Simple Slopes With One Categorical and One Continuous Variable
Interactions
- Interactions have the same meaning they did in ANOVA
- there is a synergistic effect between two variables
- However now we can examine interactions between continuous variables
- Additive Effects:
\[Y = B_1X + B_2Z + B_0\]
- Interactive Effects: \[Y = B_1X + B_2Z + B_3XZ + B_0\]
Example of Interaction
- DV: Reading Speed, IVs: Age + IQ
- Simulation: \(Y = 3X + .2Z + .4XZ +100 +\epsilon\)
- So what we mean is: \(ReadingSpeed = 3*Age + .2*IQ + .4*Age*IQ +100 +Noise\)
- Set age to be uniform distribution between 7-17
- Set IQ to be normal, \(M\)=100 and \(S\)=15
- Set \(\epsilon\) = 15 (normal distribution of noise)
- Note: For our regression simulation to match our equation, we should center our X and Z variables first, but we will save out the raw score
set.seed(42) n <- 200 # Uniform distribution of Ages X <- runif(n, 7, 17) Xc<-scale(X,scale=F) # normal distrobution of IQ (100 +-15) Z <- rnorm(n, 100, 15) Zc<-scale(Z,scale=F) # Our equation to create Y Y <- 3*Xc + .2*Zc + .4*Xc*Zc +100 + rnorm(n, sd=15) #Built our data frame Reading.Data<-data.frame(Age=X,IQ=Z,ReadingSpeed=Y) Scatterplot & correlate our predictors
- We will use a useful diagnostic tool (GGally)
- We will visualize the density, scatterplot and correlation all at once
library(GGally) DiagPlot <- ggpairs(Reading.Data, lower = list(continuous = "smooth")) DiagPlot 
- Predictors have some heteroskedastic issues, but the correlations seem to make sense
Let's test for an interaction
- Center your variables
- Model 1: Examine a main-effects model (X + Z) [not necessary but useful]
- Model 2: Examine a main-effects + Interaction model (X + Z + X:Z)
- Note: You can simply code it as X*Z in r, as it will automatically do (X + Z + X:Z)
#Center Reading.Data$Age.C<-scale(Reading.Data$Age, center = TRUE, scale = FALSE)[,] Reading.Data$IQ.C<-scale(Reading.Data$IQ, center = TRUE, scale = FALSE)[,] # Regressions Centered.Read.1<-lm(ReadingSpeed~Age.C+IQ.C,Reading.Data) Centered.Read.2<-lm(ReadingSpeed~Age.C*IQ.C,Reading.Data) # Show results library(stargazer) stargazer(Centered.Read.1,Centered.Read.2,type="html", column.labels = c("Main Effects", "Interaction"), intercept.bottom = FALSE, single.row=TRUE, notes.append = FALSE, header=FALSE) | Dependent variable: | ||
| ReadingSpeed | ||
| Main Effects | Interaction | |
| (1) | (2) | |
| Constant | 99.441*** (1.616) | 99.363*** (1.009) |
| Age.C | 2.287*** (0.555) | 2.412*** (0.346) |
| IQ.C | 0.213* (0.112) | 0.234*** (0.070) |
| Age.C:IQ.C | 0.402*** (0.023) | |
| Observations | 200 | 200 |
| R2 | 0.095 | 0.649 |
| Adjusted R2 | 0.086 | 0.644 |
| Residual Std. Error | 22.858 (df = 197) | 14.267 (df = 196) |
| F Statistic | 10.326*** (df = 2; 197) | 120.907*** (df = 3; 196) |
| Note: | p<0.1; p<0.05; p<0.01 | |
- Also, change in \(R^2\) is significant, as we might expect
ChangeInR<-anova(Centered.Read.1,Centered.Read.2) knitr::kable(ChangeInR, digits=4) | Res.Df | RSS | Df | Sum of Sq | F | Pr(>F) |
|---|---|---|---|---|---|
| 197 | 102930.36 | NA | NA | NA | NA |
| 196 | 39893.42 | 1 | 63036.93 | 309.7062 | 0 |
Examine residuals
- Main Effects Model
library(ggfortify) autoplot(Centered.Read.1, which = 1:2, label.size = 1) + theme_bw() 
- Interactions Model
autoplot(Centered.Read.2, which = 1:2, label.size = 1) + theme_bw() 
Understanding the Interaction
- Let's examine a 3d scatter plot of the raw data and plot against it the predicted results of each model (additive and interaction model)
- Once we have a "model" of the data, we can generate predictions for all possible Ages at every possible IQ (within the absolute boundary limits for our data)
- With 2 predictors that generates a "3D surface plane."
- You go past 2 predictors with these types of plots
- With 2 predictors that generates a "3D surface plane."
- Let's examine Model 1 (additive effects) as a 3D scatter/surface plot
- Scatter dots = Raw data points
- Surface = Model 1 Predicted Results
- How well does the surface fit the dots?
- Let's examine Model 2 (interaction effects) as a 3d scatter/surface plot
- Scatter dots = Raw data points
- Surface = Model 2 Predicted Results
- How well does the surface fit the dots?
How visualize this for Papers?
- Well, there are some options
- We can do our fancy-pants surface plot, but that is hard to put into a paper
- More common this is to examine slope of one factor at different levels of the other (Simple Slopes)
- What we need to decide is at which levels
Hand picking
- Manually select the what levels of each you are going to examine
- For tracking values of interest
- here I picked -3 to 3 for age (mix and max) and -15, 0 15 for IQ (theoretical 1 SD and mean)
library(effects) Inter.1a<-effect(c("Age.C*IQ.C"), Centered.Read.2, xlevels=list(Age.C=seq(-3,3, 1), IQ.C=c(-15,0,15))) knitr::kable(summary(Inter.1a)$effect, digits=4) plot(Inter.1a, multiline = TRUE) | -15 | 0 | 15 | |
|---|---|---|---|
| -3 | 106.7108 | 92.1283 | 77.5458 |
| -2 | 103.0896 | 94.5398 | 85.9900 |
| -1 | 99.4684 | 96.9513 | 94.4343 |
| 0 | 95.8471 | 99.3629 | 102.8786 |
| 1 | 92.2259 | 101.7744 | 111.3229 |
| 2 | 88.6047 | 104.1859 | 119.7671 |
