With random assignment to treatments and standard assumptions, either a one-way ANOVA of post-test scores or a two-way, repeated measures ANOVA of pre- and post-test scores provides a legitimate test of the equal treatment effect null hypothesis for latent variable Θ. In an ANCOVA for pre- and post-test variables X and Y which are ordinal measures of η and Θ, respectively, random assignment and standard assumptions ensure the legitimacy of inferences about the equality of treatment effects on latent variable Θ. Sample estimates of adjusted Y treatment means are ordinal estimators of adjusted post-test means on latent variable Θ.

Formulas for the standard error of measurement of three measures of change—simple difference scores, residualized difference scores, and the measure introduced by Tucker, Damarin, and Messick—are derived. Equating these formulas by pairs yields additional explicit formulas which provide a practical guide for determining the relative error of the three measures in any pretest-posttest design. The functional relationship between the standard error of measurement and the correlation between pretest and posttest observed scores remains essentially the same for each of the three measures despite variations in other test parameters (reliability coefficients, standard deviations), even when pretest and posttest errors of measurement are correlated.