I previously asked how to estimate the latent potential of a runner who ran the 100 metres each day for 200 days. Latent skill was defined as "the latent time it would take the individual to run if they (a) applied maximal effort; and (b) had a reasonably good run for them (i.e., no major problems with the run; but still a typical run)."
Now assume that I estimated latent skill for the 100 metres for each of the 200 days, but that I also had data on the same 200 days but this time on running the 400 metres. Obviously I could repeat whatever process I adopted for the 100 metres to form an estimate of latent skill for the 400 metres at each of the 200 time points. In both cases I would expect the time to complete the runs to generally get faster with practice, but that raw data would vary from day to day.
I want to quantify the degree of consistency of the two curves. I don't really want to quantify the consistency of the observed data.
If it makes a difference, the two methods I were considering using for estimating the effect of time, were nonlinear regression and isotonic regression.
- Thus, what is a good way to quantify and calculate the consistency of the fitted curves for the 100 and 400 metres?
Initial thoughts: I had a few initial thoughts:
- estimate the fitted values for both curves and correlate the fitted values
- Use a parametric model like $\theta_1 \exp(-\theta_2t) + \theta_3 + \epsilon$ ($t$ is an index of day) and then quantify the degree to which constraining $\theta_2$ (the parameter that determines shape) to be equal across the 100 and 400 metres would lead to poorer fit.