Module 7: Introduction to Interaction
PSYC 3032 M
Things Udi Should Remember
Before we start
- Group presentation is due in NEXT WEEK (March 25)
- To post, go to the Group Presentation topic on eClass
- Click on Presentation Videos Forum: POST VIDEOS HERE
- Add a new discussion topic and upload your video and additional files there
- Try it a little before the deadline so you won’t have any last-second surprises
- Rubric is available
- To post, go to the Group Presentation topic on eClass
About Module 7
Goals for Today:
- Interaction to Introduction
- What is an interaction/moderation
- A continuous-categorical variable interaction
- “Probing” an interaction
- Simple slope analysis
- Interaction/moderation assumptions
️ 
Me: I’m going to make one of those diagrams with overlapping circles
Dracula: Venn
Me: I don’t know, probably tomorrow
Do you have a pet? What’s their name?!
What’s the first thing you’ll do when the semester is over?
What would you title your biography / what is your DJ name?
Terminology & Definition
Interaction is moderation, these terms are synonymous
If two variables are said to interact (i.e., there’s an interaction between two variables), it means that the relationship between one predictor, \(X\), and the outcome variable, \(Y\), depends on the level of another predictor, \(Z\)
Another way to say it: in the presence of an interaction, the relationship between \(X\) and \(Y\) is different (or varies) for different values of \(Z\)
Terminology & Definition
- In psychology, we often refer to a variable that affects the relationship between a focal predictor and the outcome as a moderator
“In general terms, a moderator is a qualitative (e.g., sex, race, class) or quantitative (e.g., level of reward) variable that affects the direction and/or strength of the relation between an independent or predictor variable and a dependent or criterion variable.”
— Baron & Kenny, 1986
Example
Do Younger and Older Adults Equally Benefit from Physical Activity?
Suppose the relation between physical activity (\(X\)) and cognitive performance (\(Y\)) varies according to age group (i.e., younger adults, middle-aged adults, and older adults), such that the effect of physical activity on cognitive performance is much stronger with older age
Then we would say that age group moderates the association between physical activity and cognitive performance
Example
Do Younger and Older Adults Equally Benefit from Physical Activity?
Example
Do Younger and Older Adults Equally Benefit from Physical Activity?
Example
Do Younger and Older Adults Equally Benefit from Physical Activity?
Interactions in Psychology
We often refer to \(X\) as a focal predictor and \(Z\) as the moderator, but the choice is arbitrary
- Statistically speaking, you simply have two predictor variables (i.e., \(X\) and \(Z\)) which interact with one another in explaining their relationship to the outcome variable, \(Y\)
Moderation is very common in psychology; In fact, you’ve probably already done a moderation analysis before! Factorial ANOVAs, which is commonplace in experimental designs, is an example of moderation
In a two-way ANOVA, the model includes two categorical predictor variables and a continuous outcome variable (you could also include a three-way interaction by including three categorical predictors and testing how the three variables interact, but that’s often harder to interpret)
However, factorial ANOVA requires that all predictors are categorical, which is a real limitation
The Math of Interactions in GLM
Including a moderation in a regression framework is much more flexible, whereby you can have categorical variables, continuous variables, and their combination
Recall, an MLR model with two predictors (no interaction), looks like this:
\[\hat{y}_i= \beta_0 + {\color{darkcyan} {\beta_1}}X_i+{\color{darkcyan} {\beta_2}}Z_i\]
- For example,
\[\hat{Cognitive \ performance}_i= \beta_0 + {\color{darkcyan} {\beta_1}}Physical \ activity_i+{\color{darkcyan} {\beta_2}}Age \ group_i\]
- You can think about \({\color{darkcyan} {\beta_1}}\) and \({\color{darkcyan} {\beta_2}}\) as lower-order, main, or additive effects
The Math of Interactions in GLM
- Adding an interaction between \(X\) and \(Z\), we shall include a cross-product variable, \(X3\), representing \((X_i \times Z_i)\)
\[\hat{y}_i= \beta_0 + {\color{darkcyan}{\beta_1}}X_i + {\color{darkcyan}{\beta_2}}Z_i + {\color{deeppink} {\beta_3}}X3_i = \\ \hat{y}_i = \beta_0 + {\color{darkcyan}{\beta_1}}X_i + {\color{darkcyan}{\beta_2}}Z_i + {\color{deeppink} {\beta_3}}(X_i \times Z_i)\]
