Survival analysis to estimate mortality at a specific day and handling deaths on day 0

[Megan Moreton]

I am having trouble wrapping my head around survival analysis. I have a trial with two arms where the endpoint is mortality at day 14. The endpoint will be calculated using survival methods to account for censoring. The trial continues for longer, with patients monitored for several weeks. The data is structured with one record per subject.

Currently, I have the following code:

// setup
gen time_days = lastseen - randomdate
stset time_days, failure(death==1) exit(time 14)

// results
sts list, by(group) risktable(14) fail
sts graph, by(group)

Is this the correct set up to estimate day 14 mortality? I am getting a bit tangled up about the start of the trial being day 0 or day 1 and thus whether I'm accurately estimating the day 14 mortality.

Additionally, I have a few patients who die on the same day they are randomised (within a few hours), but we don't capture the specific time of death. These patients should be included as their true time of death isn't t=0. Does it make sense to add 0.5 to their follow-up time to account for this? Ie:

// setup
gen time_days = lastseen - randomdate
replace time_days = time_days + 0.5 if time_days==0
stset time_days, failure(death==1) exit(time 14)

Would this approach be appropriate, or is there a better way to account for early deaths? I will go on to estimate the difference in mortality at day 14 & use Cox PH to explore related factors.

[Clyde Schechter]

I am getting a bit tangled up about the start of the trial being day 0 or day 1

In principle, it is neither. In principle, survival analysis treats time as continuous, and the start of the trial is instant 0. It can get confusing because in most real settings we measure time in discrete units, such as days, and so you encounter problems such as the one you describe: how to handle deaths on the day of randomization. Setting their time of death to 0 will result in their exclusion from the analysis, so that is clearly the wrong thing to do. Using 0.5 as an approximation is reasonable. Actually, if you are truly interested only in the estimated mortality at 14 days, it doesn't matter whether you set them as 0.5 or any other number > 0 and < 1 because by the time you reach time 1, you will have the same number of deaths and survivors still at risk at all times > 1. The details of this would matter for the overall survival function estimate, but not for its value at time day = 14.

Now, if you are going on to use a Cox PH model then you are no longer just estimating mortality at 14 days. You are looking at the overall survival curves. Since your problems arising only in the first of 14 days of observation, and it is only "a few" of the subjects, the effect of how you choose to code the deaths of these people on the hazard ratios you estimate will be small, and probably negligibly so, but it cannot be said that it makes no difference at all.

If you really want to get this exactly right, you would properly regard the patients who died on day 1 as interval censored, with death occurring somewhere between time = 0 and time = 1 and you would use -stintcox- instead of -stcox-. Of course, if you are taking this approach, all of your observations are interval censored because you know only the day on which they died, not the exact time.

As a practical matter, because -stintcox- is more complicated to use and computationally more intensive, the additional degree of theoretical correctness as a model specification is usually not worth the effort. It is usually reserved for situations where there is considerably greater uncertainty about the time of the events. Pragmatically, setting the time of your first day deaths to 0.5 is probably the way to go when the study period is many days and the number of affected observations is small.

[Megan Moreton]

That's really helpful. Thank you for taking the time to reply. The primary outcome is the mortality difference at day 14 analysed using the time-to-event estimated proportions, so I will stick with adding the 0.5 to the couple of early deaths.