Applied Statistics. J. Blanchet and J. Wadsworth. Institute of Mathematics, Analysis, and Applications EPF Lausanne

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1 Applied Statistics J. Blanchet and J. Wadsworth Institute of Mathematics, Analysis, and Applications EPF Lausanne An MSc Course for Applied Mathematicians, Fall 2012

2 Outline 1 Model Comparison 2 Model Diagnostics in Proportional Hazards

3 Part I Model Comparison

4 Comparing Survival Curves Two groups Suppose that interest lies in whether two groups have different survival curves Plotting the curves gives an impression of whether they are the same What can we say more formally?

5 Comparing Survival Curves Log rank test The log-rank test allows us to compare survival curves for two groups The log-rank test works by comparing observed failures with expected failures under the null hypothesis of no difference between groups

6 Comparing Survival Curves Log rank test Recall the notation for ordered times 0 t (1) t (n) with n (i) the number at risk just prior to time t (i), and d (i) the number of failures at time t (i). Let k = 1, 2 represent the two groups and Define n k,(i) be the number at risk just prior to time t (i) in group k, so that n (i) = n 1,(i) + n 2,(i) d k,(i) be the number of failures at time t (i) in group k, so that d (i) = d 1,(i) + d 2,(i) E k = i d (i) n k,(i) n (i) O k = i d k,(i) as the expected and observed numbers of failures in group k.

7 Comparing Survival Curves Log rank test Under the null hypothesis of no difference between groups, the statistic X 2 = (O 1 E 1 ) 2 /V χ 2 1 as the sample size becomes large, where V = var(o 1 E 1 ): V = i d (i) n 1,(i) n 2,(i) (n (i) d (i) ) n 2 (i) (n (i) 1) This comes from the hypergeometric distribution (NB O 1 E 1 = (O 2 E 2 ), so either can be used in the definition)

8 Comparing Survival Curves Log rank test An alternative definition of the log-rank test statistic is 2 (E k O k ) 2 k=1 E k which is also asymptotically χ 2 1 under H 0. Either version can be used, but the second definition may be a bit more conservative (reject the null less often)

9 Log rank test in R survdiff{survival} Recall the Recidivism data; assume that fin (indicator of financial aid) is the only covariate > survdiff(surv(week, arrest) ~ fin, data=rossi) Call: survdiff(formula = Surv(week, arrest) ~ fin, data = Rossi) N Observed Expected (O-E)^2/E (O-E)^2/V fin= fin= Chisq= 3.8 on 1 degrees of freedom, p= The final column and the final line relate to the first test definition The penultimate column gives the statistics for the second test definition > [1] 3.82 > 1-pchisq(3.82,df=1) [1] The difference between groups is marginally non-significant at the 5% level

10 Comparing Survival Curves More than two groups The log-rank test can also be extended to compare more than two groups If there are G groups then the statistic asymptotically, where (O E) T V 1 (O E) χ 2 G 1 O E = (O 1 E 1,..., O G 1 E G 1 ) and V is its variance-covariance matrix. The formula G (E k O k ) 2 k=1 E k is also approximately χ 2 G 1 distributed.

11 Comparing Survival Curves Care needs to be taken when comparing groups with the log rank test, as the distribution of other covariates may not be the same within the two groups E.g. what if all the financial aid went to those over 40 years of age? This could cause us to infer a difference between groups, which is actually related to the effects of these other covariates More generally, we usually have a more complicated model than two groups, and so we want to know how to compare models in the presence of multiple covariates

12 Comparing Survival Curves Parametric models Let T denote the survival time, and z a vector of covariates Suppose that we model T z Weibull so that f (t z) = aλ a t a 1 e zβ exp{ (λt) a e zβ }, t 0. The log-likelihood for X i = min{t i, C i }, δ i = I(T i < C i ) is l(β, λ, α) = i δ i log f (x i z i, θ) + i (1 δ i ) log S(x i z i, θ). If we want to test any hypotheses about (β, λ, α), then we can use likelihood ratio tests

13 Likelihood ratio Paramteric models > wei <- survreg(surv(week,arrest) ~ fin + age + race + wexp + mar + paro + prio,data=rossi) > summary(wei) Call: survreg(formula = Surv(week, arrest) ~ fin + age + race + wexp + mar + paro + prio, data = Rossi) Value Std. Error z p (Intercept) e-21 fin e-02 age e-02 race e-01 wexp e-01 mar e-01 paro e-01 prio e-03 Log(scale) e-04 Scale= 0.712

14 Likelihood ratio Paramteric models paro does not look significant; fit the model without it: wei2 <- survreg(surv(week,arrest) ~ fin + age + race + wexp + mar + prio,data=rossi) > 2*(wei$loglik[2] - wei2$loglik[2]) [1] this likelihood ratio test has 1 df; so the critical value (at the 5% level) is We would not reject the hypothesis that paro has no effect.

15 Likelihood ratio Paramteric models Recall that the exponential model is a special case of the Weibull model Since it is nested, we can also do a likelihood ratio test for this > wei <- survreg(surv(week,arrest) ~ fin + age + race + wexp + mar + paro + prio,data=rossi) > expn <- survreg(surv(week,arrest) ~ fin + age + race + wexp + mar + paro + prio, dist="exponential", data=rossi) > 2*(wei$loglik[2]-expn$loglik[2]) [1] This is highly significant at the 5% level on 1 df.

