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I have an experiment in which there are two groups; one group that is exposed to a treatment, and one group that is not.

I have a metric I am measuring that follows each person in each group for 21 days after receiving (or not receiving) the treatment. If the person takes a desired action within those 21 days, they are assigned a 1. If they do not take a desired action within those 21 days, they are assigned a 0. Participants are not guaranteed to ever take the desired action.

However - people enter at staggered times throughout the time I run the experiment. Therefore, when I end the experiment for these individuals, I will have some people who have not completed their full 21 days. In other words, they are censored.

I am interested in estimating the average treatment effect. That is, did the people in the treatment group do the desired action more than the control group? Would a survival analysis allow me to measure the average treatment effect, given that I do not care about the actual time it takes for a person to take the desired action, but rather, if they took it at all within a fixed time period?

We can always stick to complete cases, but consider a four week experiment where patients are assigned everyday uniformly. If we require participation for a full 21 days, then everyone who enters the experiment in days 8-28 are lost - roughly 75% of the data!

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Restricting to complete cases can throw away a lot of data, as you note. Survival analysis can take advantage of all the data.

Individuals who haven't taken the action but were followed for less than 21 days provide no information about what they might have done by 21 days. You thus need to think about the "average treatment effect" (ATE) as related to something other than the binary outcome of whether the action was taken by 21 days.

For the ATE, you might evaluate whether the treatment affects the "hazard" of taking the action: the probability of taking the action at some time, given that it wasn't already taken. That's a common type of analysis. A Cox proportional hazards regression model would assume that the ratio of hazards between the treatment groups is constant over time, which is often a good enough assumption. This site has over 1800 posts related to Cox models.

If you only have daily (rather than more fine-grained) data on whether the action was taken, consider a discrete-time survival model. That would be binary model based on the individuals at risk for taking the action within each day. There are a few dozen related posts on this site. You might allow the relative hazard between groups to change over time, for example if you suspect that the treatment affects behavior soon after it's instituted but has less effect as time goes on.

These posts can provide an introduction to issues in estimating ATE from survival models.

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  • $\begingroup$ Does a CoxPH model allow for the case where some people may never take the desired action, like in cure models? $\endgroup$
    – aranglol
    Commented Mar 30 at 20:32
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    $\begingroup$ @aranglol Cox models are often applied in such cases. They are used to evaluate survival after therapy for types of cancer that are cured in some individuals. A Cox model evaluates relative hazards until the last event time (time of taking the action, in your case). What happens thereafter is irrelevant. See this page. $\endgroup$
    – EdM
    Commented Mar 30 at 21:05
  • $\begingroup$ Why is it that a Cox PH model can ignore what happens "thereafter" (either after the study is over, or whether the person takes the desired action within 21 days)? Is this also true for models that are parametric (like cure models/accelerated failure time models)? Is it because the hazard function isn't explicitly modelled in CoxPH models, so the survival function S(t) isn't explicitly modelled and therefore we don't have to make any assumptions about S(t) going to 0 as t --> infinity? $\endgroup$
    – aranglol
    Commented Mar 31 at 0:18
  • $\begingroup$ @aranglol that’s essentially correct. Covariates in an AFT model stretch or shrink the entire time scale, so you have to model the entire survival curve. If there’s a cure in an AFT model, that needs to be modeled together with the AFT for those not cured. Cox models effectively start out one event time at a time, evaluating relative hazards among cases at risk only at each separate event time, then iterating until there’s agreement on overall relative hazards. Times after the last event aren’t considered at all. $\endgroup$
    – EdM
    Commented Mar 31 at 2:21
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    $\begingroup$ @aranglol this paper examines parametric survival/cure models and Cox models in some cancers treated with immunotherapy, which leads to actual cures in a fraction of patients. Advantages of cure models generally require very long follow up, when you can be reasonably sure of a true cure. The authors conclude: "This [cure model] approach does not replace the classic Cox proportional hazards regression model for the primary analysis of randomized clinical trials but complements classic methods to evaluate treatment benefits." $\endgroup$
    – EdM
    Commented Mar 31 at 15:12

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