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.

My question:

  • 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.
  • $\begingroup$ When are two curves "consistent," Jeromy? (Without a clear definition, many different answers are possible.) Your second initial thought is suggestive, but not dispositive. For example, unless you choose just the right way to express the results--should they be times, speeds (miles per hour), or inverse speeds (hours per mile), for instance?--then you might fail to identify and quantify a "consistency" that is really present. In particular, I would expect $\theta_2$ to be smaller for the shorter race. $\endgroup$ – whuber Sep 27 '10 at 16:45
  • $\begingroup$ @whuber Thanks for the point about $\theta_2$. The analysis is motivated by a theoretical and qualitative interest in consistency. Thus, my question is asking "what is a good way to quantify consistency". I.e., what is a good definition of consistency? $\endgroup$ – Jeromy Anglim Sep 28 '10 at 0:51
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    $\begingroup$ That's not a statistical question, LOL! But (to stave off possible objections) I agree that its answer can be usefully informed by statistical thinking. "Consistency" depends on what you are studying and what use you will make of a decision that two response curves are "consistent" or not. In some applications it would be enough that they are both increasing or decreasing; in others it would amount to a test of equality of all parameters. Without more information, one can only guess where along this spectrum your needs fall. $\endgroup$ – whuber Sep 28 '10 at 15:06
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    $\begingroup$ @whuber Thanks. I see the translation of a theoretical question into a statistical question as one of the most important skills that a data analyst can acquire. In most areas of statistics, there are multiple ways of making the translation, with a body of knowledge existing on when and why you would apply one approach over another. Thus, I'm interested in knowing whether any standard methods exist for quantifying consistency of two fitted curves and when and why you would apply one approach over another. I'll have a think about how I can edit the question to make my specific aims clearer. $\endgroup$ – Jeromy Anglim Sep 29 '10 at 4:14

My understanding is that your sample size is 200 observations for each running length (i.e. the sample size is such that doubling the number of observations has a non trivial effect on accuracy).

Denote $y_1$ the 100 meters series ($y_2$ the 400 meters series). I suggest pooling both series unto $y$ and adding a dummy $l_i=I(i\in y_1)$ (where $I()$ is the indicator function) and performing an isotonic regression of $\log(y)$ on $date$ and the dummies ($l$ as well as the $z$ controling for injuries, running style ect....).

Because of the $\log()$ transformation on the left hand side, the value of the coefficient of $l$ can be understood as the average percentage increase in running time, al else equal, due to running a 4 time longer length. The doubling of sample size will improve accuracy since you have in effect only added a single parameter to the model.

Also, re-reading my previous answer, i noticed i had linked to the wrong article. By the same author, the correct reference is 'Bayesian Isotonic Regression and Trend Analysis', B. Neelon, D. B. Dunson (the previous link was to the version of there approach adapted to the case of binary dependant variable). A non gated version of the article is here

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    $\begingroup$ Thanks. The idea of modelling the two curves together seems like a good one. However, I have theoretical reasons for wanting to quantify the degree to which the shape of the curves for the 100 and 400 metres are the same. Does the approach you mention do this? $\endgroup$ – Jeromy Anglim Sep 27 '10 at 9:28
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    $\begingroup$ @Jeromy:> I'm afraid not. On the other hand, i would pay attention to the issue of sample size and model complexity as 200 observations ain't a lot. $\endgroup$ – user603 Sep 27 '10 at 11:42

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