Model the joint density of the three dimensions
The advice. “Suggested simulation set up: model the joint density of the three dimensions, and understand the density of them, determine what are the correlations between dimensions. Currently assuming linear correlation, so if you consider three dimensions, you will be able to see what is actually there. Go from joint density to the marginals to placing the cuts in the marginals; you would want them to be faithful to the joint density as you want to know the relationship from the three variables from the joint and not just assume.”
Why. The most technically substantive recommendation of the meeting.
The simulation parameterises the relationship among the three subtests as a single correlation value — 0.65, 0.70, or 0.75 — which presumes the dependence structure is linear and identical across the range. The empirical distributions in CT-PO vs SBAC Simulation Results Deck are visibly non-normal and multimodal; a correlation coefficient cannot describe dependence in that setting. Two variables can share a correlation of 0.7 and have entirely different joint behaviour in the tails, which is exactly where classification cuts fall.
The proposed sequence — estimate the joint, derive the marginals from it, place the cuts in marginals that remain faithful to the joint — keeps the dependence structure in the machinery rather than assuming it away at the start.
Companion. Establish scale, then cuts, then categorise from theta gives the operational ordering; Review item parameters and total-score correlations before Batch 3 is the prerequisite check.
Since the meeting. copula-approach-to-subtest-indicators argues this recommendation describes a copula in everything but name — Sklar’s theorem factors a joint distribution into its marginals and a copula carrying all the dependence, which is precisely “go from joint density to the marginals… faithful to the joint density.” It sets out a bounded way to act on it using data already collected.
Status. Open. Directly connected to Support Ji Zeng’s work through direct collaboration.
Related. M-STEP Subscore Reporting