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Comparative public health research makes wide use of self-report instruments. For example, research identifying and explaining health disparities across demographic strata may seek to understand the health effects of patient attitudes or private behaviors. Such personal attributes are difficult or impossible to observe directly and are often best measured by self-reports. Defensible use of self-reports in quantitative comparative research requires not only that the measured constructs have the same meaning across groups, but also that group comparisons of sample estimates (eg, means and variances) reflect true group differences and are not contaminated by group-specific attributes that are unrelated to the construct of interest. Evidence for these desirable properties of measurement instruments can be established within the confirmatory factor analysis (CFA) framework; a nested hierarchy of hypotheses is tested that addresses the cross-group invariance of the instrument's psychometric properties. By name, these hypotheses include configurai, metric (or pattern), strong (or scalar), and strict factorial invariance. The CFA model and each of these hypotheses are described in nontechnical language. A worked example and technical appendices are included.

Background: Analysis of subgroups such as different ethnic, language, or education groups selected from among a parent population is common in health disparities research. One goal of such analyses is to examine measurement equivalence, which includes both qualitative review of the meaning of items as well as quantitative examination of different levels of factorial invariance and differential item functioning. Objectives: The purpose of this essay is to review the definitions and assumptions associated with factorial invariance, placing this formulation in the context of bias, fairness, and equity. The connection between the concepts of factorial invariance and item bias (differential item functioning) using a variant of item response theory is discussed. The situations under which different forms of invariance (weak, strong, and strict) are required are discussed. Methods: Establishing factorial invariance involves a hierarchy of levels that include tests of weak, strong, and strict invariance. Pattern (metric or weak) factorial invariance implies that the regression slopes are invariant across groups. Pattern invariance requires only invariant factor loadings. Strong factorial invariance implies that the conditional expectation of the response, given the common and specific factors, is invariant across groups. Strong factorial invariance requires that specific factor means (represented as invariant intercepts) also be identical across groups. Strict factorial invariance implies that, in addition, the conditional variance of the response, given the common and specific factors, is invariant across groups. Strict factorial invariance requires that, in addition to equal factor loadings and intercepts, the residual (specific factor plus error variable) variances are equivalent across groups. The concept of measurement invariance that is most closely aligned to that of item response theory considers the latent variable as a common factor measured by manifest variables; the specific factors can be characterized as nuisance variables. Conclusions: Invariance of factor loadings across studied groups is required for valid comparisons of scale score or latent variable means. Strong and strict invariance may be less important in the context of basic research in which group differences in specific factors are indicative of individual differences that are important for scientific exploration. However, for most applications in which the aim is to ensure fairness and equity, strict factorial invariance is required. Health disparities research often focuses on self-reported clinical outcomes such as quality of life that are not observed directly. Latent variable models such as factor analyses are central to establishing valid assessment of such outcomes.