| 3 | 84.9835 | 106.5974 | 128.2114 |
Quantile
- Examine the levels based on quantiles (bins based on probability)
- We will do this into 5 equal bins based on probability cut-offs
- Does not assume normality for IV
IQ.Quantile<-quantile(Reading.Data$IQ.C,probs=c(0,.25,.50,.75,1)) IQ.Quantile<-round(IQ.Quantile,2) Inter.1b<-effect(c("Age.C*IQ.C"), Centered.Read.2, xlevels=list(Age.C=seq(-3,3, 1), IQ.C=IQ.Quantile)) plot(Inter.1b, multiline = TRUE) 
Based on the SD
- We select 3 values: \(M-1SD\), \(M\), \(M+1SD\)
- Not it does not have to be 1 SD, it can be 1.5 ,2 or 3
- Assumes normality for IV
IQ.SD<-c(mean(Reading.Data$IQ.C)-sd(Reading.Data$IQ.C), mean(Reading.Data$IQ.C), mean(Reading.Data$IQ.C)+sd(Reading.Data$IQ.C)) IQ.SD<-round(IQ.SD,2) Inter.1c<-effect(c("Age.C*IQ.C"), Centered.Read.2, xlevels=list(Age.C=seq(-3,3, 1), IQ.C=IQ.SD)) plot(Inter.1c, multiline = TRUE) 
Why center your IVs?
- The center of each IV represents the mean of that IV
- Thus when you interact them \(X*Z\), that means \(0*0 = 0\)
- Also, this can reduce multicollinearity issues
- This is because if \(X*Z\) creates a line, it means you have added a new predictor (XZ) that strongly correlates with X and Z
X<-c(0,2,4,6,8,10) Z<-c(0,2,4,6,8,10) XZ<-X*Z plot(XZ) 
- Correlations: \(r_{x,z}\) = 1, \(r_{x,xz}\) = 0.96, \(r_{z,xz}\) = 0.96 -But if you center them, now you will make a U-shaped interaction which will be orthogonal to X and Z!
X.C<-scale(X, center = TRUE, scale = FALSE)[,] Z.C<-scale(Z, center = TRUE, scale = FALSE)[,] XZ.C<-X.C*Z.C plot(XZ.C) 
-
Correlations: \(r_{x_c,z_c}\) = 1, \(r_{x_c,xz_c}\) = 0, \(r_{z_c,xz_c}\) = 0
-
Let's apply this to our class example
-
See below where I manually multiply the values and you can see a very strong positive slope
-
Left = Raw Score, Right plot = Centered Score
Let's run an uncentered regression
- as you can see the terms have changed from centered models
Uncentered.Read.1<-lm(ReadingSpeed~IQ+Age,Reading.Data) Uncentered.Read.2<-lm(ReadingSpeed~IQ*Age,Reading.Data) | Dependent variable: | ||
| ReadingSpeed | ||
| Main Effects | Interaction | |
| (1) | (2) | |
| Constant | 50.328*** (13.134) | 534.570*** (28.711) |
| IQ | 0.213* (0.112) | -4.681*** (0.287) |
| Age | 2.287*** (0.555) | -37.512*** (2.288) |
| IQ:Age | 0.402*** (0.023) | |
| Observations | 200 | 200 |
| R2 | 0.095 | 0.649 |
| Adjusted R2 | 0.086 | 0.644 |
| Residual Std. Error | 22.858 (df = 197) | 14.267 (df = 196) |
| F Statistic | 10.326*** (df = 2; 197) | 120.907*** (df = 3; 196) |
| Note: | p<0.1; p<0.05; p<0.01 | |
Multicollinearity of the Interaction
- Multicollinearity of the uncentered model
vif(Uncentered.Read.2) ## IQ Age IQ:Age ## 16.70086 43.64111 59.58237 - We lost our main effect of Age as the variances got all inflated
- We did not have this problem in the centered model
vif(Centered.Read.2) ## Age.C IQ.C Age.C:IQ.C ## 1.000440 1.000316 1.000716 - Note: Its OK to plot the uncentered version, but run the analysis on the centered data
Testing Simple Slopes
- The goal here is to figure out when the slope at a given level of another variable is different from zero
- We chop up the interaction at specific places as we did with the interactions plots (-1 SD, M, +1 SD) on the moderating variable (a third variable that affects the strength of the relationship between a dependent and independent variable)
- Next, we test the slopes on the predictor variable (IV of main interest)
- Example: Performance on a task, predicted by Level of training, moderated by how hard the challenge is
Example of Interaction
- DV: Performance, IVs: Training (X) + Challenge (Z)
- Simulation: \(Y = 2.2X + 2.46Z + .75XZ +16 +\epsilon\)
- Set Training to be uniform distribution between -10 to 10
- Set Challenge to be normal, \(M\)=0 and \(S\)=15
- Set \(\epsilon\) = 50 (normal distribution of noise)
- Note: For our regression simulation we will set means to about zero (so are not forced to center them)
set.seed(42) # 250 people n <- 250 #Training (low to high) X <- runif(n, -10, 10) # normal distribution of Challenge Difficulty Z <- rnorm(n, 0, 15) # Our equation to create Y Y <- 2.2*X + 2.46*Z + .75*X*Z + 16 + rnorm(n, sd=50) #Built our data frame Skill.Data<-data.frame(Training=X,Challenge=Z,Performance=Y) #run our regression Skill.Model.1<-lm(Performance~Training+Challenge,Skill.Data) Skill.Model.2<-lm(Performance~Training*Challenge,Skill.Data) 
| Dependent variable: | ||
| Performance | ||
| Main Effects | Interaction | |
| (1) | (2) | |
| Constant | 14.238*** (4.909) | 16.173*** (3.268) |
| Training | 0.895 (0.843) | 1.542*** (0.562) |
| Challenge | 2.855*** (0.354) | 2.600*** (0.236) |
| Training:Challenge | 0.762*** (0.043) | |
| Observations | 250 | 250 |
| R2 | 0.210 | 0.652 |
| Adjusted R2 | 0.204 | 0.648 |
| Residual Std. Error | 77.510 (df = 247) | 51.571 (df = 246) |
| F Statistic | 32.867*** (df = 2; 247) | 153.486*** (df = 3; 246) |
| Note: | p<0.1; p<0.05; p<0.01 | |
Plotting
- We will use the
rockchalklibrary to visualize our interaction and test our simple slopes
