Product of coefficients approach to mediation

Author: Dr Simon Moss

Introduction

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.

Overview of the causal steps strategy

To demonstrate and justify the product of coefficients approach, a cursory understanding of the causal steps strategy is essential. Typically, the concept of mediation is illustrated in the following diagram.

That is, 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.

According to Baron and Kenny (1986), who delineated the most popular variant of the causal steps approach, researchers 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. Specifically:

• 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 M, such as stress. In this instance, the unstandardized B value associated with X represents a in the previous diagram. If a is significantly different to zero, the researcher has confirmed that X is indeed related to M, the mediator.
• For the third analysis, the dependent variable is Y, such as dishonesty, and the independent variables are M and X, such as stress and workload respectively. In this instance, the unstandardized B value associated with M represents b in the previous diagram-that is, the extent to which M is related to Y after controlling X.
• 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 M. If c* is no longer significant, then M must fully mediate the association between X and Y.

Although these instructions might be difficult to read, the three regression analyses are simple to conduct. To conduct the product of coefficients approach, the reader merely needs to recognize that a and b in the previous diagram represent unstandardized B coefficients, both of which can be readily extracted from multiple regression analyses.

Overview of the product of coefficients approach

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

• First, this approach is not especially powerful (MacKinnon, Lockwood, Hoffman, West, & Sheets, 2002). This problem partly arises because this approach demands that three separate coefficients need to be significant, even though mediation must underpin the relationship between X and Y if only one criterion is satisfied-that if c - c* is significantly different from zero.
• 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). To conduct the product of coefficients approach, researchers need to:

• Extract a and b from the previous regression analyses, as shown in reference to the causal steps strategy.
• Although not always necessary, extract the standard error of a and b-which usually appear in the output alongside a and b in a column labeled SE.
• Multiply a and b to compute ab. The ensuing number is called the indirect effect. Note that ab = c - c*. Often, c is called the total effect, and c* is called the direct effect.
• Compute the standard error of ab. Several formulas have been developed, and these variants will be presented later.
• Divide ab by the standard error of ab. The ensuing number conforms to a z distribution. Hence, if this ratio is greater than 1.96 or less than -1.96, the researcher may conclude the indirect effect is indeed significant-which, in essence, implies mediation.

Calculation of the standard error of ab

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

Several formulas have been developed to calculate the standard error of ab. The conventional formula, as stipulated by Aroian (1944), Mood, Graybill, and Boes (1974), as well as Sobel (1982) is:

In this formula, the s values refer to the corresponding standard errors. Unfortunately, this test tends to be unduly conservative, especially when the sample size is small. Some alternative versions have been proposed:

• McKinnon and Dwyer (1993;; McKinnon, Warsi, & Dwyer, 1995) indicate the expression after the second + sign can be removed& this term is usually trivial.
• Goodman (1960) suggests this term can be subtracted rather than added.

Bootstrapping as a means to compute the standard error of ab

According to Preacher and Hayes (2004), the classical estimates of ab assume a form of normality. In particular, the ratio ab and the standard error of ab is assumed to be normally distributed. That is, if individuals computed this ratio a multiple of times with similar, but distinct, samples, the distribution of this outcome should conform to a bell shape.

However, this distribution is often positively skewed (Bollin & Stine, 1990), especially when the sample is small. To override this problem, Bollin and Stine (1990) as well as Preacher and Hayes (2004) 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)& (5, 2, 2)& (6, 4, 7)& (1, 3, 8)& (9, 5, 1).
• 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)& (1, 4, 2)& (6, 4, 7)& (6, 4, 7)& (9, 5, 1).
• Next, the researcher would compute ab from this sample.
• The researcher would then complete this process, perhaps 1000 or so times, on each occasion calculating a different ab
• The standard deviation of these estimates is equivalent to the standard error.

In principle, bootstrapping is straightforward and generates an accurate estimate of the standard error (e.g., Bollin & Stine, 1990). 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 (2004) in Appendix A present this code. They also specify a website from which this code can be downloaded.

Specifically, the researcher needs to:

• Download this code, and then execute this code in SPSS syntax. A SAS version is also available
• In SPSS syntax, type SOBEL y=dishonesty/x=workload/m=stress/boot=1000.
• The number after boot represents the number of bootstrap samples that are generated, and must be specified as multiples of 1000
• The words after y, x, and m represent the variable names in SPSS.

Complications

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 the indirect effect-a x b-rather than mediation. Mediation is established only after the researcher shows that both the 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 effect was significant, a bootstrapping procedure, with 1000 replications, was applied to estimate the standard error (Preacher & Hayes, 2004). The indirect effect was .143, z = 2.10, p < .05, implying that stress does mediate the relationship between workload and dishonesty.

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, mediator, and independent variable were included-called simple mediation. If the model does not constitute simple mediation, some refinements need to be introduced:

• 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 mediator, some adjustments, recommended by Preacher and Hayes (2008), need to be considered.
• 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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