More on Cox-regression

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1 More on Cox-regression p. 1/45 More on Cox-regression STK4080 H16 1. Repetition 2. Left truncation 3. Time-dependent covariates 4. Stratified Cox-regression 5. Residuals - Model check 6. How to handle departures from prop.haz. assumption

2 More on Cox-regression p. 2/45 Repetition Cox-regression With covariate x i the survival time T i has hazard α(t x i ) = exp(β x i )α 0 (t) where α 0 (t) is called basis (underlying) hazard and whereβ is a regressionsparameter. With D i indicator for death for individual i, T i right censoring time and R(t) = {i : T i t} = the risk set (right before) time t we estimate β by maximizing the Cox partial likelihood [ ] n exp(β Di x i ) n [ ] exp(β L(β) = Di x i ) = k R( T i ) exp(β x k ) S (0) (β,t) i=1 where S (0) (β,t) = k R( T i ) exp(β x k ) i=1

3 if the l -model hasq more parameters than theˆl-model and that ˆl-model is true (H0 -model). More on Cox-regression p. 3/45 Rep. Cox-regr., contd. In counting process notation we saw that we may express the score U(β) = log(l(β)) β = n i=1 [ ] x i S(1) (β,t) dn S (0) i (t). (β,t) Also L(β) may be treated as a regular likelihood. Thus ˆβ N(β,I(ˆβ) 1 ). where I(β) = 2 log(l(β)) and for two nested models with β 2 maximum log-partial likelihood respectively l andˆl we have LRT = 2(l ˆl) χ 2 q

4 More on Cox-regression p. 4/45 Estimation of cumulative hazarda 0 (t) = t 0 α 0(s)ds The common estimator fora 0 (t) is the Breslow-estimator  0 (t) = Ti t D i k R( T i ) exp(ˆβ x k ) = t 0 dn (s) S (0) (ˆβ,s) Note the similarity with the Nelson-Aalen estimator. Given Â0(t) it is simple to estimate cumulative hazard for an individual with hazardα(t x i ) = exp(β x i )α 0 (t) as Â(t x i ) = exp(ˆβ x i )Â0(t) The survival function S(t x i ) = P(T i > t x i ) can be estimated by Ŝ(t x i ) = exp( Â(t x i)) = exp( Â0(t)exp(ˆβ x i )) = Ŝ0(t) exp(ˆβ x i ).

5 No arguments have used the property of non-decreasing Y i (t) (might be a good exercise to check). More on Cox-regression p. 5/45 Cox-regression for left truncated and rightcens. data We may use the same likelihood expression in counting process notation. We may estimate β by maximizing the partial likelihood, or solving U(β) = log(l(β)) β = n i=1 [ ] x i S(1) (β,t) dn S (0) i (t) = 0 (β,t) where S (0) (β,t) = n i=1 Y i (t)exp(β x i ). We need only remember that Y i (t) need not be non-decreasing, but Y i (t) = I(L i < t T i ).

6 More on Cox-regression p. 6/45 R: Cox-reg. with l.trunc. and r.cens. The program need information on left truncation time L i in addition to ( T i,d i,x i ). Example: Time until death psychiatric patients. Data: x i = 1 or 2 for men / women L i = age of first time admitted to psychiatric ward (year) T i = age at death/censoring (year) D i = indicator of death. > coxph(surv(ageonset,agedeath,death) sex) coef exp(coef) se(coef) z p sex Likelihood ratio test=0.43 on 1 df, p=0.514 n= 26

7 More on Cox-regression p. 7/45 Time dependent covariates x i (t) Risk factors may depend on time: x i (t) = smoker at time t (yes/no) x i (t) = cum. no. cigarettes (pack years) smoked at age t x i (t) = no. years since quitting smoking It may furthermore be that the proportional hazards model does not fit the data well with the included covariates, but that a valid model is given by. α(t x i ) = exp(β 1 x i +β 2 x i t)α 0 (t) so that the effect of x i becomes larger (smaller)according to β 2 > 0(< 0). May code this by introducing new covariates, for instance x i2 (t) = tx i.