library(rockchalk) m1ps <- plotSlopes(Skill.Model.2, modx = "Challenge", plotx = "Training", n=3, modxVals="std.dev") m1psts <- testSlopes(m1ps) round(m1psts$hypotests,4) ## Values of Challenge OUTSIDE this interval: ## lo hi ## -3.5050807 -0.5730182 ## cause the slope of (b1 + b2*Challenge)Training to be statistically significant ## "Challenge" slope Std. Error t value Pr(>|t|) ## (m-sd) -14.50 -9.5013 0.8131 -11.6848 0.0000 ## (m) -0.61 1.0771 0.5611 1.9196 0.0561 ## (m+sd) 13.28 11.6554 0.8282 14.0737 0.0000 
- We see the slope at the mean is not significant,
- The +1SD and -1SD are significant, this is what is driving the interaction
- Well trained people at low levels of challenge get worse with more training, while high levels of challenge with more training get better
- The bonus using rockchalk is that you can change the number of lines (change \(n=3\)) you want to see or you can test quantiles as well (change modxVals="quantile")
m1pQ <- plotSlopes(Skill.Model.2, modx = "Challenge", plotx = "Training", n=5, modxVals="quantile") m1pQts <- testSlopes(m1pQ) round(m1pQts$hypotests,4) ## Values of Challenge OUTSIDE this interval: ## lo hi ## -3.5050807 -0.5730182 ## cause the slope of (b1 + b2*Challenge)Training to be statistically significant ## "Challenge" slope Std. Error t value Pr(>|t|) ## 0% -40.4989 -29.3016 1.7993 -16.2847 0.0000 ## 25% -10.5190 -6.4694 0.6990 -9.2555 0.0000 ## 50% -0.7985 0.9335 0.5610 1.6640 0.0974 ## 75% 8.8382 8.2726 0.6994 11.8278 0.0000 ## 100% 36.8939 29.6394 1.7214 17.2183 0.0000 
Notes
- You often don't need to run the t-tests on the simple slopes, but they can useful to test very specific hypotheses
- You can run simple slopes analysis on three-way interactions, but let's leave that aside for now as you would have to use a different R-package
Non-Linear interactions
- You can have non-linear terms interacting with other linear and non-linear terms
- Example: Quit smoking, X = Fear of your health, Z = moderated by Self-Efficacy for quitting
Example of Polynomial Interaction
- DV: Quit smoking, IVs: Fear (X) + Self-Efficacy (Z)
- Simulation: \(Y = -.18X - .15X^2 + Z + .16XZ + .07X^2Z +20 +\epsilon\)
- Set Fear of your health to be uniform distribution between 1 to 9 (centered)
- Set Self-Efficacy to be normal, \(M\)=0 and \(S\)=15
- Set \(\epsilon\) = 10 (normal distribution of noise)
set.seed(42) n <- 250 #Fear of Health X <- scale(runif(n, 1, 9), scale=F) #Centered Self-Efficacy Z <- scale(runif(n, 0, 15), scale=F) # Our equation to create Y Y <- -.05*X - .20*X^2 + .15*X*Z + .22*(X^2)*Z+ 35 + rnorm(n, sd=10) #Built our data frame Smoke.Data<-data.frame(Smoking=Y,Health=X,SelfEfficacy=Z) #run our regression Smoke.Model.1<-lm(Smoking~poly(Health,2)+SelfEfficacy,Smoke.Data) Smoke.Model.2<-lm(Smoking~poly(Health,2)*SelfEfficacy,Smoke.Data) Smoke.Model.1.R<-lm(Smoking~poly(Health,2, raw=T)+SelfEfficacy,Smoke.Data) Smoke.Model.2.R<-lm(Smoking~poly(Health,2, raw=T)*SelfEfficacy,Smoke.Data) - Orthogonal Polynomials
| Dependent variable: | ||
| Smoking | ||
| Main Effects | Interaction | |
| (1) | (2) | |
| Constant | 33.263*** (0.680) | 33.493*** (0.621) |
| poly(Health, 2)1 | 1.010 (10.786) | 3.083 (9.813) |
| poly(Health, 2)2 | -9.037 (10.766) | -6.505 (9.795) |
| SelfEfficacy | 1.188*** (0.155) | 1.189*** (0.141) |
| poly(Health, 2)1:SelfEfficacy | 3.042 (2.218) | |
| poly(Health, 2)2:SelfEfficacy | 16.361*** (2.288) | |
| Observations | 250 | 250 |
| R2 | 0.197 | 0.341 |
| Adjusted R2 | 0.187 | 0.328 |
| Residual Std. Error | 10.759 (df = 246) | 9.781 (df = 244) |
| F Statistic | 20.058*** (df = 3; 246) | 25.297*** (df = 5; 244) |
| Note: | p<0.1; p<0.05; p<0.01 | |
- Power Polynomials
| Dependent variable: | ||
| Smoking | ||
| Main Effects Non-Ortho | Interaction Non-Ortho | |
| (1) | (2) | |
| Constant | 33.909*** (1.027) | 33.958*** (0.934) |
| poly(Health, 2, raw = T)1 | 0.009 (0.294) | 0.070 (0.268) |
| poly(Health, 2, raw = T)2 | -0.119 (0.142) | -0.086 (0.129) |
| SelfEfficacy | 1.188*** (0.155) | 0.020 (0.217) |
| poly(Health, 2, raw = T)1:SelfEfficacy | 0.116* (0.060) | |
| poly(Health, 2, raw = T)2:SelfEfficacy | 0.216*** (0.030) | |
| Observations | 250 | 250 |
| R2 | 0.197 | 0.341 |
| Adjusted R2 | 0.187 | 0.328 |
| Residual Std. Error | 10.759 (df = 246) | 9.781 (df = 244) |
| F Statistic | 20.058*** (df = 3; 246) | 25.297*** (df = 5; 244) |
| Note: | p<0.1; p<0.05; p<0.01 | |
- Note how different the results might look when you examine that as power polynomials
Plotting the Simple Slopes when there are Curves
- These can be done with the
effectspackage as I showed you above and last week - We will work with the orthogonal polynomial models
- or we can simply use the plotCurves function from the 'rockchalk' package
SmokeInterPlot <- plotCurves(Smoke.Model.2, modx = "SelfEfficacy", plotx = "Health", n=3,modxVals="std.dev") 
- Lets plot just the main effect (and think about what it means relative the power)
plotCurves(Smoke.Model.1, plotx = "Health") 
- Does not look very quadratic, does it? In other words, you cannot see the higher order effects as they can be hidden by the moderating variable
- How to find these hidden things?
- You have to test for interactions and changes in \(R^2\)
- You have to try to add higher order terms, but they should be motivated by some theory
- What if you did not know the second order term was in the model above.
