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In addition to testing relationships with a single dependent variable and a number of independent variables, the multiple regression method can be extended to the analysis of paths of relationships (e.g. Aguinis & Pierce 1999, Baron & Kenny 1986, Cohen &

Cohen 1975). In this study, multiple regression analysis is used to test mediation effects. A variable may be considered a mediator to the extent to which it carries the influence of a given independent variable to a given dependent variable (Baron &

Kenny 1986). Illustration of mediation is presented in Figure 4-1.

Dependent variable (Y) Independent

variable (X)

Dependent variable (Y) Mediating

variable (M) Independent

variable (X)

b_XY

bXM

bXY’

bMY

Unmediated model

Mediated model

Figure 4-1 Illustration of mediation effect Steps in Establishing Mediation

Baron and Kenny (1986) have presented four steps in establishing mediation. This study follows these steps in testing mediation hypotheses. The steps are presented below for variable M mediating the relationship between independent variable X and dependent variable Y as illustrated in Figure 4-1.

Step 1. Show that the independent variable is correlated with the outcome variable (b_XY > 0). Use Y as the criterion variable in a regression equation and X as a predictor.

This step demonstrates that there is an effect that can be mediated.

Step 2. Show that the independent variable is correlated with the mediator (b_XM > 0).

Use M as the criterion variable in the regression equation and X as a predictor. This step essentially involves treating the mediator as if it were an outcome variable.

Step 3. Show that the mediator affects the outcome variable (b_MY > 0). Use Y as the criterion variable in a regression equation and X and M as predictors (estimate both b_MY and b_XY’ in same the model). It is not sufficient just to correlate the mediator with the outcome; the mediator and the outcome may be correlated because they are both caused by the independent variable X. Thus, the independent variable must be controlled in establishing the effect of the mediator on the outcome.

Step 4. To establish that M completely mediates the X→Y relationship, the effect of X on Y controlling for M should be zero (b_XY’ = 0). The effects in both Steps 3 and 4 are estimated in the same regression equation.

If all four of these steps are met, then the data are consistent with the hypothesis that variable M completely mediates the X→Y relationship. However, if the first three steps are met but Step 4 is not, then partial mediation is indicated. Moreover, Step 1 is not necessarily required for establishing mediation, because a path from the independent variable to the outcome variable is implied if Steps 2 and 3 are met. If b_XY’ is opposite in sign to b_XM * b_MY, then it could be the case that Step 1 is not met, but there is still

mediation. In this case, the mediator acts like a suppressor variable. Therefore, the essential steps in establishing mediation are Steps 2 and 3.

The amount of mediation is defined as the reduction of the effect of the initial independent variable on the dependent variable between the unmediated and mediated model. This difference in coefficients can be shown to equal exactly the product of the effect of X on M times the effect of M on Y (b_XM * b_MY = b_XY - b_XY’). The exact equality holds for multiple regression and structural equation modeling without latent variables, but it holds only approximately for structural equation model with latent variables. The amount of reduction in the effect of X on Y is not equivalent to either the change in variance explained or the change in an inferential statistic such as F or a p value. It is possible for the F from the independent variable to the outcome to decrease dramatically even when the mediator has no effect on the outcome.

Test of Mediation

If Step 2 (the test of b_XM > 0) and Step 3 (the test b_MY > 0) are met, it follows that there necessarily is a reduction in the effect of X on Y in the mediated model. An indirect and approximate test that b_XM * b_MY = 0 is to test that both b_XM and b_MY are zero (Steps 2 and 3).

Baron and Kenny (1986) provided a direct test of b_XM * b_MY which is a modification of a test originally proposed by Sobel (1982). It requires the standard error of b_XM or s_XM (which equals b_XM/t_XM where t_XM is the t-test of coefficient b_XM) and the standard error of b_MY or s_MY. Following Goodman (1960), the standard error of b_XM * b_MY can be shown to equal

Goodman I test: standard error = b_MY²*s_XM² +b_XM²*s_MY² +s_XY²*s_MY² (1) The test of the indirect effect is given by dividing b_XM * b_MY by the above standard

error and treating the ratio as a Z test (i.e., larger than 1.96 in absolute value is significant at the .05 level).

However, different versions of the above standard error have been published (Baron

& Kenny 1986, Goodman 1960, MacKinnon et al. 1995, Sobel 1982). The above formula (Goodman I) is a population formula (Baron and Kenny 1986, Goodman 1960). In the Goodman II version of the test the third term is subtracted for an unbiased estimate of the variance of the mediated effect, which can sometimes have the unfortunate effect of yielding a negative variance estimate. Sobel (1982) presented an approximation of the above formula without the last term. The formulas only differ in the last term and its size is usually trivial in that it depends on sample size squared whereas the other terms depend only on sample size. Baron and Kenny (1986) recommended using the Goodman I version of the Sobel test because it does not make an unnecessary assumption that the product of s_XM and s_MY would be negligible small.

MacKinnon et al. (1995) analyzed these tests using simulation and concluded that the Sobel test and the Goodman I test performed best in their analysis and converged

closely with sample sizes greater than 50. In this dissertation, the first version (Goodman I) of the mediation test is used (Baron & Kenny 1986, Goodman 1960).

The formula is