Simultaneous linear equating for scenarios with optional test versions or across multiple alternative anchors

This paper proposes an extension of linear equating that may be useful in one of two fairly common assessment scenarios. One is where different students have taken different combinations of test forms. This might occur, for example, where students have some free choice over the exam papers they take within a particular qualification. In this scenario we wish to transform scores on the all the various test forms to a common scale so that scores can be used interchangeably. A second scenario is where, perhaps for reasons of security, we use more than one anchor test between test forms with different students exposed to different anchors. In this second scenario, we wish to equate test forms using all of the data from different anchor versions in a coherent manner. The method proposed in this paper to address the above scenarios may be particularly useful in instances where the alternative of using item response theory is problematic.