# Regression with 3 measurement points (in R)

I have a regression in which I try to understand how much variance of the metric dependent variable each of the regressors explains. I use the package R relaimpo (Grömping, 2006) for that purpose, that allocates $R^2$ shares to each regressor using the LMG metric. The hypothesis is that the regressors differ drastically in their $R^2$ contribution.

If I had a cross-sectional sample of $N=6000$, things would be dandy. Unfortunately, I only have $N=2000$, but 3 measurement points (weeks 0, 6, 12). Both the dependent variable and regressors are time-varying, that is, they are assessed at each measurement point.

Currently I run the analyses separately for each measurement point and find large differences between relative importance estimates between regressors (0% to 25%), and I find that the ranking of regressors and the explained variance is surprisingly stable over time (at least from a qualitative perspective, plotting the explained variance of each of the 14 regressors for each measurement point).

For a scientific paper, there are two reasons why I want to do this in one regression instead of 3. (1) 3 different analyses take up a lot of space in the paper and obfuscate the main message a bit (regressors differ drastically in their relative importance). (2) $N=2000$ isn't all that much when disentangling the relative contributions of 14 correlated regressors (the CIs are rather large).

Therefore, I wondered whether there are ways to "pool" all subjects into one regression that would not lead to an outcry of anybody with some statistical background ("you severely violated the assumption of statistical independence!!"). In the best case I would simply have one regression that covers all 3 time points.

The method has to be a regression (ie., not using NLE or LME packages), because that is what the relaimpo package uses as baseline model to then calculate unique $R^2$ contributions of regressors.

What are my options?

You cannot turn a sample of 2000 into 6000, no matter how many times you measure it. So fundamentally I think you cannot do what you are trying to do within the constraints you have set out. Worse, I think even fitting three separate regressions and reporting them separately is dangerous and potentially misleading too, if it is thought these analyses are somehow independent checks on eachother.

I see two options:

1. analyse the data in one go with a mixed effects model with subject as a random effect with 2000 values and measurement time a fixed effect with 3 values, allowing for interactions between measurement time and your 14 explanatory variables. Then you will need to write your own code to "decompose" the $R^2$ (although this decomposition is not something I'd recommend in any event).
2. combine the data in some other way before conducting a single regression eg by taking the average measurement (out of the three observations) for each of your 2000 points. Then you could continue to use the relaimpo package.

The second approach would only make sense if you expect the system to be fundamentally stable, and all you are trying to do in your multiple measurements is reduce measurement error. Some careful thinking about all the sources of randomness would be necessary before you did any inference from this.

The first approach would be my preferred one, but I would recommend at least looking for some alternative way of exploring the relative contributions of explanatory variables other than decomposing the $R^2$ (not sure what - the problem might be intractable).

In either case, I have all sorts of philosophical problems with the idea of explanatory variables being allocated a share of $R^2$ in either case (it would seem to be highly vulnerable to slightly different selection of explanatory variables when there is any collinearity), but it seems to be an approach that is used at least in some quarters.

• Thank you. (1) I wouldn't know how to write an R script for that. Option (2) would also take into account all 3 times, in a less proper fashion than option (1). I'd be happy about recommendations when it comes to alternative methods of R^2 decomposition - my question is whether 14 symptoms of the disorder X are differentially associated with impairment. Obtaining individual R^2 contribution sounded like the way to go (if one has 20% variance explained and the other 1% and the CIs are 2%, that's pretty substantial). Jan 30 '13 at 11:21
• In addition, would you know of one test giving one p-value that can test the assumption that symptoms differ from each other in variance explained? I have 14 values (between 0.1% and 19.8%) of explained variance, but don't have them per subject (so I cannot use tests like Hotelling's T). Jan 30 '13 at 16:03