Commonality Analysis

Heuristic Example

To make the discussion concrete, presume a regression problem involving a single measured outcome variable (Y) and two predictor variables (X1 and X2). The results associated with the example are presented graphically in the Venn diagram in Figure 1.

In this example, each of the three measured variables has a sum of squares (SOS) of 400, corresponding to the box for each measured variable being drawn as encompassing 20 × 20 units of area. As regards the overlap of X1 and Y, because 100 units of the SOSY = 400 are predicted by X1, the r2YX1 = 100/400 = .2500 = 25.00%. X2 explains 225 of the 400 units of Y, and so the r2YX2 = 225/400 = .5625 = 56.25%.

However, the multiple CORRELATION SQUARED (R2[YX1X2]) does not equal 25.00% plus 56.25%, or 81.25%, because the predictors are correlated (r2X1X2 = 18.75%). Therefore, some of the explanatory power of X1 and X2 is common to both and should not be “double counted” in computing R2[YX1X2]. For this example, X1 and X2 together explain R2[YX1X2] = 68.75% of the variability in the Y scores (and not 81.25%).

But where does this 68.75% effect size originate? The following questions must be addressed:

1.
How much of the observed effect size is due to the unique explanatory power of X1?
2.
How much of the observed effect size is due to the unique explanatory power of X2?
3.
How much of the observed effect size is due to the common explanatory power of X1 and X2?

These are the questions addressed in a commonality analysis.

The analysis is conducted in a series of four steps. First, the R2 is computed using all possible combinations of the predictor variables. For this simple example, there are only three possible combinations, and R2[YX1X2] = 68.75%, R2[YX1] = 25.00%, and R2[YX2] = 56.25%.

Second, values obtained in Step 1 are inserted into commonality analysis formulas provided in various publications (cf. Rowell, 1996). For this example, the unique predictive contribution (U1)of X1 is computed as

The unique contribution (U2)of X2 is computed as

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The common contribution (C12)of X1 and X2 is computed as

The commonality analysis computational formulas become exponentially more complicated as there are more predictor variables (Rowell, 1996). For example, when there are three predictor variables, the formulas are as follows:

Third, as a check on the computations, the R2[YX1X2] is recomputed as