Data Collection Designs
Two main designs provide the linkage needed for equating. The common-item nonequivalent groups (CINEG) design administers each form to a different population but includes a set of anchor items $A$ common to both forms. The common-person (single-group or counterbalanced) design has the same examinees take both forms.
Linear Equating
Linear equating assumes that Form X and Form Y are linearly related. The equated score on the Y scale is:
\[y = \mu_Y + \frac{\sigma_Y}{\sigma_X}(x - \mu_X)\]Mean and variance are matched; the slope $\sigma_Y/\sigma_X$ corrects for scale differences.
Equipercentile Equating
Equipercentile equating maps $x$ to the score $y$ on Form Y with the same percentile rank:
\[e_Y(x) = F_Y^{-1}[F_X(x)]\]where $F_X$ and $F_Y$ are the cumulative score distributions. Kernel smoothing is applied to the raw distributions before computing the conversion to reduce irregularities.
IRT True-Score Equating
Under IRT, the true score on Form Y equivalent to true score $\tau_X$ on Form X satisfies:
\[\tau_X = \sum_{i \in X} P_i(\hat{\theta}), \qquad \tau_Y = \sum_{j \in Y} P_j(\hat{\theta})\]The same latent $\hat{\theta}$ drives both sums. Common-item concurrent or separate calibration places both item banks on the same metric before the mapping is applied.