I’m working with some survival data and I created linear regression models on two treatment groups to predict both RMST (restricted mean survival time) and survival probability. However, I’m unsure how to run a hypothesis test to check jointly whether these values are both the same as their counterparts across both groups.
Can anyone point me in the right direction? It’s more complicated since the RMST and survival probability have some dependence.
Say that you have a typical fully parametric survival model, for example an accelerated failure time model of the form:
$$\log T = \beta_0 + \beta_1 X + \sigma W $$
where $T$ is event time, $\beta_0$ represents the control case, $X$ is a 0/1 treatment indicator, $\beta_1$ represents the treatment-associated difference, $\sigma$ is estimated from the data but shared by both treatment and control groups, and $W$ is some standard probability distribution (e.g., minimum extreme value for Weibull, normal for log-normal). Then treatment-associated differences in both the (unrestricted) mean survival and the probability of survival at any time are just functions of $\beta_1$. See this page, for example.
If $\beta_1$ is significantly different from 0, then you have documented a significant treatment-associated difference in outcomes. It's not clear what else would be provided by a further joint statistical test of restricted mean survival time (RMST) and survival probability.
If you nevertheless want such a joint test, you could consider a permutation test in which all re-labelings of the treatment indicator among cases are modeled to provide a null distribution of the "treatment-associated" differences in both RMST and survival probability (at some specified time or times). You can then see how often, in that null distribution, you have both RMST and survival probability differences that are more extreme than what you found in your model with the correct treatment annotation. That approach can be applied to any type of model, for example a semi-parametric Cox model. If the sample size is too large for a full permutation test, you could randomize the treatment indicator in a large number of repetitions.
{}on the editing bar allows for inserting model summaries in text format. Also, when you speak of "survival probability," do you mean the probability at some particular time point, or the entire survival curve? Please provide that information by editing the question, as comments are easy to overlook and can be deleted. $\endgroup$