We have observations of individuals at various points in their life. We are curious if Variable A predicts later values of Variable B (i.e., if being high on A early on means you will be high on B later on). To give a concrete example, does the amount of calcium you drink predict later bone density?

The trick is that everyone was observed at different time points.

Here is some example data to help imagine what I mean.

Person  Age A   B
1       12  2   4
1       15  3   7
1       17  4   8
2       14  1   2
2       20  1   8

I know it will require a mixed effects model. But I'm unsure how to model the relationship between A and B.

My initial thought was to have each value of A predict the next observation of B. The problem is that there are different amounts of time passing between each observation for each person. Another thought would be to dichotomize the data. Perhaps take the median age across all individuals and then compute the averages of A and B for each person below and above that value. That's not ideal because it loses some data by averaging.

Any advice?

  • $\begingroup$ Are variables A and B likely to change with age? For example height, weight and scores on certain kinds of math tests might change greatly with age. Maybe not year of mother's birth, genetic factors and family income. $\endgroup$ – BruceET Jul 17 at 1:51

This fall into the realm of time-varying covariates, and what you describe is the functional form that is indeed of importance. Based on the description of your problem I would say that the cumulative effect of calcium intake could be a candidate functional form. Another aspect that you will also need to consider is whether the time-varying covariate is endogenous or exogenous. For both of these, you can find more information in Section 3.8 of my course notes.

Regarding the fact that the outcome and the time-varying covariate are not measured at the same time points, you can either use a step-function approximation, as in the second example in Section 3.8 (pp. 212-214). Or perhaps better you could fit first a mixed model for calcium intake, and calculate from this model the cumulative effect at the time points the outcome was recorded. In this latter case, it would be important that the mixed model for calcium intake fits the subject-specific trajectories well; for example, you could use splines in the design matrix of the random effects.


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