Trend Test Survival

Definition

Let:

• $K$ : number of ordered groups with a natural ordering (e.g., disease stage I, II, III, IV)
• $Z_j(\tau )$ : log-rank test statistic for group $j$, computed as in the k-sample log-rank test
• ${\hat{\sigma }}_{jg}$ : variance-covariance of $Z_j$ and $Z_g$

Test Hypotheses: $H_0:h_1(t)=h_2(t)=\dots =h_K(t)\forall t\le \tau$

$H_1:h_1(t)\le h_2(t)\le \dots \le h_K(t)\forall t\le \tau$

with at least one strict inequality. Equivalently: $H_1:S_1(t)\ge S_2(t)\ge \dots \ge S_K(t)\forall t\le \tau$

Test Statistic:

Assign ordered scores $a_1<a_2<\dots <a_K$ (commonly $a_j=j$). Then: $Z=\frac{{\sum }_{j=1}^K{a}_j{Z}_j(\tau )}{\sqrt{{\sum }_{j=1}^K{\sum }_{g=1}^K{a}_j{a}_g{\hat{\sigma }}_{jg}}}\sim N(0,1)$

Interpretation

The trend test detects whether hazard rates follow a monotonic pattern across ordered groups. A significant positive $Z$ indicates that groups with higher scores (e.g., later disease stages) have higher hazard. Use this test when groups have a natural ordering.

Related

• Log-Rank Test
• Stratified Test