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Lecture 20 Comparing groups Cox PHM

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Comparing two or more samples Anova type approach where τ is the largest time for which all groups have at least one subject at risk Data can be right-censored for the tests we will discuss

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Notation t 1

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Log-Rank Test Rationale Comparisons of the estimated hazard rate of the jth population under the null and alternative hypotheses If the null is true, the pooled estimate of h(t) should be an estimator for h j (t)

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Applying the Test for j = 1,…,K If all Z j (τ)’s are close to zero, then little evidence to reject the null.

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Others? LOTS! Gehan test Fleming-Harrington Not all available in all software worth trying a few in each situation to compare inferences

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2+ samples Let’s look at a prostate cancer dataset Prostate cancer clinical trial 3 trt groups (doce Q3, doce weekly, Q3 mitoxantrone) 5 PSA doubling times categories outcome: overall survival

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TAX327 results

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R: survdiff ################################# # test for differences by trt grp plot(survfit(st~trt), mark.time=F, col=c(1,2,3)) test1 <- survdiff(st~trt) test2 <- survdiff(st~factor(trt, exclude=3)) test3 <- survdiff(st[trt<3]~trt[trt<3])

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PSADT categories

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R: survdiff table(psadt) plot(survfit(st~psadt), mark.time=F, col=1:5, lwd=rep(2,5)) legend(50,1,as.character(0:4), lty=rep(1,5), col=1:5, lwd=rep(2,5)) test1 <- survdiff(st~psadt) test2 0]~psadt[psadt 0]) test3 2]~psadt[psadt>2])

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Caveat Note that we are interested in the average difference (consider log- rank specifically) What if hazards ‘cross’? Could have significant difference prior to some t, and another significant difference after t: but, what if direction differs?

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What about all those differences in our prostate cancer KM curves? Not much evidence of crossing if there isnt overlap, then tests will be somewhat consistent log-rank: most appropriate for ‘proportional hazards’

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Example K&M 1.4 Kidney infection data Two groups: patients with percutaneous placement of catheters (N=76) patients with surgical placement of catheters (N=43)

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Kaplan-Meier curves

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Log-rank

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Comparisons p 0.11 0.96 0.53 0.24 0.26 0.002 0.24 0.002 0.004

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Why such large differences?

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Notice the differences! Situation of varying inferences Need to be sure that you are testing what you think you are testing Check: look at hazards? do they cross? Problem: estimating hazards is messy and imprecise recall: h(t)= derivative H(t)

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Misconception Survival curves crossing telling about appropriateness of log-rank Not true: survivals crossing depends on censoring and study length what if they will cross but t range isnt sufficient? Consider: Survival curves cross hazards cross Hazards cross survivals may or may not cross solution? test in regions of t prior to and after cross based on looking at hazards some tests allow for crossing (Yang and Prentice 2005)

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Cox Propotional Hazards Model Names Cox regression semi-parametric proportional hazards Proportional hazards model Multiplicative hazards model When? 1972 Why? allows adjustment for covariates (continuous or categorical) in a survival setting allows prediction of survival based on a set of covariates Analogous to linear and logistic regression in many ways

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Cox PHM Notation Data on n individuals: T j : time on study for individual j d j : event indicator for individual j Z j : vector of covariates for individual j More complicated: Z j (t) covariates are time dependent they may change with time/age

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Basic Model For a Cox model with just one covariate:

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Comments on basic model h 0 (t): arbitrary baseline hazard rate. notice that it varies by t β: regression coefficient (vector) interpretation is a log hazard ratio Semi-parametric form non-parametric baseline hazard parametric form assumed only for covariate effects

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Linear model formulation Usual formulation Coding of covariates similar to linear and logistic (and other generalized linear models)

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Why “proportional”? hazard ratio Does not depend on t (i.e., it is a constant over time) But, it is proportional (constant multiplicative factor) Also referred to (sometimes) as the relative risk.

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Simple example one covariate: z = 1 for new treatment, z=0 for standard treatment hazard ratio = exp(β) interpretation: exp(β) is the risk of having the event in the new treatment group versus the standard treatment Interpretation: at any point in time, the risk of the event in the new treatment group is exp(β) times the risk in the standard treatment group

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Hazard Ratio: CAP (cyclophosphamide, doxorubicin, cisplatin) versus paclitaxel

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Hazard Ratios Assumption: “Proportional hazards” The risk does not depend on time. That is, “risk is constant over time” But that is still vague….. Hypothetical Example: Assume hazard ratio is 0.5. Patients in new therapy group are at half the risk of death as those in standard treatment, at any given point in time. Hazard function= P(die at time t | survived to time t)

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Hazard Ratios Hazard Ratio = hazard function for New hazard function for Std Makes assumption that this ratio is constant over time.

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Interpretation Again For any fixed point in time, individuals in the new treatment group are at half the risk of death as the standard treatment group.