8 Cox-regression allows for time dependentx i (t)! Model: α i (t) = exp(β x i (t))α 0 (t). Will simply solve U(β) = log(l(β)) n [ ] = x i S(1) (β,t) dn β S (0) i (t) = 0 (β,t) i=1 (as before) with the difference that S (0) (β,t) = n i=1 Y i (t)exp(β x i (t)). depend on time dependent x i (t). Similarly to left truncation all theory goes through exactly the same way as before. For the sake of repetition we check E[U(β)] = 0. However: We need to assume that x i (t) is predictable! More on Cox-regression p. 8/45

9 More on Cox-regression p. 9/45 U(β) martingale with expectation 0. As before S (0) (β,t) β score = n i=1 x i(t)y i (t)exp(β x i (t)) = S (1) (β,t). Thus the U(β) = n i=1 [x i (t) S(1) (β,t) S (0) (β,t) ]dn i(t) With predictable covariatesx i (t) the model can be written dn i (s) = Y i (t)exp(β x i (t))α 0 (t)dt+dm i (t) (with standard interpretations) which gives U(β) = n i=1 [xi (t) S(1) (β,t) ]Y S (0) (β,t) i(t)exp(β x i (t))α 0 (t)dt + n i=1 [xi (t) S(1) (β,t) ]dm S (0) (β,t) i(t) [xi (t) S(1) (β,t) ]dm S (0) (β,t) i(t) = n i=1

10 More on Cox-regression p. 10/45 because n i=1 [x i (t) S(1) (β,t) S (0) (β,t) ]Y i(t)exp(β x i (t))α 0 (t)dt = 0. This since n i=1 [x i(t) S(1) (β,t) S (0) (β,t) ]Y i(t)exp(β x i (t)) and so = S (1) (β,t) S(1) (β,t) S (0) (β,t) S(0) (β,t) = 0 U(β) = n i=1 [x i (t) S(1) (β,t) S (0) (β,t) ]dm i(t) is a sum of integrals wrt. martingales, and itself a martingale with E[U(β)] = 0. just as for time constant and right censored data!

11 Time dependent covariates in R R only allows for time dependent covariates constant on intervals, i.e. step functions. Assume thatx i (t) = x j on interval < L ij,u ij ] for j = 1,2,...,J i. Need to represent this individual J i times in the data fil as left truncated data with L ij as left trunkcation time U ij as right censoring time D ij = D i I( event in interval j) as indicator x j as covariate value Note: Did not assume L i,j+1 = U ij, and individuals may well disappear and reenter later. More on Cox-regression p. 11/45

12 More on Cox-regression p. 12/45 Stratified Cox-regression Assume that the population is divided into s = 1, 2,..., S strata so that person i = 1,2,...,n s within stratumswith covariatex is has hazard α(t x is ) = exp(β x is )α 0s (t) where α 0s (t) is the baseline in stratum s (typically α 0s (t)/α 0s (t) varies with t.) Effects of covariatesβ are the same in all strata We get a partial likelihood from each stratum: [ n s exp(β x is ) L s (β) = k R s (T is ) exp(β x ks ) i=1 ] Dis with D is = indicator for individual is in stratum s etc.

13 More on Cox-regression p. 13/45 Stratified Cox-regression, contd. All strata give information on β. If we assume that the strata are independent (weak assumption) we may then combine this information by maximizing the stratified partial likelihood L(β) = S s=1 L s (β) The corresponding score function becomes U(β) = log(l(β)) β = S s=1 log(l s (β)) β = S s=1 U s (β) where U s (β) is the score function from stratum s. These all have expectation 0, i.e. E[U(β)] = 0.