Smoke.Model.1.L<-lm(Smoking~Health+SelfEfficacy,Smoke.Data) Smoke.Model.2.L<-lm(Smoking~Health*SelfEfficacy,Smoke.Data) | Dependent variable: | ||
| Smoking | ||
| Main Effects | Interaction | |
| (1) | (2) | |
| Constant | 33.263*** (0.680) | 33.334*** (0.680) |
| Health | 0.028 (0.293) | 0.042 (0.292) |
| SelfEfficacy | 1.193*** (0.155) | 1.207*** (0.155) |
| Health:SelfEfficacy | 0.097 (0.066) | |
| Observations | 250 | 250 |
| R2 | 0.194 | 0.201 |
| Adjusted R2 | 0.188 | 0.192 |
| Residual Std. Error | 10.752 (df = 247) | 10.727 (df = 246) |
| F Statistic | 29.770*** (df = 2; 247) | 20.666*** (df = 3; 246) |
| Note: | p<0.1; p<0.05; p<0.01 | |
ChangeInR.Smoke<-anova(Smoke.Model.1.L,Smoke.Model.2.L) knitr::kable(ChangeInR.Smoke, digits=4) | Res.Df | RSS | Df | Sum of Sq | F | Pr(>F) |
|---|---|---|---|---|---|
| 247 | 28557.09 | NA | NA | NA | NA |
| 246 | 28306.63 | 1 | 250.4599 | 2.1766 | 0.1414 |
- There is no significant interaction, but let's plot it anyway.
- Replotting as linear interaction
SmokeInterPlot.Linear <- plotSlopes(Smoke.Model.2.L, modx = "SelfEfficacy", plotx = "Health", n=3, modxVals="std.dev") 
- Let's check the change in \(R^2\) from Linear interaction model to poly interaction model now
stargazer(Smoke.Model.2.L,Smoke.Model.2,type="html", column.labels = c("Linear", "Poly"), intercept.bottom = FALSE, single.row=TRUE, notes.append = FALSE, header=FALSE) | Dependent variable: | ||
| Smoking | ||
| Linear | Poly | |
| (1) | (2) | |
| Constant | 33.334*** (0.680) | 33.493*** (0.621) |
| Health | 0.042 (0.292) | |
| poly(Health, 2)1 | 3.083 (9.813) | |
| poly(Health, 2)2 | -6.505 (9.795) | |
| SelfEfficacy | 1.207*** (0.155) | 1.189*** (0.141) |
| Health:SelfEfficacy | 0.097 (0.066) | |
| poly(Health, 2)1:SelfEfficacy | 3.042 (2.218) | |
| poly(Health, 2)2:SelfEfficacy | 16.361*** (2.288) | |
| Observations | 250 | 250 |
| R2 | 0.201 | 0.341 |
| Adjusted R2 | 0.192 | 0.328 |
| Residual Std. Error | 10.727 (df = 246) | 9.781 (df = 244) |
| F Statistic | 20.666*** (df = 3; 246) | 25.297*** (df = 5; 244) |
| Note: | p<0.1; p<0.05; p<0.01 | |
ChangeInR.Smoke.2<-anova(Smoke.Model.2,Smoke.Model.2.L) knitr::kable(ChangeInR.Smoke.2, digits=4) | Res.Df | RSS | Df | Sum of Sq | F | Pr(>F) |
|---|---|---|---|---|---|
| 244 | 23341.19 | NA | NA | NA | NA |
| 246 | 28306.63 | -2 | -4965.442 | 25.9534 | 0 |
Notes
- So, model with poly is a significantly better fit
- Would it change the story by much? In this case YES, but not always!
- Sometimes the improvement in fit is minor and does not change the story
- Often ignoring the higher order term can make it easier to test simple slopes and tell a story
- However, ignoring the higher order terms can violate your assumptions when the results, let's compare our linear vs poly models residuals
- Sometimes the improvement in fit is minor and does not change the story
- Linear Model
autoplot(Smoke.Model.2, which = 1:2, label.size = 1) + theme_bw() 
- Quadtradic Model
autoplot(Smoke.Model.2.L, which = 1:2, label.size = 1) + theme_bw() 
---
title: 'Interactions and Simple Slopes'
output:
  html_document:
    code_download: yes
    fontsize: 8pt
    highlight: textmate
    number_sections: no
    theme: flatly
    toc: yes
    toc_float:
      collapsed: no
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(cache=TRUE)
knitr::opts_chunk$set(echo = TRUE) #Show all script by default
knitr::opts_chunk$set(message = FALSE) #hide messages 
knitr::opts_chunk$set(warning =  FALSE) #hide package warnings 
knitr::opts_chunk$set(fig.width=5) #Set default figure sizes
knitr::opts_chunk$set(fig.height=4.5) #Set default figure sizes
knitr::opts_chunk$set(fig.align='center') #Set default figure
knitr::opts_chunk$set(fig.show = "hold") #Set default figure
knitr::opts_chunk$set(results = "hold") #Set default figure
```

# Interactions
- Interactions have the same meaning they did in ANOVA
    - there is a synergistic effect between two variables
    - However now we can examine interactions between continuous variables
- Additive Effects:

$$Y = B_1X + B_2Z + B_0$$

- Interactive Effects: 
$$Y = B_1X + B_2Z + B_3XZ + B_0$$


## Example of Interaction
- DV: Reading Speed, IVs: Age + IQ 
- Simulation: $Y = 3X + .2Z + .4XZ +100 +\epsilon$
- So what we mean is: $ReadingSpeed = 3*Age + .2*IQ + .4*Age*IQ +100 +Noise$
    - Set age to be uniform distribution between 7-17 
    - Set IQ to be normal, $M$=100 and $S$=15
    - Set $\epsilon$ = 15 (normal distribution of noise)
    - Note: For our regression simulation to match our equation, we should center our X and Z variables first, but we will save out the raw score 
```{r, echo=TRUE, warning=FALSE}
set.seed(42)
n <- 200
# Uniform distribution of Ages 
X <- runif(n, 7, 17)
Xc<-scale(X,scale=F)
# normal distrobution of IQ (100 +-15) 
Z <- rnorm(n, 100, 15)
Zc<-scale(Z,scale=F)
# Our equation to  create Y
Y <- 3*Xc + .2*Zc + .4*Xc*Zc +100 + rnorm(n, sd=15)
#Built our data frame
Reading.Data<-data.frame(Age=X,IQ=Z,ReadingSpeed=Y)
```

## Scatterplot & correlate our predictors
- We will use a useful diagnostic tool (GGally)
- We will visualize the density, scatterplot and correlation all at once

