The relationship between mean square differences and standard error of measurement: comment on Barchard (2012)

In the discussion of mean square difference (MSD) and standard error of measurement (SEM), Barchard (2012) concluded that the MSD between 2 sets of test scores is greater than 2(SEM)² and SEM underestimates the score difference between 2 tests when the 2 tests are not parallel. This conclusion has limitations for 2 reasons. First, strictly speaking, MSD should not be compared to SEM because they measure different things, have different assumptions, and capture different sources of errors. Second, the related proof and conclusions in Barchard hold only under the assumptions of equal reliabilities, homogeneous variances, and independent measurement errors. To address the limitations, we propose that MSD should be compared to the standard error of measurement of difference scores (SEMx-y) so that the comparison can be extended to the conditions when 2 tests have unequal reliabilities and score variances.