14 More on Cox-regression p. 14/45 Stratified Cox-regression, III We find information I s from stratumsandi(β) total information becomes, due to independence, S Var[U(β)] = E[I s (β)] = E[I(β)] s=1 Thus may use the stratified partial likelihood as a regular likelihood. The stratified partial likelihood is useful when the proportional model does not hold for a categorical variable. Stratify on this variable and keep regression model for other covariates In particular this is useful when the stratification variable is a confounder and the main interest is on the other variables.

15 More on Cox-regression p. 15/45 Stratified Cox-regression i R Uses the Melanoma-data. Stratifies on grouped tumor thickness by command strata: coxph(surv(lifetime,dead) ulcer+sex+age+strata(grthick),data=mel) coef exp(coef) se(coef) z p ulcer sex age Likelihood ratio test=13.2 on 3 df, p= n= 205

16 More on Cox-regression p. 16/45 What did we gain from this? Compares with the Cox-regression where thickness is a categorical variable > coxph(surv(lifetime,dead) ulcer+sex+age+factor(grthick),data=mel) coef exp(coef) se(coef) z p ulcer sex age factor(grthick) factor(grthick) Likelihood ratio test=45.3 on 5 df, p=1.27e-08 n= 205 The results are only marginally different, but we have modeled in a much more flexible way. There is hardly a change in the standard errors, and the added flexibility did not lead to loss in efficiency.

17 More on Cox-regression p. 17/45 Baseline in each stratum in R. > nycox<-coxph(surv(lifetime,dead) ulcer+sex+age+strata(grthick),data=me > plot(survfit(nycox),fun="cum.haz") The proportional hazards assumption might be checked from these plots (or rather the log-cumulative hazard plots). Better methods are presented later in the lectures.

18 More on Cox-regression p. 18/45 How can the Cox-model fail? α(t x i ) = exp(β x i )α 0 (t)? The Cox-model is flexible wrt the baseline α 0 (t), but otherwise strict with respect to how the hazard depend on covariates, for instance We may have specified covariatex ik wrong, correct alternative may be f.ex. x ik = log(x ik) or x ik = x ik. We do not have a proportional model. The effect may vary with time, f.ex. α(t x i ) = exp(β(t) x i )α 0 (t) whereβ(t) is a function of time.

19 More on Cox-regression p. 19/45 Martingale residuals With a specified model for the hazard, say a proportional hazards modelα i (t) = exp(β x i )α 0 (t) we that M i (t) = N i (t) t 0 Y i (s)exp(β x i )α 0 (s)ds is a martingale we expectation zero. Inserting the Cox-estimator ˆβ forβ, the Breslow estimatordâ0(s) = dn (s) S (0) (ˆβ,s) forα 0(s)ds and the maximal right censored survival time τ we get the so called martingale residuals ˆM i = N i (τ) τ 0 Y i (s)exp(ˆβ x i )dâ0(s) which have the structure "Observed"-"Expected".

20 Example: Melanoma data We saw thatlog(tumorsize) was a better covariate than tumorsize directly. We will check if this can be discovered from martingale residuals. coxfit<-coxph(surv(lifetime,status==1) sex+ulcer+thickn, data=melanoma) martres<-coxfit$residuals plot(melanoma$thickn,martres) martres melanoma$thickn More on Cox-regression p. 20/45

21 Example: Melanoma data, contd. As often it is difficult to read off residual plots directly. We calculate mean mart. resid. for groups of tumor thickness > grthickn<-melanoma$grthick > lm(martres factor(grthickn)-1) Coefficients: factor(grthickn)1 factor(grthickn)2 factor(grthickn) > summary(lm(martres factor(grthickn))) Estimate Std. Error t value Pr(> t ) (Intercept) factor(grthickn) * factor(grthickn) F-statistic: on 2 and 202 DF, p-value: Note: This analysis is not strictly correct, from ABG one may figure out a correct test. But it shows that the martingale residuals are largest for the second group. More on Cox-regression p. 21/45

