Outline. Frailty modelling of Multivariate Survival Data. Clustered survival data. Clustered survival data

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1 Outline Frailty modelling of Multivariate Survival Data Thomas Scheike Department of Biostatistics University of Copenhagen Marginal versus Frailty models. Two-stage frailty models: copula modelling. Example and Exercises. November 212 Clustered survival data For k = 1,..., K, i = 1,..., n, let T ik and C ik be the failure and censoring times for the ith individual in the kth cluster and let X ik (t) be a p-vector of covariates. Put T k = ( T 1k,..., T nk ), C k = (C 1k,..., C nk ), X k (t) = (X 1k (t),..., X nk (t)). We assume that ( T k, C k, X k ( )), k = 1,..., K are independent and identically distributed variables and these variables follow the model described in the following. The right-censored failure time is denoted T ik = T ik C ik and as usual we let Y ik (t) = 1(T ik t) Clustered survival data Two basic approaches : 1) Frailty models: given random effects "Z" the survival times are independent with hazards λ (t)z exp(x T i β) 2) Marginal approach, given covariates the marginal intensity is (need intensity to construct likelihood of data) Y i (t)λ (t) exp(x T i β) This gives population regression parameters, but one can also characterize the dependende using copula (or underlying frailty distribution). Assume for example that given frailty Z, the intensities are Zh(Λ, exp(xi T β)) for a specific choice of h. More on this later.

2 Frailty models Characterizing dependence ( T k, C k, X k ( ), V k ), k = 1,, K are assumed to be i.i.d. variables. Censoring, conditional on V k and covariates, is assumed to be independent and noninformative on V k. When the frailty variable V k is gamma-distributed with mean one and variance 1 we get the. The shared frailty model is specified solely with respect to the (unobserved) filtration H t = k Hk t, where H k t = σ{n ik (s), Y ik (s), X ik (s), V k : i = 1, n, s t}, and often it is assumed to be a proportional hazards model We shall talk about estimation for the random effects models, but first some general remarks on dependence measures. Given two (T 1, T 2 ) survival times, we can compute Pearson correlation, Spearman correlation, and the Kendall s tau. Different ways of interpreting the degree of association. Spearman correlation depends on aspects of the bivariate distribution, but not the marginals. The same property is kept by Kendall s tau. These measures are rank-based. λ H ik (t) = Y ik(t)v k λ (t) exp(xik T (t)β). (1) Kendall s tau Dependence Kendall s tau is defined given an i.i.d. copy, and the degree of difference between such two i.i.d. copies E(I ((T 1 T 1 )(T 2 T 2 ) > )) E(I ((T 1 T 1 )(T 2 T 2 ) < )) which is just the concordance probability - discordance probability Given a bivariate distrubtion written on copula-form F (s, t) = C(F 1 (s), F 2 (t)) where C is the copula specifying bivariate distributions on [, 1] 2, and F j is the jth marginal. Kendall s tau is computed as τ = C(u, v)c(du, dv) Y ik (t)v k λ (t) exp(xik T (t)β). (2) with V k i.i.d. Γ with mean 1, and variance 1/, then τ = in the sense that with fixed covariates we see this Kendall s tau.

3 The frailty model When V k i.i.d. Γ with mean 1, and variance 1/, then the observed intensity given covariates become N k. Y ik (t) i exp(x ik T (t)β)λ (T ik ) λ (t) exp(xik T (t)β). (3) And the marginal intensity becomes: N ik (t ) Y ik (t) exp(xik T (t)β)λ (T ik ) λ (t) exp(xik T (t)β). (4) What is the observed hazard, and how is it different from marginal intensities? As a part of the next exercise we shall also derive these expressions. Frailty models Estimation : NPMLE for Frailty model (V k dλ (s) exp(xki T β)) N ki (s) exp ( k i s t Th V k exp(x T ki β)dλ (s) (5) and with V k Gamma distributed with mean 1, and variance 1/, giving the additional term f (V k ) (6) k For fixed write up the EM-algorithm for this model. How do we get standard errors for this model. To compute E-step we need E(Vk Data), and this conditional mean (and distribution) can be found from full likelihood. Recall that a Gamma distribution with parameters λ, α (Γ(λ, α)) has mean λα and variance λα 2 and density (Γ(λ)α λ ) 1 x λ 1 exp( x/α). Write up full EM-algorithm for the model where is not known. Frailty models, Attenuation > fit<-coxph(surv(time,status)~adult*trt+frailty(id), data=diabetes) > fit Call: coxph(formula = Surv(time, status) ~ adult * trt + frailty(id), data = diabetes) coef se(coef) se2 Chisq DF p adult trt frailty(id) adult:trt Iterations: 6 outer, 25 Newton-Raphson Variance of random effect=.926 I-likelihood = -847 Fit the model yoursel, and try to simplify the regression part, is the frailty variance significant, and what is it. What is the related Kendall s tau? Compare with the marginal esimates. How do the regression effects differ? If dim(x)=1 and gamma frailty with variance and mean 1 then the relative risk is Y (t)λ (t) exp (X T 1 β) 1 + Λ (t) ( exp (β) 1 + Λ (t) 1 + exp (β)λ (t) ), (7) where Λ (t) = t λ (s) ds. exp (β) at time t = and tends to 1 as t >. V positive stable with Laplace transform ϕ (t) = exp ( t ), < 1, then the marginal intensity is Y (t)λ (t)λ (t) 1 exp (X T β),

