## Mediation Term Paper

### Analyzing the Models Simultaneously

By simultaneously investigating mediation and moderation, the effects may not only be disentangled and analyzed separately but can also be evaluated together. There have been two primary effects analyzed in the literature: (a) the mediation of a moderator effect, and (b) the moderation of an indirect effect. The mediation of a moderator effect involves exploring mediating mechanisms to explain an overall interaction of XZ in predicting Y, whereas the moderation of an indirect effect involves investigating whether a mediated relation holds across levels of a fourth, moderating variable. These effects have previously been referred to as mediated-moderation and moderated-mediation in the literature, respectively. These alternative descriptions may enhance the distinction between the two.

Previous models to simultaneously test mediation and moderation effects have been presented with varying notation (e.g., Edwards and Lambert 2007; James and Brett 1984; Muller et al. 2005; Preacher et al. 2007) or without testable equations (e.g., Baron and Kenny 1986; Wegener and Fabrigar 2000), making it difficult to understand similarities and differences among the methods. Moreover the criteria for testing the effects have varied across sources, making it hard to extrapolate recommendations for use. It is possible to create a general model to test these effects, however, that subsumes several previous frameworks by including all possible interactions between variables in the mediation and moderation models (MacKinnon 2008). Such a model unifies the methods into a single presentation where different models are represented as special cases of the larger framework. Three regression equations form the model:

*Y* = *i*_{6} + *c*_{1}*X* + *c*_{2}*Z* + *c*_{3}*X**Z* + *e*_{6}

(6)

*M* = *i*_{7} + *a*_{1}*X* + *a*_{2}*Z* + *a*_{3}*X**Z* + *e*_{7}

(7)

(8)

where all predictors in the model are centered at zero to improve interpretation of the lower order coefficients. In Eq. 6, *c*_{1} is the effect of the independent variable on the outcome when *Z* = 0 (also the average effect of X on Y because the mean of *Z* = 0), *c*_{2} is the effect of the moderator variable on the outcome when *X* = 0 (also the average effect of Z on Y because the mean of *X* = 0), *c*_{3} is the effect of the interaction between the independent variable and the moderator on the outcome, and *i*_{6} and *e*_{6} are the intercept and the residual in the equation, respectively. In Eq. 7, *a*_{1} is the effect of the independent variable on the mediator when *Z* = 0 (also the average effect of X on M because the mean of *Z* = 0), *a*_{2} is the effect of the moderator variable on the mediator (also the average effect of Z on M because the mean of *X* = 0), *a*_{3} is the effect of the interaction between the independent and moderator variables on the mediator, and *i*_{7} and *e*_{7} are the intercept and the residual in the equation, respectively. In Eq. 8, *c*′_{1} is the effect of the independent variable on the outcome when *M* = 0 and *Z* = 0 (the average effect of X on Y), *c*′_{2} is the effect of the moderator on the outcome when *X* = 0 and *M* = 0 (the average effect of Z on Y), *c*′_{3} is the effect of the interaction between the independent and moderator variables on the outcome when *M* = 0 (the average effect of XZ on Y), *b*_{1} is the effect of the mediator on the outcome when *X* = 0 and *Z* = 0 (the average effect of M on Y), *b*_{2} is the effect of the interaction between the moderator and mediator variables on the outcome when *X* = 0 (the average effect of MZ on Y), *h* is the effect of the interaction between the independent and mediator variables on the outcome when *Z* = 0 (the average effect of XM on Y), and *j* is the effect of the three-way interaction of the mediating, moderating, and independent variables on the outcome. The intercept and residual in Eq. 8 are coded *i*_{8} and *e*_{8}, respectively. A path diagram for the model is presented in Fig. 6.

Fig. 6

MacKinnon (2008) General Joint Analysis Mode. *Note.* X= the independent variable, Y= the dependent variable, Z= the moderator variable, M= the mediating variable, XZ= the interaction of X and Z, MZ=the interaction of M and Z, XM= the interaction of X and**...**

Assumptions of the general model include assumptions of the mediation and moderation models as described earlier. Issues of causal inference in non-additive models may also require additional stipulations for estimation. Note that the presence of any significant two-way interactions in the model implies that the main effects of X and M do not provide a complete interpretation of effects. The presence of a significant three-way interaction in the model also implies that lower order two-way interactions do not provide a complete interpretation of effects. If there are significant interactions, point estimates can be probed with plots and tests of simple effects to probe the interaction effects. Edwards and Lambert (2007), Preacher et al. (2007), and Tein et al. (2004) provide methods to perform these analyses.

#### Testing effects: Criteria for the moderation of an indirect effect

To examine whether an indirect effect is moderated, it is of interest to investigate whether the mediated effect (*ab*) differs across levels of a fourth, moderating variable. Previous sources have argued that this effect can be defined by either a moderated *a* path, a moderated *b* path, or both moderated *a* and *b* paths in the mediation model (James and Brett 1984; Muller et al. 2005; Preacher et al. 2007; Wegener and Fabrigar 2000), such that if there is moderation in either path of the indirect effect then the mediated relation depends on the level of a moderator variable. There are circumstances, however, in which a heterogeneous *a* or *b* path does not imply a heterogeneous *ab* product term.

