visualizing confidence intervals in longitudinal models

Question

I have a longitudinal dataset with continuous variables for 6 different time points nested within each ID. I prepared a longitudinal Poisson model with ID as a random effect for the intercepts, which in itself is working fine. In preparing the figures for the report, I prepared a figure with the predicted values for each of the 6 timepoints and the confidence intervals derived from the model. Then I grew uncertain and started looking in the literature. I could not really decipher what other people had done for their confidence intervals since it was not shared in the methods section or figure legend. Looking at some of these figures I am left wondering whether some have just reported either means or medians with 95%CI or IQR. I guess median/IQR could be defended, but if means and CI´s are reported, wouldn't the confidence interval be biased due to within-subject correlation?

So I guess my question is if I should use my calculated confidence intervals from the model (with ID as a random intercept) or if there is another universally accepted way of doing this that I have missed?

Thanks for the input regarding confidence and prediction intervals. What I am looking for is the most appropriate way of demonstrating the uncertainty of our measurements over time in three groups - see figure - whether I should include the fact that the measurements are correlated over time or not.

Left is Poisson based fitted population level values and to the right is means and confidence intervals for each time point disregarding the within-subject correlation

Answer 1

The question is whether you want to display "the uncertainty of our measurements over time" (essentially the raw data) or the uncertainty of the model estimates over time.

If all you have is a simple Poisson model of counts versus time and random intercepts for individuals, and you have complete data, then it might make sense to take the original data and show means with standard errors. Boxplots can be even more useful for illustrating the variability among measurements. Then your model evaluates whether any differences apparent in the raw data are statistically reliable.

Particularly if there are other covariates in your model, however, plots of raw data alone can be misleading. Then you should report means estimated from the model for each condition; showing 95% confidence intervals (CI) for those estimates is frequent practice. Those would be determined from the model coefficients and the variance-covariance matrix of the estimates. For your data, a logarithmic y-axis might make sense. Make it clear to your audience precisely what you are displaying.

A few warnings.

First, as comments indicate, a CI for a model estimate is not the same as a prediction interval. Use of the phrase "prediction interval" isn't always consistent, but it typically includes both the error in the model estimate and the additional error in sampling: for example the Poisson variance for observations here, or the residual variance in an ordinary least squares model.

Second, it's tempting to equate overlap of CI with lack of significance. That's incorrect. This page explains the relationship for simple t-tests, but the principles hold in general: non-overlap of CI is too stringent a test for "significance." The emmeans package can provide a visual display for which non-overlap of arrows is approximately related to significance of differences among model estimates.

Third, @usεr11852 raises an interesting point about displaying pointwise CI (like you seem to have) versus "simultaneous" CI. The latter involve a correction for multiple comparisons so that the CI represent the entire family of comparisons. Provided that you are clear in describing what you show, one could argue either way about what to plot.