On mean-sigma estimators and bias

When two test forms measure the same construct but are independently modelled using item response theory, the two forms' respective metrics cannot be assumed to be equivalent. Thus, before comparing parameter estimates across forms, a linear transformation must be applied to at least one form's scale. The mean-sigma method is a well-known procedure for estimating this adjustment when a common set of items appears on both forms. In this paper, I show both analytically and empirically (through a small simulation study) that the mean-sigma estimators of the transformation constants are biased. While this systematic error was modest relative to random error under the conditions studied here, it is nevertheless intrinsic and its magnitude is conditional on extrinsic design features that include the number of anchor items and the quality of their difficulty estimates.