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This article offers a gentle introduction to the measurement invariance (MI) literature with a focus on its relevance to comparative political research. It reviews 1) the conceptual foundations of MI; 2) standard procedures of testing for MI in practical applications within the multiple-group confirmatory factor analysis (MGCFA) paradigm; and 3) two novel approaches to MI, Bayesian approximate measurement invariance, and MGCFA alignment optimization, which are especially suitable for dealing with extremely heterogeneous data from large-scale comparative surveys typical for modern political science. It then provides an empirical illustration of the key concepts and methods from the MGCFA-MI literature by applying them to testing for MI of two recently introduced measures of democracy attitudes, so-called liberal and authoritarian notions of democracy, across 60 countries in the sixth round of the World Values Survey. These analyses show that both measures can be considered reliable comparative measures of democratic attitudes, although for different reasons. Finally, this study emphasizes that some survey-based constructs, e.g., authoritarian notions of democracy, do not follow the reflective (correlation-based) logic of construct development. These alternative measures, known as formative measures, do not assume strong correlations between their indicators, for which reason it is inappropriate to test their comparability using the reflective MGCFA approach. Instead, their comparability can be tied to their correlations with theoretically relevant external variables.

This article reports evidence of misspecification of the measurement model for the index of emancipative values, a value construct used as a key explanatory variable in many important contributions to political science. It shows that the scale on which the index is measured is noninvariant across cultural zones and countries in the World Values Survey. In addition, it demonstrates that the current index composition mixes different value dimensions and their actual associations with various political outcomes, in particular the index of effective democracy. However, an analysis using a novel approximate Bayesian approach shows that at least one specific subdimension of emancipative values, known as pro-choice values, truly exists and may be validly measured and compared cross-nationally. The article also contributes to the recent discussion on whether emancipative values are a reflective or a formative construct by providing thought experiments and empirical evidence supporting the former interpretation.