Test that at least one hazard is greater than hazard in control

Question

I have an experiment with one control and three experimental arms, A, B and C. My experiment generates survival data. I have survival times for each group, $T_{tj}$, where $t = \{ control, A, B, C\}$ and $j = 1, \dots, t_j.$

I want to know if at least one arm has a lower hazard than the control arm. Denote the hazards with $\lambda_t$ I know hazards can be functions, so I can assume the hazards are proportional (preferred) or constant over time. The hypothesis test I'm interested in is: $H_0 \lambda_j = \lambda_k$ for all $(j,k)$ against, $H_a: \lambda_{control} < \lambda{j}$ for some $j$. The alternative can be rewritten as $H_a: max\{ \lambda_{j} - \lambda_{control} \} > 0$ or $H_a: max\{ \lambda_{j}/ \lambda_{control} \} < 1$

I am not sure how to test this. Things I've considered that don't work:

1. Fit a model with maximum likelihood. The log hazard ratio estimates are asymptotically normal, so try to construct a test based on normal random variables. I'm not sure how to construct a test statistic in this case or calculate a p-value.

2. Use a likelihood ratio test. This works if all the log hazard ratio estimates are negative (hazard ratio less than 1). You could have hazard ratios of 0.99, 0.98, and 200, in which case you'd reject the null when you have no strong evidence you have a hazard ratio less than 1.

EDIT: A user asked why I don't look at the estimates give by R's coxph. I want a single test of the maximum to avoid correcting for multiple comparisons. If I take multiple contrasts simultaneously, I have to adjust the confidence intervals and p-values. If I look at just the smallest hazard ratio, and see if it's different from 0, I inflate the Type I error rate.

Answer 1

Cox models use maximum partial likelihood, since the baseline hazard function is not known. Nevertheless, the Partial-Wald, Partial-Score, and Partial-LR test statistics can be compared to standard normal distributions for inference. This is just the usual method of significance testing for covariates, but doesn't do exactly what you're trying to do.

If you want to make the method robust, just use a multiple testing procedure, like Benjamin-Hochberg to conserve FWER, and perform the individual test for each A vs control, B vs control, likewise C and D. You can make the test more general by stratifying on treatment at the cost of lower power when the proportional hazards assumption actually holds.