: Estimated random eﬀects from univariate lognormal frailty model and 95% conﬁdence intervals (CGD data). (a) without adjustment for covariates other than treatment; (b) with adjustment for covariates.

Notes on recurrent events

Both the marginal model and the random eﬀects model can be used for recurrent event data. When the random eﬀects model is used, the assumption is a common baseline hazard function for the events of the same individual, like in the CGD example.

14 • Other methods have been studied that are aimed speciﬁ- cally for recurrent event data. One popular such method is PWP (Prentice, Williams and Peterson 1981), which is a conditional approach that takes into account the ordering of the events, and was the recommended method by Oakes (1992, in Survival Analysis, State of the Art, ed. Klein and Goel). • Another simple approach that is sometimes used is the Anderson-Gill model, or the counting process approach. This has the favor of the simple Cox model, and requires the relatively strong assumption that the times between successive events be conditionally independent given the history. The covariate ‘prior number of events’ is sometimes included in such a model. • Therneau and Hamilton (1997, Stat in Med, p.2029- 2047) compared the above methods using example datasets, showing how they are implemented using popular statis- tical software. Similar material can also be found in Therneau and Grambsch’s book.

15 The marginal approach

The marginal approach (WLW method) was proposed by Wei, Lin and Weissfeld (1989).

An example: in an AIDS trial to evaluate the drug rib- avirin, there are 3 treatment groups: placebo, low dose, and high dose. To study the antiretroviral capability of the drug, blood samples were collected at 4, 8 and 12 weeks. For each sample, the ‘viral load’ was measured by the number of days until virus positivity was detected. So potentially each patient in the study should have 3 event times.

In general, suppose that there are m types of failure. Let Tij be the failure time of the jth type for the ith subject, j = 1, ..., m, i = 1, ..., n. We observe

(Xij, δij) where Xij = min(Tij,Cij), δij = I(Tij ≤ Cij), and Cij is the censoring time.

Let Zij(t) be a p × 1 vector of possibly time-dependent covariates for subject i with respect to failure type j. Note that some covariates can be common for diﬀerent types of failure. In the example above, Zij = Zi =treatment assign- ment for patient i, stays the same for all three event times of the same patient.

16 The WLW model is 0 λij(t) = λ0j(t) exp{βjZij(t)}.

Here βj is the failure-speciﬁc regression parameter. This is a marginal model. It models the marginal hazard of type j failure time, and says nothing about the joint distribution of diﬀerent types of failures.

The assumption regarding censoring is that given Zij, the Tij’s and the Cij’s are independent.

Inference under the WLW model is carried out via the failure-speciﬁc partial likelihood

n ( 0 )δij Y exp[βjZij(Xij)] Lj(βj) = P 0 , exp[β Zlj(Xij)] i=1 l∈Rj(Xij) j

where Rj(t) = {l : Xlj ≥ t} is the set of subjects at risk for the jth type of failure just prior to time t. ˆ The MPLE (not a MLE) βj is solution to ∂ log Lj(βj)/∂βj = 0.

ˆ • βj is consistent for βj (this is exactly the same as ﬁt the Cox model to one failure type at a time); ˆ • βj’s are generally correlated; ˆ ˆ • (β1, ..., βm) are asymptotically normal with covariance matrix that can be consistently estimated by a sandwich estimator.

This allows simultaneous inference about the βj’s.

17 In the AIDS example, if β1j indicates the diﬀerence between low-dose and placebo, we may want test H0 : β11 = β12 = β13 = 0. This can be done using ˆ ˆ ˆ −1 ˆ ˆ ˆ 0 (β11, β12, β13)Ψˆ (β11, β12, β13) ˆ ˆ ˆ where Ψˆ is the estimated covariance matrix of (β11, β12, β13).

Furthermore, if we can assume that β11 = β12 = β13 = β1, ˆ we can estimate β1 by a linear combination of the β1j’s. The vector of weights that gives the smallest aymptotic variance is (eΨˆ −1e0)−1Ψˆ −1e0, where e = (1, ..., 1) is the vector of 1’s.

18 Notes:

• The marginal model is suitable for multivariate survival data that are correlated and have well-defined failure types, when we are interested in relating the marginal survival to the covariates, while seeing the correlation among the survival times as a nuisance parameter. The coefficients in the model has a ‘population-averaged’ interpretation (to be contrasted with the random effects approach later). • This model is sometimes used for recurrent event data. The interpretation, however, has some difficulty. This is because the model does not take the ordering of the events into account so that a subject can be at risk for the second event even before the first event has occurred. The method is also not efficient in the recurrent events setting (because it does not take into account the ordering of the events) though can be ‘convenient and useful’. coxph() with option cluster (see its relation with robust)

Therneau and Grambsch book p.186-189 also discusses ways of creating ‘new’ data set to ﬁt the model.

19 Random eﬀects vs. marginal models

• Marginal models are often simpler to fit (computation- ally faster). • One advantage of the random effects model over the marginal (WLW) model is that it allows ‘estimation’ of the random effects; this usually gives us more insight into the data and the scientific problem. – In E1582 data, estimated institutional treatment effects would allow the researchers to further investi- gate why certain institutions have larger treatment effects than others. – Estimation of the random effects and their variance is also of interest in genetic studies, where the variance components can be translated to the correlation. – The magnitudes of the random effects across clusters reflect the ‘heterogeneity’ among the clusters. In a recurrent events (CGD) example in Vaida and Xu (2000), where cluster = patient, and survival times = time to first infection, from the first infection to the second infection, .... The fact that people with more infections have larger random effects from fitting a univariate frailty model, reflects that these people are different from those with fewer infections, in a way not captured by the covariates that were collected.

20 A relation with robust estimation under the Cox model

‘Robust’ refers to when the assumption(s) are wrong. There are several possibilities:

1. the i.i.d. assumption is wrong: this becomes the WLW approach, which parallels the generalized estimating equations (GEE) approach for generalized linear models (GLM). In this case the original partial likelihood estimator is still consistent for the true regression coefficients, but its variance needs to be estimated by the robust sandwich estimator. 2. the Cox model is wrong, eg. non-PH; in this case the partial likelihood estimator converges in probability to some ‘least- false’ parameter defined by a population estimating equation, eg. an average regression effect under non-PH. Again its variance needs to be estimated by a sandwich estimator. (Lin and Wei, 1989; Xu and O’Quilgley, 2000) 3. missing covariate: this is specific to non-linear models like the Cox or logistic regression, i.e. if a true covariate is not included the model, then the resulting model is no longer a Cox or logistic model. In that sense the model is wrong, and 2) applies. 4. In addition, sometimes we might wish to use the weighted Cox model, eg. inverse probability weighting (IPW) for causal inference. (Lin, 1991; Sasieni, 1993)

All of the above fall under estimating equations instead of the likelihood approach, and use the ‘robust’ sandwich estimator for variance.