```{r,fig.width=4.5,fig.height=4.5}
library(GGally)
DiagPlot <- ggpairs(Reading.Data,  
              lower = list(continuous = "smooth"))
DiagPlot
```

- Predictors have some heteroskedastic issues, but the correlations seem to make sense


## Let's test for an interaction
1. Center your variables
2. Model 1: Examine a main-effects model (X + Z) [not necessary but useful]
3. Model 2: Examine a main-effects + Interaction model (X + Z + X:Z)

- Note: You can simply code it as X*Z in r, as it will automatically do (X + Z + X:Z)

```{r, echo=TRUE, results='asis'}
#Center
Reading.Data$Age.C<-scale(Reading.Data$Age, center = TRUE, scale = FALSE)[,]
Reading.Data$IQ.C<-scale(Reading.Data$IQ, center = TRUE, scale = FALSE)[,]
# Regressions
Centered.Read.1<-lm(ReadingSpeed~Age.C+IQ.C,Reading.Data)
Centered.Read.2<-lm(ReadingSpeed~Age.C*IQ.C,Reading.Data)
# Show results
library(stargazer)
stargazer(Centered.Read.1,Centered.Read.2,type="html",
          column.labels = c("Main Effects", "Interaction"),
          intercept.bottom = FALSE, single.row=TRUE, 
          notes.append = FALSE, header=FALSE)
```

- Also, change in $R^2$ is significant, as we might expect

