RCT analysis using ANCOVA for rates I have a question based on the following approach for the analysis of RCT's. 
The following works well for the outcome (and baseline) being continuous with normal errors.

Expanding upon this, I was curious what would be the required form if one was assessing rates. Assume Yt was with respect to some denominator of relevance, N. Also, the baseline Y0 was with respect to some denominator of relevance, N0.
In Generalized Linear Modelling parlance, would either of the following be correct?:


*

*Yt = beta0 + beta1X + beta2*ln(Y0/N0)  

*Yt = beta0 + beta1X + beta2*ln(Y0) + beta3*ln(N0)


In both cases; link=log, errors=Poisson, offset=ln(N)
My main query is the term Y0/N0. Should it be calculated directly and ln(Y0/N0) be a single variable as an independent (case 1), or, split into two vars as using the log rule: ln(Y0/N0) = ln(Y0) - ln(N0)    (case (2))?
Data format is:

 A: If you have individual patient data for rates (something like disease episodes in observation time $t_i$ with a baseline for the past 1 time unit e.g. 1 year), you would traditionally do
$$\log EY_i = \beta_0 + \beta_1 x_i +\beta_2 \log\text{base}_i + \log t_i$$
(taking the logarithm of the baseline is not totally standard, but makes sense, just make sure to have some rule in case of zeroes; you could also divide by the baseline observation time). Poisson is not used much in practice and Negative Binomial is much more popular (it accounts for unexplained between patient variation).
If you have something with $Y$ number of successes out of $N$ tries, then logistic regression (binomial outcome,  logit link function) would seem like a better fit. You'd do much the same as above, but for the link function,  you'd probably write it for each individual post-baseline try and for the baseline term $\log (Y_{0i}/(N_{0i}-Y_{0i}))$ (again with some solution needed in case of all zeroes or ones). I.e. for one trial out of the many for a person the logit of getting a 1 in that trial is
$$\text{logit} EY_i = \beta_0 + \beta_1 x_i +\log (y_{0i}/(N_{0i}-y_{0i})),$$
i.e. logit link and binomial distribution. 
