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I am still trying to find a model for a large dataset, approximately 1-5 measurements per patient (over time), one is the baseline value at t=0. The researcher is interested in the change over time and in the effect of the baseline value on that change. I want to set up a LMEM with random intercept+slope and I want to account for the baseline by adding this as a covariate.

However, I have read some literature and it seems that it is not correct due to the dependency which is created by this. Nevertheless I read more than one paper where this model was performed.

So basically I mean something like this:

$$ z_{i,j}=y_{i,j}-y_{i,0}=(\beta_0+b_{i,0})+(\beta_1+b_{1,0})\cdot t_{i,j}+\beta_3 y_{i,0}+...(\text{other covariates})+\epsilon_{i,j} $$

and it appears strange to me. So basically change scores are used for pre/post experiments but for several patients I have more than one post value, this might also be a problem? Furthermore I'm not sure whether we should really consider the change as the response because I also read that these types of model in general have some undesired properties.

Maybe anyone can help? Thanks!

EdM
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Kathrin
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1 Answers1

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The fact that many studies erroneously model change scores as a function of baseline does not remove the inherent problem with that approach. See this answer among many links on this site. It's important for the researcher to understand that.

If you are expecting a linear change in the measurement values over time, as your model implies, then random effects with the actual measurement as the outcome variable nicely accommodate both the baseline value and multiple measurements over time. You code the baseline value to have a measurement time of 0. Depending on how you set up the model, it can incorporate correlations between random intercepts and slopes. The model then can provide any desired estimates of value differences between measurement times.

All the data for a patient will be used this way, partially pooled with information from other patients, to estimate the random intercept and slope. That avoids putting undue extra weight on any (potentially uncertain) single baseline value.

A potential practical problem will be having too few data points to fit a model of desired complexity. For example, if there's only 1 value for some patients then you can't estimate a slope over time for them. But that would be a problem even if the researcher were to insist on modeling change scores, as such patients wouldn't have any.

EdM
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    Thanks a lot. So you also propose to model the response directly instead of the change. And further, you propose to NOT include a baseline covariate but use these values (with t=0) together with the other measurements for the model fit. Is that right? Actually this would also work because in this case I definitely have at least two measurements per patient, meaning that I CAN estimate a random slope which I couldn't if I use these values for the baseline covariate and not coded in the complete data set. – Kathrin Aug 08 '20 at 14:11
  • @Kathrin that's correct. And you can test for the correlation between baseline values (intercepts in the mixed model) and changes in values (slopes with respect to time in the model), which seems to be what the researcher is interested in. See examples on [this page](https://www.ssc.wisc.edu/sscc/pubs/MM/MM_MultRand.html) for how to code a mixed model to either exclude or allow for correlations and how to compare results to test for the significance of the correlation. – EdM Aug 08 '20 at 15:08
  • Perfect, I see what you mean. As I'm using lmer from the lme4 package I usually assume an unstructered covariance matrix (there is no possibility for implementing something else...) for the random effects, so this should yield exactly what I am interested in? – Kathrin Aug 08 '20 at 15:33
  • @Kathrin yes, although when the model allows for intercept/slope correlations it necessarily imposes and estimates a covariance between slopes and intercepts. Comparing that model against a model that does not allow for those correlations, as in the page linked from my other comment, provides the test you want. – EdM Aug 08 '20 at 15:41
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    (+1) nice explanation. I wish the problem of regressing change on baseline was given some space on introductory applied stats courses/modules. – Robert Long Aug 09 '20 at 04:30
  • @EdM thanks again, I looked at this page and fit these two models. For the one allowing correlations I got -0.56 between random intercept and slope. I think this is the value I am interested in? You suggested to fit both models, is this because I should compare which one provides the better fit? I only know the ICC for getting an idea of the model performance. For both models the (fixed effect) estimates themselves are quite close to each other. The ICC is a bit higher for the one allowing for correlation (adj.ICC 0.63). Sorry for all these questions, I am rather new to real world problems. – Kathrin Aug 09 '20 at 12:03
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    @Kathrin the -0.56 correlation between random intercepts and slopes is what you are interested in, as I understand the statement of the problem. That's the principled equivalent of what the researcher seems to be interested in: how do baseline values (now represented as intercepts) correlate with changes over time (slopes)? Comparing the model with correlated slopes against one without correlations is simply a test of the statistical significance of the correlation. The page linked in my first comment above shows how to do that comparison correctly. – EdM Aug 09 '20 at 14:23