Although significant heterogeneity in either the *a* or *b* path may imply significant heterogeneity in the *ab* product term in some cases, examining moderation of the product term or moderation of both paths versus examining moderation of single paths in the mediation model are not conceptually identical. Consider the following example where a moderated *a* path in the mediation model means something different from both moderated *a* and *b* paths in the model. Presume that X is calcium intake, M is bone density, Y is the number of broken bones, and Z is gender. Calcium intake is known to have an effect on the bone density of women, and the relation between calcium intake and bone density is stronger in women than it is in men (i.e., heterogeneity in the *a* path in the model). Specifically, men have greater bone density in general and thus yield fewer gains from supplemental calcium intake. However, bone density affects the fragility of bones in a constant way across males and females, such that low bone density leads to more broken bones (i.e., no heterogeneity of the *b* path in the model). Previous models would deem this scenario as moderation of the indirect effect, arguing that moderation of the *a* path suffices as a test for the effect. There are two problems with this argument. First, testing the heterogeneity of only the *a* or *b* path in the mediation model is not a test of mediation because only a single link in the mediated effect is tested in each case. Second, a heterogeneous *a* path in this model suggests something different from both heterogeneous *a* and *b* paths or a heterogeneous *ab* product. Heterogeneity in both paths of the mediated effect would suggest that gender not only moderates the effect of calcium intake on bone density, but that gender also moderates the effect of bone density on broken bones. Heterogeneity of the product estimate of the mediated effect would suggest that gender moderates the mechanism by which calcium intake affects bone loss; this may or may not be true based on the research literature. Although the moderation of a single path may imply moderation of the product term in some cases, it is critical to differentiate the scenarios as they correspond to different research hypotheses.

There are also numerical examples that show instances when heterogeneity in individual paths of the mediation model does not imply heterogeneity of the product term. Consider the following mediated effect scenarios in two moderator-based subgroups:

Mediated Effect in Group 1 | Mediated Effect in Group 2 | |

Case 1: | (a = −2)(b = −2) | (a = 2)(b = 2) |

Case 2: | (a = 1)(b = 2) | (a = 2)(b = 1) |

In both scenarios the *a* and *b* paths are heterogeneous across groups thus satisfying criteria for the moderation of a mediated effect as defined by Edward and Lambert (2007), James and Brett (1984), Morgan-Lopez and MacKinnon (2006), Preacher et al. (2007), and Wegener and Fabrigar (2000). However the *ab* product is identical across groups, indicating that there is no moderation of the indirect effect. Although tests for the moderation of a mediated effect based on the heterogeneity of individual path coefficients in the mediation model will be more powerful than a test based on the heterogeneity of the product term (given the usual low power to detect interactions), these tests may also have elevated Type 1 error rates.

Despite potential problems for making inferences on moderation of the *ab* product from information on the moderation of individual paths, initial simulation work suggests that extending a test of joint significance (where the test for mediation is based on the significance of component paths in the model such that if both *â* and are significant then the mediated effect *â* is deemed significant) to models for mediation and moderation may be acceptable. Specifically, if conclusions about the moderation of *â* are based on whether both *â* and are significantly affected by the moderator variable, Z, Type 1 error rates never exceed .0550 (Fairchild 2008). Effects of the moderator variable on component paths of the mediation model are examined using Eq. 7 and 8, where *â*_{3} quantifies the effect of Z on the *a* path and _{2} quantifies the effect of Z on the *b* path, respectively (See Fig. 6). If both coefficients are significant, it may be claimed that there is significant moderation of the indirect effect. To obtain either a point estimate or confidence limits for the effect, a product of coefficients test can be used.

To estimate a product of coefficients test for moderation of the indirect effect in the case of a dichotomous moderator variable, separate mediation models can be estimated for each group and equivalence of the *â* point estimates can be compared across moderator-based subgroups. An example of a dichotomous moderator variable might be gender or clinical diagnosis. The null hypothesis associated with the test is that the difference between the two mediated effect point estimates in each group is zero:

(9)

If the point estimates in each group are statistically different from one another, there is significant moderation of the indirect effect (i.e., heterogeneity in the *ab* product), such that the mediated effect is moderated by group membership. To test the estimate in Eq. 9 for statistical significance, the difference is divided by a standard error for the estimate to form a *z* statistic. If the groups are independent, the standard error of the difference between the two coefficients is:

(10)

