How to transform time-varying covariate measure of response time in a multi-level model of longitudinal data? I am trying to fit a multi-level model to some longitudinal data that I have. As an example, let's pretend participants had to make 10 basketball free throws, and I measured how long it took them to make each one (in seconds). In this case, the time variable is severely positively skewed because although there is a minimum amount of time it takes to make 10 throws, some participants could take much longer than others.
When trying to fit the model with the original data, I ran into all sorts of convergence errors using lmer in R. After transforming the time variable (in seconds) by applying a fourth-root transformation, the model fits fine, and the coefficients are interpretable.
I'm just wondering if (a) this is a valid thing to do, and (b) what I should be aware of/careful about when doing this.
edit: If someone could direct me to further reading on why the non-transformed model might fail to converge, that would be great!
 A: I don't think the multilevel nature of the data changes the issue too much.
You are basically just dealing with the general problematic issues associated with response time data (i.e., outliers and skewed distribution). I guess because you are not aggregating in anyway, these issues are amplified.
A few  suggestions:


*

*I think a log or a related transformation of your response time data is more likely to capture any  effect of interest on performance in the data. I think as long as you transform it to something vaguely normally distributed, you'll be able to find the effect if it is there.

*Even after log transforming you may still get outliers. Have a think about the data generating process and how that effect is meant to influence the dependent variable. For example, at free throws, a certain time may imply rushing, other times may be a sweet spot, other times may imply over thinking, and furthermore really long times might imply that some external factor has delayed the shot. In particular, you may want to handle extremely long times differently (e.g., transform times longer than $c$ (e.g., 20 seconds) to $c$. You can often decide on a appropriate time by looking at a histogram and looking for discontinuities combined with domain specific knowledge.

*Finally, given the phenomena you may want to include a linear and a quadratic effect if you are expecting an "optimal response time" type effect, where slower and longer times result in a drop in performance.


While it is focused on cognitive psychology data, I think you may find useful this question on cogsci on transforming reaction time data. In particular see Whelan (2010).
Whelan, R. (2010). Effective analysis of reaction time data. The Psychological Record, 58(3), 9.
