For this posting, I will be providing an overview of Process model 8 in SPSS. To begin, let us pick up from my previous post (link) on Process model 7 using SPSS. In that posting, I began with a brief introduction to the model using conceptual and statistical diagram as provided in Hayes (2018).
In Process model 7, the proposed moderator (W) is specified as conditioning the effect of the focal independent variable (X) on the mediator.
In order to test this model statistically, we need to estimate parameters using a pair of equations. The first equation involves regressing the mediator (M) onto the independent variable (X), the proposed moderator (W), and the product of X and W (the interaction term). The second equation involves regressing the dependent variable (Y) onto M and X (denoted as path c-prime).
Process model 8 simply extends model 7 by adding in an additional specification, where W also moderates the path from X to Y.
To test this model statistically, it must be specified in the form of the following equations...
Notice that the first equation in Process model 8 is the same as that for Process model 7. It is the second equation that differs between the two models since Process model 8 specifies moderation of the c' path.
SPSS Example
The example we will rely on will actually be a continuation of the example from my previous post (link) on Process model 7 using Stata. We will be testing the effect of student subject matter interest (X) on achievement (Y) via the mediator (M), engagement using this simulated data [DOWNLOAD HERE]. We will test whether self-efficacy moderates the mediation of the effect of interest on achievement is itself moderated by self-efficacy (W). However, rather than relying on Process model 7, we will use Process model 8. This means that we will also be testing whether efficacy (W) moderates the direct effect of interest (X) on achievement (Y). Finally, we are going to included socioeconomic status (SES) as a covariate in the model.
Here is the conceptual diagram of our model:
Here is our statistical diagram...
...representing the following equations...
Now, let's use the Process macro in SPSS to generate our output.
Select Model number 8 (from the dropdown) and then specify the model (as shown below).
Under Options, click Generate code for visualizing interactions (useful for visualizing simple slopes). For this first demo, I will request mean-centering of the focal independent variable (X) and proposed moderator (W) prior to forming the product term. The dropdown for Probe Interactions is set by default at "if p<.10". What this means is that if the interaction term is significant in the first regression model, then you will obtain simple slopes tests at mean-1sd, mean, and mean+1sd of the moderator (efficacy). If you would like to would generate simple slopes and to test them, irrespective of whether the interaction term is significant, you can set this to "Always" (or some other p-value that is > .10). For this example, I will leave it set to p<.10. Finally, for this example, I will rely on conditioning values at the 16th, 50th, and 84th percentiles of the X and W variables.
Making sense of the results
The first set of results are based on the first regression of the mediator onto interest (X), efficacy (W), their interaction (XW), and SES (the covariate).
The interaction between interest and self-efficacy is significant (b=.0288, s.e.=.0099, p=.0037). This finding is consistent with our assumption that efficacy moderates the effect of interest on engagement.
Since we mean-centered our interest and efficacy variables, we can interpret the slopes for these variables as the predicted relationship between one of the variables at the mean of the other. For hypothetical cases falling at the mean on efficacy, interest has a positive and significant effect on engagement (b=.3471, s.e.=.0478, p<.001). Those hypothetical cases falling at the mean on interest, efficacy is also a positive and significant predictor of engagement (b=.2297, s.e.=.0341, p<.001).
Finally, we see that SES is a positive and significant predictor of engagement (b=.4198, s.e.=.0443, p<.001) of engagement, holding the remaining predictors constant.
Under Options, we requested conditioning values based on the 16th, 50th, and 84th percentiles of our X and W variables. Below, we have simple slopes and tests computed for the effect of interest on engagement at the 16th, 50th, and 84th percentiles of efficacy. We see that at the 16th percentile of efficacy, the slope for interest was positive and significant (b=.2219, s.e.=.0659, p=.0008). At the 50th percentile, the slope was positive and significant (b=.3461, s.e.=.0478, p<.001). Finally, the slope computed at the 84th percentile of efficacy was positive and significant (b=.4766, s.e.=.0637, p<.001). Notice that the slopes appear to be increasing as we move from lower relative levels of efficacy to higher levels: .222<.346<.477.
Next in the output is syntax for generating a plot of the simple slopes. You can generate the figure following the same steps as described
HERE.
The plot on contains lines depicting the simple slopes from the regression of the mediator onto X, W, and XW. The red line represents the effect of interest on engagement at the 84th percentile (relatively high) for efficacy, whereas the light blue line represents the effect of interest on engagement (relative low) for efficacy. The green line represents the simple slope at the 50th percentile of efficacy. Consistent with what we saw with the simple slopes tests above, the positive effect of interest on engagement becomes increasingly positive as we move from lower to higher levels of efficacy.
The next part of our output contains the regression of achievement (Y) onto interest (X), efficacy (W), their interaction (XW), and SES (covariate).
We see here that the interaction between interest and self-efficacy is not statistically significant (b=.0029, s.e.=.0113, p=.8012). Failure to reject the null (as in this case) leads us to assume that the moderator (efficacy) is not conditioning the direct effect of the independent variable (interest) on the dependent variable (achievement).
Given no evidence of statistical moderation, there is no real value-added in probing simple slopes. However, let's generate this output for the sake of demonstration.
Notice that the slopes do not appear to vary much from the 16th percentile of the moderator (b=.3962) to the 84th percentile (b=.4214) of the proposed moderator.
Finally, we see the Index of Moderated Mediation (IMM) and conditional indirect effects (that is, the indirect effects computed at the conditioning values of the moderator, efficacy).
The IMM (a3b1) represents the linear change in the conditional indirect effect per one unit increase on the moderator based on the function (for the current model, depicted below): f(W) = a1b1+ a3b1W.
For our model the IMM = .0054, and the percentile bootstrap confidence interval is (.0018,.0101). Since the null value of 0 does not fall between the lower and upper bound, then we have evidence that efficacy (W) is moderating the effect of interest on achieve by way of the mediator engagement. The positive value of the IMM indicates that as we move from lower levels of the moderator (efficacy) to higher relative levels, the conditional indirect effect becomes increasingly positive.
Now, let us probe the conditional indirect effects given evidence of moderated mediation. The conditional indirect effect at the 16th percentile of efficacy is .0417, and the 95% bootstrap confidence interval is (.0153,.0753). The conditional indirect effect at the 50th percentile of efficacy is .0651, and the 95% bootstrap confidence interval is (.0372,.0989). The conditional indirect effect at the 84th percentile of efficacy is .0896, and the 95% bootstrap confidence interval is (.0524,.1345). As you can see, as we move from the 16th to the 50th to the 84th percentile of the moderator, the conditional indirect effect is increasing (i.e., .0417<.0651<.0896).
References and suggested readings
Darlington, R.B, & Hayes, A.F. (2017). Regression analysis and linear models: Concepts, applications, and implementation. New York: The Guilford Press.
Hayes, A.F. (2015). An index and test of linear moderated mediation. Multivariate Behavioral Research, 50, 1-22.
Hayes, A.F. (2018). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach (2nd edition). New York: The Guilford Press.
Hayes, A.F. & Rockwood, N.J.(2020). Conditional process analysis: Concepts, computation, and advances in the modeling of the contingencies of mechanisms. American Behavioral Scientist, 64, 19-54. [Download from Sage site here: link]
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