psychometrika—vol. 82, no. 3, 637–647
ON THE FINITENESS OF THE WEIGHTED LIKELIHOOD ESTIMATOR OF ABILITY
UNIVERSITY OF LIÈGE
The purpose of this note is to focus on the ﬁniteness of the weighted likelihood estimator (WLE) of
ability in the context of dichotomous and polytomous item response theory (IRT) models. It is established
that the WLE always returns ﬁnite ability estimates. This general result is validfor dichotomous (one-, two-,
three- and four-parameter logistic) IRT models, the class of polytomous difference models and divide-by-
total models, independently of the number of items, the item parameters and the response patterns. Further
implications of this result are outlined.
Key words: item response theory, dichotomous models, polytomous models, weighted likelihood estima-
tion, Bayesian estimation, ﬁniteness.
The weighted likelihood estimator (WLE) of ability (Warm, 1989) has become a popular
ability estimation method in the framework of dichotomous item response theory (IRT). By
weighting the likelihood function with some appropriate function, it reduces the bias of the
maximum likelihood estimator (MLE). The WLE method was also generalized to polytomous
IRT models (Samejima, 1998) and to multidimensional IRT models (Wang, 2015).
Despite these promising aspects, the WLE remains somewhat unstudied and consequently
undiscovered. First, the uniqueness of the WLE cannot be guaranteed (this issue is further
addressed in the ﬁnal comments). Second, the standard error (SE) of the WLE is not well-known.
Most commonly it is taken as the SE of the MLE, computed at the point WLE estimate (Warm,
2007) because Warm (1989) established the asymptotic equivalence between MLE and WLE.
A proper formula for the SE of the WLE is, however, barely available (Magis, 2016). Finally,
and this is the purpose of this note, there remains some confusion about the ﬁniteness (or not) of
the WLE, that is: can the WLE values take only ﬁnite values (unlike the MLE for which inﬁnite
estimates may occur)? The latter issue is not insigniﬁcant. For instance, in the polytomous IRT
context, Boyd, Dodd, and Choi (2010, p. 242) wrote that “… WLE can be estimated after the
ﬁrst item is administered if the response is not in the ﬁrst or last category.” Though written in the
speciﬁc context of computerized adaptive testing (CAT), the meaning of this sentence is clear:
the WLE should be avoided if the item responses are all equal to the ﬁrst or to the last category
(for each respective item). This corresponds to the well-known situation of dichotomous items
for which responses are all correct or all incorrect.
As such, the WLE has some drawbacks (or undiscovered properties) with respect to other
ability estimators that might perhaps explain why it has not become yet a standard estimator in
Electronic supplementary material The online version of this article (doi:10.1007/s11336-016-9518-9) contains
supplementary material, which is available to authorized users.
Correspondence should be made to David Magis, Research Unit on Childhood, University of Liège, Building B32,
Quartier Agora, Place des Orateurs 2, 4000 Liege, Belgium. Email: email@example.com
© 2016 The Psychometric Society