Normal gains: estimators of learning rates in pretest-posttest settings

Nonlinear transformations of pretest and posttest scores are widely used in educational and psychological measurement to estimate group-level change, yet the statistical behavior of estimators derived from such transformations under measurement error remains poorly understood. We examine this problem in the context of normalized gains (ngains), a ratio-based transformation used to estimate group-level "learning rates" in pretest-posttest designs. Two standard estimation methods - the average ngain of the group ( $\overline{ng}$ ) and the ngain of the average learner ( $\widehat{ng}$ ) - routinely produce different results. A prior study established a mathematical relationship between this discrepancy and the pretest-ngain correlation, interpreting it as a characterization of the learning process. The pretest-ngain correlation has itself sparked debate: researchers have argued it indicates that ngains favor high-pretest populations, undermining their validity as a measure of student growth.

Using Classical Test Theory along with a rencently proposed statistical framework to analize ngains, we show that measurement error is one common cause behind both phenomena.

When measurement errors are absent, both $\overline{ng}$ and $\widehat{ng}$ are unbiased and any discrepancy between them reflects only sampling variation. When measurement errors are present, $\overline{ng}$ acquires a systematic negative bias - consistently underestimating the true learning rate - while $\widehat{ng}$ remains asymptotically unbiased. We further prove that measurement errors induce a spurious negative correlation between pretest scores and ngains, even when prior knowledge and learning capacity are statistically independent.

Such correlations may reflect insufficient instrument reliability rather than any inherent flaw in the transformation. These findings generalize beyond ngains: any nonlinear derived score computed from fallible instruments is susceptible to the same bias structure, and the analytical approach developed here offers a methodological template applicable to other ratio-based metrics in educational and psychological measurement. For applied researchers, we recommend computing both estimators and treating a large discrepancy as a warning sign, reporting instrument reliability alongside ngain estimates, and interpreting pretest-ngain correlations conditionally on reliability.