# Full vs. partial equivalence

When the parameters for all items in the measurement model are equal across groups, we speak of full measurement equivalence. However, Byrne et al. (1989) [Byr89] have argued that full equivalence is not a necessary condition for comparisons to be valid. If at least two items per latent variable (namely, the item that is fixed at unity to identify the model and one other item) are equivalent, comparisons can be validly made across countries and time points. Thus, partial equivalence does not necessarily require the invariance of all loadings and intercepts. This idea is also supported by Steenkamp and Baumgartner (1998) [Ste98].

In this study, we want to study the evolution of anti-immigration attitudes in 17 different European countries. This boils down to comparing the means of the latent variable over the 51 groups. Before such comparisons can be made in a valid way, we need to test whether the scale possesses the characteristic of partial scalar equivalence. In other words, we need to assess whether, for at least two out of three items, factor loadings and indicator intercepts are equal across groups.

In practice, the different levels of measurement equivalence can be tested by fitting various, increasingly restrictive multi-group models. The first model will have the same factor structure across groups, but with no constraints on the parameters. In other words, factor loadings and intercepts can vary across countries (= configural equivalence). In a second model, we will constrain the factor loadings to be equal across groups (= metric equivalence). The third model will, besides loadings, also have equal intercepts across groups (= scalar equivalence). The level of measurement equivalence can then be determined by judging the fit of these various models. In addition, we will look at modification indices (cf. infra) to determine the sources of possible misfit.

#### References

- [Byr89] Byrne, B. M., Shavelson, R. J. and Muthén, B. (1989). Testing for the equivalence of factor covariance and mean structures: the issue of partial measurement invariance. Psychological Bulletin, 105, 456-466.
- [Ste98] Steenkamp, J. E. and Baumgartner, H. (1998). Assessing measurement invariance in cross-national consumer research. Journal of Consumer Research, 25, 78-90.