Using Hayes’ Process macro to perform moderated multiple regression Mike Crowson, Ph.D.
(last updated August 26, 2019)
In this presentation, I demonstrate how to use the Process macro to carry out moderated multiple regression involving a two-way interaction among independent variables. I will demonstrate models incorporating
continuous and/or binary IV’s as predictors of an outcome variable (Y). [Note that in the models that follow, the X variable will be treated as the focal independent variable and the W variable is treated as the moderator
variable, reflecting the manner which Process is set up to input these variables. Nevertheless, due to the underlying “symmetry in moderation” (Hayes, 2018), the designation of X and W variables as focal IV or moderator is more of a conceptual or theoretical issue than a statistical one.]
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downloading it, as it goes into substantially more detail than I can provide in a timely fashion in this video. Also, a link to the data is provided under the video description as well.
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NegLifeEvts Hope Grit
Example 1: We are predicting Hope (Y) from Negative Life Events (X) with Grit (W) serving as a moderator of that relationship. In other words, are hypothesis is that the relationship between
negative life events (our focal IV) and hope (our DV) is moderated by grit. All three variables are being treated as continuous. Below is the conceptual diagram of the relationships for the model. Data can be downloaded from https://drive.google.com/open?id=1H-hetsYww-v-UC41dVohq2Whd6qyZL9A .
This example and data is based in part on that found in:
Jose, P.E. (2013). Doing statistical
mediation and moderation. New York: The Guilford Press.
Model number must be set to “1”.
Additional options can be found under the ‘Options’ tab. Here, I’ve clicked on ‘Generate code for visualizing interactions’ and ‘mean center for construction of products’.
Mean centering involves subtracting the mean from the raw scores on a variable. In our case, we will have
NeglifeEvts(centered) = and Grit(centered) = . So the model will be run with the IV’s being the centered Negative life events and Grit variables, as well as the interaction between the centered values on these variables.
*A common misconception is that mean-centering is a requirement in testing interactions using regression in order to alleviate multicollinearity among the regression terms (Jose, 2013). This is not the case. Nevertheless, although mean centering is not a requirement when carrying out moderated multiple regression, it can
facilitate interpretation of the regression parameters (Hayes, 2018). It is particularly useful in interpretation when a value of 0 does not fall within the range of values on your IV(s).
I’ve also clicked under ‘Probe interactions’ on ‘-1sd,Mean,+1sd’, which are three points along the scale of the (continuous) moderator variable (W) conventionally chosen to represent “low”, “medium”, and “high” values on that variable (see e.g., Aiken & West, 1991). In other words, the relationship between X and Y is tested at those three levels. By setting the ‘Probe interactions to…if p<.10’, only those conditional relationships where p<.10 will appear. If you want the conditional relationships to always show up, you can click the arrow so that it says “always”.
Finally, by clicking on ‘Johnson-Neyman output’ we can probe the relationship between X and Y for regions of significance across levels on the (assumed continuous) moderator variable (W). Another way to think of this is an extension of the ‘Probe interactions’ above, but instead of obtaining tests of conditional relationships at only three levels you obtain tests of conditional relationships across ranges of the moderator variable.
The interaction term was statistically significant (b=2.6055, s.e.=1.0509, p=.0136) in our model, indicating that grit was a significant moderator of the effect of negative life events on hope.
The box below contains the same test result as that in the table for the interaction effect. However, the value-added is that it contains an index of the R-square change due to the moderation effect (Hayes, 2018). This is akin to the change in R-square one would observe if running a hierarchical multiple regression with the unconditional effects of the IV’s in Model 1 and adding in the interaction term in Model 2. The R-square change from Model 1 to Model 2 (adding in the interaction term) was .0138, indicating the interaction effect accounted for 1.38% added variation in Y.
Keep in mind, however, that since moderation is symmetric we could have also theoretically interpreted the interaction as indicating that negative life events moderated the effects of grit on hope. Recall that the designation of variables as being the focal IV versus moderator is a theoretical or conceptual issue and not a statistical one.