22 More on Cox-regression p. 22/45 GAM: Generalized Additive Models (Hastie & Tibshirani (1990): By smoothing techniques we may fit Linear model: Y = α+f(x)+ε Logistic model: log( p 1 p ) = α+f(x) where f(x) is some smooth function. This may be done in R with the library gam that may need to be downloaded from CRAN. Similar models may be fitted for survival data under specifications α(t x) = exp(f(x))α 0 (t)

23 More on Cox-regression p. 23/45 Example: Melanoma data, GAM-plot We add a gam-curve with 95% CI to the scatter plot of martingale residuals vs. thickness. thick<-melanoma$thickness) library(gam) plot(gam(martres s(thick)),se=t,ylim=c(min(martres),max(martres))) points(thick,martres) s(thick) thick

24 More on Cox-regression p. 24/45 GAM for Cox-regression in R Regression-spline: R-syntax coxph(surv(time,status) ns(z,df=4)) Cubic smoothing-spline (penalized partial likelihood): R-syntax: coxph(surv(time,status) pspline(z,df=4))

25 Penalized log-partial-likelihood: Maximize, for given smoothing parameterλ, log(l(f)) λ (f (x)) 2 dx where L(f) = Partial likelihood λ = "penalty"-term for curvature of f(x) In particular: λ = : No curvature,f(x) = βx straight line λ = 0 : No smoothing Maximation problem actually has a simple numerical solution. The penalty term λ corresponds to a certain degree of freedom and may be interpreted as df = no. covariates in mod., though df may be any real number > 0. More on Cox-regression p. 25/45

26 More on Cox-regression p. 26/45 Example: Penalized log-partial-likelihood: Melanoma-data, thickness, R-commands: newcoxfit<-coxph(surv(lifetime,status==1) pspline(thickn,df=4),data=mela newmel<-melanoma thckspacing<-min(melanoma$thickn) +(1:205)*(max(melanoma$thickn)-min(melanoma$thickn))/205 newmel$thickn<-thckspacing newpred<-predict(newcoxfit,newmel,type="terms",term=1,se=t) mi<-min(newpred$fit-1.96*newpred$se.fit) ma<-max(newpred$fit+1.96*newpred$se.fit) plot(thckspacing,newpred$fit,type="l",ylim=c(mi,ma),xlab="thickness") lines(thckspacing,newpred$fit+1.96*newpred$se.fit,lty=2) lines(thckspacing,newpred$fit-1.96*newpred$se.fit,lty=2)

27 Example: Penalized log-partial-likelihood: Melanoma-data, thickness, Plot of ˆf(x) against x = thickness: newpred$fit thickness The plot shows what we also found with grouping of thickness, it is the smallest tumors that have smaller risk. More on Cox-regression p. 27/45

28 May smooth with more covariates α(t x) = exp(β 1 x 1 +β 2 x 2 +β 3 x 3 +f 4 (x 4 ))α 0 (t) Ex: Melanoma,x 1 = sex,x 2 = ulc., x 3 = age,x 4 = tumorth. newcoxfit<-coxph(surv(lifetime,status==1) sex+ulcer+age +pspline(thickn,df=4),data=mel newpred<-predict(newcoxfit,newmel,type="terms",term=4,se=t) newpred$fit spline log thickness More on Cox-regression p. 28/45

29 More on Cox-regression p. 29/45 Non-proportional hazards The old-fashioned way of checking departure from proportionality is based on the following: With α(t x) = exp(βx)α 0 (t) we have log(a(t x)) = β x+log(a 0 (t)) i.e. log(a(t x)) for differentxand log(a 0 (t)) should be parallel lines. Thus if x is the only covariate and is categorical we may plot log of Nelson-Aalen for every level ofx If the lines are parallel then proportionality OK

30 More on Cox-regression p. 30/45 Ex. Melanoma: Checks tumor-thickness and ulceration Ulcer Tumortykkelse log(h) log(h) tid tid

31 More on Cox-regression p. 31/45 Multivariable models If we have decided thatx 1,x 2,...,x p should be included in the model and want to check if the categorical covariatex p+1 satisfies proportionality we should rather Fit a stratified Cox-model with x p+1 -levels as strata and x 1,x 2,...,x p as covariates. Plotlog(Âs(t, x)) against t for different levelssof x p+1 Are the lines parallel? Again, it might be an advantage to look at log(âs(t, x)) log(â1(t, x)).

