Bock-Aitkin (1981) Marginal Maximum Likelihood Estimation of Item Parameters

Summary

Bock and Aitkin make item-response-theory (IRT) estimation practical by estimating item parameters from the marginal likelihood — integrating the person ability out of the likelihood — and maximizing it with an EM algorithm. This sidesteps the two failures of earlier approaches: conditional ML works only for the Rasch model (which has a sufficient statistic for ability), and joint ML of person + item parameters is inconsistent (the number of nuisance ability parameters grows with the sample — the incidental-parameters problem). By characterizing the ability distribution empirically (on quadrature nodes) they avoid arbitrary distributional assumptions, and the method applies to general models — two- and three-parameter logistic/normal-ogive, multi-category, and multidimensional IRT.

Key Claims

• Why the general models were stuck. Ability $\theta$ is a nuisance parameter that cannot be conditioned away except in the Rasch (one-parameter) model; and joint ML of $(\theta_i)$ and item parameters is not consistent because ability parameters proliferate with subjects, so standard limit theorems fail (Neyman-Scott incidental-parameters problem).
• Marginalize, don't condition or jointly maximize. Treat ability as a random effect with distribution $g(\theta)$ and integrate it out: the marginal probability of a response pattern is $P(\mathbf u_i)=\int \prod_j P_j(\theta)^{u_{ij}}(1-P_j(\theta))^{1-u_{ij}}\,g(\theta)\,d\theta$; item parameters are estimated by maximizing $\sum_i \log P(\mathbf u_i)$.
• EM algorithm. Treat the unobserved abilities as missing data. E-step: at each quadrature node $\theta_k$ compute the posterior "expected number tested" $\bar N_k$ and "expected number correct" $\bar r_{jk}$ for item $j$ (posterior-weighted counts). M-step: re-estimate each item's response function by fitting $\bar r_{jk}/\bar N_k$ — a weighted GLM/probit fit at the nodes. Iterate; the integral is approximated by Gauss-Hermite quadrature.
• Empirical ability distribution. Rather than assume $g(\theta)$ normal, the weights on the quadrature nodes can be estimated from the data (a nonparametric characterization), giving robustness to the shape of the latent-ability distribution.
• Generality. The procedure works for models lacking simple sufficient statistics for ability, including 2PL/3PL, normal ogive, Samejima/Bock multi-category models, and more than one latent dimension (dichotomous factor analysis). Demonstrated on the classic LSAT item data.

Concepts Introduced or Extended

• Rasch Model
• EM Algorithm

Entities Mentioned

• R. Darrell Bock
• Murray Aitkin

Quotes

"Maximum likelihood estimation of item parameters in the marginal distribution, integrating over the distribution of ability, becomes practical when computing procedures based on an EM algorithm are used."

"By characterizing the ability distribution empirically, arbitrary assumptions about its form are avoided. The EM procedure is shown to apply to general item-response models lacking simple sufficient statistics for ability."

My Take

This is the paper that made modern IRT computable, and it is a textbook case of the right way to handle a latent nuisance variable: don't try to estimate every person's ability (that way lies the incidental-parameters inconsistency), and don't restrict yourself to the one model where you can condition it away (Rasch) — instead integrate it out and let EM do the missing-data bookkeeping. The E-step's "expected number correct at each ability level" is a beautifully concrete instance of the EM philosophy, and the empirical/quadrature treatment of $g(\theta)$ prefigures nonparametric mixing and the Bock-Aitkin "empirical prior" that the wiki's multivariate probit page already uses as a normality check. It sits directly upstream of the Bayesian data-augmentation alternative (Albert 1992, Béguin-Glas 2001): where Bock-Aitkin marginalize ability and maximize by EM, the Gibbs samplers impute it and draw — two routes to the same latent-trait likelihood. For the wiki it is the marginal-maximum-likelihood foundation under the IRT and EM threads (and the estimation engine of BILOG/TESTFACT).