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I have data from several different measurements of physical performance, each done on the same individual, and I'm looking for ones that correlate with each other. A typical question might be, "does weight-adjusted lean muscle mass correlate with grip strength" ...and so on for each pair of measurements.

The problem is that some types of measurements were done multiple consecutive times on the same individual. For example, balance might be measured in four consecutive trials per individual, while grip strength might be measured every day for one week. To see how much balance correlates with grip strength, I assume that I need to collapse both series into a single score for each individual. I could take a mean, or I could fit a linear model (i.e. lmList(grip ~ day|subject) and lmList(balance ~ trial|subject)) and then separately calculate the correlations of the respective slopes and intercepts. But in either case, I would also have variance estimates that I shouldn't just throw away.

From my searches so far it seems like this might be either an error-in-variance problem or a structural equation model problem. I'm completely new to both approaches, and would be grateful if someone can point me to an introduction or tutorial, hopefully with some example code, for going from a something like lm(grip ~ balance) to a model that takes into account the variances in both the grip and balance scores of each individual.

Or in symbolic terms, how to go from $$y_i = \beta_0 + \beta_1 x_i+\epsilon$$ to $$\hat y_i = \beta_0 + \beta_1 (\hat x_i+ \hat\sigma_{x_i}) + \hat\sigma_{y_i} + \epsilon$$ Thanks.

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  • $\begingroup$ I'm starting to think that if I do ignore the standard errors for each individual's slope or intercept and just use the raw slope or intercept estimate as a predictor variable in a new model, there might be bias but it will be conservative bias. After all, why should the fact that we chose to be diligent and collect several replicates for each individual and record the order they were collected result in a less reliable model than if we had collected one measurement per individual like everyone else does? $\endgroup$
    – f1r3br4nd
    Oct 4, 2012 at 19:00

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Your situation sounds very much like the situation I described with the last two questions I wrote here. I think those questions and the responses will help if you haven't already read them. If there are two variables that are linearly related and each is observed with measurement error then you have an error-in-variables problem. OLS is not appropriate because it considers one of the variables to be fixed and known.

As I mentioned in my question the "interference" or "correlation" in errors for the repeated samples is bogus because they are applied to the same points but the error in each of the variables is still independent. This means that the original error-in-variables method is valid even with repeated measurement.

Bill Huber indicated that a more standard approach to showing equivalence of two measurement techniques is something different that is referred to as inverse regression in books such as Draper and Smith. That method is apparently different from error-in-variables/Deming regression and it doesvariance component analysis that Deming regression does not do. I plan to look into it but have not yet. Your problem may not be the same as mine and Deming regression may be fine for you.

You will note that the R package mrc was recommended to me as tool that does Deming regression. Bill Hiiber mentioned it to me either in comment or chat.

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