Several different effect-size measures for mediation may be calculated from the two regression equations presented in

Equations 1 and

2, and some of these effect-size measures have been applied in the mediation literature. The focus of effect-size measures in mediation analysis concentrates on comparing the magnitudes of different effects in the model—the indirect effect, the direct effect, and the total effect—in order to assess the relative contribution of each (

Sobel, 1982). One frequently used effect-size measure for mediation is the proportion mediated. This measure indicates what proportion of the total effect is mediated by the intervening variable, and it has been cited in substantive research (e.g.,

Chassin, Pitts, De-Lucia, & Todd, 1999;

Ouimette, Finney, & Moos, 1999;

Wolchik et al., 1993). The proportion mediated also provides a means to assess the relative contribution of single mediators in multiple mediator models by indicating what proportion of the total effect is attributable to individual mediational pathways. The measure is unstable in several parameter combinations, however, and has excess bias in small sample sizes (

MacKinnon, Fairchild, Yoon, & Ryu, 2007;

Taborga, 2000). Specifically, the proportion-mediated measure only performs well with samples of greater than 500. This large sample-size requirement may limit the utility of the measure, given the prevalence of research with smaller sample size.

Other effect-size measures for mediation, such as the partial

*r*^{2} and standardized regression coefficients, have been applied from multiple regression analysis and cited in substantive research (

Taborga, 2000). These measures are qualitatively different from other mediation effect-size measures, such as the proportion mediated, in that they focus on the relation between two variables in the mediation model. The partial

*r*^{2} provides information on the amount of variance in a criterion variable that can be uniquely explained by an independent variable once other variables in the model have been accounted for. In the mediation model, there are two possible partial

*r*^{2} measures corresponding to variance explained in the

*β* and

*τ*′ paths of the model, respectively: (1)

, or the variance in

*Y* that is explained by

*M* but not

*X*; and (2)

, or the variance in

*Y* that is explained by

*X* but not

*M*. The squared correlation between

*X* and

*M*,

, is the variance in

*M* that is explained by

*X*. This measure is not a partial correlation, since

*M* is predicted by a single independent variable.

There are three possible standardized regression coefficients in the mediation model, each corresponding to one of the three unstandardized paths in . Specifically,

*β*_{standardized} represents the change in

*Y* for every 1 standard deviation change in

*M*,

*α*_{standardized} represents the change in

*M* for every 1 standard deviation change in

*X*, and

represents the change in

*Y* for every 1 standard deviation change in

*X*. The standardized regression coefficients provide information on the strength of individual paths in the mediation model in a standardized metric. When testing mediation in a structural equation modeling or path-analysis framework, the standardized regression coefficients are referred to as

*standardized structure coefficients*, but the interpretation of the weights is the same. In both frameworks, the magnitude of the weights is relative to the variables involved in their computation. Both the standardized regression (or structure) coefficients and

*r*^{2} measures provide information only on component parts of the mediation model. Neither of these measures is able to provide information on the mediated effect as a whole.

Given the weaknesses of the proportion-mediated effect-size measure and the limited application of those component effect-size measures from regression analysis, alternative measures of effect size in mediation analysis are still needed. Overall

*R*^{2} measures that quantify variance explained in an outcome provide a useful tool for this goal.

*R*^{2} measures have been proposed in

*commonality analysis*, also known as

*elements analysis* (or

*components analysis*). Originally presented by

Newton and Spurrell (1967) and later refined by others (e.g.,

Mood, 1969,

1971;

Seibold & McPhee, 1979), commonality analysis partitions variance explained in a criterion variable into unique and nonunique parts using multiple squared partial correlation estimates from the model. In addition to assessing the unique contribution of each predictor with squared partial correlations and the total variance explained in a model with the overall

*R*^{2} from regression analysis, commonality analysis also provides estimates of

*common effects* (

Seibold & McPhee, 1979). These quantities provide a promising

*R*^{2} measure of effect size for mediation analysis, since they quantify the proportion of variance in an outcome variable that is common to a set of predictors but not to a predictor alone.

