I am not entirely sure whether I am interpreting a within-subject treatment effect produced by a Poisson mixed effects regression model correctly, so I am hoping to get some help on this forum.

In the study that generated the data for this model, half the patients were assigned to Treatment A in Period 1 and Treatment B in Period 2. (There was a washout period between treatments.) The other half of the patients were assigned to Treatment B in Period 1 and Treatment A in Period 2.

The outcome in each period was the number of days (out of the number of completed treatment days) patients were able to achieve a pre-defined clinical goal.

The Poisson mixed effects regression model used to analyse the data included fixed effects for Treatment and Period, as well as random intercepts and random slopes for Treatment across patients. The model used a log link and an offset term accounting for the fact that different patients completed a different number of treatment days in each period.

Since the model controls for Period and both Treatment and Period are dynamic (or time-varying predictors), I want to make sure I interpret the fixed effect of Treatment in this model correctly.

I know this effect is a subject-specific effect of Treatment in a particular period. After exponentiating the reported effect, can I interpret it as quantifying the multiplicative factor by which the expected number of days (per number of completed treatment days) differs for the typical patient on Treatment B relative to Treatment A in a particular treatment period?

I guess what confuses me is that we’re typically interested in how the same subject responds to switching treatments from one period to another, but here we are controlling for period. Does that mean that we would hypothetically give the typical subject both treatments in the same period (in either order?) and then monitor how well they respond on Treatment B relative to Treatment A?