Moderation

Klinkenberg

University of Amsterdam

28 sep 2022

Moderation

In statistics and regression analysis, moderation occurs when the relationship between two variables depends on a third variable. The third variable is referred to as the moderator variable or simply the moderator. The effect of a moderating variable is characterized statistically as an interaction.

Source WIKIPEDIA

Model

\[\definecolor{red}{RGB}{255,0,0} \definecolor{black}{RGB}{0,0,0} \color{black}Out_i = b_0 + b_1 Pred_i + b_2 Mod_i + \color{red}b_3 Pred_i \times Mod_i \color{black}+ \epsilon_i\]

Example

Video games are among the favourite online activities for young people: two-thirds of 5–16-year-olds have their own video games console, and 88% of boys aged 8–15 own at least one games console. Although playing violent video games can enhance visuospatial acuity, visual memory, probabilistic inference, and mental rotation, compared to games such as Tetris, these games have also been linked to increased aggression in youths Another predictor of aggression and conduct problems is callous-unemotional traits such as lack of guilt, lack of empathy, and callous use of others for personal gain.

Imagine a scientist wanted to look at the relationship between playing violent video games such as Grand Theft Auto, MadWorld and Manhunt and aggression. She gathered data from 442 youths. She measured their aggressive behaviour, callous unemotional traits, and the number of hours per week they play video games.

Source: Field 11.3.1

Simulate data

set.seed(283) ## Mooie seed :-)
predictor <-             rnorm(100, 0, 3)
moderator <- predictor + rnorm(100, 0, 3) ## multicollineariteit
error     <-             rnorm(100, 0, 3) ## error
intercept <- 5
outcome   <- intercept + .5*predictor + 
                        -.4*moderator + 
                         .4*predictor*moderator + 
                         error

Mean un centring

predictor <- predictor + 15 
moderator <- moderator + 7

data <- data.frame(predictor, moderator, outcome)
head(data)

  predictor moderator   outcome
1  18.02061  7.746939  9.797006
2  15.60117  5.937336  8.638962
3  14.02281  6.477516  8.336367
4  16.47405  9.520816  5.396567
5  11.99709  5.404452  8.246535
6  13.00026  2.415985 11.893936

View data

Scatterplots

plot(data)

Correlations

cor(data)

          predictor moderator   outcome
predictor 1.0000000 0.7220140 0.2151668
moderator 0.7220140 1.0000000 0.1094856
outcome   0.2151668 0.1094856 1.0000000

3D plot

Take it for a spin (does not work on tablet)

Fit model

#fit2 <- lm(outcome ~ predictor + moderator)
fit <- lm(outcome ~ predictor + moderator + predictor*moderator); summary(fit)


Call:
lm(formula = outcome ~ predictor + moderator + predictor * moderator)

Residuals:
    Min      1Q  Median      3Q     Max 
-7.1725 -2.0186  0.2975  2.2995 10.0602 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)         40.70870    3.38786  12.016  < 2e-16 ***
predictor           -2.18586    0.24825  -8.805 5.47e-14 ***
moderator           -5.91087    0.42821 -13.804  < 2e-16 ***
predictor:moderator  0.37000    0.02649  13.967  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.086 on 96 degrees of freedom
Multiple R-squared:  0.6869,    Adjusted R-squared:  0.6771 
F-statistic:  70.2 on 3 and 96 DF,  p-value: < 2.2e-16

Regression equation

\(\definecolor{red}{RGB}{255,0,0} \definecolor{black}{RGB}{0,0,0} \color{black}\widehat{Out_i} = b_0 + b_1 Pred_i + b_2 Mod_i + \color{red}b_3 Pred_i \times Mod_i \color{black}\)

\(\definecolor{red}{RGB}{255,0,0} \definecolor{black}{RGB}{0,0,0} \color{black}\widehat{Out_i} = 40.71 + -2.19 \times Pred_i + -5.91 \times Mod_i + \color{red} 0.37 \times Pred_i \times Mod_i \color{black}\)

regeq <- function(model, predictor, moderator) { 
    fit$coefficients[1] + 
    fit$coefficients[2]*predictor + 
    fit$coefficients[3]*moderator + 
    fit$coefficients[4]*predictor*moderator
}

x.pre <- seq(10,21,length.out=30)
y.mod <- seq( 0,16,length.out=30)

z.pre <- outer(x.pre, y.mod, FUN='regeq', model=fit)

Simple Slopes

We use the interactions packaage

if(!"interactions" %in% installed.packages()) { install.packages("interactions") };
if(!"sandwich" %in% installed.packages()) { install.packages("sandwich") };
library("sandwich")

library("interactions")
sim_slopes(fit, pred = predictor, modx = moderator, johnson_neyman = FALSE)

SIMPLE SLOPES ANALYSIS 

Slope of predictor when moderator =  3.684843 (- 1 SD): 

   Est.   S.E.   t val.      p
------- ------ -------- ------
  -0.82   0.18    -4.52   0.00

Slope of predictor when moderator =  7.807686 (Mean): 

  Est.   S.E.   t val.      p
------ ------ -------- ------
  0.70   0.15     4.54   0.00

Slope of predictor when moderator = 11.930528 (+ 1 SD): 

  Est.   S.E.   t val.      p
------ ------ -------- ------
  2.23   0.20    11.35   0.00

Visual

interact_plot(fit, pred = predictor, modx = moderator, johnson_neyman = FALSE)

Johnson-Neyman intervals

sim_slopes(fit, pred = predictor, modx = moderator, johnson_neyman = T)

JOHNSON-NEYMAN INTERVAL 

When moderator is OUTSIDE the interval [5.02, 6.74], the slope of predictor
is p < .05.

Note: The range of observed values of moderator is [-0.70, 18.20]

SIMPLE SLOPES ANALYSIS 

Slope of predictor when moderator =  3.684843 (- 1 SD): 

   Est.   S.E.   t val.      p
------- ------ -------- ------
  -0.82   0.18    -4.52   0.00

Slope of predictor when moderator =  7.807686 (Mean): 

  Est.   S.E.   t val.      p
------ ------ -------- ------
  0.70   0.15     4.54   0.00

Slope of predictor when moderator = 11.930528 (+ 1 SD): 

  Est.   S.E.   t val.      p
------ ------ -------- ------
  2.23   0.20    11.35   0.00

Visual

johnson_neyman(fit, pred = predictor, modx = moderator, alpha = .05)

JOHNSON-NEYMAN INTERVAL 

When moderator is OUTSIDE the interval [5.02, 6.74], the slope of predictor
is p < .05.

Note: The range of observed values of moderator is [-0.70, 18.20]

Plot 3D

1 SD planes

quantiles <- as.vector(quantile(moderator,seq(.1,.9,.1)))
sds       <- c(mean(moderator)+(sd(moderator)*c(-1,0,1)))

SD Planes visual

Expected surface

Total variance

Explained variance

End

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