```{r}
ChangeInR<-anova(Centered.Read.1,Centered.Read.2)
knitr::kable(ChangeInR, digits=4)
```

### Examine residuals
- Main Effects Model
```{r, fig.width=6}
library(ggfortify)
autoplot(Centered.Read.1, which = 1:2, label.size = 1) + theme_bw()
```

- Interactions Model

```{r, fig.width=6}
autoplot(Centered.Read.2, which = 1:2, label.size = 1) + theme_bw()
```

## Understanding the Interaction 
- Let's examine a 3d scatter plot of the raw data and plot against it the predicted results of each model (additive and interaction model)
- Once we have a "model" of the data, we can generate predictions for **all possible Ages** at **every possible IQ** (within the absolute boundary limits for our data)
    - With 2 predictors that generates a "3D surface plane."
        - You go past 2 predictors with these types of plots
- Let's examine Model 1 (additive effects) as a 3D scatter/surface plot
    - Scatter dots = Raw data points
    - Surface = Model 1 Predicted Results
        - How well does the surface fit the dots? 

```{r, echo = FALSE}
library(plotly)
library(dplyr)
Read.1<-lm(ReadingSpeed~Age+IQ,Reading.Data)
Read.2<-lm(ReadingSpeed~Age*IQ,Reading.Data)
# predict over sensible grid of values
graph_reso <- .5
Age.PL <- seq(min(Reading.Data$Age), max(Reading.Data$Age), by = graph_reso)
IQ.PL <- seq(min(Reading.Data$IQ), max(Reading.Data$IQ), by = graph_reso)
d <- setNames(data.frame(expand.grid(Age.PL, IQ.PL)),c("Age", "IQ"))
vals1 <- predict(Read.1, newdata = d)
vals2 <- predict(Read.2, newdata = d)
# form matrix and give to plotly
m1 <- matrix(vals1, nrow = length(unique(d$Age)), ncol = length(unique(d$IQ)))
m2 <- matrix(vals2, nrow = length(unique(d$Age)), ncol = length(unique(d$IQ)))
```

```{r, echo = FALSE, fig.width=5.5,fig.height=5.5}
p1 <- plot_ly(Reading.Data, x = ~IQ, y = ~Age, z = ~ReadingSpeed) %>%
  add_markers(marker = list(size = 4))  %>% 
  add_surface(x = ~IQ.PL, y = ~Age.PL, z = ~m1,showscale = FALSE, opacity=.75)
p1
```

- Let's examine Model 2 (interaction effects) as a 3d scatter/surface plot
    - Scatter dots = Raw data points
    - Surface = Model 2 Predicted Results
        - How well does the surface fit the dots? 

```{r, echo = FALSE, fig.width=5.5,fig.height=5.5}
p2 <- plot_ly(Reading.Data, x = ~IQ, y = ~Age, z = ~ReadingSpeed) %>%
  add_markers(marker = list(size = 4))  %>% 
  add_surface(x = ~IQ.PL, y = ~Age.PL, z = ~m2,showscale = FALSE, opacity=.75)
p2
```


## How visualize this for Papers?
- Well, there are some options
- We can do our fancy-pants surface plot, but that is hard to put into a paper
- More common this is to examine slope of one factor at different levels of the other (Simple Slopes) 
- What we need to decide is at which levels

### Hand picking
- Manually select the what levels of each you are going to examine
- For tracking values of interest
- here I picked -3 to 3 for age (mix and max) and -15, 0 15 for IQ (theoretical 1 SD and mean)

```{r}
library(effects)
Inter.1a<-effect(c("Age.C*IQ.C"), Centered.Read.2,
                           xlevels=list(Age.C=seq(-3,3, 1), 
                                        IQ.C=c(-15,0,15)))
```