- where \(X3_i = X_i \times Z_i\) is the higher-order, interaction term, and its effect is captured by \({\color{deeppink} {\beta_3}}\)
- For example:
\[\hat{Cognitive}_i = \beta_0 + {\color{darkcyan}{\beta_1}}Activity_i + {\color{darkcyan}{\beta_2}}Age_i + {\color{deeppink}{\beta_3}}(Activity_i \times Age_i)\]
Effects of Interest in Moderation
- There will be three types of effects that we will discuss in the context of moderation:
1. Main Effects
2. Interaction Effects
3. Simple Effects
Specifying a moderation model involves estimating main effects and an interaction
Because an interaction implies that the relationship between the focal predictor and outcome changes at different levels of the moderator, researchers typically follow up a moderation model by examining simple effects (often called “probing” an interaction)
A simple effect is the magnitude of the relationship between the focal predictor and outcome variable at a particular level of the moderator; e.g., the strength of association between physical activity and cognitive performance only for older adults
Research Example
- To demonstrate how to conduct and interpret such a moderation analysis, we will use the following research example:
Parental alcoholism, externalizing behaviours, and alcohol use
Identify the variables
A) alcohol use = moderator, CoA = outcome, ext. behav. = focal pred.
B) alcohol use = outcome, CoA = moderator, ext. behav. = focal pred.
C) alcohol use = CoA, CoA = ext. behaviours, ext. behav. = alcohol use
D) It’s a trick question because life as we know it is a simulation
E) New phone, who dis?
Binary \(\times\) Continuous Variable Interaction
We have learned that it’s easy to incorporate binary/dichotomous variables in regression; recall that a 1-unit increase on the dummy-coded variable describes the mean difference between the two groups
CoA (child of alcoholism) is a dichotomous/binary variable, which, in our example, serves as a moderator in a regression model explaining alcohol use based on externalizing behaviours
\[\hat{Alcohol}_i = \beta_0 + {\color{darkcyan}{\beta_1}}Externalizing_i + {\color{darkcyan}{\beta_2}}CoA_i+{\color{deeppink} {\beta_3}}(Externalizing_i \times CoA_i)\]
- The interaction term’s coefficient, \({\color{deeppink} {\beta_3}}\), describes the extent to which CoA moderates, or interacts with, externalizing behaviour in explaining alcohol use
At the same time and equally valid, we can also say that…
Externalizing behaviour moderates the relation between CoA and alcohol use (two sides of the same coin). Mathematically, it’s exactly the same.
Research Example: Visualizing
- Before fitting the model to the data, we should always begin by graphing the data:
library(haven);library(e1071); library(kableExtra); library(tidyverse)
drink <- read_sav("drink.sav")
cols <- c("#F76D5E", "#72D8FF")
ggplot(drink, aes(x = as_factor(coa), y = alcuse, fill = as_factor(coa))) +
geom_boxplot( alpha = 0.85) +
geom_jitter(width = 0.15, color = "black", size = 1.5, alpha = 0.7) +
labs(x = "Child of Alcoholism", y = "Alcohol Use", fill = "Child of Alcoholism") +
theme_classic() + scale_fill_manual(values = cols)
Research Example: Visualizing
- And, of course…
drink$CoA <- as_factor(drink$coa)
ggplot(drink, aes(x = ext, y = alcuse, color = CoA)) +
geom_smooth(method = "lm", se = TRUE, aes(fill = CoA), alpha = 0.25) +
geom_point(size = 1.5, alpha = 0.8) +
labs(x="Externalizing Behaviour", y = "Alcohol Use") +
theme(legend.position = "none") +
theme_classic()
Research Example: Modeling
- Let’s estimate the model
Call:
lm(formula = alcuse ~ ext * CoA, data = drink)
Residuals:
Min 1Q Median 3Q Max
-1.2213 -0.2754 -0.1382 0.1372 3.4282
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.104718 0.106675 0.982 0.3277
ext 0.101535 0.173048 0.587 0.5582
CoACoA 0.004128 0.155173 0.027 0.9788
ext:CoACoA 0.564626 0.224237 2.518 0.0128 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.6121 on 161 degrees of freedom
Multiple R-squared: 0.1937, Adjusted R-squared: 0.1787
F-statistic: 12.89 on 3 and 161 DF, p-value: 1.364e-07 Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.104717893 0.1066750 0.98165395 0.32774320
ext 0.101535027 0.1730475 0.58674655 0.55819576
CoACoA 0.004128219 0.1551733 0.02660392 0.97880859
ext:CoACoA 0.564626007 0.2242367 2.51799144 0.01277935Do the results indicate that the interaction between externalizing and CoA significantly explains additional variability in alcohol use over and above the individual effects?”
