Larynx Data Trend Test Example

Example

Worked example of trend test on larynx cancer data with 4 disease stages.

Data

Larynx cancer patients, $n=90$, grouped by disease stage (I, II, III, IV). Variable $T$ = time to death (months). Goal: determine if hazard of death increases with disease stage.

Step 1: Descriptive Analysis

KM curves by stage show a clear ordering: Stage I (highest survival) → Stage IV (lowest survival). Boxplots of survival time show decreasing median survival with increasing stage.

Step 2: K-Sample Test (Omnibus)

First, test if any difference exists:

fit <- survdiff(Surv(time, status) ~ as.factor(stage), data = dat)
fit

If $H_0$ is rejected, proceed to trend test for the ordered alternative.

Step 3: Trend Test with Scores

Assign scores $a_j=j$ for stages I–IV ($a=1,2,3,4$):

$Z=\frac{{\sum }_{j=1}^4{a}_j{Z}_j(\tau )}{\sqrt{{\sum }_{j=1}^4{\sum }_{g=1}^4{a}_j{a}_g{\hat{\sigma }}_{jg}}}\sim N(0,1)$

R Implementation (using Cox model for trend):

# Treat stage as numeric for trend
fit_trend <- coxph(Surv(time, status) ~ as.numeric(stage), data = dat)
summary(fit_trend)

Step 4: Results Interpretation

Coefficients (Cox model with stage as factor, stage I as reference):

Stage	Coefficient	$\exp (\hat{\beta })$	Hazard Ratio
I (ref)	0	1.000	—
II	0.0648	1.067	6.7% higher than stage I
III	0.6148	1.849	84.9% higher than stage I
IV	1.7349	5.667	466.7% higher than stage I

Trend pattern: $0<0.0648<0.6148<1.7349$ — confirming increasing hazard with stage.

Likelihood ratio test: p-value = 0.0009 → reject $H_0$, conclude a significant increasing trend.

Step 5: Partial Testing

Identify which specific stages differ significantly:

Comparison	Coefficient	p-value	Significant ($\alpha =0.05$)?
Stage II vs I	0.0648	0.8876	No
Stage III vs I	0.6148	0.0949	At $\alpha =0.10$
Stage IV vs I	1.7349	< 0.05	Yes

Conclusion: Hazard of death increases with larynx cancer stage. Stage IV patients have significantly higher hazard than stage I. The increase from stage II to III is borderline significant. Stage I and II are not significantly different.

Interpretation

The trend test is more powerful than the omnibus k-sample test when groups have a natural ordering. Use the omnibus test to establish that some difference exists, then the trend test to confirm the direction of the difference.

Related

• Trend Test
• Log-Rank Test
• K-Sample Test Procedure