Various levels of measurement equivalence, part 2
A second and higher level of equivalence is called metric equivalence [Ste98], which has also been referred to as construct equivalence [Van97]. Operationally, metric equivalence presupposes that factor loadings in the measurement model are equal across groups:
|Λ1 = Λ2 = ... = ΛG||(5)|
where Λ stands for the factor loading vector and G for the group number (country at a specific time point). Metric equivalence implies the cross-cultural equality of the intervals of the scale on which the latent concept is measured. In other words, an increase of one unit on the measurement scale of the latent variable has the same meaning across groups. However, latent variable scores can still be uniformly biased upwards or downwards. Because of this possibility of additive bias, metric equivalence does not guarantee that latent means are comparable over groups. Nevertheless, metric equivalence is a necessary and sufficient condition for comparing statistics that are based on mean-corrected scores (such as regression coefficients and covariances) across groups 1.
An even higher level of equivalence, scalar invariance, should be established to justify comparing the means of the latent variables across countries or over time [Mee97] [Ste98]. Scalar equivalence holds if, in addition to factor loadings, the intercepts of the indicators in the measurement model also have to be equal across groups:
|τ1 = τ2 = ... = τG||(6)|
where τ stands for the indicator intercept vector and G for the group number (country at a specific time point). Scalar equivalence implies that the measurement scales not only have the same intervals, but also share origins. This makes it possible to compare raw scores in a valid way, which is a prerequisite for latent mean comparisons across countries or over time.
To summarise, if we want to compare attitude means over countries and time points, factor loadings as well as intercepts should be equal across groups.
-  Steenkamp and Baumgartner 1998: 80.
- [Mee97] Meertens, R. W. and Pettigrew, T. F. (1997). Is subtle prejudice really prejudice? Public Opinion Quarterly, 61(1), 54-71.
- [Ste98] Steenkamp, J. E. and Baumgartner, H. (1998). Assessing measurement invariance in cross-national consumer research. Journal of Consumer Research, 25, 78-90.
- [Van97] van de Vijver, F. and Leung, K. (1997). Methods and data-analysis for cross-cultural research. Sage, London.