Multivariate Survival Models

Multivariate Survival Models GermánRodr´ıguez [email protected] Spring, 2001; revised Spring 2005 In this unit we study models for multivariate survival (in the statistical sense of many outcomes, not just many predictors). 1 Areas of Application We start by reviewing four main areas of applications of these models. 1.1 Series of Events One area of interest is processes where each individual may experience a succession of events. Examples include birth intervals and spells of unem- ployment. Because the various events occur to the same individual, the waiting times will in general not be independent. Some couples tend to have short birth intervals while others have long ones. Observed covariates such as contraceptive use may explain some of the association. In general, however, there will remain some correlation due to unobserved individual traits. Because the events occur one after the other, it will generally be the case that only the last interval can be censored. This introduces some simplifica- tion in estimation. In particular, it makes it possible to study the sequence using successive conditioning. To fix ideas consider an example with three intervals. The joint density of T1,T2 and T3 f123(t1, t2, t3) can always be written as the product of the marginal of T1, the conditional distribution of T2 given T1, and the conditional distribution of T3 given T1 and T2: f1(t1) f2|1(t2|t1) f3|12(t3|t1, t2). 1 These two equations are an identity: any joint distribution can be factored in this way. A typical contribution to the likelihood function, given the fact that T3 is the only waiting time that can be censored, will look like d3 f1(t1) f2|1(t2|t1) λ3|12(t3|t1, t2) S3|12(t3|t1, t2). where the last term is, as usual, the conditional survival function for censored cases and the conditional density for deaths. As long as we model the conditional distributions using different parameters, the likelihood will factor into separate components. Typically, one would model the conditional distributions by introducing the previous waiting times as covariates, for example we could write the three-equation model 0 x β1 λ1(t1|x) = λ01(t1)e 0 x β1+s1(t1) λ2(t2|t1, x) = λ02(t2)e 0 x β1+s1(t1)+s2(t2) λ3(t3|t1, t2, x) = λ03(t2)e where s(t) denotes a smooth term on t, such as a smoothing or regression spline. My own work on birth intervals (Rodr´ıguez et al. 1984) used this approach. It turns out that only the previous interval turned out to be rele- vant, so our models were simplified. Some of the advantages of this approach are • It is easy, because it breaks down into a series of univariate analyzes. • It is consistent with sequential decision making, where the actual val- ues of t1, . , tj−1 may affect behavior influencing tj. On the other hand it has the disadvantage of using separate parameters for each spell. 1.2 Kindred Lifetimes A second area where we may use multivariate survival models consists of related lifetimes, such as the survival of husband and wife, siblings, or other kin. Following Vaupel I will call these kindred lifetimes. In general there is reason to believe that these lifetimes are correlated, because of common unobserved characteristics of the couple (in the case of husband and wife survival) or the family (in the case of sibling survival). 2 An important feature of kindred lifetimes is that any (or all) of the waiting times may be censored. With three children, for example, you may observe 8 different patterns of censoring. This means that we cannot adopt the simple sequential approach outlined earlier for series of events, as we will often lack the information needed. For example we can’t very well model T2 as a function of T1 when T1 is censored, at least not in the simple way we have described. Thus, we need a more general approach. 1.3 Competing Risks We have already encountered a third type of multivariate data in our discus- sion of competing risks, where T1,T2,...,Tk represent latent survival times to different causes of death. As noted earlier, estimation of these models is complicated by the fact that we only observe T = min{T1,...,Tk} and even this can be censored. Keep in mind, however, that the models that follow could be used, at least conceptually, in this context. 