Where is the variance of the mediated effect in group 1 and is the variance of the mediated effect in group 2.To test heterogeneity of the indirect effect in the case of a continuous moderator variable, variance in the estimates of the *ab* product across levels of the moderator variable is examined. An example of a continuous moderator variable might be individual motivation to improve. Because a *z* test as shown above can only accommodate moderator variables with two levels (or a small number of levels with contrasts between two variables) and because levels of continuous moderators often do not represent distinct groups, tests with continuous moderators are more complicated. The question becomes how to assess differences in the *ab* product across a large number of levels of the moderator variable, and the answer to that question is incomplete at this time. Random coefficient models assess variance in regression coefficients across multiple levels such as multiple levels of a moderator. If the moderator is thought of as a higher order variable across which lower order effects (such as *ab*) may vary, the random coefficient modeling framework may be suitable to assess variance in the indirect effect across levels of a continuous moderator. Kenny, Korchmaros, and Bolger (2003) describe an estimate of the variance of *ab* when *a* and *b* are correlated for the case of random effects based on Aroian (1947):

(11)

Bauer, Preacher, and Gil (2006) make an important distinction between the variance of *ab* and the variance of the average value of *ab* in the multilevel model. It is the variance of *ab* that is relevant to the question of whether an indirect effect is moderated as we would like to know if there is substantial variability in *ab* across levels of the moderator variable.

It is also possible to test whether individual paths in the mediation model differ across levels of a moderator variable. These tests can investigate moderation of the direct effect of X on Y (*c*′) or evaluate the generalizability of action and conceptual theory for a program. The null hypothesis to test homogeneity of a program’s direct effect is:

This hypothesis is tested by examining the significance of in Eq. 8; if the regression coefficient is significant, there is significant moderation of the program’s direct effect.

Recall that conceptual theory for a program corresponds to the *b* path of the mediation model, which defines the theory that links variables or psychological constructs (e.g., M) to behavioral outcomes. The relationships examined in this piece of the model are driven by previous models or theories presented in the literature that explain motivations for behavior. The null hypothesis to test homogeneity of a program’s conceptual theory across levels of the moderator variable is:

This hypothesis is tested by examining the significance of _{2} in Eq. 8; if the regression coefficient is significant there is significant moderation of the program conceptual theory.

Action theory for a program corresponds to the *a* path of the mediation model, which defines what components of the program are designed to manipulate the mechanisms of change. This piece of the model illustrates how the program intervenes to modify hypothesized mediators. The null hypothesis to test homogeneity of a program’s action theory across levels of the moderator variable is:

This hypothesis is tested by examining the significance of *â*_{3} in Eq. 7; if the regression coefficient is significant there is significant moderation of the program action theory. See Chen (1990) for more details on program action and conceptual theory.

#### Testing effects: Criteria for the mediation of a moderator effect

A test for the mediation of a moderator effect examines whether the magnitude of an overall interaction effect of the independent variable (X) and the moderator variable (Z) on the dependent variable (Y) is reduced once the mediator is accounted for in the model (Muller et al. 2005). Thus, the examination of the mediation of a moderator effect considers the mediation model as a means to explain why a treatment effect of X on Y is moderated by a third variable, Z. In this way, the mediation of a moderator effect hypothesis probes mediation as a possible process that accounts for the interaction of the treatment and the outcome. There is no need to differentiate methods for categorical and continuous moderator variables here.

One way to test whether an XZ interaction in the data is accounted for, at least in part, by a mediating relation is to examine whether the magnitude of the regression coefficient corresponding to the overall interaction effect, *c*_{3}, is reduced once the mediator is added to the model. Using coefficients from Equations 6 and 8, where *ĉ*′_{3} represents the direct effect of the interaction effect on Y once the mediator is included in the model, a point estimate and standard error of this difference can be computed:

(15)

(16)

Dividing an estimate of the difference in Eq. 15 by its standard error in Eq. 16 provides a test of significance for the estimate. If there is a significant difference in the coefficients then the overall moderation effect of XZ on Y is significantly explained, at least in part, by a mediated relation. The null hypothesis corresponding to this effect is that there is no reduction in the overall interaction once accounting for the mediator:

A test of the mediation of a moderator effect with a product of coefficients estimator circumvents the need for testing the overall interaction of XZ on Y as shown above. Morgan-Lopez and MacKinnon (2006) presented a point estimator and standard error for the product of coefficients method:

(19)

The product of coefficients estimator in Eq. 18 illustrates that the effect of the interaction of the independent and moderator variables on the outcome (represented by *â*_{3}) is transmitted through a mediating variable (represented by _{1}). Dividing the estimate of the product by an estimate of its standard error in Eq. 19 provides a test of significance for the estimate based on normal theory. However, as described for the product of coefficients estimator in the single mediator model, asymmetric confidence intervals for the distribution of the product are more accurate than tests based on the normal distribution and should be implemented (note also that a test of joint significance of the coefficients could be conducted). If the product is significantly different from zero, then the moderator effect is explained, at least in part, by a mediating mechanism. The null hypothesis corresponding to this test is that the product of the coefficients is equal to zero:

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