Profile-likelihood Confidence Intervals in Item Response Theory Models

ABSTRACT

Confidence intervals (CIs) are fundamental inferential devices which quantify the sampling variability of parameter estimates. In item response theory, CIs have been primarily obtained from large-sample Wald-type approaches based on standard error estimates, derived from the observed or expected information matrix, after parameters have been estimated via maximum likelihood. An alternative approach to constructing CIs is to quantify sampling variability directly from the likelihood function with a technique known as profile-likelihood confidence intervals (PL CIs). In this article, we introduce PL CIs for item response theory models, compare PL CIs to classical large-sample Wald-type CIs, and demonstrate important distinctions among these CIs. CIs are then constructed for parameters directly estimated in the specified model and for transformed parameters which are often obtained post-estimation. Monte Carlo simulation results suggest that PL CIs perform consistently better than Wald-type CIs for both non-transformed and transformed parameters.

KEYWORDS:

• Item response theory
• large-sample confidence intervals
• likelihood inference
• profile-likelihood confidence intervals
• Wald confidence intervals

Notes

1 For reference, the profile-likelihood CI (described in the next section) for the f parameter in this example is [.800,.976].

2 The log-likelihood is not always easy to obtain for some models that are estimated with the EM algorithm (McLachlan & Krishnan, 2008). However, for IRT models estimated with MML this is not an issue.