# Chapter 2: Factor Analysis

### Inherent identifiability of the model parameters

Identification of the model: Numbers of factors vs. Numbers of items

The second and more important issue of identifiability for a factor analysis model is that once we have fixed the latent scales and their directions as discussed above, the model should then be fully identified. If it is not, the model must be changed.

The most basic condition for the model to be identified is that it should not have more parameters than there are distinct pieces of information in the joint distribution of the observed variables from which the parameters will be estimated. For factor analysis, these observed pieces of information are the sample means, variances and covariances of the p observed items. The means are matched by the p freely estimable values of the factor means κj and item intercepts νj, so these parameters will be identified if the rest of the model is. The main question of identification is then whether the factor variances φκ and covariances φκl, the factor loadings λ, the error variances θj, and the error covariances θ are identified. The total number of freely estimable parameters of these kinds should not exceed the total number of distinct variances and covariances of the observed items, which is

p(p + 1)
2
(where p is the number of the observed items). This condition is necessary, meaning that a model which does satisfy it is not identified. It is, however, not sufficient, meaning that there are models which satisfy this condition but are nevertheless not identified.

For exploratory factor analysis (EFA) models, this kind of identifiability is determined simply by the number of factors (letâ€™s denote it q) compared to the number of observed indicators for the factors (p). An EFA model is identified if its degrees of freedom

(p - q)2 - (p + q)
2
are greater than or equal to 0. This means, for example, that no EFA model is identified if we have only p=1 or p=2 indicators, only a model with q=1 factors is identified if there are 3 or 4 indicators, and a 2-factor model requires at least 5 indicators.

For a confirmatory factor analysis (CFA) model, identifiability depends both on how many indicators are treated as measures of each individual factor, and on other details of the specification of the model. The following conditions, which are sufficient but not necessary, can be used to determine identification for a very commonly used class of CFA models:

• If a CFA model is such that (i) it has 2 or more factors, (ii) all the factors are correlated with each other, (iii) there are no correlations between the measurement errors, (iv) each observed indicator measures only one factor, and (v) each factor is measured by at least 2 indicators, then the model is identified.
• If the model has 1 factor, there are no correlations between the measurement errors, and the factor is measured by at least 3 indicators, then the model is identified. This is simply a 1-factor EFA model, so the identification condition is just the general EFA condition on the number of indicators.

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