How should I include age in this mixed-effects model?

I've got repeated measures of hearing thresholds for a set of patients who each have a different inner ear measurement. This data is unbalanced, there are a varied number of measurements at varied intervals for each subject. My goal is to build a random-intercept random-slope (longitudinal) model that describes the impact of inner ear size (an unchanging value) and change in repeated measure over time, and I'm using lme4.

At the moment, I'm having difficulty choosing between two models. These models differ in how age is accounted for in the model.

The first model (M1) is written in R as:

M1 <- lmer(repeatedMeasure ~ innerEarSize + ageAtMeasurement + (1 + ageAtMeasurement | subjectName), data = data)

And the second model has age split into age at first repeated measure test, versus time since first repeated measure test.

To be clear, for each subject:

ageAtFirstMeasure = ageAtMeasurement - min(ageAtMeasurement)

and

timeSinceFirstMeasure = ageAtMeasurement - ageAtFirstMeasure

so that

ageAtMeasurement = ageAtFirstMeasure + timeSinceFirstMeasure

I've written the second model in R as:

M2 <- lmer(repeatedMeasure ~ innerEarSize + ageAtFirstMeasure + (1 + yearsSinceFirstMeasure | subjectName), data = data)

Is this a reasonable specification for a longitudinal repeated-measures problem?

The second model accounts for ageAtFirstMeasure as a fixed effect, and restricts the longitudinal component to the number of years after the first test. This teases out the effect of age on the repeated measure, while still accounting for increasing age.

• This is an interesting question that I hope gets an answer some day! – theforestecologist May 2 '18 at 23:38
• In retrospect, this is a question of whether to (a) use age at measurement as both a fixed and random effect as opposed to (b) centering the random effect and reducing the fixed effect of age at first measurement to its minimum value. I think (b) is the better approach because it avoids repeating the same information (a time covariate) in the model. Considering the interpretation of either model, I also think it makes more sense to include a random effect that reflects an index event and a fixed effect that reflects an initial event. – Mustafa Ascha Oct 16 '18 at 2:44