Stratified Test Survival

Tests whether hazard function of different groups are equal, controlling for confounding covariates by partitioning into homogeneous strata.

Definition

Let:

• $M$ : number of strata defined by a set of covariates
• $K$ : number of groups being compared within each stratum
• $Z_{js}(\tau )$ : log-rank statistic for group $j$ in stratum $s$
• ${\hat{\sigma }}_{jgs}$ : variance-covariance in stratum $s$

Test Hypotheses

$$\begin{array}{rl}{H}_0& :h_{1s}(t)=h_{2s}(t)=\dots =h_{Ks}(t)\forall s=1,\dots ,M,\forall t\le \tau \\ H_1& :\text{not all }{h}_{js}(t)\text{ are equal}\end{array}$$

Procedure

For each stratum $s$, compute $Z_{js}(\tau )$ and ${\hat{\Sigma }}_s$ as in the k-sample log-rank test.

Global Test

Pool across strata:

$$\begin{array}{rl}{Z}_{j\cdot }(\tau )& =\sum _{s=1}^M{Z}_{js}(\tau )\\ {\hat{\sigma }}_{jg\cdot }& =\sum _{s=1}^M{\hat{\sigma }}_{jgs}\end{array}$$

${\chi }^2=(Z_{1\cdot },\dots ,Z_{K-1\cdot }){\hat{\Sigma }}_{\cdot }^{-1}(Z_{1\cdot },\dots ,Z_{K-1\cdot }{)}^T\sim {\chi }_{K-1}^2$

For $K=2$ (two groups): $Z=\frac{{\sum }_{s=1}^M{Z}_{1s}(\tau )}{\sqrt{{\sum }_{s=1}^M{\hat{\sigma }}_{11s}}}\sim N(0,1)$

Interpretation

The stratified test controls for confounding covariates by testing within homogeneous strata. For example, comparing treatments while controlling for disease stage: compute the log-rank statistic separately within each stage, then pool the results. This produces a treatment effect estimate adjusted for stage.

Example: Controlling for Sex in Larynx Cancer

Goal: Compare survival across larynx cancer stages while controlling for sex.

# Stratified log-rank: stage effect adjusted for sex
fit_strat <- survdiff(Surv(time, status) ~ as.factor(stage) + strata(sex), data = dat)
fit_strat

Procedure:

1. Partition data into strata (male, female)
2. Within each stratum, compute $Z_{js}$ and ${\hat{\Sigma }}_s$ as in the k-sample test
3. Pool across strata: $Z_{j\cdot }={\sum }_s{Z}_{js}$, ${\hat{\sigma }}_{jg\cdot }={\sum }_s{\hat{\sigma }}_{jgs}$
4. Global test: ${\chi }^2\sim {\chi }_{K-1}^2$

Interpretation: A significant result means that stage differences in survival persist after controlling for sex. The stratified test answers: “Is the stage effect still there, even when accounting for sex differences?”

Interpretation

The stratified test controls for confounding by testing within homogeneous strata, then pooling results globally. It’s a bridge between non-parametric methods (stratification) and regression (Cox PH).

Related

• Log-Rank Test
• Trend Test