1.4 Event History Models The fourth and final type of multivariate data involves transitions among several types of states. This combines elements of competing risk models with models for series of events. Consider for example the analysis of nuptiality. You start in the single state. From there you can move to cohabiting or married. From cohabiting you can move to married or to separated. And so on. If you distinguish separations from marriage or cohabiting as well as widowhood and divorce, you probably have about 15 possible transitions of interest. Analysts often study one type of transition, for example age at first marriage or marriage dissolution. With event history data, however, one may study the complete process, allowing for inter-dependencies among the different kinds of transitions. The nature of the data allows conditioning each move on the entire history of previous moves. A closely related subject in demography are multi-state models. A lot of work in that area assumes a homogeneous population with constant transition rates and independent moves, and emphasizes analytic results, such as the steady-state proportion in each state. In some ways event history models are to multi-state models what Cox regression models are to the traditional life table. 3 2 Bivariate Survival Models We consider first the case of only two survival times, T1 and T2. This section follows Cox and Oakes (1984, Chapter 10) and Guo and Rodr´ıguez (1992). 2.1 Basic Definitions Interest will focus on the joint survival S12(t1, t2) = Pr{T1 ≥ t1,T2 ≥ t2}. Note that S12(t, t) is the probability that both units are alive at t. We also have the marginal survival function S1(t1) = Pr{T1 ≥ t1} = S12(t1, 0), and similarly for t2. If T1 and T2 were independent the joint survival function would be the product of the marginals. We might also be interested in the conditional survival function, which has two variants S1|2(t1|T2 = t2) = Pr{T1 ≥ t1|T2 = t2}, giving the survival probabilities given that the other unit failed at time t2, and S1|2(t1|T2 ≥ t2) = Pr{T1 ≥ t1|T2 ≥ t2}, given that the other unit survived to just before time t2. Associated with each of these survival functions there will be a cumulative hazard function, which can be obtained by taking minus the log of the survival function. There will also be a hazard function, which can be obtained by taking derivatives of the cumulative hazard. Specifically, we can define the joint hazard function as 2 λ12(t1, t2) = lim Pr{T1 ∈ [t1, t1 +dt),T2 ∈ [t2, t2 +dt)|T1 ≥ t1,T2 ≥ t2}/dt . dt→0 This is the instantaneous risk that one unit fails at t1 and the other fails at t2 given that they were alive just before t1 and t2, respectively. We can also define a marginal hazard, λ1(t1) = lim Pr{T1 ∈ [t1, t1 + dt)|T1 ≥ t1}/dt. dt→0 Under independence, the joint hazard is the sum of the marginal hazards. Can you prove this result? 4 We can also define conditional hazards λ1|2(t1|T2 = t2) = lim Pr{T1 ∈ [t1, t1 + dt)|T1 ≥ t1,T2 = t2}/dt dt→0 which tracks the risk for one unit given that the other failed at t2, and λ1|2(t1|T2 ≥ t2) = lim Pr{T1 ∈ [t1, t1 + dt)|T1 ≥ t1,T2 ≥ t2}/dt dt→0 given that the other unit survived to just before t2. The cause-specific hazard considered in the context of competing risks (tracking the risk of death due to cause j among survivors to time t) is a special case of the latter, namely the case where t1 = t2: λ1|2(t|T2 ≥ t) = lim Pr{T1 ∈ [t, t + dt)|T1 ≥ t, T2 ≥ t}/dt. dt→0 Knowledge of the two types of conditional hazards completely determines a joint distribution, see Cox and Oakes (1984, p. 157) for an expression linking the joint density to the conditional hazards. 2.2 Frailty Models One way to model a joint survival function is to assume the existence of a random effect θ such that given θ, T1 and T2 are independent. Depending on the context, θ may represent traits that persist across spells or are common among kin, and which account for the lack of independence. In symbols, we can write the assumption of conditional independence as S12(t1, t2|θ) = S1(t1|θ)S2(t2|θ), where all survival functions are conditional on θ. Usually the random effect is assumed to act multiplicatively on the hazard, so that θ Si(ti|θ) = S0i(ti) for some baseline survival function S0i(t). Under this assumption the cumulative hazards are Λi(ti) = θΛ0i(ti) and the individual hazards are λi(ti) = θλ0i(ti). 5 The conditional joint survival function is then θ θ S12(t1, t2|θ) = S01(t1) S02(t2) = e−θΛ01(t1)e−θΛ02(t2) = e−θ(Λ01(t1)+Λ02(t2)). This is not very useful because θ is not observed. To obtain the uncon- ditional survival function we need to ‘integrate out’ θ. Suppose that θ has density g(θ).

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