# How can I model optimal time at which a treatment becomes effective?

In a nutshell:

Two treatments are thought to result in comparatively different outcomes depending on when they are administered, with one possibly being better for patients who are treated soon after symptom onset, and the other being better for patients who are treated later.

How can I determine what the cut-off time is ?

More detail:

Interest is centred on the effect of a new treatment for an emergency medical condition.

Of primary concern is the time elapsed between the onset of a patient's symptoms, and the time at which the treatment is given.

The outcome is binary - death.

The time to treatment is measured in minutes and its log is approximately normally distributed.

Data are available over five years, on all patients admitted who received either treatment. The treatment status is binary coded - 0 for the old treatment, 1 for the new treatment.

In year 1, approximately 10% of patients received the new treatment. In year 5, approximately 70% of patients received the new treatment and there was an approximately linear change in the proportion during the study period.

There are approximately 15,000 cases per year over the 5 year period.

Preliminary analysis using a GLM with death as the outcome, time-to-treat and treatment status as the exposures, shows statistically significant estimates for both, also controlling for several confounders.

              Est    SE
Treatment    -0.72  0.04
TimeToTreat   0.13  0.03


It has been suggested that I split the dataset according to the time to treatment, for example into 2 subsets (short and long times to treatment), and look for opposite signs in the treatment effect estimate for each subset, then narrow it down by splitting at different time points until, magically, some "ideal" is reached. Apparently this would be appealing to clinicians. This does not seem like a great way forward to me - for one thing, it could be quite time consuming, and for another it seems it would give "average" effects in the short and long time groups, rather than instantaneous effects.

So instead I thought about including the interaction between time-to-treat and the treatment status.

              Est    SE
Treatment    -0.06  0.35
TimeToTreat   0.19  0.05
Interaction  -0.14  0.07


0.19 is the slope of the regression line for the Treatment=0 group. 0.19-0.14=0.05 is the slope of the regression line for the Treatment=1 group. Since the Interaction term has a negative estimate, it means that the regression line for the Treatment=1 group is below that for the Treatment=0 group, and when the time-to-treat variable is 0, the difference in the estimates is 0.14. Given this, it means that the regression lines for each group diverge above zero for increasing time to treat. Furthermore, since the Treatment effect is not significant, there is no significant difference between the estimates for the two groups, when time-to-treat is zero. Therefore the new treatment should be preferred to the old treatment for all time-to-treat.

Does this interpretation make sense ?

Are there better ways I could analyse these data ?

• It looks to me like it should be possible to estimate survival (or death) rate as a function of time-to-treatment and the type of treatment, yielding two curves ... and that the primary interest is in 'at what time do the two curves cross?'. Would that be a reasonable way to describe what you're after? – Glen_b Oct 8 '13 at 20:16
• @Glen_b to be honest, I'm not completely sure what you mean. I thought my approach with the interaction was something like that ?...A line for one treatment group and a line for the other, and where they cross is the point at which both treatments have the same outcome, but when I ran this model on the data, this point would be in negative time-to-treat (if I have interpreted the output correctly) – P Sellaz Oct 9 '13 at 8:34
• An interaction model should do as I suggested, yes (though for a preliminary analysis I'd probably look at a display of two separate nonparametric estimates just to get an idea whether my model made any sense). Can you plot the two fitted curves (that is, get the output fitteds)? I think you have interpreted the model correctly, but it never hurts to ask the GLM code what it thinks it fitted. – Glen_b Oct 9 '13 at 9:24