A) No, we need to conduct hierarchical regression to know that
B) Yes, the p value for the interaction term is significant, identical to the F test from a two model comparison differing only by this term
C) Yes, because interactions are always justified, regardless of statistical significance
D) Having a dog named “Shark” at the beach was a mistake
E) I’m not sure, can I call a friend?
Research Example: Results
Table X
Explaining Alcohol Use from Parental Alcoholism and Exeternalizing Behaviour
\[\scriptsize \begin{array}{lcccccc} \hline \text{Variable} & \hat{\beta} & SE(\hat{\beta}) & t & p & \text{95% CI} & sr^2 \\ \hline \text{(Intercept)} & 0.105 & 0.107 & 0.98 & .33 & [-0.11, 0.31] & \\ \text{Externalizing} & 0.102 & 0.173 & 0.59 & .56 & [-0.24, 0.44] & 0.08 \\ \text{CoA} & 0.004 & 0.155 & 0.03 & .98 & [-0.31, 0.31] & 0.05 \\ \text{Externalizing} \ \times \text{CoA} & 0.565 & 0.224 & 2.52 & .01 & [0.12, 1.01] & 0.03 \\ \hline R^2 = .194, \ F(3,161) = 12.94, \ p < .001 \end{array}\]
- Recall that we defined our model earlier as:
\[\scriptsize \hat{Alcohol}_i = \beta_0 + {\color{darkcyan}{\beta_1}}Externalizing_i + {\color{darkcyan}{\beta_2}}CoA_i+{\color{deeppink} {\beta_3}}(Externalizing_i \times CoA_i)\]
- Therefore, our estimated MLR equation is
\[\scriptsize \hat{Alcohol}_i = 0.105 + {\color{darkcyan}{0.102}}Externalizing_i + {\color{darkcyan}{0.004}}CoA_i+{\color{deeppink} {0.565}}(Externalizing_i \times CoA_i)\]
Research Example: Results
Just like in a two-way ANOVA, our first and most interesting parameter is the one capturing the interaction term (\(\text{Externalizing} \ \times \text{CoA}\)), why?
Given that the research hypothesis pertained to an interaction between externalizing and being a child of an alcoholic parent in predicting alcohol use, our primary interest lies in the interaction effect
The parameter estimate for the interaction term, \(\hat{\beta}_3=0.567\) reflects the difference between the two slopes in Slide 22
That is, the difference in the two groups’ (CoA and non-CoA) strength of association between the focal predictor (externalizing) and outcome variable (alcohol use)
We can look at the results from the significance test to further infer if this difference in slopes is unlikely to be zero (i.e., whether the interaction is statistically significant)
Research Example: Results
These results suggests that there is some evidence that CoA moderates the relationship between externalizing behaviour and alcohol use
But how, or in what way, does parent alcoholism affect the relationship between externalizing and alcohol use? Given that there is an interaction term in the model, how should we interpret the parameter estimates?
