# Multiple Group Confirmatory Factor Analysis, part 2

When correctly identified, this model results in the following mean structure **μ** and covariance structure
**Σ**:

μ = τ + Λκ | (2) |

Σ = ΛΦΛ + Θ | (3) |

where **μ** equals a p*1 vector with observed item means and **κ** an m*1 vector
with means of latent variables **ξ**_{j}; **Σ** is the p*p covariance
matrix of the observed indicators, **Φ** an m*m covariance matrix of the latent variables and
**Θ** a p*p matrix with the error (co)variances [Ste98]. Comparing these implied mean and covariance structures with the item means
and covariances observed in the dataset makes it possible to assess how well the measurement model fits. The most straightforward way of assessing model fit is the chi-square test. However, the
chi-square value is known to be very sensitive to large sample sizes. As an alternative, several goodness-of-fit indices have been developed, such as the Root Mean Square Error of Approximation (RMSEA)
or the Comparative Fit Index (CFI) [Ben90] [Bro92].

In order to be useful for measurement equivalence testing, the CFA model described above has to be extended to a multi-group setting [Jör71]. Specifically, this means that a CFA model is estimated separately but simultaneously for different groups g (g = 1,...,G) of respondents.

x^{g} = τ^{g} + Λ^{g}ξ + δ^{g} | (4) |

In this case, we have 51 different groups, namely the inhabitants of 17 countries at three time points (17 x 3 = 51).

#### References

- [Ben90] Bentler, P.M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 238-246.
- [Bro92] Browne, M.W., and Cudeck, R. (1992). Alternative ways of assessing model fit. Sociological Methods & Research, 21(2), 230-258.
- [Jör71] Jöreskog, K. G. (1971). Simultaneous factor analysis in several populations. Psychometrika, 36(4), 408-426.
- [Ste98] Steenkamp, J. E. and Baumgartner, H. (1998). Assessing measurement invariance in cross-national consumer research. Journal of Consumer Research, 25, 78-90.