Mathematical coupling: a simpler approach

Jon Rasbash and Harvey Goldstein

Institute of Education, 20 Bedford Way, London, WC1H 0AL, UK. E-mail: j.rabash{at}ioe.ac.uk

Our view is that neither approach1,2 really clarifies the key issues for the journal readers. We have some specific queries and a general suggestion to make to the authors.

Gunnell et al.1 base their re-analysis on the null hypothesis assumption, but it is not clear why this should hold—as Gilthorpe and Tu2 say. If it does not, then the results from the reanalysis would also seem to be biased.

We also feel that while Gilthorpe and Tu make some valid points and is an adequate mathematical treatment, their exposition does little to clarify interpretations. If you want to get the unbiased covariance term between the mean and the difference then the simplest way is just to run it as a bivariate response model: Gilthorpe and Tu complicate this unnecessarily.

Our general view is that there is a much simpler and easier to understand approach to this. If you regress (possibly non-linearly) self-reported height (y) on measured height (x), then you can determine the exact form of the relationship. If your null hypothesis is that y = x + e, where e is simple random error, then if this is true you will finish up with a (straight) line with coefficient = 1 (but note correlation is <1). Departures from this are then interpreted directly as deviations from the null hypothesis. For example, if the empirical fitted line lies below the null hypothesis line for short people then there is a negative bias for these people, etc. You might want to fit a non-linear relationship or even some kind of smoothed (spline) model to get the relationship right. In addition we suggest that the variation about the line is modelled as a function of x since one might expect the variability of self-reported height to be a function of true height—software such as M1wiN will allow you to do this.


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1 Gunnell D, Berney L, Holland P et al. Does the mispreporting of adult body size depend upon an individual's height and weight? Methodological debate. Int J Epidemiol 2004; 33:1398–99.[Free Full Text]

2 Gilthorpe MS, Tu Y-K. Mathematical coupling: a multilevel approach. Int J Epidemiol 2004; 33:1399–400.[Free Full Text]