We answer these questions by probing the interaction!
Probing the Interaction: Simple Effects
- Again, the two-predictor MLR with lower- and higher-order effects is:
\[\hat{y}_i = \beta_0 + {\color{darkcyan} {\beta_1}}X_i + {\color{darkcyan} {\beta_2}}Z_i + {\color{deeppink} {\beta_3}}(X_i \times Z_i)\]
- With some algebra, we can rearrange the terms in the equation to equivalently write the regression equation as:
\[\hat{y}_i = \left( \beta_0 + {\color{darkcyan} {\beta_2}}Z_i \right) + ({\color{darkcyan} {\beta_1}} + {\color{deeppink} {\beta_3}}Z_i)X_i\]
The component in the first parentheses \(\left( \beta_0 + {\color{darkcyan} {\beta_2}}Z_i \right)\) produces the simple intercept for \(X\) and the second component in parentheses, \(({\color{darkcyan} {\beta_1}} + {\color{deeppink} {\beta_3}}Z_i)\), produces the simple slope coefficient for \(X\)
Why are we complicating things with math?!
Because we want to examine the relationship between \(X\) and \(Y\) at each level of the moderator
Probing the Interaction: Simple Effects
- We are going to plug the actual values for \(Z\) into the equation above; this means that \(Z\) will ‘disappear’ from the equation and instead represent a constant (i.e., a number) that impacts the intercept and slope for \(X\)
- Consequently, we can define the simple intercept for \(X\) as Omega-knot
\[{\color{darkorange} {\omega_0}}=\left( \beta_0 + {\color{darkcyan} {\beta_2}}Z_i \right)\]
- And, the simple slope for \(X\) as Omega-1:
\[{\color{darkorange} {\omega_1}}=({\color{darkcyan} {\beta_1}} + {\color{deeppink} {\beta_3}}Z_i)\]
- We can, therefore, re-write this equation:
\[\scriptsize\hat{y}_i = \left( \beta_0 + {\color{darkcyan} {\beta_2}}Z_i \right) + ({\color{darkcyan} {\beta_1}} + {\color{deeppink} {\beta_3}}Z_i)X_i\]
- as:
\[\hat{y}_i = {\color{darkorange} {\omega_0}} + {\color{darkorange} {\omega_1}}X_i\]
\[\hat{y}_i = {\color{darkorange} {\omega_0}} + {\color{darkorange} {\omega_1}}X_i\]
What type of model is this?
A) Bootstrap
B) ANOVA/ANCOVA
C) Meowdel (i.e., Meow + model)
D) Simple linear regression
E) Multiple linear regression
Interpreting Simple Effects
From our new simple linear regression equation, we interpret \({\color{darkorange} {\omega_0}}\) and \({\color{darkorange} {\omega_1}}\) the same way that you would interpret an intercept and slope terms in a regression model…
For example, \({\color{darkorange} {\omega_0}}\), the simple intercept for \(X\), is the predicted value of \(Y\) when \(X_i = 0\),
And, \({\color{darkorange} {\omega_1}}\), the simple slope for \(X\), is the predicted change in \(Y\) associated with a one-unit change in \(X\), remember?
- That’s nice…but wait! Actually, both \({\color{darkorange} {\omega_0}}\) and \({\color{darkorange} {\omega_1}}\), include \(Z\)
\[{\color{darkorange} {\omega_0}}=\left( \beta_0 + {\color{darkcyan} {\beta_2}}Z_i \right) \quad \quad \quad {\color{darkorange} {\omega_1}}=({\color{darkcyan} {\beta_1}} + {\color{deeppink} {\beta_3}}Z_i)\]
- Which means that the simple intercept and slope of \(X\) depend on the value of \(Z\)
Interpreting Simple Effects
So the simple intercept and slope for \(X\) change when \(Z\) takes on different values
And, if you recall, \(Z\) is the categorical moderator variable which can take only certain kind of values (e.g., CoA and non-CoA, Male and Female, younger/middle/older adults)
That is, for a categorical moderator with \(K\) levels, we will need \(K\) simple regression equations to capture the relationships between \(X\) and \(Y\) at all \(K\) levels of the moderator!