32 More on Cox-regression p. 32/45 Schoenfeld residuals The Schoenfeld-residuals are given by, for T j such thatd j = 1 x jk n i=1 x iky i ( T j )exp(ˆβ x i ) n i=1 Y i( T j )exp(ˆβ x i ) There is thus one residual for event time and for each component of the covariates. Since Y i ( T j )exp(ˆβ x i ) ni=1 Y i ( T j )exp(ˆβ x i ) sums to one they may be thought of as point mass probabilities for some distribution. The interpretation of the distribution is that of the covariatesx j given that individual j experienced the event. And so n i=1 x iky i ( T j )exp(ˆβ x i ) n j=1 Y = x k ( T j ) i( T j )exp(ˆβ x i )

33 More on Cox-regression p. 33/45 More Schoenfeld becomes the expectation in this distribution. The Schoenfeldt-residuals at time T j may also be written x j x( T j ). Furthermore, with U(β) = the scorefunction, 0 = U(ˆβ) = [x j x( T j )] T j :D j =1 a sensible property for a residual. Sometimes component k for x jk x k ( T j ) shows a clear tendency for positive values over intervals (and negative over others). In the "positive" intervals there is the greater risk connected to component k than in the "negative".

34 More on Cox-regression p. 34/45 Schoenfeld, contd. We could have done a local Cox-regression within the "positive" interval. In such case the x(t) would tend to be larger. Thus the Schoenfeld-residuals give information as to whether the prop.haz. assumption holds. They may even be used to estimate how the hazard ratio varies over time. Let V(t) = S(2) (ˆβ,t) S (0) (ˆβ,t) [ S (1) (ˆβ,t) S (0) (ˆβ,t) The observed information is given as V(t)dN(t) and in particular V(t) can be interpreted as the variance of x given event att. ] 2

35 More on Cox-regression p. 35/45 Scaled Schoenfeld-residual If the true model equalsα(t x) = exp(β(t)x)α 0 (t) for some function β(t) we have as a 1.order approximation β( T j ) ˆβ +V( T j ) 1 (x j x( T j )) i.e. varying around the scaled Schoenfeld-residual V( T j ) 1 (x j x( T j )) that measures the departure from mean risk ˆβ. Plot of this quantity - smoothed over time - may give a picture of how the risk varies.

36 More on Cox-regression p. 36/45 Ex: Melanoma data Calculates and smooths scaled Schoenfeld-residuals for all 4 covariates in the melanom data. coxfit<-coxph(surv(lifetime,dead) sex+age+ulcer+logthick,data=mel) plot(cox.zph(coxfit)) Beta(t) for sex Beta(t) for age Time Time Beta(t) for ulcer Beta(t) for logthick Time Time

37 More on Cox-regression p. 37/45 Test for prop. assumption can be based on Schoenfeld-residuals: The tests are directed to departures from the model given by α(t x) = exp((β 0 +θg(t))x)α 0 (t) for specified functions g(t). Actually they are score-tests for a new covariate x (t) = g(t)x. If the model is extended by a new time-dependent covariate x p+1 (t) = x p g(t) the score for the coefficientβ p+1 is given U p+1 = j g( T j )[x p x p ( T j )] as a weighted sum of the Schoenfeld-residuals.

