Goodness-Of-Fit for Cox s Regression Model. Extensions of Cox s Regression Model. Survival Analysis Fall 2004, Copenhagen

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1 Outline Cox s proportional hazards model. Goodness-of-fit tools More flexible models R-package timereg Forthcoming book, Martinussen and Scheike. 2/38 University of Copenhagen Goodness-Of-Fit for Cox s Regression Model. Extensions of Cox s Regression Model. Survival Analysis Fall 24, Copenhagen Torben Martinussen and Thomas Scheike 1/38

2 Cox s proportional hazards model In practice one has covariates: X i (p-dimensional). Hazard conditional on covariates: α i (t, X i ). The absolute dominant model is Cox s proportional hazards model: α i (t) = α (t) exp (β T X i ) where α (t) is unspecified baseline hazard (hazard for X i = ). Flexible model. Model is easily fitted using for example SAS or R (S-Plus). Primary model for survival data because of its nice properties. Suppose X is 1-dim. (or fix other covariates) then the relative risk α(t, X + 1) α(t, X) = exp (β) 4/38 is not depending on time (key assumption)! This assumption is often violated! Survival analysis Standard setup for right-censored survival data. IID copies of (T, D) where T = T C D = I(T C) with T being the true survival time and C the (potential) censoring time and possibly covariates X i (t). Hazard-function Counting process Martingale 1 α(t) = lim h h P (t T < t + h T t, F ). N i (t) = I(T i t, D i = 1) M i (t) = N i (t) Λ i (t) 3/38 where Λ i (t) = Y i(s)α(s) ds (compensator), Y i (t) = I(t T i ) (at risk process).

3 > library(survival) > data(pbc) > attach(pbc) > cbind(time,status,age,edema,bili,protime,alb)[1:5,] time status age edema bili protime alb [1,] [2,] [3,] [4,] [5,] > sum(status) [1] 161 > fit.pbc<-coxph(surv(time/365, status) age+edema+log(bili)+log(protime)+log(alb)) > fit.pbc Call: coxph(formula = Surv(time/365, status) age + edema + log(bili) + log(protime) + log(alb)) coef exp(coef) se(coef) z p age e-7 edema e-3 log(bili) e+ log(protime) e-3 log(alb) e-4 Likelihood ratio test=234 on 5 df, p= n= 418 6/38 PBC-data PBC data (primary biliary cirrhosis): 418 patients are followed until death or censoring. PBC is a fatal chronic liver disease. Important explanatory variables: Age Albumin Bilirubin Edema Prothrombin time Fitting Cox s model in R. 5/38

4 Graphical GOF for Cox s regression model 8/38 Figure 1: Estimated log-cumulative hazards difference along with 95% pointwise confidence intervals. The straight lines (dashed lines) are based on the Cox model. Cox s proportional hazards model Traditional goodness-of-fit tools. Model: α i (t) = α (t) exp(β 1 X i β p X ip ) Investigate if each of the covariates are consistent with the proportional hazards assumption. Stratify based on a grouping (k=1,...,k) based on X i1 s values: α i (t) = α k (t) exp(β 2 X i β p X ip ); ifx i1 A k 7/38 Now, if the underlying full Cox-model is true the baseline estimates α k (t) should be proportional, as α k (t) = α (t) exp( K β 1k I(X i1 A k )). k=1 Graphical model-check of proportionality by making graphs of estimates of log( α k(s)ds). Plotted against t they should be parallel.

5 Cumulative martingale residuals Alternative: Cumulative martingale residuals, (Lin et al., 1993). The martingales under the Cox regression model can be written as M i (t) = N i (t) Y i (s) exp(x T β)dλ (s); ˆMi (t) = N i (t) Y i (s) exp(x T ˆβ)dˆΛ (s) The score function, evaluated in the estimate ˆβ, and seen as a function of time, can for example be written as U( ˆβ, t) = n i=1 X i (s)d ˆM i (s) = n i=1 (X i (s) E(t, ˆβ))d ˆM i (s). and is asymptotically equivalent to a Gaussian process (not a martingale) that can easily be simulated (LWY,93). Can now proceed to suggest some appropriate test statistic like sup U( ˆβ, t) t [,τ] 1/38 This is essentially a test for time-constant effects of all covariates!! Cox s proportional hazards model Traditional goodness-of-fit tools. Make tests against specific deviations: Replace X 1 with (X 1, X 1 (log (t))), say (β 1 β 1 + β p+1 log (t)). Test the null β p+1 =. These methods are quite useful but also have some limitations : Graphical method: Not parallel. What is acceptable? What if a given covariate is continuous? Test: Ad hoc method. Which transformation to use? Both methods: They assume that model is ok for all the other covariates. 9/38