Mathematically, it’s completely arbitrary as to which variable is referred to as the moderator
If \(X\) and \(Z\) interact to explain \(Y\), then we can say that \(X\) moderates the relation between \(Z\) and Y
OR, we can say that \(Z\) moderates the relation between \(X\) and \(Y\)—tomayto, tomahto
So, just as we can define the simple slope for \(X\) as a function of \(Z\), we can also define the simple slope for \(Z\) as a function of \(X\)
The original, substantive research question should be your guide as to which variable is the moderator and which simple slopes are interesting
Back to Our Research Example
In our current example, the conceptual moderator, CoA, is a binary variable
Thus, there are only two possible simple regression equations for the relation between externalizing and alcohol use, one for CoA (
coa=1) and one for non-CoA (coa=0)Let’s now calculate the two simple slope equations, recall that we re-organized our model to
\[\hat{y}_i = \left( \beta_0 + {\color{darkcyan} {\beta_2}}Z_i \right) + ({\color{darkcyan} {\beta_1}} + {\color{deeppink} {\beta_3}}Z_i)X_i\]
- And we can now plug in our estimates for \(\beta_0\), \(\beta_1\), \(\beta_2\), and \(\beta_3\)
\[\hat{Alcohol}_i = \left( 0.105 + {\color{darkcyan} {0.004}}CoA_i \right) + ({\color{darkcyan} {0.102}} + {\color{deeppink} {0.565}}CoA_i)Externalizing_i\]
Class Activity 🪩
\[\hat{Alcohol}_i = {\color{darkorange} {\omega_0}} + {\color{darkorange} {\omega_1}}Externalizing_i= \\ \hat{Alcohol}_i = \left( 0.105 + {\color{darkcyan} {0.004}}CoA_i \right) + ({\color{darkcyan} {0.102}} + {\color{deeppink} {0.565}}CoA_i)Externalizing_i\]
On your computer or piece of paper, calculate the two simple slope equations, when when \(CoA=0\) and one when \(CoA=1\)
That is, calculate the \({\color{darkorange} {\omega_0}}\) and \({\color{darkorange} {\omega_1}}\) for the two CoA categories
Look at the value of \({\color{darkorange} {\omega_1}}\) for the non-COA, where did you see it before?
What is the difference between the \({\color{darkorange} {\omega_1}}\) for CoA and that of non-CoA?
Where did you see this value before?
Extracting Simple Slopes in R
- Sure, we can use hand calculations. But, why not let R do it?
SIMPLE SLOPES ANALYSIS
When CoA = not CoA:
Est. S.E. 2.5% 97.5% t val. p
--------------------------- ------ ------ ------- ------- -------- ------
Slope of ext 0.10 0.17 -0.24 0.44 0.59 0.56
Conditional intercept 0.10 0.11 -0.11 0.32 0.98 0.33
When CoA = CoA:
Est. S.E. 2.5% 97.5% t val. p
--------------------------- ------ ------ ------- ------- -------- ------
Slope of ext 0.67 0.14 0.38 0.95 4.67 0.00
Conditional intercept 0.11 0.11 -0.11 0.33 0.97 0.34Plottting Simple Slopes
- We can also plot the simple slopes using the same package
Interpreting Simple Slopes
- The simple slope coefficient when
CoA=0is a small effect and not significantly different from zero (\(\hat{\beta}_{1_{non-CoA}} =0.10\), p=.56), indicating that the effect of externalizing behaviour on alcohol use is likely negligible (0.10 points out of a observed range of 4.09 points) among individuals who do not have an alcoholic parent- The intercept when
CoA=0tells us the expected alcohol use for children of non-alcoholics with no externalizing behavior (i.e.,ext=0; \(\hat{\beta}_{0_{non-CoA}}=0.10\)), but intercepts are usually not so helpful (unless we mean-centre our predictor/s…but more on this next week)
- The intercept when
Interpreting Simple Slopes
- The simple slope coefficient when
CoA=1has a larger effect which also happens to be statistically significant (\(\hat{\beta}_{1_{CoA}} =0.67\), p<.01), suggesting that the effect of externalizing behaviour on alcohol use is more prominent among individuals with at least one alcoholic parent- Specifically, among persons with alcoholic parent/s, for every one-unit difference on the externalizing behaviour scale, we expect a 0.67-point increase on the alcohol use scale, a rather small effect (~16.5%), yet likely still meaningful