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Hazard ratio is not always valid …. Hazard Ratio =.71

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Refresher of coding covariates This should be nothing new Two kinds of ‘independent’ variables quantitative qualitative Quantitative are continuous need to determine scale units transformation? Qualitative are generally categorical ordered nominal coding affects the interpretation

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Tests of the model Testing that β k =0 for all k=1,..,p Three main tests Chi-square/Wald test Likelihood ratio test score(s) test All three have chi-square distribution with p degrees of freedom

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Example: TAX327 Randomized clinical trial of men with hormone- refractory prostate cancer three treatment arms (Q3 docetaxel, weekly docetaxel, and Q3 mitixantrone) other covariates of interest: psa doubling time lymph node involvement liver metastases number of metastatic sites pain at baseline baseline psa tumor grade alkaline phosphatase hemoglobin performance status

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Some of the covariates

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Cox PHM approach st <- Surv(survtime, died) attach(data, pos=2) reg1 <- coxph(st ~ trtgrp) reg2 <- coxph(st ~ factor(trtgrp)) summary(reg2) attributes(reg2) reg2$coefficients summary(reg2)$coef

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Results > summary(reg2) Call: coxph(formula = st ~ factor(trtgrp)) n= 1006 coef exp(coef) se(coef) z p factor(trtgrp)2 0.105 1.11 0.0882 1.19 0.2300 factor(trtgrp)3 0.245 1.28 0.0863 2.84 0.0045 exp(coef) exp(-coef) lower.95 upper.95 factor(trtgrp)2 1.11 0.900 0.935 1.32 factor(trtgrp)3 1.28 0.783 1.079 1.51 Rsquare= 0.008 (max possible= 1 ) Likelihood ratio test= 8.12 on 2 df, p=0.0173 Wald test = 8.16 on 2 df, p=0.0169 Score (logrank) test = 8.19 on 2 df, p=0.0167

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Multiple regression In the published paper, the model included all covariates included in previous list

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Fitting it in R reg3 <- coxph(st ~ factor(trtgrp) + liverny + numbersites + pain0c + pskar2c + proml + probs + highgrade + logpsa0 + logalkp0c + hemecenter + psadtmonthcat) reg4 <- coxph(st ~ factor(trtgrp) + liverny + numbersites + pain0c + pskar2c + proml + probs + highgrade + logpsa0 + logalkp0c + hemecenter + factor(psadtmonthcat))

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> reg3 Call: coxph(formula = st ~ factor(trtgrp) + liverny + numbersites + pain0c + pskar2c + proml + probs + highgrade + logpsa0 + logalkp0c + hemecenter + psadtmonthcat) coef exp(coef) se(coef) z p factor(trtgrp)2 0.1230 1.131 0.1099 1.12 2.6e-01 factor(trtgrp)3 0.3784 1.460 0.1070 3.54 4.0e-04 liverny 0.4813 1.618 0.2168 2.22 2.6e-02 numbersites 0.4757 1.609 0.1430 3.33 8.8e-04 pain0c 0.3708 1.449 0.0925 4.01 6.1e-05 pskar2c 0.3167 1.373 0.1339 2.37 1.8e-02 proml 0.3132 1.368 0.1125 2.78 5.4e-03 probs 0.2568 1.293 0.0991 2.59 9.5e-03 highgrade 0.1703 1.186 0.0922 1.85 6.5e-02 logpsa0 0.1549 1.168 0.0312 4.96 7.0e-07 logalkp0c 0.2396 1.271 0.0483 4.96 7.0e-07 hemecenter -0.1041 0.901 0.0351 -2.96 3.1e-03 psadtmonthcat -0.0884 0.915 0.0430 -2.05 4.0e-02 Likelihood ratio test=205 on 13 df, p=0 n=641 (365 observations deleted due to missingness) >

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proportional? recall we are making strong assumption that we have proportional hazards for each covariate we can investigate this to some extent via graphical displays but, limited for quantitative variables

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“Local” Tests Testing individual coefficients But, more interestingly, testing sets of coefficients Example: testing the psa variables testing treatment group (3 categories) Same as previous: Wald test Likelihood ratio Scores test

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TAX327 reg5 <- coxph(st ~ liverny + numbersites + pain0c + pskar2c + proml + probs + highgrade + logpsa0 + logalkp0c + hemecenter + factor(psadtmonthcat)) lrt.trt <- 2*(reg4$loglik[2] - reg5$loglik[2]) p.trt <- 1-pchisq(lrt.trt, 2) #` to compare, you need to have the same dataset liverny1 <- ifelse(is.na(psadtmonthcat),NA,liverny) reg6 <- coxph(st ~ factor(trtgrp) + liverny1 + numbersites + pain0c + pskar2c + proml + probs + highgrade + logpsa0 + logalkp0c + hemecenter) lrt.psadt <- 2*(reg4$loglik[2] - reg6$loglik[2]) p.psadt <- 1-pchisq(lrt.psadt, 4)

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