# Multiple Group Confirmatory Factor Analysis, part 1

Several techniques have been proposed to test measurement equivalence. Multiple group confirmatory factor analysis (MGCFA) [Jör71] is one of the most popular techniques to assess
measurement equivalence [Bil03] [Byr89] [Ren98] [Ste98]. MGCFA is a quite straightforward extension from conventional confirmatory factor analysis (CFA). In CFA, observed items are
considered to be indicators of an unobserved -or latent- concept. These indicators are imperfect, in the sense that, besides measuring the intended concepts, they are also affected
by measurement error. Systems of equations are used to describe the relations between observed items and the latent concepts these items are supposed to measure. In our case, we have
three items measuring one latent concept, ‘anti-immigration attitudes’. More in general, let us assume that we have p items measuring m latent variables. The observed items
scores x_{i} (i = 1,..,p) can then be written as linear functions of latent variables **ξ**_{j} (j = 1,...,m):

x = τ + Λξ + δ | (1) |

In expression (1), **x** refers to a p*1 vector containing the observed item scores. This vector is modelled as the sum of three components. **Λξ**
is the product of a p*m matrix containing the factor loadings (**Λξ**) and an m*1 vector with the latent variable scores
(**ξ**). The factor loadings can be seen as the slopes of a regression of x_{i} on **ξ**_{j}.
**τ** is a p*1 vector with the intercepts of the functions. These intercepts refer to the expected value of the observed items when the latent variable score is
equal to zero. Finally, **δ** is a p*1 vector containing stochastic error terms that are assumed to follow a multivariate normal distribution and to have the
expected value 0.

#### References

- [Bil03] Billiet, J. (2003). Cross-cultural equivalence with structural equation modeling. In: Harkness, J., Van de Vijver, F., Mohler, P. (Eds.), Cross-cultural survey methods. John Wiley and Sons, Hoboken, NJ, pp. 247-264.
- [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.
- [Jör71] Jöreskog, K. G. (1971). Simultaneous factor analysis in several populations. Psychometrika, 36(4), 408-426.
- [Ren98] Rensvold, R. B. and Cheung, G. W. (1998). Testing measurement models for factorial invariance: a systematic approach. Educational and Psychological Measurement, 58, 1017-1034.
- [Ste98] Steenkamp, J. E. and Baumgartner, H. (1998). Assessing measurement invariance in cross-national consumer research. Journal of Consumer Research, 25, 78-90.