Principal Component Analysis >

In PCA and Factor Analysis, a variable’s **communality **is a useful measure for predicting the variable’s value. More specifically, it tells you what proportion of the variable’s variance is a result of either:

- The principal components or
- The correlations between each variable and individual factors (Vogt, 1999).

In Factor Analysis, communality may be denoted as **h ^{2}.**

## Values for Communality

A variable’s communality ranges from **0 to 1.**

In general, one way to think of communality is as the proportion of common variance found in a particular variable.

- A variable that doesn’t have any unique variance at all (i.e. one with explained variance that is 100% a result of other variables) has a communality of 1.
- A variable with variance that is completely unexplained by any other variables has a communality of zero (Field, 2013).

## Relation to Error and Specific Variance

Communality (aka common variance) is intertwined with *unique variance.* The two types of variance make up 100% of the variable’s variance.

**Unique variance** is in turn made up of *specific variance* and *error variance*. Specific variance has something specific to do with your model/survey. For example, if you were performing factor analysis on results from a questionnaire to evaluate depression, a recent divorce would certainly account for some variance. Something fairly unrelated (like the subject didn’t get any sleep the night before the test because they were binge watching *Friends*) would be error variance.

The relationship between all of these different types of variance is best explained with an image:

## Communality in PCA

One way to define communality is in terms of the sum of squared loadings on each principal component (McGarigal et al., 2013):

**Where**:

- c
_{j}= communality of the jth variable - s
_{ij}= loading (or correlation) between the ith component and the jth variable.

## References

Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. SAGE.

McGarigal et al. (2013). Multivariate Statistics for Wildlife and Ecology Research. Springer Science & Business Media.

Sapp, M. (2006). Basic Psychological Measurement, Research Designs, and Statistics Without Math. Charles C Thomas Publisher.

Vogt, W. P. (1999). Dictionary of Statistics and Methodology A Non-Technical Guide for the Social Sciences (2nd ed.). London Sage Publications