psychometrika—vol. 82, no. 3, 648–659
LOWER BOUNDS TO THE RELIABILITIES OF FACTOR SCORE ESTIMATORS
David J. Hessen
Under the general common factor model, the reliabilities of factor score estimators might be of more
interest than the reliability of the total score (the unweighted sum of item scores). In this paper, lower
bounds to the reliabilities of Thurstone’s factor score estimators, Bartlett’s factor score estimators, and
McDonald’s factor score estimators are derived and conditions are given under which these lower bounds
are equal. The relative performance of the derived lower bounds is studied using classic example data
sets. The results show that estimates of the lower bounds to the reliabilities of Thurstone’s factor score
estimators are greater than or equal to the estimates of the lower bounds to the reliabilities of Bartlett’s and
McDonald’s factor score estimators.
Key words: reliability, classical test theory, common factor model, factor scores.
In the psychometric literature, the description and the assessment of the reliability of the
total score (the unweighted sum of the item scores) has received a lot of attention (e.g., ten Berge
&Soˇcan, 2004; Zinbarg, Revelle, Yovel, & Li, 2005; Sijtsma, 2009; Bentler, 2009; Revelle &
Zinbarg, 2009). For the assessment of the reliability of the total score, many coefﬁcients have been
proposed (for an overview, see Revelle and Zinbarg, 2009). All these coefﬁcients are mathematical
lower bounds to the reliability of the total score. One of these coefﬁcients is the popular coefﬁcient
alpha (Guttman, 1945; Cronbach, 1951). Coefﬁcient alpha is widely used but other coefﬁcients
have been shown to be greater lower bounds to the reliability of the total score. Therefore, Sijtsma
(2009) advised against using coefﬁcient alpha and suggested to report the coefﬁcient referred
to as the greatest lower bound (Woodhouse & Jackson, 1977; ten Berge, Snijders, & Zegers,
1981). Bentler (2009), however, made some critical observations about the greatest lower bound
and suggested to consider reliability coefﬁcients based on a factor or structural equation model.
Revelle and Zinbarg (2009) showed that for some classic example data sets the estimate of the
greatest lower bound was systematically less than the estimate they obtained using the factor
model-based coefﬁcient ω (Heise & Bohrnstedt, 1970; McDonald, 1978).
Under a factor model, however, the reliability of the total score might be of less interest than
the reliabilities of factor score estimators. Factor score estimators are the random variables used
to estimate the true values of the unobserved common factors assumed to underlie the item scores.
The estimated values of the common factors are called factor scores. When factor scores are used
for diagnostic purposes, or as inputs to subsequent analyses, then the reliabilities of the factor
score estimators are of interest.
Many factor score estimators have been proposed (for an overview, see Grice, 2001). Three
well-known types of factor score estimators that are deﬁned in terms of the parameters of a factor
model are the regression estimators proposed by Thurstone (1935), the weighted least squares
estimators proposed by Bartlett (1937), and the correlation-preserving estimators proposed by
McDonald (1981). In this paper, lower bounds to the reliabilities of these three types of factor
Correspondence should be made to David J. Hessen, Department of Methodology and Statistics, Utrecht University,
Padualaan 14, PO Box 80.140, 3508 TC Utrecht, The Netherlands. Email: D.J.Hessen@uu.nl
© 2016 The Psychometric Society