Proficiency order invariance of MLE, MAP, EAP, and WLE in item response theory

When high-stakes decisions depend on test scores, it is natural to ask whether examinees' relative standing is determined by the response evidence or by the scoring rule. This paper characterizes when four IRT proficiency estimators-maximum likelihood, maximum a posteriori, expected a posteriori, and Warm's weighted likelihood estimator-can change the rank ordering of examinees under the 1PL (Rasch), 2PL, and 3PL models with fixed item parameters. We show that, under the 1PL and 2PL, and restricting attention to response patterns for which the relevant estimators are well-defined, these four estimators agree on every strict comparison implied by the model's evidence order. The reason is structural: with fixed item parameters, the person likelihood in both the 1PL and 2PL depends on the response pattern through a single evidence statistic, yielding a monotone likelihood-ratio order in that statistic. Under the stated conditions, MLE, EAP, MAP, and WLE all increase in this same statistic. By contrast, the 3PL does not retain this same single-statistic structure in the examinee latent trait ( $\theta$ ), so global rank invariance is not guaranteed without additional restrictions. We identify suborders where invariance still holds and clarify where disagreements are possible, along with the implications for measurement specialists and policymakers.