6 Resampling technique The observed score process is given as U( ˆβ, t) and its asymptotic distribution is equivalent to the asymptotic distribution of i (Ŵ1i (t) + I( ˆβ, t)i 1 ( ˆβ, τ)ŵ1i(τ)) G i where G 1,..., G n are independent standard normals, and independent of the observed data. 12/38 Score process The score process evaluated at β can be written as U(β, t) = i W 1i (t) (1) where with W 1i (t) = ( Zi Z T Y (β )(Y T (β )W Y (β )) 1 Y i ) dmi (s), Y (β, t) = (Y 1 exp(z T 1 β),..., Y n exp(z T n β)) W (t) = diag(y i exp( Z T i β)) Variance can then be estimated robustly by { } ˆΣ β = ni 1 ( ˆβ, τ) Ŵ 1i (t) 2 I 1 ( ˆβ, τ). i 11/38 where Ŵ1i is obtained by replacing M i by ˆM i.

7 PBC-data 14/38 PBC-data > library(timereg) > fit<-cox.aalen(surv(time/365,status) prop(age)+prop(edema)+ + prop(log(bilirubin))+prop(log(albumin))+prop(log(protime)), + weighted.score=,pbc); Right censored survival times Cox-Aalen Survival Model Simulations starts N= 5 > summary(ourcox) Cox-Aalen Model Score Test for Proportionality sup hat U(t) p-value H_: U(t) Proportional prop(age) prop(edema) prop(log(bilirubin)) prop(log(albumin)) prop(log(protime)) > plot(fit,score=t,xlab="time (years)") 13/38

8 Cumulative Residuals Now, ˆM(t) = G(β, s)dm(s) + { Y (β, s)y (β, s) Y ( ˆβ, s)y ( ˆβ, } s) dn(s). The second term can be Taylor series expanded [ G(β, s)diag { Y (β, s)y (β, s)dn(s) } ] Z(s) ( ˆβ β ) [ = G(s)diag { Y (β, s)y (β, s)dn(s) } ] Z(s) I 1 (β, τ)u(β, τ) where β and β are on line segment between ˆβ and β. Therefore ˆM(t) is asymptotically equivalent (see below) to 16/38 M(t) + B(β, t)m(τ) Cumulative Residuals Model can be written as the n 1 vector N(t) = Y (t)λ (t) + M(t) where M(t) is the mean-zero martingale, and then the martingales are estimated by ˆM(t) = N(t) Y ( ˆβ, s)y ( ˆβ, s)dn(s) = G(s)dN(s). where and G( ˆβ, s) = I Y ( ˆβ, s)y ( ˆβ, s) Y (β, s) = (Y T W Y ) 1 Y T W. 15/38

9 Cumulative Residuals M U (t) is asymptotically equivalent to n i=1 [ U i (t) U T (s)y (β, s) { Y T (β, s)w (s)y (β, s) } 1 Yi (s)dm i (s) U T (s)g(s)diag { Y (β, s)y (β, s)dn(s) } ] n Z(s) = W i (t) + o p (n 1/2 ), where W i (t) are i.i.d. an W 1,i is an i.i.d. decomposition of ˆβ β. Resample construction Ŵi (t)g i, i=1 W 1,i + o p (n 1/2 ), and G 1,..., G n are standard normals have the same asymptotic distribution. 18/38 Where Ŵ i (t) is obtained by using ˆM i instead of M i. Cumulative Residuals A cumulative residual process is then defined by M U (t) = U T (t)d ˆM(s) and this process is asymptotically equivalent to [ U T (t)g(β, s)dm(s) U T (s)g(s)diag { Y (β, s)y (β, s)dn(s) } ] Z(s) ( ˆβ β ). Denote the second integral in the latter display by B U (β, t). The variance of M U (t) can be estimated by the optional variation process [M U ] (t) = U T (s)g( ˆβ, s)diag(dn(s))u(s)g( ˆβ, s) + B U ( ˆβ, t) [U] (τ)b T U ( ˆβ, t) B U ( ˆβ, t) [M U, U] (t) [U, M U ] (t)b T U ( ˆβ, t). 17/38