These are the effects of negative life events (X) and grit (W) on the dependent variable (Y), conditional on the other IV being 0 (Field, 2018). According to Hayes (2018), the language researchers should be using when referring to the effects of X and W is not “main effects” or the “effects of these variables, controlling for the interaction”. Rather, since the slopes represent the effect of X (or W) on Y, conditional on the other variable being 0, then it is more appropriate to think of the conditional effect as akin to a simple effect (using ANOVA terminology, Hayes, 2018) or as a simple slope (using regression terminology). As such, we can interpret the effects of grit and negative life events as follows: (a) The effect of negative life events on hope was negative and significant (b=-1.276, s.e.=.6305, p=.0437), conditional on grit = 0; (b) the
conditional effect of grit was positive and significant (b=5.7082, s.e.=.7205, p<.001), conditional on negative life events = 0.
Now, because we mean centered the IV’s we can now interpret these coefficients as the relationship between each IV and the DV at the grand mean of the other IV. In other words, the effect of negative life events is -1.276 for those individuals scoring at the grand mean on grit. The effect of grit is 5.7082 for those individuals scoring at the grand mean on negative life events.
Since the interaction term in our model (see previous slides) was statistically significant, we want to probe the interaction to better interpret the nature of the moderated relationship between negative life events and hope. These are tests of simple slopes, which tests the relationship between negative life events (X) and hope (Y) at three levels of the moderator (W; grit).
At -1 sd (i.e., at -.6145) on the centered grit variable (representing low grit), the relationship between
negative life events and hope was negative and significant (b=-2.8772, s.e.=.8303, p=.0006). Similarly, at the mean (i.e., at 0) on the centered moderator variable (representing medium grit), the relationship was
negative and significant (b=-1.276, s.e.=.6305, p=.0437). Finally, at +1sd (i.e., +.6145) on the centered grit variable (represent high grit), the relationship was positive, but non-significant (b=.3251, s.e.=.9695,
p=.7375).
As noted previously, the effect of negative life events (X) on Y was conditional on grit (W) = 0.
Because we used mean centering for our IV’s, a value of 0 on each centered IV is interpreted as the grand mean on that variable.
So due to the use of mean
centering, the effect of negative life events on hope as shown in the original regression output (left) is also repeated in table containing the simple effects tests (above).
We see from the Johnson-Neyman output that the slope between negative life events (X) and hope (Y) becomes increasingly positive over levels of the grit moderator (W) variable.
Conditional slopes
The regions on the Grit variable where the relationship between negative life events and hope is statistically significant (p’s ≤ .05) are shown in the box.
To obtain a plot of the interaction of negative life events (X) and grit (W) on hope (Y), copy this syntax out of the output file and paste it into a new syntax file.
Here I have pasted the syntax into an open syntax file. Next I clicked the Green Arrow.
A plot of the conditional means on Y (hope) at each combination of negative life events (X) and the moderator (W).
You can see “roughly” that the relationship between negative life events and hope is more strongly negative at -1sd on the centered Grit variable (see blue dots) and less strongly negative at the mean on that variable (see red dots).
When you run the syntax, a new dataset is
created in SPSS, reflecting the conditional means on Y (hope) at each combination of low (-1sd), medium (mean) and high (+1sd) levels on the moderator (W).
To plot out the slopes of the regression lines at low, medium, and high levels of the moderator…
These are the simple slopes for the relationship between negative life events (X) and hope (Y) at different levels of the moderator (W=grit).
The simple effects tests (reviewed previously) are tests of these slopes (the b’s are found in the Effect
column).
Example 2: If you desire to include covariates in your model, just add them to the Covariate(s) box. Here, I am adding a measure of life satisfaction as a covariate in the model & leaving the previous options clicked
(including mean centering).
Here, we again see the interaction term is statistically significant
(b=3.4697, s.e.=.9207, p=.0002).