Writing Up the Results
- With a statistically significant interaction in the model, I do not recommend reporting or interpreting the main effects
- In a model with a dummy-coded moderator with one group coded 0 (e.g., non-CoA), the main effect of the focal predictor (e.g., externalizing) is actually the simple slope for those who are in the 0 group
- The main effect of the dummy-coded moderator is the difference between the categories (e.g., CoA vs. non-CoA) on the outcome variable (e.g., alcohol) when the focal predictor is 0 (e.g., externalizing behaviour =0)
- Regardless, interpreting the main effects can be misleading because the relationships between each predictor and the outcome are not constant across the levels of the other predictor
Writing Up the Results
Instead, I recommend focusing on the coefficient for the interaction term and reporting the simple effects
Here’s a write up of the results from our research example:
Example Results Write-up
Externalizing behaviour, having parental alcoholism, and their interaction accounted for a substantial amount—about 20%—of the variability in alcohol use among adolescents, \(R^2=.19\), 95% CI [.09, .30], F(3, 161) = 12.94, p < .001.
The relationship between externalizing behaviour and alcohol use was significantly moderated by parental alcoholism, \(\hat{\beta}_3 = 0.56\), 95% CI [.12, 1.01], t(161) = 2.52, p = .013, suggesting that the simple slope of adolescents with at least one alcoholic parent was steeper than those with no alcoholic parents by about 0.56-points on alcohol use per one-unit on externalizing.
Tests of simple slopes indicated that for those with non-alcoholic parents, the effect of externalizing behaviour on alcohol use was small and not statistically significant \(\hat{\beta}_{1_{non-CoA}} =0.10\), 95% CI [-.24, .44], t(161) = 0.59, p = .56. Given an observed range of alcohol use (4.09 points), a 0.10 points—roughly 2.5%—is likely negligible. However, among adolescents with an alcoholic parent, the effect of externalizing behaviour on alcohol use was larger and statistically significant, \(\hat{\beta}_{1_{CoA}} =0.67\), 95% CI [.38, .95], t(161) = 4.69, p<.01; that is, among persons with alcoholic parent/s, we expect a 0.67-point increase on alcohol use for every one-unit difference on externalizing behaviour. This effect, although rather small (about 16.5% of the observed scale range), is still likely to be meaningful in real-world contexts. See Figure X for a plot of the interaction.
Writing Up the Results
Moderation Assumptions
The assumptions of OLS regression apply equivalently to models with an interaction term and the same diagnostic procedures presented in earlier modules can be used
Recall, LINE:
- Linear relationship (applies to covariate and outcome only)
- Independence
- Normally distributed residuals
- Equal variance (homoscedasticity)
And, of course:
- Multicollinearity
- Influential cases/outliers
…But, there are two practical considerations that are a bit different from what we’ve seen thus far in the course
Moderation Assumptions
Re: Multicollinearity
- Multicollinearity will be present when there’s an interaction because the interaction term is based off of the two other predictor variables in the model; this is called structural multicollinearity, take a look:
- We will discuss this in more detail next week when we touch on mean-centring
Moderation Assumptions
Re: Linearity
In addition to the methods previously discussed for examining linearity, I’d also recommend looking at the relationship between the focal predictor and outcome at each level of the moderator
Luckily, the plots in the interactions package provide an easy way to get this by simply adding the argument
linearity.check=TRUE, have a look:
Moderation Assumptions
Re: Linearity
In the Next Episode of PSYC3032…
- Moderation between two continuous predictors