4 Frailty models, Attenuation Frailty models Assume that V is a positive stochastic variable with Laplace transform ϕ (t), that the covariate is one-dimensional, and that β >. The relative risk in the marginal model is exp (β) (D log ϕ )(exp ((X + 1)β)Λ (t)) (D log ϕ )(exp (X β)λ (t)) = exp (β)k(t), and we see that k(t) 1 if and only if (D log ϕ )(exp ((X + 1)β)Λ (t)) (D log ϕ )(exp (X β)λ (t)). The latter inequality holds if log (ϕ ) is convex, which is the case since D 2 log (ϕ )(t) = E(V 2 h(t, V )) E(Vh(t, V )) 2 with h(t, V ) = exp ( tv )/E(exp ( tv )). Two sets of frailty models quite similar: Frailty model Frailty model has subject specific regression effects, that relies on choice of frailty (ofcourse). Frailty parameter model can in principle be identified solely on univariate data because marginal depends on. Frailty parameter primarily identified from correlation in practice. Frailty parameter has Kendall s tau interpretation. Frailty model more easy to extend to various other settings. Two-stage model, Later today Marginals fixed and regression parameters gives population effects. Frailty accounts solely for dependence. Frailty parameter has Kendall s tau interpretation. Two-stage modelling Two-stage modelling To be explicit, assume that the marginal and conditional intensities are λ F ik ik (t) = λ ik(t), λ H ik (t) = V kλ ik (t), where we assume that λ ik (t) is predictable with respect to the marginal filtration. One may show that the relationship between the above two intensities is Establish the connection between the the two sets of intensities: Hint : compute the marginal survival for both models! λ ik (t) = Y ik (t)( λ ik (t))(d log ϕ )( t λ ik (t) = Y ik(t)( λ ik (t)) exp ( t λ ik (s) ds), t λ ik (s) ds)(dϕ 1 )(exp ( λ ik (s) ds)),

5 Marginal models Right-censored failure times T ik = T ik C ik, Y ik (t) = 1(T ik t), N ik (t) = 1(T ik t, T ik = T ik ) Marginal (intensity) model is a model given Cox model: F ik t = σ{n ik (s), Y ik (s), X ik (s) : s t}, (8) λ F ik ik (t) = Y ik(t)λ (t) exp (Xik T (t)β). (9) It is important to note that (9) is not the intensity with respect to the observed filtration F t = Ft k, (1) k where F k t = σ{n ik (s), Y ik (s), X ik (s) : i = 1, n, s t} is the information generated by observing all the individuals in the kth cluster. Now, with respect to (unobserved) filtration where is H t = k H k t, (11) H k t = σ{n ik (s), Y ik (s), X ik (s), V k : i = 1, n, s t}, λ H ik (t) = V kλ ik (t,, λ ( )), (12) referred to as the, and so that the marginal intensities are on Cox form λ F ik ik (t) = Y ik(t)λ (t) exp (Xik T (t)β), (13) Characterizing dependence Copula models P( T 1 > t 1,..., T n > t n ) = C (S 1 (t 1 ),..., S n (t n )), where S j, j = 1,..., n, denotes the marginal survivor functions. All multivariate distribution has this form. Archimedean copula model family, C (u 1,..., u n ) = ϕ (ϕ 1 (u 1) + + ϕ 1 (u n)) for some non-negative convex decreasing function ϕ with ϕ () = 1. Then λ ik is λ ik (t,, λ ( )) = Y ik (t)λ (t) exp(xik T (t)β) t exp( 1 exp(xik T (s)βt )λ (s)ds), It is of interest to find the intensities with respect to the observed filtration F t given in (1). It can be shown that these are where λ F ik (t) = Y ik(t)λ (t) exp(x T ik (t)β)f ik(t), (14) f ik (t) = ( + N k (t ) n t f k (t) = 1 + (exp( 1 j=1 )(exp( 1 t λ (s) exp(x T f k (t) ik (s)β)ds) ), Y jk (s)λ (s) exp(xjk T (s)β)ds) 1).