```{r}
knitr::kable(summary(Inter.1a)$effect, digits=4)
plot(Inter.1a, multiline = TRUE)
```

### Quantile
- Examine the levels based on quantiles (bins based on probability)
- We will do this into 5 equal bins based on probability cut-offs 
- Does not assume normality for IV

```{r}
IQ.Quantile<-quantile(Reading.Data$IQ.C,probs=c(0,.25,.50,.75,1))
IQ.Quantile<-round(IQ.Quantile,2)

Inter.1b<-effect(c("Age.C*IQ.C"), Centered.Read.2,
                 xlevels=list(Age.C=seq(-3,3, 1), 
                              IQ.C=IQ.Quantile))
plot(Inter.1b, multiline = TRUE)
```

### Based on the SD 
- We select 3 values: $M-1SD$, $M$, $M+1SD$
- Not it does not have to be 1 SD, it can be 1.5 ,2 or 3
- Assumes normality for IV

```{r}
IQ.SD<-c(mean(Reading.Data$IQ.C)-sd(Reading.Data$IQ.C),
         mean(Reading.Data$IQ.C),
         mean(Reading.Data$IQ.C)+sd(Reading.Data$IQ.C))
IQ.SD<-round(IQ.SD,2)

Inter.1c<-effect(c("Age.C*IQ.C"), Centered.Read.2,
                 xlevels=list(Age.C=seq(-3,3, 1), 
                              IQ.C=IQ.SD))
plot(Inter.1c, multiline = TRUE)
```

## Why center your IVs?
- The center of each IV represents the mean of that IV
- Thus when you interact them $X*Z$, that means $0*0 = 0$
- Also, this can reduce multicollinearity issues
- This is because if $X*Z$ creates a line, it means you have added a new predictor (XZ) that strongly correlates with X and Z

```{r}
X<-c(0,2,4,6,8,10)
Z<-c(0,2,4,6,8,10)
XZ<-X*Z
plot(XZ)
```

- Correlations: $r_{x,z}$ = `r round(cor(X,Z),2)`, $r_{x,xz}$ = `r round(cor(X,XZ),2)`, $r_{z,xz}$ = `r round(cor(Z,XZ),2)`
-But if you center them, now you will make a U-shaped interaction which will be orthogonal to X and Z!

```{r, echo=TRUE, warning=FALSE}
X.C<-scale(X, center = TRUE, scale = FALSE)[,]
Z.C<-scale(Z, center = TRUE, scale = FALSE)[,]
XZ.C<-X.C*Z.C
plot(XZ.C)
```

- Correlations: $r_{x_c,z_c}$ = `r round(cor(X.C,Z.C),2)`, $r_{x_c,xz_c}$ = `r round(cor(X.C,XZ.C),2)`, $r_{z_c,xz_c}$ = `r round(cor(Z.C,XZ.C),2)`

- Let's apply this to our class example
- See below where I manually multiply the values and you can see a very strong positive slope
- Left = Raw Score, Right plot = Centered Score

```{r, echo=FALSE, fig.width=3.25, fig.height=3.25,fig.show='hold',fig.align='center'}
library(car)
Reading.Data$Age.X.IQ<-Reading.Data$Age*Reading.Data$IQ
Reading.Data$Age.X.IQ.C<-Reading.Data$Age.C*Reading.Data$IQ.C
scatterplot(ReadingSpeed~Age.X.IQ, data= Reading.Data, reg.line=FALSE, smoother=loessLine)
scatterplot(ReadingSpeed~Age.X.IQ.C, data= Reading.Data, reg.line=FALSE, smoother=loessLine)
```

### Let's run an uncentered regression
- as you can see the terms have changed from centered models

```{r, echo=TRUE}
Uncentered.Read.1<-lm(ReadingSpeed~IQ+Age,Reading.Data)
Uncentered.Read.2<-lm(ReadingSpeed~IQ*Age,Reading.Data)
```

```{r, echo=FALSE, results='asis'}
stargazer(Uncentered.Read.1,Uncentered.Read.2,type="html",
          column.labels = c("Main Effects", "Interaction"),
          intercept.bottom = FALSE, single.row=TRUE, 
          notes.append = FALSE, header=FALSE)
```

#### Multicollinearity of the Interaction
- Multicollinearity of the uncentered model

```{r, echo=TRUE, warning=FALSE}
vif(Uncentered.Read.2)
```

- We lost our main effect of **Age** as the variances got all inflated
- We did not have this problem in the centered model

```{r, echo=TRUE, warning=FALSE}
vif(Centered.Read.2)
```

- Note: Its OK to plot the uncentered version, but run the analysis on the centered data

# Testing Simple Slopes 
- The goal here is to figure out when the slope at a given level of another variable is different from zero
- We chop up the interaction at specific places as we did with the interactions plots (-1 SD, M, +1 SD) on the *moderating* variable (a third variable that affects the strength of the relationship between a dependent and independent variable) 
- Next, we test the slopes on the *predictor* variable (IV of main interest)
- Example: Performance on a task, predicted by Level of training, moderated by how hard the challenge is

## Example of Interaction
- DV: Performance, IVs: Training (X) + Challenge (Z)
- Simulation: $Y = 2.2X + 2.46Z + .75XZ +16 +\epsilon$
    - Set Training to be uniform distribution between -10 to 10 
    - Set Challenge to be normal, $M$=0 and $S$=15
    - Set $\epsilon$ = 50 (normal distribution of noise)
    - Note: For our regression simulation we will set means to about zero (so are not forced to center them)
    
```{r}
set.seed(42)
# 250 people
n <- 250
#Training (low to high)
X <- runif(n, -10, 10)
# normal distribution of Challenge Difficulty
Z <- rnorm(n, 0, 15)
# Our equation to  create Y
Y <- 2.2*X + 2.46*Z + .75*X*Z + 16 + rnorm(n, sd=50)
#Built our data frame
Skill.Data<-data.frame(Training=X,Challenge=Z,Performance=Y)
#run our regression
Skill.Model.1<-lm(Performance~Training+Challenge,Skill.Data)
Skill.Model.2<-lm(Performance~Training*Challenge,Skill.Data)
```