16 Likelihood ratio Cox PH model Likelihood ratio tests are also applicable to the Cox Proportional Hazards partial likelihood The asymptotic distributions are the same, i.e. a likelihood ratio test with q constraints, asymptotically follows a χ 2 q distribution ## test for significance of paro under the Cox PH model > mod0 <- coxph(surv(week, arrest) ~ fin + age + race + wexp + mar + paro + prio,data=rossi) > mod1 <- coxph(surv(week, arrest) ~ fin + age + race + wexp + mar + prio, data=rossi) > 2*(mod0$loglik[2]-mod1$loglik[2]) [1] Not significant at the 5% level on 1 df

17 Cox PH model Other test statistics The output for a fitted Cox PH model gives three test statistics which compare the fitted model to a null model These are the likelihood ratio, Wald, and score tests all χ 2 under H 0 > mod5 <- coxph(surv(week, arrest) ~ age + prio,data=rossi) > summary(mod5) Call: coxph(formula = Surv(week, arrest) ~ age + prio, data = Rossi) n= 432, number of events= 114 coef exp(coef) se(coef) z Pr(> z ) age *** prio *** --- Signif. codes: 0 *** ** 0.01 * exp(coef) exp(-coef) lower.95 upper.95 age prio Rsquare= (max possible= ) Likelihood ratio test= on 2 df, p=2.657e-06 Wald test = on 2 df, p=4.766e-06 Score (logrank) test = on 2 df, p=2.723e-06 The score test is the same as the log rank test when there are only two groups

18 Part II Model Diagnostics in Proportional Hazards

19 Checking Proportional Hazards Schoenfeld residuals for the ith subject on the kth covariate is ˆr ik = δ i (z ik ˆ z xi k), where ˆ z xi k is given by j R(x i ) z jk e z j ˆβ j R(x i ) ez j ˆβ Scaled Schoenfeld residuals: ˆr ik = ˆr ik mean(ˆr ik, i = 1,..., n) sd(ˆr ik, i = 1,..., n) For PH model, the (scaled) residuals ˆr ik should exhibit a random (i.e. unsystematic) pattern at each failure time. Otherwise it suggests that as time passes, the covariate effect is changing.

20 Checking Proportional Hazards cox.zph{survival} mod2 <- coxph(surv(week, arrest) ~ fin + age + prio, data=rossi) cox.zph(mod2) rho chisq p fin age prio GLOBAL NA The function tests proportionality of all the predictors by looking at their interactions with time. The column rho is the Pearson correlation between the scaled Schoenfeld residuals and time for each covariate. The last row contains the global test for all the interactions tested at once. A p-value less than 0.05 indicates a violation of the proportionality assumption.

21 Checking Proportional Hazards plot, cox.zph {survival} Graphs of the scaled Schoenfeld residuals against time: par(mfrow=c(2,2)) plot(cox.zph(mod2)) Beta(t) for fin Time Beta(t) for age Time Beta(t) for prio Time The curve is a smoothing spline with ±2 standard-error envelopes around the fit. Systematic departures from a horizontal line are indicative of non-proportional hazards. Here, there appears to be a trend in the plot for age, with the age effect declining with time.

22 Checking linearity We assume in PH models that λ(t; z) = λ 0 (t)e zβ, This means that log λ(t; z) is linearly dependent on the covariates z. Is this true? The martingale residual for subject i is ˆM i = δ i e z i ˆβ xi 0 ˆλ 0 (u)du For each k, the plot of ˆM i against z ik, i = 1,..., n, should exhibit a random pattern with mean 0.

23 Checking linearity cox.zph{survival} res <- residuals(mod2, type="martingale") X <- as.matrix(rossi[,c("age", "prio")]) # matrix of covariates for (j in 1:2) { # residual plots plot(x[,j], res, xlab=c("age", "prio")[j], ylab="residuals") abline(h=0, lty=2) lines(lowess(x[,j], res, iter=0)) } residuals age residuals prio Nonlinearity is slight here.

24 Influential observations We want to check the influence of each observation on the estimate ˆβ. Let ˆβ i denote the estimated vector of coefficients computed on the sample with the ith subject deleted. The idea is to check for each i which component of the vector ˆβ ˆβ i has large absolute values. This involves fitting n + 1 Cox regression models, which can be computationally expensive. There is an approximation based on the fit obtained from the whole data: ˆβ ˆβ i can be approximated by dfbeta i = I( ˆβ) 1 (ˆr i1,..., ˆr ik ), where I( ˆβ) is the observed Fisher information matrix and, for k = 1,..., K, ˆr ik is a function of ˆβ and of Schoenfeld residual ˆr ik. Plots of the quantities ˆr ik against i are used to gauge the influence of the i subject on the k covariate.

25 dfbeta <- residuals(mod2, type="dfbeta") for (j in 1:3) { plot(dfbeta[,j], ylab=names(coef(mod2))[j]) abline(h=0, lty=2) } fin Index age Index prio Index Comparing the magnitudes of the largest dfbeta values to the regression coefficients ( 0.35, 0.07, 0.1) suggests that none of the observations is terribly influential individually.