10 Cumulative residuals > # our PBC version with no ties!!!!!!!!!!!!!!!!!!!! > fit<-cox.aalen(surv(time/365,status) prop(age)+ prop(edema)+prop(log(bilirubin))+prop(log(albumin))+ prop(log(protime)),pbc, + weighted.score=,resid.mg=1); Cox-Aalen Survival Model Simulations starts N= 5 > > X<-model.matrix( -1+cut(Bilirubin,quantile(Bilirubin),include.lowest=T),pbc) > colnames(x)<-c("1. quartile","2. quartile","3. quartile","4. quartile"); > > resids<-mg.resids(fit,pbc,x,n.sim=1,cum.resid=1) > summary(resids) Test for cumulative MG-residuals Grouped Residuals consistent with model sup hat B(t) p-value H_: B(t)= 1. quartile quartile quartile quartile int ( B(t) )ˆ2 dt p-value H_: B(t)= 1. quartile quartile quartile quartile /38 Residual versus covariates consistent with model sup hat B(t) p-value H_: B(t)= prop(age) prop(log(bilirubin)) prop(log(albumin)) prop(log(protime)) Cumulative Residuals (Lin et al., 1993) suggest to cumulate the residuals over the covariate space as well as over time, and thus considers the double cumulative processes M j (x, t) = = K T (j, x, s)d ˆM(s) K T (j, x, s)g(s)dm(s) for j = 1,.., p where K(j, x, t) is an n 1 vector with ith I(Z i,j (t) x) for i = 1,.., n. Integrating over time we get a process in x M j (x) = M j (x, τ) (2) 19/38

11 Cumulative martingale residuals 22/38 Cumulative martingale residuals 21/38

12 Cumulative martingale residuals 24/38 Cumulative residuals > nfit<-cox.aalen(surv(time/365,status) prop(age)+prop(edema)+prop(bilirubin)+prop(log(albumin))+prop(log(protime)),pbc, weighted.score=,resid.mg=1); > nresids<-mg.resids(nfit,pbc,x,n.sim=1,cum.resid=1) > summary(nresids) Test for cumulative MG-residuals Grouped Residuals consistent with model sup hat B(t) p-value H_: B(t)= 1. quartile quartile quartile quartile int ( B(t) )ˆ2 dt p-value H_: B(t)= 1. quartile quartile quartile quartile Residual versus covariates consistent with model sup hat B(t) p-value H_: B(t)= prop(age) prop(bilirubin) prop(log(albumin)) prop(log(protime)) /38

13 Cox s model with time-dependent effects A typical deviation from Cox s model is time-dependent covariate effects. Treatment is effective for some time, but then effect levels off. Takes some time before treatment has an effect. Model α i (t) = exp (β(t) T X i ) where coefficients β(t) are now depending on time! Score-equation: X(t) T (dn(t) λ(t)dt) = (3) Cannot solve (3). Taylor expansion and integration of (3) yields an algorithm. 26/38 Cumulative martingale residuals 25/38

14 Cox s model with time-dependent effects Algorithm: g( B)(t) = β(s) ds + Ã(s) 1 X(s) T dn(s) Ã(s) 1 X(s) T λ(s) ds, (4) with Ã(t) = A β(t) = i Y i(t)e x i(t) T β(t) x i (t)x i (t) T and β(t) obtained from B(t) by smoothing. Theorem (4) has a solution g( ˆB) = ˆB n 1/2 ( ˆB B) D U ˆΣ(t) = n Â(s) 1 ds ˆB is efficient 28/38 Reference:(Martinussen et al., 22). Cox s model with time-dependent effects If consistent estimate, β(t), is present for estimating β(t) Newton-Raphson suggests that λ(t)dt = Ã(s) 1 X(s) T dn(s) Ã(s) 1 X(s) T λ(s) ds, where Ã(t) = A β(t) = i Y i(t)e x i(t) T β(t) x i (t)x i (t) T 27/38

15 PBC-data > fit<-timecox(surv(time/365,status) Age+Edema+log(Bilirubin)+log(Albumin)+log(Protime),pbc, + maxtime=3/365,band.width=.5); Right censored survival times Nonparametric Multiplicative Hazard Model Simulations starts N= 5 > plot(fit,ylab="cumulative coefficients",xlab="time (years)"); > summary(fit) Multiplicative Hazard Model Test for nonparametric terms 3/38 Test for non-siginificant effects sup hat B(t)/SD(t) p-value H_: B(t)= (Intercept) Age Edema log(bilirubin) log(albumin) log(protime) Test for time invariant effects sup B(t) - (t/tau)b(tau) p-value H_: B(t)=b t (Intercept) Age Edema log(bilirubin) log(albumin) log(protime) Cox s model with time-dependent effects Important: can also handle the semi-parametric model λ(t) = Y (t)λ (t) exp(x T (t)β(t) + Z T (t)γ) (5) Can investigate the important H : β p (t) γ q+1 of non-time-dependency; Notice that it may be done in a model allowing other covariates to have time-dependent effects! 29/38

16 PBC-data > fit.semi<-timecox(surv(time/365,status) semi(age)+edema+semi(log(bilirubin))+semi(log(albumin))+log(protime),pbc, maxtime=3/365,band.width=.5) Right censored survival times Semiparametric Multiplicative Risk Model Simulations starts N= 5 > summary(fit.semi) Multiplicative Hazard Model Test for nonparametric terms Test for non-siginificant effects sup hat B(t)/SD(t) p-value H_: B(t)= (Intercept) Edema log(protime) Test for time invariant effects sup B(t) - (t/tau)b(tau) p-value H_: B(t)=b t (Intercept) Edema log(protime) /38 Parametric terms : Coef. Std. Error Robust Std. Error semi(age) semi(log(bilirubin)) semi(log(albumin)) PBC-data 31/38

