## 4.3 Strategy to form and test equate groups

An equate group is a collection of items. Content matter experts may form equate groups by evaluating the contents of items and organising them into groups with similar meaning. The modelling phase takes this set of equate groups (which may be hundreds) as input. Based on the analytic result, we may activate or modify equate groups. It is useful to distinguish between active and passive equate groups. What do we mean by these terms?

• Active equate group: The analysis treats all items within an active equate group as one super-item. The items obtain the same difficulty estimate and are assumed to yield equivalent measurements. As the items in an active equate group may originate from different instruments, such a group acts as a bridge between instruments.
• Passive equate group: Any non-active equate groups are called passive. The model does not restrict the difficulty estimates, i.e., the milestones within a passive equate group will have separate difficulty estimates.

Since active equate groups bridge different instruments, they have an essential role in the analysis. In general, we will set the status of an equate group to active only if we believe that the milestones in that group measure the underlying construct in the same way. Note that this does not necessarily imply that all items need to be identical. In Table 3.2, for example, small differences exist in item formulation. We may nevertheless believe that these are irrelevant and ignore these in practice. Reversely, there is no guarantee that the same milestone will measure child development in the same way in different samples. For example, a milestone like “climb stairs” could be more difficult (and more dangerous) for children who have never seen a staircase.

The data analysis informs decisions to activate equate groups. The following steps implement our strategy for forming and enabling equate groups:

• Content matter experts compare milestones from different instruments and sort similar milestones into equate groups. It may be convenient to select one instrument as a starting point, and map items from others to that (see section 3.2);
• Visualise age profiles of mapped items (see section 3.3). Verify the plausibility of potential matches through similar age profiles. Break up mappings for which age profiles appear implausible. This step requires both statistical and subject matter expertise;
• Fit the model to the data using a subset of equate groups as active. Review the quality of the solution and optimise the quality of the links between tools by editing the equate group structure. The technical details of this model are explained in section 4.4. Refit the model until (1) active equate groups link all cohorts and instruments, (2) active equate groups are distributed over the full-scale range (rather than being centred at one point);
• Assess the quality of equate groups by the infit and outfit (see section 4.6).
• Test performance of the equate groups across subgroups or cohorts by methods designed to detect differential item functioning (see section 4.7).

The application of equate groups is needed to connect different instruments to a universal scale. The technique is especially helpful in the situation where abilities differ across cohorts.

If the cohort abilities are relatively uniform (for example as a result of experimental design) and if the risk of misspecification of the equate groups is high, a good alternative is to rely on the equality of ability distribution. In our application, this was not an option due to the substantial age variation between cohorts.