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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

Notation

- t1<t2<…tDbe distinct death times in all samples being compared
- At time ti, let dij be the number of deaths in group j out of Yij individuals at risk. (j=1,..,K)
- Define

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 hj(t)

Applying the Test

for j = 1,…,K

If all Zj(τ)’s are close to zero, then little evidence to reject the null.

Others?

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

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

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])

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 <- survdiff(st[psadt<3 & psadt>0]~psadt[psadt<3 & psadt>0])

test3 <- survdiff(st[psadt>2]~psadt[psadt>2])

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?

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’

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)

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)

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)

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

Cox PHM Notation

- Data on n individuals:
- Tj : time on study for individual j
- dj : event indicator for individual j
- Zj : vector of covariates for individual j
- More complicated: Zj(t)
- covariates are time dependent
- they may change with time/age

Basic Model

For a Cox model with just one covariate:

Comments on basic model

- h0(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

Linear model formulation

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

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.

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

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

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)

Hazard Ratios

- Hazard Ratio =

hazard function for New

hazard function for Std

- Makes assumption

that this ratio is

constant over time.

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.

Hazard ratio is not always valid ….

Hazard Ratio = .71

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

Tests of the model

- Testing that βk=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

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

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

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

Multiple regression

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

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))

> 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)

>

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

“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

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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