Moderation with continuous predictors
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 (Wikipedia, 2024).
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 (Field, 2017, sec. 11.3.1)
View data
mean centering
\(\text{pred}_\text{c} = \text{pred} - \bar{\text{pred}} = \text{pred} - 15.39\)
\(\text{mod}_\text{c} = \text{mod} - \bar{\text{mod}}= \text{mod} - 7.808\)
SPSS syntax
COMPUTE pred = pred - 15.39.
COMPUTE mod = mod - 7.808.
EXECUTE.
Correlations
predictor moderator outcome
predictor 1.000 0.722 0.215
moderator 0.722 1.000 0.109
outcome 0.215 0.109 1.000Scatterplots
3D plot
Take it for a spin (does not work on tablet)
1 SD planes
Fit model
| Dependent variable | |||
| Predictors | Estimates | CI | p |
| (Intercept) | 5.38 | 4.62 – 6.14 | <0.001 |
| predictor | 0.70 | 0.40 – 1.01 | <0.001 |
| moderator | -0.22 | -0.43 – -0.00 | 0.050 |
| predictor:moderator | 0.37 | 0.32 – 0.42 | <0.001 |
| Observations | 100 | ||
| R2 / R2 adjusted | 0.687 / 0.677 | ||
Regression equation
\(\definecolor{red}{RGB}{255,0,0} \definecolor{black}{RGB}{0,0,0} \color{black}\widehat{Out_i} = b_0 + b_1 Pred_i + b_2 Mod_i + \color{red}b_3 Pred_i \times Mod_i \color{black}\)
\(\definecolor{red}{RGB}{255,0,0} \definecolor{black}{RGB}{0,0,0} \color{black}\widehat{Out_i} = 5.38 + 0.7 \times Pred_i + -0.22 \times Mod_i + \color{red} 0.37 \times Pred_i \times Mod_i \color{black}\)
Determine slopes
To determine the slopes of the regression plane at different levels of the moderater you can add the required moderator value to the regression equation and simplify.
\(5.38 + 0.7 \times Pred + -0.22 \times Mod + 0.37 \times Pred \times Mod\)
Expected surface
Total variance
Explained variance
Variance decomposition 3D
End
Contact
References
Field, A. (2017). Discovering statistics using IBM SPSS statistics, 5th edition. Retrieved from https://sussex.figshare.com/articles/book/Discovering_statistics_using_IBM_SPSS_statistics_5th_edition/23460641
Wikipedia. (2024). Moderation (statistics) — Wikipedia, the free encyclopedia. http://en.wikipedia.org/w/index.php?title=Moderation%20(statistics)&oldid=1165385148.