6 The observed (partial) log-likelihood function is K k=1 log(1 + N k(t ) )dn k (t) + K [ + N k (τ) ] n log(1 + 1 k=1 i=1 K n k=1 i=1 log(y ik (t) λ ik (t))dn ik (t) Y ik (t) λ ik (t)dt), (15) Terms depending on in (15) gives 1 K ( K k=1 log(1 + 1 N k (t ))dn k (t) + K k=1 i=1 n 1 N ik (τ)h ik K ) ( + N k (τ)) log(r k ()), (16) k=1 where λ ik (t) = λ (t)e X T ik (t)β exp( 1 t e X T ik (s)β λ (s)ds). H ik = Y ik (t)e X T ik (t)β dλ (t), R k () = 1 + n (exp( 1 H ik ) 1). i=1 Now, by replacing H ik with Ĥ ik = Y ik (t) exp(x T ik (t) ˆβ I )d ˆΛ I (t) in (16), we obtain the pseudo log likelihood for, and maximizing this function in gives the two-stage estimator of. Under some regularity conditions, [?] showed consistency and asymptotic normality of this estimator. This is equivalent to fitting directly the MLE for the Clayton-Oakes copula form C (u 1,..., u n ) = ϕ (ϕ 1 (u 1) + + ϕ 1 (u n)) Derive this Copula form for the Clayton-Oakes Gamma frailty model with Cox marginals. Write up the likelihood for censored bivariate survival data based on this represenation.

7 Two-stage modelling Two-stage modelling data(diabetes) # Marginal Cox model with treat as covariate marg <- cox.aalen(surv(time,status) ~prop(treat)+cluster(id),data=diabetes) fit<-two.stage(marg,data=diabetes,theta=1) summary(fit) Dependence parameter for Clayton-Oakes-Glidden model Variance SE z P-val Kendall s tau intercept Marginal Cox-Aalen model fit Proportional Cox terms : Coef. SE Robust SE D2log(L)^-1 z P-val prop(treat) e-6 # Stratification after adult theta.des<-model.matrix(~-1+factor(adult),diabetes); fit.s2<-two.stage(marg,data=diabetes,nit=4,theta=1,theta.des=theta.des) summary(fit.s2) Dependence parameter for Clayton-Oakes-Glidden model Variance SE z P-val Kendall s tau factor(adult) factor(adult) Marginal Cox-Aalen model fit Proportional Cox terms : Coef. SE Robust SE D2log(L)^-1 z P-val prop(treat) e-6 Two-stage modelling Likelihoods # test for same variance among the two strata theta.des<-model.matrix(~factor(adult),diabetes); fit.s3<-two.stage(marg,data=diabetes,nit=4,theta=1,theta.des=theta.des) summary(fit.s3) Dependence parameter for Clayton-Oakes-Glidden model Variance SE z P-val Kendall s tau (Intercept) factor(adult) Marginal Cox-Aalen model fit Proportional Cox terms : Coef. SE Robust SE D2log(L)^-1 z P-val prop(treat) e-6 Generally given af bivariate survival distribution G(x, y) and censored bivariate survival data (X i, δ i, Y i, ν i ), the likelihood is written as G(X i, Y i ) (1 δ i )(1 ν i ) (D x G)(X i, Y i ) δ i (1 ν i ) (17) (D y G)(X i, Y i ) (1 δ i )ν i (D X D y G)(X i, Y i ) δ i ν i (18) # to fit model without covariates, beta.fixed=1, but still need prop term! fit<-two.stage(marg,data=diabetes,theta=.95,detail=,beta.fixed=1) summary(fit) Dependence parameter for Clayton-Oakes-Glidden model Variance SE z P-val Kendall s tau intercept

8 Goodness of fit of Gamma Frailty Different frailt distribution have different consequences for the type of dependence. GOF One idea (Shih, 1998, Shih and Louis 1995) Compare likelihoods from different models. Score process. Under Gamma-distribution we earlier computed E(Z k History) = γ k (t) = N k i exp(x T ik (t)β)λ (T ik ) Idea is to look at γ k (t) that has mean 1 under the gamma distribution. Let W (t) = K 1/2 k (γ k (t) 1) that is asymptotically Gaussian with a variance we can also estiamte under the null. This can be resampled and a supremum test can be constructed. Frailty models Considering the twin.csv data of menarche ages for pairs of twins. Look at the two.stage help pages to see different examples of its use. First estimate the marginal effect of cohort, and assess wheter or not the zygosity affects the marginal models. Fit a frailty model for the overall data, to asses the effect of cohort. Use the phmm and coxph program and compare the estimates of the cumulative baseline. Compare also the baseline with that from the marginal model for mono and dyzygotic twins. Fit separate models for monozygotic and dizygotic twins. Estimate also the marginal baselines using this model, and compare them. Is there a genetic effect on the timing of menarche? Report a Kendall s tau for this data for mono- and dizygotic twins and overall. Is the dependence different for mono- versus dizygotic twins. How can you make survival predictions for the monozygotic and dyzygotic twins? Consider the (simulated) twin data of menarche age for twins. Fit marginal models for the mono- and the dizygotic twins. Report a Kendall s tau. How does this Kendall s tau compare with that from the standard frailty model? Validate the fit of the model, by validating the marginal model. What about the frailty part? Compare formally the dependence between mono- and dyzygotic twins.

Outline. Frailty modelling of Multivariate Survival Data. Clustered survival data. Clustered survival data

Outline. Frailty modelling of Multivariate Survival Data. Clustered survival data. Clustered survival data Outline Frailty modelling of Multivariate Survival Data Thomas Scheike ts@biostat.ku.dk Department of Biostatistics University of Copenhagen Marginal versus Frailty models. Two-stage frailty models: copula