Because it was significant, we will want to probe the interaction (using tests of simple slopes, the Johnson- Neyman technique, and/or or
plotting simple slopes).
The effect of grit, conditional on negative life events = 0 (i.e., grand mean), was statistically significant in the model (b=3.8674, s.e.=.6521, p<.001).
Since the effect of life satisfaction on hope is not treated as conditional (on any moderating variables), then we can interpret the effect as follows:
Life satisfaction was a positive and significant predictor of hope,
controlling for the other effects in the model (b=.2656, s.e.=.2049, p<.001).
Probing the interaction with simple slopes tests of the relationship
between negative life events at -1sd, mean, and +1sd on grit.
Regions on the moderator where the relationship
between negative life events and hope are significant.
At the end of the output is information concerning which variables were centered. Notice here that Grit and Negative Life Events were both centered. Obviously, life satisfaction was not. If we wanted to include life satisfaction in the model after centering, then we’d have to manually center the variable first by obtaining the mean on the variable and then constructing a new variable that involves deviating the mean from each of the raw scores. [An example of this type of approach is presented in the next example; although not with the life satisfaction variable]
Example 3: We are running the same model as before but incorporating the use of a binary moderator
variable (W=grit.ms, dummy coded 0=low, 1=yes). [I took the original grit variable and performed a median split. Keep in mind this was for demonstrative purposes only. It is not something I would generally
recommend.] We’ll also leave life satisfaction in the model as a covariate. I will also manually center the negative life events variable and include it in the model so that it is the only IV that is centered.
Step 1: Centering negative life events. Obtain mean for the original variable and then use compute function to create new centered variable.
Here, we have most of the same options clicked as before, with the exception of ‘Johnson-Neyman output’. Since our moderator has only two levels, there are no “regions of significance” to be examined.
Step 2: I will run the analysis without requesting mean centering. This provides meaningful values of 0 for the Grit.ms variable (coded 0 and 1) and the centered negative life events variable (where 0 is the grand mean of the variable).
Here, we see the interaction term is statistically significant (b=3.8484, s.e.=1.1426, p=.0008), consistent with the hypothesis that the binary grit variable would moderate the effect of negative life events on hope.
We see that the effect of (centered) negative life events on hope was negative and significant (b=-1.6402, s.e.=.7462, p=.0286). The regression slope for negative life events represents the relationship between
negative life events and hope in the low grit group (coded 0). In short, among those lower in grit, there was a significant negative predictive relationship between negative life events and hope.
The slope for individuals high in grit (coded 1) is: b1+b3 = -1.6402+3.8484 = 2.2082.
Grit was a positive and significant predictor of hope (b=3.2513, s.e.=.8176, p=.0001). This is interpreted as the difference in conditional means (on hope) between the low (coded 0) and high (coded 1) grit groups for
persons scoring at the grand mean on negative life events and controlling for life satisfaction.
Life satisfaction was positive and significant predictor in the model (b=.2823, s.e.=.0253, p<.001),
controlling for the remaining IV’s in the model. This result indicates that persons scoring higher on life satisfaction were more likely to experience hope.
The slope for (centered) negative life events shown previously (see above) appears in the table of conditional effects (left).
Moreover, we previously computed the simple slope for the high grit group as:
b1+b3 = -1.6402+3.8484 = 2.2082.
These results indicate that there is a negative predictive relationship between negative life events and hope among those persons low (originally coded 0) in grit (b=-1.6402, s.e.=.7462, p=.0286). On the other hand, there is a positive predictive relationship among those persons high (originally coded 1) in grit
(b=2.2082, s.e.=.8919, p=.0138).
NegLifeEvt.ms Hope Grit.ms
Example 4: We are including a binary independent variable (X=NegLifeEv.ms, coded 0=low, 1=high) and a binary moderator variable (W=grit.ms, coded 0=low, 1=high) in the model predicting Hope (Y).
[Again, median splits were performed. I still don’t recommend this generally!] For simplicity, we’ll remove life satisfaction from the model.
