Interest in the invariance of the factor model with respect to different populations of

individuals, different time periods, or even different variables from a domain appeared

early in the history of factor analysis. Thurstone (1947) studied how the factor structure

for a set of measured variables changes when additional variables are included in the set

being analyzed (Little, Lindenburger, & Nesselroade, 1999). Change in the factor

structure of a measure taken longitudinally has also been studied, but population

differences in factor structure have received the most attention. Technical developments

in methods for studying group differences in factor structure have now advanced to the

point where studies of invariance can be completed by anyone with a personal computer

and access to confirmatory factor-analytic software. As noted by McDonald (1999), the

factor model can even be applied to the study of group differences in the factor structure

of dichotomously scored items. However, these technical developments have not fully

resolved some conceptual or practical problems facing researchers who study factorial

invariance. This chapter focuses on four of these unresolved problems and the prospects

for their solutions. We begin with a review of definitions of factorial invariance and their

relation to broader notions of measurement invariance (Meredith & Millsap, 1992), and

follow with sections describing each problem in turn.