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Every achievement originates from the seed of determination.

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Survival Analysis

Nonparametric Methods for Comparing Survival Distributions

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Abbreviated Outline

How to formally compare 2 or more survival distributions using hypothesis tests

These tests look at weighted differences between the observed and expected hazard rates, allowing us to put more emphasis on certain parts of the curves

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Hypotheses

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Notation

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Test Statistics

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Test Statistics

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Test Statistics

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Test Statistics

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Test Statistics

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Test Statistics

Reject Ho if U is too large.

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Log-rank Test

Constant weight function: Treat all observed failure times equally.

It has optimum power to detect alternatives where the hazard rates in the M populations are proportional to each other

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Proportional Hazard Assumption

An underline assumption of many methods

Suppose there are 2 groups of survival data. Then h1(u)=c*h2(u) where hi(u) is the hazard function of group i and c is a constant

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Wilcoxon Test

Survival time t(j) is weighted by nj, the number of individuals at risk at time t(j).

This test is less sensitive than the log-rank test to deviation of the observed to the expected in the tail of the distribution of survival times.

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Example: 6-MP

To compare the survival distributions of the placebo group and the 6-MP group using the log-rank test

Test of Equality over Strata

Pr > Test Chi-Square DF Chi-Square

Log-Rank 16.7929 1 <.0001 Wilcoxon 13.4579 1 0.0002 -2Log(LR) 16.4852 1 <.0001

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Stratified Tests

Previously, we assumed that the various groups of individuals under comparison are homogeneous with respect to other factors which may affect survival time

One way of detecting differences in survival between groups, while accounting for the effects of other factors is to stratify.

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Stratified Tests

When the number of strata is large, a test typically has low power to detect treatment differences.

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Stratified Tests

Hypothesis:

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Stratified Tests

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Example: 6-MP

The patients are stratified according to remission status (partial or complete).

Consider a test of Ho of no treatment effect, adjusting for the patient’s remission status.

The stratified log-rank test (chisq=17.9 and p-value = 2.28x10^-5) indicates that the distribution of survival times is significantly different between 6-MP and placebo groups.