Communality

Communality, Shared Variance and Heywood Cases in Health Measurement Instruments

Communality tells an analyst how much of one questionnaire item's variance is shared with the other items through the factors of a factor analysis. For a standardised item it normally lies between 0 and 1, so a communality of 0.65 means that 65% of the item's variance is common variance and 35% is unique to that item. In health economics it matters most when patient-reported outcome measures and health-related quality of life instruments are developed and validated, because the preference-based instruments among them supply the health state values behind quality-adjusted life years. The calculation differs for uncorrelated and correlated factors, and a low value need not remove an item from a preference-based measure.

Common and unique variance in the factor model

The common factor model explains each observed variable as a weighted sum of a few unobserved common factors plus one unique factor. The SAS/STAT documentation defines a common factor as a hypothetical latent variable that contributes to the variance of at least two observed variables, and a unique factor as one that contributes to the variance of only one. Two assumptions are critical. The unique factors are uncorrelated with each other, and they are uncorrelated with the common factors.

When the observed variables are standardised to unit variance, those assumptions split each variable's variance into two parts, as follows.

$$1 = h_j^2 + \psi_j$$

where $h_j^2$ is the communality of variable $j$ and $\psi_j$ is its uniqueness. The SAS documentation describes the communality as the extent of overlap with the common factors, the proportion of variance accounted for by them. The R documentation for the factanal function writes the same model for the correlation matrix as $\Sigma = \Lambda\Lambda^{\mathsf T} + \Psi$, with the uniquenesses on the diagonal of $\Psi$.

Uniqueness is a residual, so it mixes variance specific to the item with measurement error. Factor analysis cannot separate the two from a single administration of the instrument, which is why uniqueness should not be read as the error share of an item.

Calculating communality from loadings and factor correlations

Communality is computed from the item's row of the loading matrix. The formula depends on whether the factors are allowed to correlate. With uncorrelated factors, which is how factors are first extracted and how they remain after an orthogonal rotation, it is the sum of the item's squared loadings, as shown below.

$$h_j^2 = \sum_{k=1}^{m} \lambda_{jk}^2$$

where $h_j^2$ is the communality of item $j$, $\lambda_{jk}$ is the factor loading of item $j$ on factor $k$, and $m$ is the number of common factors retained.

After an oblique rotation the factors are correlated, and the shared variance includes cross-products of the pattern loadings weighted by the factor correlations, as follows.

$$h_j^2 = \boldsymbol{\lambda}_j^{\mathsf T} \Phi \boldsymbol{\lambda}_j$$

where $\boldsymbol{\lambda}_j$ is the column vector of item $j$'s pattern loadings and $\Phi$ is the factor correlation matrix. When the factors are uncorrelated, $\Phi$ is the identity matrix and the formula reduces to the sum of squared loadings. Summing squared pattern loadings after an oblique rotation therefore misstates the communality whenever the factors are correlated.

Rotation does not change communalities. The SAS documentation states that rotating a set of factors does not change their statistical explanatory power, and that all rotations, orthogonal or oblique, are equally good statistically. A factor rotation redistributes an item's common variance among the factors.

Prior and final communality estimates

A communality is unknown before the analysis, so many extraction methods begin with a prior estimate and refine it. The SAS FACTOR procedure offers several priors, including each variable's squared multiple correlation with all the other variables, its maximum absolute correlation with any other variable, and a value of 1.0. It uses squared multiple correlations by default for the maximum likelihood, unweighted least squares and alpha methods, and for iterated methods iteration stops when the largest change in the communalities falls below a convergence criterion.

With the principal method, a prior of 1.0 for every variable turns the analysis into a principal component analysis. Costello and Osborne note that principal components analysis does not separate unique from shared variance and so sets all item communalities at 1.0, while maximum likelihood estimation estimates the shared variance; in their example the maximum likelihood communalities ranged from 0.39 to 0.70. A principal component therefore carries unique variance along with common variance, and a table of extracted communalities means something different under the two methods.

In exploratory factor analysis software usually reports both the initial and the extracted communalities. In confirmatory factor analysis the same quantity appears as the proportion of an indicator's variance explained by its factor, which for an item loading on a single factor equals its squared standardised loading.

Worked example of an illustrative five-item health status scale

The figures below are illustrative and are not from a real instrument. Five standardised items are analysed with two uncorrelated factors, labelled physical function (F1) and emotional wellbeing (F2).

Step 1 (communality of each item). For item A, 0.80 squared plus 0.10 squared is 0.64 + 0.01 = 0.65. For item E, 0.35 squared plus 0.25 squared is 0.1225 + 0.0625 = 0.185. Uniqueness is 1 minus the communality, so item E has uniqueness 0.815.

Step 2 (variance explained by each factor). Summing the squared loadings down a column gives the variance each factor explains. For F1 this is 0.64 + 0.5625 + 0.36 + 0.01 + 0.1225 = 1.695, and for F2 it is 0.01 + 0.0025 + 0.16 + 0.7225 + 0.0625 = 0.9575. The two factors together explain 1.695 + 0.9575 = 2.6525, which equals the sum of the five communalities. Dividing by the five units of total variance gives 2.6525 / 5 = 0.5305, so the two factors account for about 53% of the total variance of the five items.

Step 3 (rotation leaves communality unchanged). Rotating the factors by 45 degrees moves item A's loadings to about 0.6364 and minus 0.4950. The communality is 0.4050 + 0.2450 = 0.65, unchanged.

