Many papers have delineated the procedures that researchers should follow to test models that comprise a single mediator. To illustrate, consider a researcher who wants to examine why two variables, such as workload and dishonesty, are related to one another. They might, for example, want to assess the proposition that workload promotes stress, which in turn tends to provoke dishonesty. That is, in this example, they want to ascertain whether stress mediates the relationships between workload and dishonesty.

Two main approaches are commonly applied to assess mediation. The first approach is sometimes called the causal steps strategy and was promulgated by Baron and Kenny in 1986. In essence, this strategy comprises a coordinated series of three or four tests, such as multiple regression analyses.

The second approach is sometimes called the product of coefficients approach. Researchers need to conduct only two regression analyses. Two of the B values are then multiplied together, representing the indirect effect-for example, the extent to which workload relates to stress and the degree to which stress relates to dishonesty. Then, researchers apply a formula or process, such as the Sobel test, to ascertain the standard error of this indirect effect. The indirect effect divided by the standard error is used to assess whether or not mediation can be inferred.

Typically, the concept of mediation is illustrated in the following diagram. The researcher wants to examine why variable X, such as workload, is related to variable Y, such as dishonesty. In particular, they want to ascertain whether variable M, a mediator, mediates this relationship between variable X and variable Y.

In many instances, however, the model comprises more than one mediator. To demonstrate, the researcher might feel that workload induces not only stress, but other psychological states, such as excitement and anger. These psychological states might all provoke dishonesty. Hence, the model entails three mediators: stress, excitement, and anger. These mediators are represented in the following diagram.

Unfortunately, most discussions that center on mediation assume that researchers often want to examine one mediator only. Several complications arise when researchers need to assess several mediators (see Preacher & Hayes, 2008). Preacher and Hayes (2008) recommend that researchers apply the product of coefficients approach to examine mediation in this context.

To demonstrate and justify the product of coefficients approach to examine multiple mediators, a cursory understanding of the causal steps strategy is essential. Baron and Kenny (1986) delineated the most popular variant of the causal steps strategy but did not generalize this approach to instances in which the researcher wants to examine more than one mediator.

Nevertheless, Preacher and Hayes (2008) showed how the causal steps approach could be extended to multiple mediators. To fulfill this objective, they need to conduct a series of multiple regression analyses-often by selecting "Analyze", "Regression", and "Linear" in SPSS, and then merely choosing the appropriate dependent variable and independent variables.

- For the first analysis, the dependent variable is X, such as workload, and the independent variable is Y, such as dishonesty. In this instance, the unstandardized B value associated with X represents c in the previous diagram. If c is significantly different to zero, the researcher has confirmed that X is indeed related to Y.
- For the second analysis, the independent variable is X, such as workload, and the dependent variable is M1, such as stress. In addition, the other mediators, such as excitement and anger, are also included as independent variables. In this instance, the unstandardized B value associated with X represents a1 in the previous diagram. If a1 is significantly different to zero, the researcher has confirmed that X is indeed related to M1, after controlling the other mediators. This process is then repeated to examine the other mediators in turn.
- For the third analysis, the dependent variable is Y, such as dishonesty, and the independent variables are M1, M2, and M3, as well as X, such as stress, excitement, anger, and workload respectively. In this instance, the unstandardized B value associated with M1 represents b1 in the previous diagram-that is, the extent to which M1 is related to Y after controlling X, M2, and M3. Likewise, the unstandardized B value associated with M2 represents b2 in the previous diagram-that is, the extent to which M2 is related to Y after controlling X, M1, and M3, and so forth.
- In addition, the unstandardized B value associated with X in the same analysis represents c*--that is, the extent to which X is related to Y after controlling M1, M2, and M3. If c* is no longer significant, then M1, M2, and M3 must fully mediate the association between X and Y.

