Survival Analysis: Hazard Curves When Treatment Is Expected To Both “Stretch” And “Shift” It

[Sadish D]

Hi folks,

I am going to try and present a simplified version of a problem I have encountered in working with a time-to-event dataset to study a natural experiment. I am absolutely new to survival analysis, so hope you have some insights.

A policy change affects half a population. In the period after the policy announcement (Post), the group affected by the policy (Treated) are expected to be more likely to experience an event of interest. But, they are expected to take longer to experience the event. So, my hypothesis is twofold. The Treated in the Post period are:

1. more likely to experience the event
2. likely to have a delayed experience of the event

My intuition is to plot the hazard curves for the treated and some comparison group. I expect that the hazard function of the Treated people in the Post period to “stretch” vertically, and “shift” horizontally, somewhat as in the accompanying figure. The peak of the Treated group (solid line) in the Post period will be higher (hypothesis 1) and the peak will occur at a later time (hypothesis 2) compared to that of the comparison group (dotted line).

Question 1: Am I correct in understanding that this is a case of non-proportional hazards?

Question 2: How can I plot these hazard functions while adjusting for covariates in the fashion of a difference-in-difference?

Here is what I think the answer might be using sts graph. I define the binary variables Treatment, and Post to use as covariates, and the interaction TreatmentXPost to use as strata. I think what I should do is this:

sts graph, hazard strata(TreatmentXPost) adjustfor(Treatment Post)

Following the explanation here (towards the bottm of the webpage) of what “strata” in sts graph does, I think what I would be plotting are the baseline hazard functions obtained by estimating:
log[ h(t,X) ] = log[h0(t)] + b1*Treatment + b2*Post
log[ h(t,X) ] = log[h1(t)] + b1*Treatment + b2*Post

Where h1 is the baseline hazard for the stratum TreatmentXPost=0, and h2 is the one for TreatmentXPost=1.

Question 3: Would I be plotting the two baseline hazard functions with the values of the covariates held at zero?

Question 4: Are there non-graphical ways to think about the dual hypotheses?

I am using Stata 15.1.

[Andrew Lover]

You might look into comparing the restricted mean survival times, which have no proportional hazard assumption, and would allow you to compare survival separately at both early and late time intervals- but you'll obviously need to think carefully and justifiably about those time points.

https://www.stata-journal.com/article.html?article=st0451

https://www2.le.ac.uk/Members/pl4/workshop2011-1/Royston-Stockholm-10nov2011b.pdf

[Sadish D]

Thanks for the resources Andrew. I will also welcome answers from others to my specific questions. They would help me understand survival analysis.

[Clyde Schechter]

Re Question 1: Definitely a case of non-proportional hazards.

Re Question 2: I don't have an actual answer, but I think your approach is wrong. The model you show

log[ h(t,X) ] = log[h0(t)] + b1*Treatment + b2*Post
log[ h(t,X) ] = log[h1(t)] + b1*Treatment + b2*Post

is a proportional hazards model. Think about it. Comparing Treatment = 1 to Treatment = 0, the log hazards will differ by b1, which means that the hazards will be proportional with a ratio b1.

Another analytic approach you might consider is just doing simple comparisons of the proportion in each group who have not yet experienced an event as of some selected, relevant, times.

[Sadish D]

Clyde,

Maybe your response to Question 2 makes sense. I think I am looking at the model as somehow "partialing out" problems with the proportional hazards assumption. If the hazard functions shift only because of the policy, then it should only be the TreatmentXPost group who do not conform to the assumption. In my mind, estimating different baseline hazards for TreatmentXPost=0 and TreatmentXPost=1 would solve the issue, and I could continue assuming proportional hazards for Treatment and Post. But maybe this is not the right way to think about it.

The approach I am currently using might be a version of what you suggested-- except I'm looking at having experienced the event rather than not having experienced it. I create binary variables "experienced event at t=1", "experienced event at t=2" and so on, and use OLS for a difference-in-difference as follows:

[experienced event at t=i] = b0 + b1*Treatment + b2*Post + b3*TreatmentXPost, where i = 1, 2, 3, ...

Plotting the various values of b3 over t shows me the t at which TreatmentXPost have higher or lower probability of experiencing the event.

Thanks for responding.

[Clyde Schechter]

I think this approach makes sense.