Step 4 (correlated factors). Suppose an oblique solution gives item C the same pattern loadings of 0.60 and 0.40 with a factor correlation of 0.30. The squared loadings give 0.36 + 0.16 = 0.52, the cross-product term adds 2 × 0.60 × 0.40 × 0.30 = 0.144, and the communality is 0.52 + 0.144 = 0.664, leaving a uniqueness of 0.336. Summing the squared pattern loadings alone would report 0.52 and understate the shared variance by 0.144. This compares two hypothetical solutions; in real data a change in the factor correlation would also change the fitted loadings.

Interpretation. Items A, B, C and D share a reasonable part of their variance with the two factors. Item E shares less than a fifth of its variance, so the two-factor model says little about it. That flags the item for review; the next sections explain why review does not automatically mean removal.

Reading communalities when instruments are developed

Communalities inform judgements about whether items belong together and whether enough factors were retained. Costello and Osborne describe communalities of 0.8 or more as high but unlikely in real data, and report that low to moderate values of 0.40 to 0.70 are more common in the social sciences. In their account an item with a communality below 0.40 may be unrelated to the other items or may point to an additional factor that should be explored, and the researcher then has to decide whether to drop it or add similar items.

Communality also governs how much data a factor analysis needs. Preacher and MacCallum report that MacCallum and colleagues found communality to be the most important determinant of how well a sample recovers the population factor structure. High communalities offset the effects of small samples and few items per factor, while at lower communalities recovery suffers greatly when the sample is small. They also warn that retaining too few factors lowers communalities. A validation study for a new patient-reported outcome measure in a small patient group is more likely to recover a stable structure when the items have high communalities.

For construct validity, communalities are one piece of structural evidence alongside the loading pattern, residual correlations and fit indices.

Preference-based measures and causal indicators

Generic health-related quality of life instruments that produce a health state utility value need particular care. Gamst-Klaussen, Gudex and Olsen describe such health state utility instruments as having an important role in cost-effectiveness analyses, and their analysis of EQ-5D-5L data from 7,933 respondents in six countries supported a model in which pain/discomfort, anxiety/depression and mobility act as causal indicators that drive change in the construct, with self-care and usual activities as effect indicators that reflect it. They note that factor analysis usually depends on a set of homogeneous items and is often not appropriate when causal and effect items are both present.

This matters for how a low communality is read. A symptom item such as pain can share little variance with functioning items and still be one of the dimensions people care about most. The same authors observe that in the EQ-5D-5L value sets for Canada, England, Spain and the Netherlands the two symptom dimensions are on average 50% stronger than each of the three functioning dimensions. Deleting item E in the example above because of its communality would remove variance that a QALY calculation may weight heavily. In a utility instrument, decisions about dimensions rest on content and preference evidence as well as on the factor structure.

Heywood cases and other warning signs

The SAS documentation notes that communalities are squared correlations and would be expected to lie between 0 and 1, but that final estimates can exceed 1. A communality of exactly 1 is called a Heywood case, and one above 1 is an ultra-Heywood case, which implies that some unique factor has negative variance and renders the solution invalid. The documentation lists the following possible causes.

• poor prior communality estimates
• too many or too few common factors
• too little data to give stable estimates
• a common factor model that does not suit the data

The SAS documentation adds that a communality should not exceed the variable's reliability, and calls a violation a quasi-Heywood case to be regarded with the same suspicion. It describes maximum likelihood as especially susceptible to quasi-Heywood and ultra-Heywood cases, because a variable with high communality receives a high weight during iteration, which pushes its communality higher. The R documentation for factanal observes that Heywood cases, with one or more uniquenesses near zero, are much more common than most texts would lead one to believe. Software options that cap communalities at 1 allow iteration to continue, but the causes listed above still need investigating.

Communality and nearby quantities

Communality sits beside several quantities that are easily confused with it. An eigenvalue, which in principal factor extraction equals an unrotated factor's sum of squared loadings, summarises one factor across all items, while communality summarises one item across all factors; Step 2 of the example shows that the two totals agree. Reliability concerns consistency of measurement, such as test-retest reliability across occasions, and the reliability of an item sets an upper limit for its communality rather than equalling it.

Where communality estimates mislead

Communality is an estimate that depends on the sample, the extraction method, the number of factors retained and the items included in the analysis. Adding or removing items changes every communality.

Common errors include the following.

• reading the communalities from a principal component analysis as common variance, when components absorb unique variance as well
• summing squared pattern loadings after an oblique rotation without the factor correlations
• treating uniqueness as pure measurement error
• dropping an item from a preference-based instrument on its communality alone
• treating a guide such as 0.40 as a fixed rule rather than a prompt to examine the item
• accepting a solution with a communality of 1 or more without investigating the cause

Sources

• Costello AB, Osborne JW. Best practices in exploratory factor analysis: four recommendations for getting the most from your analysis. Practical Assessment, Research and Evaluation. 2005;10(7):1-9. https://doi.org/10.7275/jyj1-4868
• Gamst-Klaussen T, Gudex C, Olsen JA. Exploring the causal and effect nature of EQ-5D dimensions: an application of confirmatory tetrad analysis and confirmatory factor analysis. Health and Quality of Life Outcomes. 2018;16:153. https://doi.org/10.1186/s12955-018-0975-y
• Preacher KJ, MacCallum RC. Exploratory factor analysis in behavior genetics research: factor recovery with small sample sizes. Behavior Genetics. 2002;32(2):153-161. https://doi.org/10.1023/A:1015210025234
• R Core Team. factanal: Factor Analysis. R documentation, stats package (R-devel, accessed 2026). https://stat.ethz.ch/R-manual/R-devel/library/stats/html/factanal.html
• SAS Institute. SAS/STAT 9.3 User's Guide: The FACTOR Procedure (Background; PROC FACTOR Statement; Heywood Cases and Other Anomalies about Communality Estimates). Cary, NC: SAS Institute; 2011. https://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/statug_factor_sect002.htm