Can person-time be treated as an independent variable? I have a count outcome and a treatment vs. comparison group independent variable. I would like to model how many times patients return following their first visit (I might actually make this outcome binary - TBD). However, the patients all have different study enrollment lengths. Rather than lose data by subsetting based on standardized enrollment length, I would like to treat enrollment length as an integer (# months enrolled - my smallest unit). This would sort of be a modified person-time variable but it would be adjusted for, rather than used to express a rate. 
Does anyone know if this method is acceptable? If so, do you know how much precedent there is for this method?
 A: Two answers, based on whether you are analyzing the effect of the treatment, or simply predicting number of times returning.
If analyzing treatment effect:
You effectively have a regression
$$
y = \alpha + \beta T + \epsilon
$$
The coefficient $\beta$ measures the average effect of the treatment on the outcome.
You could control for the enrollment length:
$$
y = \alpha + \beta_1 T +\beta_2 L +  \epsilon
$$
However, if $L$ and $T$ are uncorrelated, then the estimate of $\beta_1$ is invariant to the inclusion of $L$.  The inclusion of $L$ will however reduce the variance of $\epsilon$, and this in turn will reduce the standard errors on $\beta_1$, because the variance-covariance matrix is defined as $\sigma^2_\epsilon(X^TX)^{-1}$ (in the usual case -- variants are analogous).  If that $\sigma$ gets smaller, the SE's, which are the diagonals on the resultant matrix, also get smaller.  So you're more sure of your estimate $\beta_1$, even though it shouldn't affect the magnitude of $\beta_1$ directly.
Alternatively, you might believe that $L$ influences the effect of $T$.  In this case you want
$$
y = \alpha + \beta_1 T +\beta_2 L + \beta_3 T\times L+  \epsilon
$$
The treatment effect (call it $\tau(L)$) is thus a function of $L$:
$$
\tau(L) = \beta_1 + \beta_3 L
$$
If simply predicting:
Just throw all of your variables in a random forest (after reading about how they work, if not already familiar).  Generally the more information you include within them, the better.  
A: It rather standard to use the natural logarithm of the follow-up as an offset in Poisson or negative-binomial regression, if you want to model events per time unit. This is a pretty common and widely accepted approach. There are lots of precedents e.g. in clinical trials of asthma/COPD exacerbations, multiple sclerosis relapses etc. Btw. it needs to be a log(time) offset (variable with coefficient fixed to 1), because a log-link function is used.
Doing so makes an implicit assumption that data after the end of follow-up are missing at random. I.e. whether the follow-up becomes truncated can depend on the model covariates and on the observed number of events, but not on other things (e.g. a bad case would be if patients feel an oncoming hospitalization and quit your study because of that without the event getting recorded). Also there is an assumption that afterwards patients behave / are treated as before.
