IRT Test Equating in Complex Linkage Plans

Abstract

Linkage plans can be rather complex, including many forms, several links, and the connection of forms through different paths. This article studies item response theory equating methods for complex linkage plans when the common-item nonequivalent group design is used. An efficient way to average equating coefficients that link the same two forms through different paths will be presented and the asymptotic standard errors of indirect and average equating coefficients are derived. The methodology is illustrated using simulations studies and a real data example.

Appendices

Appendix A. Partial Derivatives of the Equating Coefficients with Respect to the Item Parameters

A.1 Indirect Coefficients

Irrespective of the method used to obtain direct equating coefficients, the partial derivatives of indirect equating coefficients used in Equation (5) are as follows:

$$\frac{\partial A_{0, \dots, l}}{\partial a_{gj}}= A_{0,\dots,g-1} \frac{\partial A_{g-1, g}}{\partial a_{gj}} A_{g, \dots, l} + A_{0,\dots,g} \frac{\partial A_{g, g+1}}{\partial a_{gj}} A_{g+1, \dots, l}. $$

Note that $\frac{\partial A_{{g-1, g}}}{\partial a_{gj}}$ and $\frac{\partial A_{{g, g+1}}}{\partial a_{gj}}$ are both different from zero only if item j of form g is present in both form g−1 and in form g+1. Similarly,

$$\frac{\partial A_{0, \dots, l}}{\partial b_{gj}}= A_{0,\dots,g-1} \frac{\partial A_{g-1, g}}{\partial b_{gj}} A_{g, \dots, l} + A_{0,\dots,g} \frac{\partial A_{g, g+1}}{\partial b_{gj}} A_{g+1, \dots, l} , $$

while

$$\frac{\partial B_{0, \dots, l}}{\partial a_{gj}}= \sum_{h=1}^l \biggl( \frac{\partial B_{h-1, h}}{\partial a_{gj}} A_{h, \dots, l} + B_{h-1, h} \frac{\partial A_{h, \dots, l}}{\partial a_{gj}} \biggr) , $$

where $\frac{\partial B_{h-1, h}}{\partial a_{gj}}$ is equal to zero if g≠h−1 and g≠h and $\frac{\partial A_{h, \dots, l}}{\partial a_{gj}}$ is equal to zero if g<h. Finally,

$$\frac{\partial B_{0, \dots, l}}{\partial b_{gj}}= \sum_{h=1}^l \biggl( \frac{\partial B_{h-1, h}}{\partial b_{gj}} A_{h, \dots, l} + B_{h-1, h} \frac{\partial A_{h, \dots, l}}{\partial b_{gj}} \biggr). $$

A.2 Bisector Equating Coefficients

The partial derivatives of the bisector coefficients with respect to direct and indirect equating coefficients relative to one of the paths that link to forms used in Equation (7) are as follows:

Appendix B. Equating Coefficients for Common-Item Equating to a Calibrated Pool

It is assumed that a _gj =1 for all g and j. Then the equating coefficient for converting the parameters of Form 2 on the scale of Form 1 is given by

$$B_{21}=\frac{1}{n_{12}}\sum_{j\in I_{12}}b_{1j}-\frac{1}{n_{12}}\sum _{j\in I_{12}}b_{2j}, $$

where I ₁₂ is the set containing the items in common between forms 1 and 2. I ₁₃ and I ₂₃ are defined similarly. We denote by I _p3=I ₁₃∪I ₂₃ the set of items in common between the pool and Form 3, by n _p3=n ₁₃+n ₂₃ (assuming that I ₁₃∩I ₂₃=⊘) the cardinality of I _p3 and by b _pj the j-th difficulty item parameter of the pool. The equating coefficient for converting the parameters of Form 3 on the scale of the pool is given by