The model you propose can be written:
$$(\log Y - \log Y_{bl}) \sim \log Y_{bl} + \text{Treatment} + \text{Visit} + \text{Visit : Treatment} .$$
This is modeling the differences in $\log Y$ values from corresponding baseline values as a function of the baseline values, $\log Y_{bl}$. That type of "change score" model is typically not a good choice, as change scores are typically correlated with baseline values, thus posing problems with interpreting regression models.
For longitudinal data analysis with a pre-treatment baseline value of the outcome, it's generally good practice to use the baseline value as a predictor in the model, as you propose, but to model the later outcome values themselves as a function of combinations of treatment and post-baseline observation times. Frank Harrell discusses that in Chapter 7 of Regression Modeling Strategies. From that perspective it does make sense to start the modeling from the second visit, with the first visit's outcome value as a predictor for all subsequent times. Furthermore, the generalized least squares approach that Harrell recommends is an alternative way to deal with correlations within individuals, a way that might get around problems you are having with random effects in a mixed model.
Those considerations suggest the following linear regression structure for fixed effects, for outcomes after the baseline visit:
log(Y)~ log(Ybl) + Treatment + Visit + Visit : Treatment
You would include random effects or use generalized least squares to deal with the correlations within individuals.
Note that regression coefficients from this type of model represent differences on the log scale, which are logs of the ratios that you say are standard in your field. You thus end up where you want to be.
In response to comment
A random intercept makes sense in the model I propose. The fixed-effect intercept in that model would be an estimate of "the mean level of log(Y)
at the second visit in the reference treatment group when Ybl=1
," but not the actual mean value within the reference treatment group when their values are Ybl=1
. Think of a random intercept primarily as being one way to account for within-individual correlations and try to account for un-modeled differences among individuals. The random-intercept estimates for those in the treatment group in your situation are estimates of what their outcomes (log(Y)
) would have been if they had been in the reference group instead. Although those values aren't themselves observable, the model with that form of random intercept uses information from all individuals to maximize the (possibly restricted) likelihood of the data if the form of your model is correct.
For a simple situation like yours, without multiple levels of random effects, you can alternatively account for within-individual correlations via generalized least squares, as explained in Chapter 7 of the Harrell reference above. That's a marginal model rather than the conditional model provided by random-effect terms in a mixed model, but with linear regression (on previously log-transformed values in this situation) the coefficient interpretation will be the same.
Finally, with a linear regression model you can specify particular structures of the within-subject correlations if you use the nlme
package in R, which allows for both generalized least squares and mixed models. That might be advantageous with your 7 visits after baseline. See this page and the Harrell reference for the differences between those correlation structures and what you get with the R lme4
package .