38 More on Cox-regression p. 38/45 Test for proportionality Example: Melanoma data: KM transform. Sex Ulceration Age χ 2 1 = 0.5 p=0.46 χ 2 1 = 0.7 p=0.40 χ 2 1 = 2.8 p=0.09 log(tumor thickness) χ 2 1 = 4.1 p=0.04 Indication for departure for tumor thickness R-syntax: coxfit<-coxph(surv(lifetime,dead) sex+age+ulcer+logthick,data=mel) cox.zph(coxfit) rho chisq p sex age ulcer logthick GLOBAL NA

39 More on Cox-regression p. 39/45 Strategies when proportional hazard fails Stratified Cox-regression Separate analyzes on disjoint time intervals Time-dependent covariates Alternative regression models Accelerated failure time models Additive models

40 More on Cox-regression p. 40/45 Ex. Stratified Cox-regression Weak departure wrt. thickness. Stratifies on grthick: > coxstrat<-coxph(surv(lifetime,dead) sex+age+ulcer+strata(grthick),data > coxstrat coef exp(coef) se(coef) z p sex age ulcer Likelihood ratio test=13.2 on 3 df, p= n= 205 > cox.zph(coxstrat) rho chisq p sex age ulcer GLOBAL NA But possibly the stratification changed other estimates somewhat?

41 Separate intervals We may split the time interval i 2 and make separate Cox-regressions within each interval: Ex: Melanoma data. Half of death before τ = 3 Analysis on [0,3 >: Uses events only if lifetime < 3 coxph(surv(lifetime,dead*(lifetime<3)) sex+age+ulcer+logthick,data=mel) coef exp(coef) se(coef) z p sex age ulcer logthick Analysis on [3, >: Uses only events with lifetime> 3 coxph(surv(lifetime,dead*(lifetime>3)) sex+age+ulcer+logthick,data=mel) coef exp(coef) se(coef) z p sex age ulcer logthick More on Cox-regression p. 41/45

42 More on Cox-regression p. 42/45 Time dependent covariats If similar parameter estimates for age, sex and ulceration on these intervals we may fit a common model log( α(t x) α 0 (t) ) = β 1x 1 +β 2 x 2 +β 3 x 3 +β 4 x 4 I(t < 3)+β 5 x 4 I(t 3) i.e. Cox-regression with time dependent covariates x 4 I(t < 3) x 4 I(t 3)

43 Cox-regression with time dependent covariates Need to set up data-frame with time dependent covariates: mel2 <- data.frame(intime = c(rep(0,205),rep(3,167))) mel2$outtime <- c(pmin(mel$lifetime,3),mel$lifetime[mel$lifetime>3]) mel2$indi <- c(mel$dead*(mel$lifetime<3),mel$dead[mel$lifetime>3]) mel2$sex mel2$ulcer mel2$age <- c(mel$sex,mel$sex[mel$lifetime>3]) <- c(mel$ulcer,mel$ulcer[mel$lifetime>3]) <- c(mel$age,mel$age[mel$lifetime>3]) mel2$logtha <- c(mel$logthick,rep(0,167)) mel2$logthb <- c(rep(0,205),mel$logthick[mel$lifetime>3]) coxph(surv(intime,outtime,indi) sex+ulcer+age+logtha+logthb,data=mel2) coef exp(coef) se(coef) z p sex ulcer age logtha logthb Likelihood ratio test=49 on 5 df, p=2.17e-09 n= 372 More on Cox-regression p. 43/45

44 More on Cox-regression p. 44/45 Advantages/disadvantages with strategies 1. Stratification Easy More difficult to show effect of stratification variable Allows for only a few problem covariates 2. Separate intervals Relatively easy Choice of interval difficult/arbitrary Looses power for covariates where the assumption is OK Many parameter estimates

45 More on Cox-regression p. 45/45 Advantages/disadvantages with strategies 3. Time dependent covariates Somewhat awkward to arrange (in R(?)) Difficult choice of interval Only helpful when prop.haz. OK for most covar. Consequences of departure from proportionality biased estimates of coefficients both for covariates where the assumption hold and fail biased survival estimates