Here, we are not asking for mean centering.
The interaction term was not statistically significant (b=2.2237, s.e.=1.8818, p=.2381), suggesting that the relationship between negative life events and hope is not conditional on the level of grit.
The slope for negative life events (ms) is negative and non-significant. The slope, however, would be
interpreted as the difference in marginal means on hope among those low in grit (coded 0), and the t-test is the same as a simple effects test of the difference (following factorial ANOVA) in this group. The
difference in marginal means for those high in grit (coded 1) is: -1.5422+2.2237 = .6815. The slope for grit.ms is positive and significant (b=4.4425, s.e.=1.2996, p=.0007). This slope is the difference in marginal means on hope for persons falling into group 0 on negative life events (i.e., those low in negative life
events). The t-test is the same as a simple effects tests (following factorial ANOVA) in this group. The difference in means between grit groups in the high negative life events group (coded 1) is computed as:
4.4425+2.2237 = 6.6662. For a thorough discussion on this, see Hayes (2018).
Simple effects tests of mean differences between grit groups in the low and high negative events groups.
Recall, the slope for the effect of grit in the high negative events group is: b1+b3
= 4.4425 + 2.2237 = 6.6662.
Simple effects tests of mean differences
between negative life event groups at low and high levels of grit.
The slope for the effect of grit in the high negative events group is: b2+b3 = -1.5422 + 2.2237 = .6815.
Final notes
1. According the Hayes (2018) if the interaction term is not statistically significant, then one should consider removing the term and re-analyzing relationships using a reduced model with only the
unconditional effects of X and W estimated (see e.g., p. 237). This seems completely logical to me as a possible solution to the problem of collinearity when you have uncentered or centered (note: centering generally does not reduce the correlations among the IV’s and the interaction to 0, although it often reduces the magnitude of the correlations) IV’s included in your model alongside the interaction term (comprised of those variables). If the interaction effect is non-significant, then for all practical purposes you can treat the effects of your IV’s (X and W) as unconditional on the level of the other IV. Using this logic, we might have removed the interaction term from the final model in Example 4 and re-analyzed the data using a reduced model. [Field (2018) argues, on the other hand, that if you intend to include the non-significant interaction term in the model, then you should definitely use centering to facilitate interpretation of the effects of the individual IV’s.]
2. Your decision to center or not center your IV’s does not matter with respect to the test of the
interaction effect. The slope and test for the interaction term will be the same, irrespective of your decision regarding centering. Nevertheless, you DO need to pay attention to centering as it concerns interpretation of the individual IV’s in the model. [I strongly recommend getting a copy of Hayes (2018) book on Conditional process modeling, as he goes into terrific detail on the implications of centering decisions on interpretation of regression coefficients!]
Final notes
3. As I have shown you in this Powerpoint, it is possible to center predictors on your own and to add them to a model (see e.g., Example 3). Hayes (2018), however, points out that Process relies on listwise
deletion. You will need to ensure that any centering you do manually comes after dealing with the missing data problem (either through listwise deleting cases with missing data on any of the variables in your analysis or some other approach) so that that your centering (or standardization) is performed only on those cases that will be included in the final analysis.
4. It is worth noting that under the Options box, you can click on ‘Heteroskedasticity-consistent inference’
to obtain heteroskedasticity consistent standard errors if you suspect violation of the constant variance assumption (see Hayes & Cai, 2007 for discussion).
References
Hayes, A.F., & Cai, L. (2007). Using heteroskedasticity-consistent standard error estimators in OLS regression: An introduction and software implementation. Behavior Research Methods, 39, 709-722.
Hayes, A.F. (2018). Introduction to mediation, moderation, and conditional process analysis: A regression- based approach. New York: The Guilford Press.
Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed). Los Angeles: Sage.
Jose, P.E. (2013). Doing statistical mediation and moderation. New York: The Guilford Press.
Thanks for watching!
Obtain Process macro here: https://processmacro.org/download.html Process macro