17 Mix of Aalens and Cox s models Cox-Aalen model α i (t) = α(t) T X i (t) exp (β T Z i (t)), Gives a mix of Aalens and Cox s models Flexible modelling in additive part and multiplicative relative risk parameters for Z. Reference : (Scheike and Zhang, 22; Scheike and Zhang, 23) Model can also be fitted in timereg. Gives excess risk interpretation of additive part and relative risk interpretation for multiplicative part. Cox s regression model and stratified versions of it is a special case of the model. 34/38 Cox s model with time-dependent effects Model with timedependent and constant effects is from a theoretically point of view much more satisfactory Model is available using the R-package timereg. Practical experience is needed. May need quite a bit of data to get reliable inference. Needs to choose a bandwidth. Optimal, cross-validation. Additional reference: (Scheike and Martinussen, 24) Alternative models that allow for timedependent effects without the unpleasant bandwidth choice are for example : Aalens s additive hazards model Cox-Aalen model 33/38 Proportional Excess hazard models These can also be fitted using timereg!

18 Cox-Aalen model 36/38 PBC-data > logbili.m<-log(pbc$bilirubin)-mean(log(pbc$bilirubin)); > logalb.m<-log(pbc$albumin)-mean(log(pbc$albumin)); > Age.m<-pbc$Age-mean(pbc$Age); > fit<-cox.aalen(surv(time/365,status) prop(age.m)+edema+ + prop(logbili.m)+prop(logalb.m)+log(protime),resid.mg=1, + max.time=3/365,pbc) Cox-Aalen Survival Model Simulations starts N= 5 > summary(fit) Test for non-siginificant effects sup hat B(t)/SD(t) p-value H_: B(t)= (Intercept) Edema log(protime) Test for time invariant effects sup B(t) - (t/tau)b(tau) p-value H_: B(t)=b t (Intercept) Edema.269. log(protime) Proportional Cox terms : Coef. Std. Error Robust SE D2log(L)ˆ-1 prop(age.m) prop(logbili.m) prop(logalb.m) /38 Score Tests for Proportionality sup hat U(t) p-value H_ prop(age.m) prop(logbili.m) prop(logalb.m)

19 Summary Cox s proportional hazards model. Are the relative risks really not depending on time? Check model carefully. More flexible models Multiplicative model with timevarying covariate effects and also constant effects. Inference. Other flexible models: Cox-Aalen model and excess risk models. Aalens additive hazards model (and the semiparametric version). Can all be fitted in the R-package timereg: ts/timereg.html 38/38 Mix of Aalens and Cox s models Excess-risk type model α i (t) = α(t) T X i (t) + ρ i λ (t) exp (β T Z i (t)), ρ i = 1, all i, gives a mix of Aalens and Cox s models Model is perhaps most naturally seen as an excess risk model: ρ i is excess indicator eg I(d i > ) with d i dosis for ith subject. Has proven useful in cancer studies, see (Zahl, 23). Notice α i (t) = α(t) T X i (t) + ρ i λ (t) exp (β T Z i (t)) = ψ(t) T X i where ψ(t) = (α(t), λ (t)), XT i = (X T i, φ i (β)), φ i (β) = ρ i exp (β T Z i ). 37/38 May derive estimators of unknown parameters and also their large sample properties, see (Martinussen and Scheike, 22) Model can also be fitted in timereg.

20 References Lin, D.Y., Wei, L.J., and Ying, Z. (1993). Checking the Cox model with cumulative sums of martingale-based residuals. Biometrika 8, Martinussen, T. and Scheike, T.H. (22). A flexible additive multiplicative hazard model. Biometrika 89, Martinussen, T., Scheike, T.H., and Skovgaard, I.M. (22). Efficient estimation of fixed and time-varying covariate effects in multiplicative intensity models. Scandinavian Journal of Statistics 28, Scheike, T.H. and Martinussen, T. (24). On efficient estimation and tests of time-varying effects in the proportional hazards model. Scandinavian Journal of Statistics 31, Scheike, T.H. and Zhang, M.J. (22). An additive-multiplicative Cox-Aalen model. Scandinavian Journal of Statistics 28, Scheike, T.H. and Zhang, M.J. (23). Extensions and applications of the Cox-Aalen survival model. Biometrics 59, Zahl, 38/38 P. (23). Regresion analysis with multiplicative and timevarying additive regression coefficients with examples from

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