Confirmatory Factor Analysis

Sources of Variability

Confirmatory factor analysis partitions variability in scores on indicators according to sources specified in a measurement model. The basis for this partitioning is illustrated in Figure 1. On the left is a Venn diagram that shows six possible sources of variability in three indicators—V1, V2, and V3—that could be specified in a measurement model. In this informal depiction, the indicators are represented by circles, each comprising three parts. The shaded area, labeled F1, is variability in each indicator shared with the other two indicators. The crosshatched area is variability shared with only one of the remaining indicators, and the unshaded area is variability shared with neither of the remaining indicators.

Figure 1 Venn and Path Diagram Depictions of the Partitioning of Variability in the Score on Three Variables Into Shared and Unique Components

On the right in Figure 1 is a more formal depiction of this partitioning of variability. In this path diagram, the indicators are represented by squares. The large shaded circle, labeled F1, corresponds to the shaded area in the Venn diagram. It is the latent variable the three indicators are purported to measure and reflects the variability they share. The directional paths between the latent variable and the squares represent the assumption that the latent variable is a cause of variability in the indicators. The small circles represent variability unique to each indicator and correspond to the unshaded areas in theVenn diagram. They represent measurement error because they are unintended causes of variability on the indicators. The crosshatched areas from the Venn diagram are not represented in the path diagram. This is because the typical specification of a measurement model assumes unrelated errors of measurement. If the strength of these relations is statistically nontrivial, a satisfactory measurement model will need to include covariances between the errors of measurement.

Two additional features of the path diagram warrant mention. First, notice that attached to each circle is a sharply curved two-headed arrow. This path indicates a variance and illustrates the fact that variability in each indicator has been allocated either to the latent variable or to measurement error. Second, notice that every path is accompanied by either a 1 or an asterisk. These represent parameters, numeric values that reflect the strength of the paths. Parameters indicated by a value such as 1 reflect a fixed assumption about the association between two variables or a variance. Parameters indicated by an asterisk reflect a parameter whose value is not assumed but will be estimated from the observed data. A subset of these parameters of particular interest in confirmatory factor analysis is associated with paths between the latent variable and the indicators—the loadings. The larger these values, the greater the portion of variability in the indicators that can be attributed to the latent variable(s) as opposed to measurement error.

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