Mood (1969) provided a general equation for determining the

*n*th

*-*order common effect for a set of

*n* predictors, where the −1 that results from expanding any given product is dropped from computation:

All resulting terms following expansion of the product in

Equation 3 become

*R*^{2} quantities to represent the contribution of the variable(s) to variance explained in the outcome. We can apply this general framework to the mediation model to find the variance in

*Y* that is common to both

*X* and

*M* but that can be attributed to neither alone. Applying

Mood’s (1969) formula to the first mediation regression equation (

Equation 1), where

*Y* is predicted from

*X* and

*M*, we find that the variance common to both

*X* and

*M* in predicting

*Y* is

Expanding

Equation 4, rearranging terms, and introducing

*R*^{2} values yields

where

is the portion of the variance in

*Y* explained by

*M* (i.e., the squared raw correlation between the dependent variable and the mediator),

is the portion of the variance in

*Y* explained by

*X* (i.e., the squared raw correlation between the dependent variable and the independent variable), and

is the overall model

*R*^{2} from

Equation 1.

Equation 5 shows that the second-order common effect for the mediation model partitions the observed variance in

*Y* into several parts to isolate that part of the system that is uniquely attributable to the mediated effect. Specifically, the measure subtracts those portions of observed variance in

*Y* that are explained uniquely by

*M* or uniquely by

*X* from the overall observed variance in

*Y*, leaving the portion of variance that is explained by the predictors together. By illustrating the extent to which the combination of

*X* and

*M* together explain variance in

*Y*,

Equation 5 is able to combine component parts of the mediational chain into a single effect-size measure. The meaning of this overlap or redundancy in prediction in the mediation model is a bit different from what it would mean in a single multiple regression equation, since the mediation model is defined by a second equation in which

*M* is predicted by

*X* (

Equation 2). Because

*M* takes on the unique quality of being both a predictor and an outcome in the mediation model, the second-order common effect is able to provide information about the magnitude of the mediated effect, or the extent to which

*X* predicts variance in

*M*, which subsequently predicts variance in

*Y*. Thus, the second-order common effect for the mediation model—or what we will call the

—informs the researcher about the practical significance of the overall mediation relation.

Decomposition of the component parts in the

measure illustrates how the effect-size measure estimates the portion of variance uniquely associated with the mediated effect in a single mediator model, represented by

in . Recasting the squared raw correlations and the overall

*R*^{2} from

Equation 5 in terms of pieces from the Venn diagram presented in helps illustrate this point. Specifically,

where all terms from

Equation 5 retain their original meaning,

is the portion of variance in

*Y* explained by the mediated effect,

is the squared partial correlation of

*Y* and

*M* partialed for

*X*, and

is the squared partial correlation of

*Y* and

*X* partialed for the influence of the mediator. Substituting terms from

Equations 6–

8 into

Equation 5 further demonstrates how the

measure isolates the portion of variance in

*Y* explained by the mediated effect:

By accounting for variance in the dependent variable explained by the independent variable and the mediator variable together, the

*R*^{2} effect-size measure for mediation estimates the portion of that is labeled

. Note that because the three variables in the diagram are standardized, they all have a variance of 1. Note also that because the

measure does not square the resulting quantity from the difference computed in the equation, it is possible to have negative values of the

estimate. This quality of second-order common effects is what sets them apart from primary effects in multiple regression models; they are not sums of squares and therefore may take on negative values under some circumstances (

Newton & Spurrell, 1967).

Seibold and McPhee (1979) noted that negative values of the estimates indicate that suppression effects may be present; consideration of a particular pair or group of variables in predicting variance in an outcome may lead to the reduction in prediction from either variable alone.

Pilot research on the bias of the

measure has been conducted on a small set of parameter values and sample sizes (

Taborga, 2000). Simulation results suggested that the measure was unbiased at varying sample size for the medium and large effect sizes studied in the simulation. The purpose of the present study was to further investigate

*R*^{2} effect-size measures for mediation analysis. To that end, we evaluated the overall

measure for mediation in a comprehensive set of parameter combinations to expand empirical evidence for its use and to examine component

*r*^{2} measures for the mediation model. Investigating the

measure in a wider set of parameters will demonstrate its performance in a variety of contexts. Simulation results for the accompanying component

*r*^{2} measures in the mediation model (i.e.,

, and

) will complete the presentation of

*R*^{2} measures for mediation. Although previous simulation work has been published on

*r*^{2} measures (

Algina & Olejnik, 2003;

Wang & Thompson, 2007), these studies do not preclude further examination of the measures in the present article. Not only did the sample sizes, effect sizes, and simulation outcome measures in the present study differ from those in previous research, but no study has examined the special case of

*r*^{2} for mediation models either. Following the presentation of the simulation work, the

*R*^{2} effect-size measures for the mediation model are applied to a real mediation example using data from a team-based program to improve the nutrition and exercise habits of firefighters.