# Chapter 3: Multigroup Factor Analysis

### Models with some non-invariance of measurement

To introduce the key concepts related to non-invariance of measurement in factor analysis models, we focus on the simple case of a model with one factor η. In a multigroup context, the measurement model for any item *y*_{j} (*j* = 1, ..., *p*) for a respondent in group *g* = 1, ..., *G* can then be expressed as

*y _{j} = ν_{j}^{(g)} + λ_{j}^{(g)}η + ε_{j}*

where *ε _{j}* is normally distributed with mean 0 and variance

*θ*

_{j}

^{(g)}. In other words, such a measurement model allows any or all of the measurement parameters for an item (intercept

*ν*

_{j}

*, loading*

^{(g)}*λ*

*and/or the error variance*

_{j}^{(g)}*θ*

*) to have different values in different groups*

_{j}^{(g)}*g*.

The first question we should ask about comparability of measurement is whether the overall structure of the measurement model for the items is the same in all groups. In the one-factor case this is the question of whether a one-factor model is indeed adequate in all groups. More generally, it is the question of whether a measurement model with the same number of factors and the same pattern of zero and non-zero factor loadings is adequate in all groups. Example 2 in Chapter 2 was an illustration of this kind of analysis, there for a particular 2-factor confirmatory factor analysis model. If the same model is adequate in all groups in this sense, the measurement model is said to possess *configural invariance* (or construct invariance) across the groups. In essence this means that the items can be thought to measure the same latent constructs in each group, even if possibly with different exact values of the measurement parameters. If construct invariance does not hold, the items cannot really be used for meaningful comparisons of constructs between the groups. If it does hold, we can proceed to examine whether the parameters of the common measurement model also have equal values across groups.

If any of the parameters of the measurement model for item *y*_{j} do vary across the groups, the item is non-invariant across the groups. If, in contrast, each of the measurement parameters has the same value in all groups (i.e. *ν** _{j}^{(g)}* =

*ν*

_{j},

*λ*

*=*

_{j}^{(g)}*λ*

*and*

_{j}*θ*

*=*

_{j}^{(g)}*θ*

*for*

_{j}*g*= 1, ...,

*G*), full (or "strict") invariance of measurement holds for that item. When an item is fully invariant, it thus functions as a measure of the factor in exactly the same way in all of the groups. If full invariance holds for all items

*y*

_{1}, ...,

*y*

_{p}which are treated as measures of the factor η, the measurement of the factor itself is fully invariant.

We may also consider models which possess partial invariance, meaning that some but not all items for a factor and/or some but not all measurement parameters for an item are non-invariant. The following terms are often used to refer to specific kinds of partial invariance in terms of types of parameters:

*Scalar*invariance (also known as "strong" factorial invariance) holds for an item if the intercepts*ν*_{j}and loadings*λ*_{j}are invariant across the groups, but the error variances*θ*are not. If this is the case for all the items, scalar invariance holds for the whole scale of measurement for the factor._{j}^{(g)}*Metric*invariance (also known as "weak" factorial invariance) holds for an item if the loadings*λ**j*are invariant across the groups, but the intercepts*ν*and the error variances_{j}^{(g)}*θ*are not. If this is the case for all the items, metric invariance holds for the whole scale of measurement for the factor._{j}^{(g)}