# Analyzing impact of baseline value on change score using lmm

I'm using a big clinical data set with a the evolution of a continuous lab value over time being the dependent variable and I am interested in the impact of the baseline of this value on this evolution. I have come across some similar qustions in CrossValidated and have learned that baseline cannot be incorporated as a covariate (quite locgical). Instead, it is possible to analyze the correlations between random intercepts (which actually represent the baseline) and slopes to answer this question. In this case, at least as I understand it, the subjects are treated as random effetcs. However, I wonder whether it is also possible (and sensible) to incorporate the baseline value as a random effect (instead of or in addition to subject) and either change score or the measured values directy as dependent variable. Time would be one of the covariates. Does that make any sense?

You seem to be on the right track and thinking of the right things. Regressing change scores on baseline is a very bad idea as you have already learned. For the same reasons and more, don't include baseline as a random effect. That doesn't make much sense at all. Also, change scores themselves are not a good way to analyse change, as the estimand in such a model is often not the estimand that you want. A much better approach is what you described in the first part of your question: the measurements are the dependent variable, along with time. Time should be centered, and you should fit random intercepts and slopes as you suggest.

For some background reading on this, see:

Senn, S., 2006. Change from baseline and analysis of covariance revisited. Statistics in medicine, 25(24), pp.4334-4344. https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.2682

Tennant, P.W., Arnold, K.F., Ellison, G.T. and Gilthorpe, M.S., 2019. Analyses of'change scores' do not estimate causal effects in observational data. arXiv preprint arXiv:1907.02764 https://arxiv.org/pdf/1907.02764.pdf

Tu, Y.K. and Gilthorpe, M.S., 2007. Revisiting the relation between change and initial value: a review and evaluation. Statistics in medicine, 26(2), pp.443-457. https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.2538

Edit to address the question of why time should be centered. This is related to the correlation $$r$$ between the random slopes and intercepts, and is explained in the paper by Tu & Gilthorpe above. Here is the relevant quote:

It should be noted that different parameterizations of Time will yield different results [43]. For instance, when Time is coded as 0 (initial) and 1 (post-treatment), testing the correlation between intercept and slope is equivalent to testing $$r_{x−y, x}$$ , because the intercept variance is the variance of x and the slope variance is the variance of x − y; when Time is centred, such as − 0:5 (initial) and 0.5 (post-treatment), testing the correlation between intercept and slope is equivalent to Oldham’s method because the intercept variance is the variance of (x + y)=2 and the slope variance is the variance of x − y; when Time is coded as − 1 (initial) and 0 (post-treatment), testing the correlation between intercept and slope is equivalent to testing $$r_{x−y, y}$$ because the intercept variance is the variance of y and the slope variance is the variance of x − y. An advantage of multilevel modelling over other approaches is that this method can be applied to more than two measurement occasions; details can be found in our previous study [43].

Reference [43] from the paper is, if interested:

Blance A, Tu Y-K, Gilthorpe MS. A multilevel modelling solution to mathematical coupling. Statistical Methods in Medical Research 2005; 14:553–565.

Also, as mentioned in the comments, you intend to include time in an interaction with treatment. It is usually a good idea to centre time in that case because the main effect of treatment is conditional on time being zero, so if time is centred zero then become the mid point, and this usually helps with interpretation.

• Very helpful, thank you. Could you please add why time should be centered. One more question: since I am interested in the impact of covariates on the change score over time I have to include the interaction terms in the model. Some covariates (e.g., treatment) are influenced by the baseline value (simply because more serious deterioration of that value will prompt physicians to choose a more effective treatment). Thus, I get a significant positve estimate for treatment while the treatment*time term gives me a negative estimate (or vice versa). Is there a way of reducing complexity here? – user311327 Feb 15 at 18:33
• @user311327 I'm not sure what you mean about reducing complexity, but I've tried to address the point about centering time with the edit in my answer. – Robert Long Feb 15 at 20:01
• Thanks again. I completely unterstand that entering baseline as random effect won't give me reliable information on the impact of baseline on response variable. But would it help me to adjust for the impact of the fixed effects? – user311327 Feb 17 at 11:53
• You're welcome. No, it won't help adjust for the fixed effects. – Robert Long Feb 17 at 14:37
• So to sum up: there is not really a straight forward way to analyze the impact of baseline of a lab value on its timely evolution (= change score) and at the same time to analyze the effect of the fixed effects on the same change score. I think I am stuck, then. Anyways, thanks a lot for your help. – user311327 Feb 17 at 16:57