Although these instructions might be difficult to read, these regression analyses are simple to conduct. To conduct the product of coefficients approach, the reader merely needs to recognize that a1, a2, a3, b1, b2, and b3 in the previous diagram represent unstandardized B coefficients, all of which can be readily extracted from multiple regression analyses. Of course, analogous regression analyses are conducted if the number of mediators is 2, 4, 5 and so forth rather than 3.

The causal step strategy, although popular, does present some difficulties.

- First, this approach is not especially powerful (MacKinnon, Lockwood, Hoffman, West, & Sheets, 2002).
- Second, this approach cannot be as readily adjusted accommodate complications, such as violations of normality (Preacher & Hayes, 2004).

To override these issues, the product of coefficients approach might be preferable (MacKinnon, Lockwood, Hoffman, West, & Sheets, 2002;; Preacher & Hayes, 2004, 2008). To conduct the product of coefficients approach, researchers need to:

- Extract a1, a2, a3, b1, b2, and b3 from the previous regression analyses, as shown in reference to the causal steps strategy.
- Although not always necessary, extract the standard error of all these terms-which usually appear in the output alongside the corresponding coefficients in a column labeled SE.
- Calculate the total direct effect, which equals a1 x b1 + a2 x b2 + a3 x b3 .... Note this total direct effect = c - c*. Often, c is called the total effect, and c* is called the direct effect.
- Compute the standard error of this total direct effect. Several formulas have been developed, and these variants will be presented later.
- Divide the total indirect effect by its standard error. The ensuing number roughly conforms to a z distribution. Hence, if this ratio is greater than 1.96 or less than -1.96, the researcher may conclude the total indirect effect is indeed significant-which, in essence, implies mediation.

To reiterate, the product of coefficients approach is straightforward, at least after the appropriate regression analyses are conducted. The only complication revolves around estimating the standard error of this indirect effect.

Several formulas have been developed to calculate this standard error. The simplest formula, which is asymptotically correct-that is, accurate with very large samples-is:

b12 sa12 + a12 ba12 + b22 sa22 + a22 sb22 + b32 sa32 + a32 sb32 + 2 (a1 a2 sb1 sb2 + a1 a3 sb1 sb3 +

a2 a3 sb2 sb3 + b1 b2 sa1 sa2 + b1 b3 sa1 sa3 + b2 b3 sa2 sa3 )

In this formula, which was derived by Bollen (1987, 1989) and promulgated by Preacher and Hayes (2008), the s values refer to the corresponding standard errors. Although more precise formulas, applicable to smaller samples, have been derived, these adjustments are usually negligible (Preacher, Rucker, Hayes, 2007) or unsuitable when the mediators are correlated.

The previous set of procedures are conducted to ascertain whether the total indirect effects-in this instance, the three pathways-together mediate the relationship between X and Y. Researchers also often want to examine which of these pathways are significant. That is, they might want to assess which of a1 x b1, a2 x b2, or a3 x b3 reach significance. The same procedure that is used to compute the total indirect effect can be applied to compute specific indirect effects, except:

- A specific indirect effect comprises only one term, like a1 x b1, rather than a1 x b1 + a2 x b2 + a3 x b3.
- The standard error comprises only the terms that correspond to the pathway. To compute the standard error for a1 x b1, any terms in the previous formula that include a 2 or 3 are dropped.

According to Preacher and Hayes (2008), the ratio of indirect effects and their standard errors are assumed to be normally distributed. That is, if individuals computed this ratio many times with similar, but distinct, samples, the distribution of this outcome should conform to a bell shape. Indeed, this formula assumes multivariate normality-a more stringent assumption-which is seldom fulfilled unless the sample is especially large.

