Communality

In factor analysis, the proportion of a variable's total variance explained by the underlying common factors, distinct from variance specific to it alone.

Source: Spearman 1904

In factor analysis, communality is the share of a variable's variance that the underlying common factors account for, as opposed to the part unique to that variable alone. A high communality means the variable is well explained by the shared factors and fits the model, while a low one means it stands largely apart, driven by something the factors do not capture. Examining it shows how well each measured variable belongs to the factor structure. How much a variable shares with the common factors is what it captures. Kline (2015) describes this concept.

Source: Kline 2015

Communality is calculated as the sum of the squared factor loadings of a variable across the common factors, since each squared loading gives the proportion of the variable's variance explained by that factor, and their sum gives the total explained by all the common factors. So communality is the sum of a variable's squared loadings on the retained factors, expressing the share of its variance accounted for by the common factors, and the remaining variance, one minus the communality, is the uniqueness, comprising variance specific to the variable and measurement error, which the common factors do not explain.

Source: Spearman 1904

Communality indicates how well the common factors account for a measured variable: a high communality means most of the variable's variance is explained by the factors, so it is well represented in the factor solution, while a low communality means the factors explain little of it, suggesting the variable shares little with the others or is poorly captured. So communality indicates the degree to which a variable is explained by the underlying factors, helping to judge which variables fit the factor structure well and which do not, and low communalities may signal variables that should be reconsidered or removed from the analysis.

Source: Spearman 1904

Communality relates to uniqueness as its complement: for a standardised variable, the communality is the proportion of variance explained by the common factors, and the uniqueness is the remaining proportion, one minus the communality, which comprises variance specific to the variable plus measurement error. Together they account for all the variable's variance. So communality and uniqueness partition a variable's variance into the part shared with the common factors and the part particular to the variable, and this division is central to factor analysis, which seeks to explain the shared variance among variables through a smaller set of common factors.

Source: Spearman 1904

Communality is important in factor analysis because it shows how much of each variable's variance the common factors explain, informing judgements about how well the factor solution represents the variables and whether particular variables belong in it. Low communalities flag variables poorly captured by the factors. So communality matters for evaluating and refining a factor analysis, since it indicates the adequacy of the factor structure for each variable, guiding decisions about retaining or dropping variables and about how many factors are needed, which helps ensure the extracted factors meaningfully account for the shared variation among the measured variables.

Source: Spearman 1904