Solved – How to analyse a moderator effect in hierarchical multiple regression with covariates

interactionmultiple regressionstata

I am trying to find an appropriate method to examine the significant interaction effect in my hierarchial multiple regression analyses.
I am looking at moderation analysis but most of of the methods (macros etc) do not take into account the covariates in my regression equations.

I was advised to try out the Stata programme but at the moment, I have no access to it.

Is there another statistical program that I can perform moderator regression with covariates?
How can I perform simple slopes analysis with covarites?

Best Answer

Terminology and Overview

In the context of multiple regression:

a moderator effect is just an interaction between two predictors, typically created by multiplying the two predictors together, often after first centering the predictors.
a covariate is just a predictor that was not used in the formation of the moderator and that is conceptualised as something that needs to be controlled for.

Thus, you should be able to run a hiearchical regression with moderators and covariates in just about any statistical software that supports multiple regression.

Typical approach to testing moderator effect after controlling for covariates

SPSS: If you are doing the hierarchical regression in SPSS, you'd probably enter the predictors in blocks. Here's a tutorial.
R: If you are doing this in R, you'd probably define separate linear models lm each adding additional predictors and use anova to compare the models. Here's a tutorial.

Once you understand hierarchical regression in your chosen tool a simple recipe would be as follows. Let's assume that you have the following variables

main effect precitors: IV1 IV2
interaction effect: multiplication of IV1 and IV2
covariates CV1 CV2

In some cases you may need to create the moderator

If you are using SPSS, you will need to multiply the two predictor variables together (e.g., compute iv1byiv2 = iv1 * iv2.). If you want to interpret the regression coefficients, you may find it useful to center iv1 and iv2 before creating the interaction term.
If you are using R, you can just use the notation iv1*iv2 in the linear model notation.

You can then estimate the models

Block/model 1: Enter covariates m1 <- lm(DV~CV1+CV2)
Block/model 2: Enter main effect predictors m2 <- lm(DV~CV1+CV2+IV1+IV2)
Block/model 3: Enter interaction effect m3 <- lm(DV~CV1+CV2+IV1*IV2)

You can then interpret the significance of the r-square change between block 2 and 3 as a test of whether there is an interaction effect: anova(m2, m3)

Simple slopes analysis

If you want to perform simple slopes analysis, you can take the regression formula provided by the final multiple regression and calculate some appropriate values to plot.

You can do this by hand or you can use predict in R. For example, you might calculate the values predicted by the regression equation using the following values

IV1   IV2   CV1   CV2
-2sd  -2sd  mean  mean
-2sd  +2sd  mean  mean
+2sd  -2sd  mean  mean
+2sd  +2sd  mean  mean

You can then plot these values using whatever plotting tool that you like (e.g., R, SPSS, Excel).

Personally, I find Conditioning Plots a better option than simple slopes analysis. R has the coplot function. The idea is to show a scatter plot of the relationship between IV and DV in a set of arranged scatterplots defined by ranges of the moderator. When I searched, I found an example of using conditioning plots for moderator regression on page 585 of Handbook of Research Methods in Personality Psychology

Overview of simple slopes analysis

The points used to generate a graph for a simple slopes analyses are predicted values of the dependent variable given the model and various values of the predictor variables (see here for an example).

Are negative predicted values possible?

Yes. A regression equation can give rise to negative predicted values.
For example, if the equation was $Y = 1 - 2X -3Z - 1XZ$, then predicted $Y$ at $X=1$ and $Z=1$ is $Y = 1 - 2(1) -3(1) - 1(1)(1)$ or $1 -2 -3 - 1=-5$.

Do negative predicted values imply that you've done something wrong?

Possibly. If some or all of the following apply then negative predicted values might be a red flag on your analysis:
1. the range of the dependent variable is all positive, and particularly if the values are a long way from 0.
2. the correlation between predictors is not huge
3. you have chosen appropriate values of predictors.
When performing a simple slopes analysis typical values for predictor values include:
- continuous predictors that are part of the moderator effect: values drawn from plus or minus one or often two standard deviations above and below the mean or something similar.
- categorical predictors that are part of the moderator effect: each of the values that the categorical predictor takes
- continuous covariates: the mean of the covariate
- categorical covariates: one of the categories

Thus, choosing predictor values consistent with the above prescriptions will generally give rise to predictions on the dependent variable that are in the ballpark of the range of the dependent variable. Of course, there are situations that could legitimately give rise to negative values even when the range of the dependent variable includes all positive values. This might be related to non-normal errors, correlated predictors, etc.

Solved – Test whether simple slopes are different from zero in 3-way interaction in multiple regression

I know this is really late, but you can do it using the pequod package.

library(pequod)
model.peq <- lmres(outcome ~ predictor*moderator1*moderator2, centered =  c("predictor","moderator1","moderator2"), data=mydata)
S_slopes<-simpleSlope(model.peq, pred="predictor", mod1="moderator1", mod2="moderator2")
S_slopes

In the example, it seems like the error is really small so all the slopes are significant. If you change the data slightly, you get only one significant slope:

set.seed(123)
predictor <- rnorm(1000, 10, 5)
moderator1 <- rnorm(1000, 100, 25)
moderator2 <- rnorm(1000, 50, 20)
outcome <- predictor*moderator1*moderator2*rnorm(20, 0)/10000 # I changed this to zero from 30
mydata <- data.frame(predictor, moderator1, moderator2, outcome)
model.peq <- lmres(outcome ~ predictor*moderator1*moderator2, centered =  c("predictor","moderator1","moderator2"), data=mydata)
S_slopes<-simpleSlope(model.peq, pred="predictor", mod1="moderator1", mod2="moderator2")
S_slopes