To override this problem, Preacher and Hayes (2008) recommend a bootstrapping technique. In principle, researchers should:

- Extract a sample of n cases from the data, by randomly sampling with replacement.
- For example, although unrealistic suppose the sample comprised 5 participants only, with scores of (1, 4, 2, 6, 7)& (5, 2, 2, 6, 7)& (6, 4, 7, 6, 7)& (1, 3, 8, 6, 7)& (9, 5, 1, 6, 7).
- A computer would then randomly extract 5 of these cases, permitting one case to be selected more than once. The computer might extract (1, 4, 2, 6, 7)& (1, 4, 2, 6, 7)& (6, 4, 7, 6, 7)& (6, 4, 7, 6, 7)& (9, 5, 1, 6, 7).
- Next, the researcher would compute the indirect effects from this sample.
- The researcher would then complete this process, perhaps 1000 or so times, on each occasion calculating a different indirect effect
- This process would generate a distribution of indirect effects
- This distribution can be used to compute 95% confidence intervals of the indirect effect
- Briggs (2006) showed that minor adjustments to this process, which yield estimates called the bias corrected (BC) and bias corrected and accelerated (BCa) optimize the results-and are certainly more effective than application of formulas to calculate standard errors.

In practice, bootstrapping is time consuming, of course, unless the researcher constructs a macro or computer code to conduct this process. Fortunately, if researchers use SPSS, Preacher and Hayes (2008) recommend that researchers access www.quantpsy.org to download this code. When this code is applied to the data, several tables are generated. A subset of this output is presented below

Lower | Upper | |

TOTAL | .06 | .12 |

Stress | .02 | .14 |

Excitement | -.03 | .14 |

Anger | -.23 | .10 |

C1 | .04 | .12 |

C2 | -.12 | .01 |

C3 | -.02 | .05 |

According to this table:

- The 95% confidence interval for the total indirect effect is (.06 to .12). This interval does not include 0. Hence, the total indirect effect differs significantly from 0
- The 95% confidence interval for the indirect effect that relates workload to stress and ultimately to dishonesty (.02 to .14). This interval does not include 0. Hence, this specific indirect effect differs significantly from 0
- The other indirect effects do not differ significantly from zero
- C1, C2, and C3 are comparisons between the indirect effects-which are not applicable to this example.

Once the code is downloaded, the products of coefficients approach, together with the bootstrapping technique to estimate the standard error, is straightforward. A few complications need to be considered.

First, strictly speaking, the products of coefficients approach tests indirect effects-such as a1 x b1-rather than mediation. Mediation is established only after the researcher shows that both the total indirect effect is significant and X is related to Y. If the researcher does not show that X is related to Y, mediation cannot be inferred from the indirect effect& that is, M cannot mediate the relationship between X and Y, because this relationship might not exist.

Therefore, researchers often report the relationship between X and Y before presenting the indirect effect. They might write "A regression analysis showed that workload was significantly associated with dishonesty, B = .213, p < .01. Furthermore, to establish whether the indirect effects were significant, a bootstrapping procedure, with 5000 replications, was applied to estimate the standard error (Preacher & Hayes, 2004). The total indirect effect was .143, z = 2.10, p < .05, implying that stress, excitement, and anger do mediate the relationship between workload and dishonesty. However, only the indirect effect associated with stress was significant at .23, z = 1.99, p < .05.

Second, in the previous examples, no control variables were included. Researchers might want to estimate all of these B values or coefficients after controlling age, for example. These control variables can merely be included as additional independent variables in each of the regression analyses& no other changes are needed to the core procedure.

Third, in the previous examples, only one dependent variable and independent variable were included. However:

- If the model comprises more than one dependent variable, the entire process can be repeated for each of these criteria separately.
- If the model comprises more than one independent variable, all of these predictors can be examined simultaneously in the regression equations. Then, a separate indirect effect is computed for each independent variable.

Nevertheless, a problem arise when several independent variables are utilized. The bootstrapping macro, recommended before, can examine only one independent variable at a time. Yet, the researcher might want to assess each independent variable after controlling other independent variables or confounds.

To fulfill this objective, individuals can examine the unique component of each independent variable, one at a time. That is, they can undertake a regression in which one of the independent variables is designated as the dependent variable& the other independent variables are indeed designated as the independent variables. They should save the "unstandardized residuals". The ensuing column represents the unique component of the variable and can be applied in subsequent analyses.

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