When Good Loadings Go Bad: Robustness in Factor Analysis. Issue 4 (3rd July 2020) - Bollen, Kenneth A.
ABSTRACT: Structural misspecifications in factor analysis include using the wrong number of factors and omitting cross loadings or correlated errors. The impact of these errors on factor loading estimates is understudied. Factor loadings underlie our assessments of the validity and reliability of indicators. Thus knowing how structural misspecifications affect a factor loading is a key issue. This paper develops analytic conditions of when misspecifications affect Bollen's (1996) model implied instrumental variable, two stage least squares (MIIV-2SLS) estimator of a factor loading. It shows that if an indicator equation is correctly specified, then correlated errors among other measures, mixing up causal indicators with reflective, omitting cross loadings, and omitting direct effects between indicators leave the MIIV-2SLS estimator of the factor loading unchanged. Alternatively, if the indicator or the scaling indicator equation is misspecified, then the loading is unlikely to be robust. The results are illustrated with hypothetical and empirical examples.