Like a lot of fathers, my dad is fond of ugly ties, bad jokes, and meant-to-be-funny sayings. When stuck behind a slow car, “Drive it or milk it” is an odds-on favorite to come out of his mouth. If you are talking to him and say, “I’m not sure but I guess…” about a topic, his response will likely be, “You know what happens when you assume, don’t you?” referring to the adage, “When you assume, you make an ASS of U and ME.” It’s lucky for him (and probably the world) that my father didn’t go into statistics. The most familiar statistical techniques, from regression, correlation, and ANOVA to item response theory (IRT), are based on assumptions that are needed to make the model “work.” For instance, in correlation there’s an assumption that the observed data for a given variable has a certain shape when looked at altogether. In IRT, it’s commonly stated that there is an assumption that the items (which together make a scale/assessment/ metric/ test) all only measure one common thing, where the “thing” is the construct/concept of interest, often called a latent (meaning it can’t be directly seen) variable. In articles and textbooks, this is often called the assumption of unidimensionality and, SURPRISE, it’s not actually an assumption that is required for IRT models to work.

While a lot of the most commonly used IRT models do only allow for a single latent variable (that is, they are unidimensional models), this is not something fundamental to all IRT models. In fact, for a long time the theory existed for IRT models with more than one dimension (termed multidimensional IRT or MIRT models), but there were limitations to what computers were able to accomplish in fitting such models, due to available memory, processing speed, etc. Because of this computational limitation, it was not really feasible to use an IRT model with more than just a couple dimensions for even hard-core nerds who wrote their own IRT programs and, for the average joe, the commercial IRT software available only allowed for one dimension. Stemming from these facts is the commonly misstated IRT assumption of unidimensionality.

To include all types of IRT models (uni- and multi-dimensional), the assumption is more correctly termed as one of “adequate dimensionality.” So if you have a set of items that measures 3 different things (general math ability, specific geometry ability, and reading ability), then your IRT model needs to have 3 dimensions to be able to account for all inter-relationships among the items. This assumption of adequate dimensionality is closely tied to another IRT assumption, that of local independence. The idea behind local independence is that once you have statistically accounted for the dimensions included in the model, the unaccounted-for bits of the items should be unrelated to each other. For example, if you have a set of items meant to measure depression and fit them with a unidimensional model, the “depression” information in the items is accounted for by model. After you have accounted for depression, the remaining information in the items (information not accounted for by depression) should be unrelated. If, for instance, your item set had somatic items (changes in sleep patterns or eating) and emotional items (feeling blue, suicidal ideation), it may be that after accounting for depression the items related to somatic topics are still related to each other (by an unaccounted for “somatic” dimension) and the emotional items are still related to each other (by an unaccounted for “emotional” dimension). In this case, when the depression items are fit with a unidimensional model the unaccounted-for somatic and emotional aspects of the items will lead to a violation of the assumption of local independence. If you used a 3-dimensional MIRT model (depression, somatic symptoms, emotional symptoms) for the same data, you would likely meet both the assumption of local independence and adequate dimensionality because your model is a more accurate depiction of what is actually found in the data and is able to account for more of the intricacies in the relationships among the items.

Point being, IRT is not limited to single-dimensional models and, in a lot of cases, a MIRT model will provide a more accurate representation of the data. That MIRT models exist and are now more feasible for estimation (such as in commercial programs like flexMIRT^{®}) is a good thing, given the complexities of the concepts that exist when we try to describe human thoughts and behaviors with equations.