Solved – Longitudinal data: baseline effect versus random intercept

The variable $Y$ is measured at time points $t_1$, $ldots$, $t_9$ for each of five objects. Also available for each object is the value of $Y$ at time $t_0 = 0$ (baseline). Thus, the sample size is $n = 50$. I would like to fit a regression line of $Y$ versus time. Further, it is important for me to include the baseline measure in the model.

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Is it okay to include a random slope but no random intercept? Thus, my model would be: $$y_{ij} = (beta_0 + beta_1 t_{0i}) + (beta_2 + gamma_i) t_{ij} + epsilon_{ij}$$
with $beta_0$ an overall intercept, $beta_1$ the baseline effect, $beta_2$ an overall slope, and $gamma_i$ the random deviation from the overall slope for object $i$.


My concern is that $beta_1 t_{0i}$ already results in an object-specific intercept and therefore I do not see the point to further include a random intercept in the model.

I assume $j=1,…,9$.

Yes, you can treat $beta_1t_{0i}$ as object-specific intercept. The difference from random intercept is that $beta_1t_{0i}$ is fixed. It is kind of like the difference between fixed and random effects in econometrics. All objects share the same $beta_1$, which is then weighted by the baseline value of each object. If the baseline measurement is important, it is fine to do so.

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