Commonality analysis
Commonality is a statistical technique within linear regression modelling with multiple independent variables[1][2].
Commonality analysis decomposes the variance explained (R-squared) by all independent variables on a dependant variable in a multiple linear regression model in commonality coefficients, which are variance components that are uniquely explained by each independent variables (unique effects, equivalent to the square of the semipartial correlation), and variance components that are shared in each possible combination of the independent variables (common effects). These commonality coefficients sum up to the total variance explained (r-squared) of all the independent variables on the dependent variable. Commonality analysis produces 2k -1 commonality coefficients, were k is the number of the independent variables.
Example
As an illustrative example, in the case of three independent variables (A, B and C), commonality returns 7 (23 -1) coefficients:
· The unique contributions of A, B and C (3 coefficients)
· The contribution common to each possible pair of variables (A&B, B&C, A&C)
· The contribution common to all three variables (A&B&C)
The unique coefficient indicates to which degree the variable is independently associated with the dependent variable. Positive commonality coefficients indicate that a part of the explained variance of the dependent variable is shared between independent variables. Negative commonality coefficients indicate that there is a suppressor effects between independent variables.
Calculation
The calculation of commonality coefficients can be done in principle with any software that calculates R-squared (for example SPSS; see [3]), however, this becomes quickly burdensome as number of independent variable increases. For example, with 10 independent variables, there are 210 -1 = 1023 commonality coefficients to be calculated. The yhat package[4] in R can be used to calculate commonality coefficients, and to produce bootstrapped confidence intervals for commonality coefficients.
References
- ^ Nimon, Kim F.; Oswald, Frederick L. (October 2013). "Understanding the Results of Multiple Linear Regression: Beyond Standardized Regression Coefficients". Organizational Research Methods. 16 (4): 650–674. doi:10.1177/1094428113493929. ISSN 1094-4281.
- ^ Nimon, Kim; Reio, Thomas G. (2011-06-22). "Regression Commonality Analysis: A Technique for Quantitative Theory Building". Human Resource Development Review. 10 (3): 329–340. doi:10.1177/1534484311411077. ISSN 1534-4843.
- ^ "Commonality analysis: Demonstration of an SPSS solution for regression analysis" (PDF).
{{cite web}}: Cite has empty unknown parameter:|dead-url=(help) - ^ Nimon, Kim; Lewis, Mitzi; Kane, Richard; Haynes, R. Michael (May 2008). "An R package to compute commonality coefficients in the multiple regression case: An introduction to the package and a practical example". Behavior Research Methods. 40 (2): 457–466. doi:10.3758/BRM.40.2.457. ISSN 1554-351X.