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In a model where subjects are evaluated over time and a baseline (time=0) covariate is used (eg, score ~ baseline + group + time + group*time), it seems that baseline data are often suggested to be excluded from the dependent variable (eg, score). But excluding baseline data would seem to decrease the model accuracy.

The above model is statistically equivalent to a change from baseline analysis (eg, change from baseline ~ baseline + group + time + group*time). If time = 0 is not excluded, then all groups have change from baseline of 0 at time = 0, and the argument is that the 100% homogeneity at time = 0 biases the results.

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An additional point not mentioned in the other answers is that when you include the baseline measurement as a covariate into the model you assume a constant correlation of this measurement with all subsequent measurements, which most often is not logical to assume.

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    $\begingroup$ This is a very good point, Dimitris, and is another reason tcb2 may want to move over to an SEM framework if they are intent on including baseline in their model. In SEM one can test whether a model in which that correlation is constrained to be the same for all subsequent measurements is a better fit to the data relative to a model in which you freely estimate the correlation at each occasion. $\endgroup$
    – Erik Ruzek
    Commented Jul 24, 2019 at 13:52
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    $\begingroup$ Agree with Dimitris and perhaps Erik is right, tho Model 2 in User158565's answer is the de facto standard in most clinical trials, which often correctly assume that the higher the baseline (up to a point where the disease is too severe to modify), the more likely the drug is to effect change from baseline over time. Conversely, the lower the baseline, the drug is less likely to have a large effect over time given proximity to the scale floor. $\endgroup$
    – tcb2
    Commented Jul 29, 2019 at 16:23
  • $\begingroup$ In clinical trials we for this reason always include a baseline by time (as a factor) interaction. The model above would never be used. $\endgroup$
    – Björn
    Commented Jul 24, 2023 at 9:22
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The two main model that get used in practice for randomized controlled trials in this setting are fit to only the post-baseline time points (with time as a factor variable):

score ~ baseline + group + time + group*time + time*baseline

or

change in score ~ baseline + group + time + group*time + time*baseline

with an unstructured covariance matrix (often allowed to be different for each level of group). I'll outline reasons for the modeling choices below. Most software has good implementations of these so-called mixed models for repeated measures (MMRM): e.g. the mmrm package in R or using PROC MIXED in SAS/STAT.

The reasons for the modeling choices include

  • Using score or change in score is equivalent in this setting (it would e.g. not be if you didn't include baseline and baseline*time interaction in the model). You can do either and it makes no differences for between group comparisons.
  • If you modelled baseline as another timepoint, this could be an alternative (in which case you'd not include baseline + time*baseline in the model) as long as you use an unstructured covariance matrix. Through the correlations between visits such a model would end up doing similar (but not identical) things as adjusting for a linear baseline values. However, this makes much stronger multivariate-normal-residual distributional assumptions that are often heavily violated (e.g. baseline often has a truncated distribution due to inclusion/exclusion criteria, which really messes up the properties of the model). The two models are not equivalent, at all, without additional assumptions.
  • baseline tends to matter a lot for outcomes, but the less so the further away you are from the baseline (unsurprisingly there's usually a declining correlation of observations over time). That's why you really want a time*baseline term in addition to the main effects.
  • You cannot have baseline + time*baseline as covariates and at the same time model the baseline, because the baseline value perfectly predicts the baseline value. I.e. describing it as a random quantity conditioning on its own value does not make sense.
  • time should be a factor, because you don't want to assume a linear curve over time, unless you have strong prior data that this would be so.
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The advice is to not include lagged dependent variables as fixed effects predictors in a longitudinal mixed model. See here.* Technically, you are not doing that with your formulation, however I would still be careful with this. One workaround might be to switch to a structural equation modeling (SEM) framework, particularly if you have few time points. This would allow you to specify regression paths from your baseline score to each of the subsequent scores. The extent to which your theory is contingent on the baseline effect being present would determine whether the switch to SEM makes sense.

*It may be worth posting this question to Paul Allison in that blog post. I know from firsthand experience that Paul is very responsive to questions. If you do so, please report back on what he says!

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You have 3 choices:

Model 1: score ~ group + time + group*time) including baseline measurement

Model 2: score ~ baseline + group + time + group*time) excluding baseline measurement

Model 3: (score - baseline)~ group + time + group*time) excluding baseline measurement

Of course, you need to incorporate the correlation of the error terms into the model indirectly by specifying random effect, or directly by specifying covariance matrix of error terms, both.

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  • $\begingroup$ Model 2 is the question at hand. Why exclude baseline measurement? $\endgroup$
    – tcb2
    Commented Jul 23, 2019 at 23:47
  • $\begingroup$ Basically, one variable cannot be the response variable and covariate, even for part of the observations. Mixed model accepts unequal number of observations from subjects. Suppose 90% subjects only have baseline, then what will happen if baseline appears in both sides of model. $\endgroup$
    – user158565
    Commented Jul 24, 2019 at 0:52

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