# Random intercept in mixed model w/ post baseline measurements

I'm running a LMM analysis for a clinical trial (two treatment conditions, five visits) and I can't understand the exact role of a random intercept.

The baseline score is not included in the outcome (Y) and is instead retained for use as a covariate. A random intercept is also included in the model. My question is: Is this logical? Or does the inclusion of a random intercept at a post-baseline visit 'control out' some possible effect of treatment?

In other words, is it possible to use baseline visit as a covariate in a LMM while also including a random intercept?

Any advice would be much appreciated

Using baseline score as a covariate is reasonable, especially when the follow-up measurements on response variable have relation with baseline value.

Because there are 5 measurements of response variable from a single patient, so the response variable is not independent from each other, which is required by general(ized) linear model. To overcome this difficulty, we need to specify the pattern of the relationship between response variables. One pattern is compound symmetry. This pattern specify that the variance of measurements at the different time is the same, the correlations between any two measurements from the same patient are the same. The measurements from different patients are independent. You can specify this pattern by two different ways. One of them is patient level random intercept. It means each patient has his own intercept. The measurements from the same patient share the same intercept, so they are correlated by the same correlation coefficient.

In fact, baseline visit as a covariate and including a random intercept have no relation.

It depends on the data structure.

I'm assuming that, although the baseline measurement isn't counted among the possible outcomes, the post-baseline assessment occurs more than once in each patient. So you have correlated data, by repeated measures in each subject. If that's the case, then including the random intercept is a good approach to analyzing the data. The effect is kind of like producing a score for all their unmeasured predictors of the outcome based on between-and-within subject residual effects so that, when the random intercept is subtracted off, the data are independent-like and the analysis is unbiased and efficient.

By the way, although the baseline assessment is like all the post-baseline assessments in that it is part of a statistical process, you would never want to include the baseline assessment in the model unless you recode treatment to a time-varying covariate. That's because a treatment by definition could never affect a baseline value. Doing this inadvertently will bias your analyses. Including the baseline value as a covariate is called an ANCOVA analysis and is like a more general way of modeling post-pre differences as an outcome.

• Interesting response! Could you give a specific example of how you would code treatment as time-varying if you did want to include the baseline value of the outcome in the vector of response values? Let's say someone is assigned to either Treatment A post-baseline and followed up 2 times. I can understand that, for that person, Treatment will take on the values A at times 1 and 2. What about at time 0 (i.e., baseline)? What value will Treatment take? Some arbitrary value like "No Treatment"? – Isabella Ghement Dec 1 '18 at 13:29
• @IsabellaGhement that's exactly right. You would code treatment as 0 at all baseline observations, since treatment has not yet been administered in any group, then recode to 1. This of course makes it impossible to control for baseline as a covariate. So the bonus is that if there are any confounders you estimate them with a touch greater precision, although the overall efficiency is reduced. It is not a biased analysis most importantly. – AdamO Dec 2 '18 at 3:54