Parameter Invariance

Although parameter invariance may at first glance appear to be an arcane mathematical or statistical concept, in practice, it is far from that. Partly because of parameter invariance, statistical model–based measurement using item response theory (IRT) is one of the most popular current methodological frameworks for modeling data from assessments.

As is widely appreciated in statistics courses, the word parameter indicates that parameter invariance refers to population quantities, whose values are to be estimated with data collected within a random sampling design. Parameters can in this [Page 1202]context refer to the set of item parameters (item difficulty, discrimination, and/or guessing) and the set of examinee parameters (the examinee test scores, or theta [θ] scores, implied by the IRT model) that are tied to a particular measurement model. The word invariance indicates that parameter values are identical in separate examinee populations or across separate measurement conditions, which is commonly investigated through estimated parameter values from different calibration samples. What this implies is that parameter invariance is only relevant when comparing groups or measurement conditions. If there is only one population or condition, invariance is not relevant. That is, the matter of parameter invariance addresses the question of whether test scores or item parameters are equally valid for different populations of examinees or different measurement conditions. If parameters are not invariant, the statistical foundation for inferences is not identical across the populations or measurement conditions and, hence, the inferences are not generalizable across those to the same degree. Note, however, that parameter invariance denotes an absolute ideal state that holds only for perfect model fit, and any discussion about whether there are “degrees of invariance” or whether there is “some invariance” is technically inappropriate. And as noted earlier, the question of whether parameter invariance exists in any single population is illogical because parameter invariance requires at least two examinee populations or two measurement conditions for parameter comparisons to be possible and meaningful.