# Baseline adjustment in mixed models

I am doing mixed model analysis to evaluate (Y=) fruit intake (continuous variable) between two groups (intervention versus control) over time (baseline, year 1, year 2, year 5, year 7 and year 15). My model look like this: Y = group + time + group*time. My question regards the baseline measurements. In my first analysis I was told to transform my data to a long format without the baseline measurements, and include this as a fixed effect to adjust for it. However, I then struggle as I have to interpret the results differently. If I include baseline data when I convert my data to long format,

1. Do I need to adjust for baseline - assume there are baseline differences between groups? I read somewhere that I don't need to adjust for baseline differences in mixed models with interaction terms.

2. Can I then include baseline measurements as a fixed effect or will this create collinearity?

My question is basically, how can I adjust for baseline measurements in mixed model analysis?

I assume this is a parallel group experiment (randomized assignment to just one group or the other, not some kind of cross-over).

Assuming this is a continuous outcome, the purpose of baseline adjustment is to reduce the variability of the treatment difference. Some discussion around this and the conditions under which it is helpful are found in Senn, S. (2006). Change from baseline and analysis of covariance revisited. Statistics in medicine, 25(24), 4334-4344. It does not really make a difference whether it is a mixed model or not. However, if one does include the baseline, it is usually recommended to have a time (as a factor) by baseline interaction, because the importance of the baseline will usually decrease over time. Personally, I would by default include the baseline in any model in the type of trial you are describing (and for the type of mixed model you describe always include the time by baseline interaction).

The most usual way is to indeed include the baseline as a fixed covariate. Using it as a random effect is of course also possible, but less common.

The final option is to use the baseline as yet another observation instead of as a model term. This assumes joint (multivariate-)normality (assuming you are using a normal model) of the error terms across the visits including the baseline. This can work nicely, but tends to be problematic if subjects were only included in the study, if their values were above (or below) a certain threshold (because that induces very strong non-normality due to the truncated distribution of the baseline values). For that reason including it as a model term is more common.

• Thank you for answering Björn. By the way, you assume correct about the study design. Just to make sure I do this correctly when doing mixed models with data in long format; I should include baseline measurements in Y in addition to baseline measurements as a fixed effect? Bente Mar 5, 2018 at 11:47
• Yes, the usual thing would be a model that is fit to each post-baseline visit (no extra record for baseline visit) with a model like factor(treatment) + factor(visit) + factor(treatment)*factor(visit) + covariate(baseline) + covariate(baseline)*factor(visit). Mar 5, 2018 at 13:08
• Just trying to get better understanding,@Bjorn, if baseline is treated only as a predictor, then for participants who only have baseline, their data are not used in any way since they have no observations in the response vector; I am thinking this would be against an Intention to Treat principle. Do you know what the issue is if baseline is included on both sides of the mixed model equation? Thanks Apr 12, 2022 at 18:12
• You can of course do multiple imputation for the baseline first before analysis to deal with missing baseline. That's much safer than including baseline in the model as another observation - especially if inclusion criteria are applied on the variable under analysis or variables at least somewhat correlated with it. Such inclusion cut offs produce truncated/skewed distributions that lead to bad violations of joint multivariate normality (which you really on heavily when modelling baseline as just another observation, but much less so with MI). See e.g. here doi.org/10.1002/pst.1705 Apr 13, 2022 at 6:08
• Regarding ITT: I guess nowadays we'd talk about a treatment policy estimand and then the implicit imputation that mixed models/MMRM do for missing observations when you have post baseline data is in line with a hypothetical estimand (as if everyone had stayed on their treatment, which we know they haven't, because they quit the trial = not treatment policy estimand, unless you can argue that some specific aspects make it so from some perspective). Apr 13, 2022 at 6:13