Errors, Variances and Correlations

We are so accustomed to thinking in terms of raw scores, with their apparent pin-point precision, that we find the concept of measurement error at first abhorrent and later perplexing. We understand, finally, that measurement error always exists, but we don't know how to handle it when reporting our results.

Measurement error always affects our findings. When reporting our results we need to decide whether the result is supposed to describe the local details of this data collection, or to infer some more general "reality". Local results, such as an individual's performance on a test and the test's reliability, must contain their measurement error. General inferences, however, such as the math level of third graders, are intended to be estimated without bias from measurement error. To ascertain what, if anything, of global import we have discovered, we need to deduct the effect of measurement error from our final estimates.

Consider person n with measure bn of standard error sn. The more observations we make, the more we know about this person, and the smaller sn will be. What about the dispersion of a sample of N people? When measurement error is uncorrelated with what is measured, the variance of the sample of observed measures contains the sample variance and also the measures' error variance. But it is only the sample variance that estimates the population sampled.

The estimation for this corrected sample variance is:

s^2 = sum( bn^2 - b.^2 - sn^2 ) / (N-1)

where b. = sum(bn)/N

Correlation coefficients are also affected. Measurement error reduces the maximum observable correlation. Adjusting observed correlation coefficients for measurement error isolates how much sample variance is left unexplained.

Consider the typical regression analysis, in which dependent measures are mistaken for points without error. Let R^2 be the observed multiple squared correlation (proportion of explained variance), E be the observed RMS residual from the regression, and S be the observed standard deviation of the measures. Then E contains both unexplained variance and measurement error, and R^2 has been reduced accordingly. R^2 is defined as

R^2 = (S^2 -E^2) / S^2

If r^2 is the squared multiple correlation corrected for measurement error in the dependent variable and e is the RMS measurement error, then

r^2 = (S^2 - E^2) / (S^2 - e^2)

= R^2 / [ 1 - (e/E)^2 (1 - R^2)]

e<=E; r>=R; when e=0, then r=R; when e=E, then r=1.

It is this larger correlation r, and not R, which tells us how successfully our independent variables have explained the variance of the dependent measure.

Errors, Variances and Correlations, B Wright … Rasch Measurement Transactions, 1991, 5:2 p. 147

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Rating Scale Analysis, Wright & Masters	Introduction to Rasch Measurement, E. Smith & R. Smith	Introduction to Many-Facet Rasch Measurement, Thomas Eckes	Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr.	Statistical Analyses for Language Testers, Rita Green
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