```{r,echo=FALSE,fig.width=4.5,fig.height=4.5}
DiagPlot2 <- ggpairs(Skill.Data,  
              lower = list(continuous = "smooth"))
DiagPlot2
```

```{r,echo=FALSE, results='asis'}
stargazer(Skill.Model.1,Skill.Model.2,type="html",
          column.labels = c("Main Effects", "Interaction"),
          intercept.bottom = FALSE, single.row=TRUE, 
          notes.append = FALSE, header=FALSE)
```


## Plotting 
- We will use the `rockchalk` library to visualize our interaction and test our simple slopes

```{r}
library(rockchalk)
m1ps <- plotSlopes(Skill.Model.2, modx = "Challenge", plotx = "Training", n=3, modxVals="std.dev")
m1psts <- testSlopes(m1ps)
round(m1psts$hypotests,4)
```

- We see the slope at the mean is not significant, 
- The +1SD and -1SD are significant, this is what is driving the interaction
- Well trained people at low levels of challenge get worse with more training, while high levels of challenge with more training get better
- The bonus using rockchalk is that you can change the number of lines (change $n=3$) you want to see or you can test quantiles as well (change modxVals="quantile")

```{r}
m1pQ <- plotSlopes(Skill.Model.2, modx = "Challenge", plotx = "Training", n=5, modxVals="quantile")
m1pQts <- testSlopes(m1pQ)
round(m1pQts$hypotests,4)
```

### Notes
- You often don't need to run the *t*-tests on the simple slopes, but they can useful to test very specific hypotheses
- You can run simple slopes analysis on three-way interactions, but let's leave that aside for now as you would have to use a different R-package

# Non-Linear interactions
- You can have non-linear terms interacting with other linear and non-linear terms
- Example: Quit smoking, X = Fear of your health, Z = moderated by Self-Efficacy for quitting

## Example of Polynomial Interaction
- DV: Quit smoking, IVs: Fear (X) + Self-Efficacy (Z)
- Simulation: $Y = -.18X - .15X^2 + Z + .16XZ + .07X^2Z  +20 +\epsilon$
    - Set Fear of your health to be uniform distribution between 1 to 9 (centered) 
    - Set Self-Efficacy to be normal, $M$=0 and $S$=15
    - Set $\epsilon$ = 10 (normal distribution of noise)
    
```{r}
set.seed(42)
n <- 250
#Fear of Health
X <- scale(runif(n, 1, 9), scale=F)
#Centered Self-Efficacy
Z <- scale(runif(n, 0, 15), scale=F)
# Our equation to  create Y
Y <- -.05*X - .20*X^2 + .15*X*Z + .22*(X^2)*Z+ 35 + rnorm(n, sd=10)
#Built our data frame
Smoke.Data<-data.frame(Smoking=Y,Health=X,SelfEfficacy=Z)
#run our regression
Smoke.Model.1<-lm(Smoking~poly(Health,2)+SelfEfficacy,Smoke.Data)
Smoke.Model.2<-lm(Smoking~poly(Health,2)*SelfEfficacy,Smoke.Data)
Smoke.Model.1.R<-lm(Smoking~poly(Health,2, raw=T)+SelfEfficacy,Smoke.Data)
Smoke.Model.2.R<-lm(Smoking~poly(Health,2, raw=T)*SelfEfficacy,Smoke.Data)
```

- Orthogonal Polynomials
```{r, echo=FALSE, results='asis'}
stargazer(Smoke.Model.1,Smoke.Model.2,type="html",
          column.labels = c("Main Effects", "Interaction"),
          intercept.bottom = FALSE, single.row=TRUE, 
          notes.append = FALSE, header=FALSE)
```

- Power Polynomials

```{r, echo=FALSE, results='asis'}
stargazer(Smoke.Model.1.R,Smoke.Model.2.R,type="html",
          column.labels = c("Main Effects Non-Ortho", "Interaction Non-Ortho"),
          intercept.bottom = FALSE, single.row=TRUE, 
          notes.append = FALSE, header=FALSE)
```

- Note how different the results might look when you examine that as power polynomials


## Plotting the Simple Slopes when there are Curves
- These can be done with the `effects` package as I showed you above and last week
- We will work with the orthogonal polynomial models
    - or we can simply use the plotCurves function from the 'rockchalk' package

```{r}
SmokeInterPlot <- plotCurves(Smoke.Model.2, 
                             modx = "SelfEfficacy", plotx = "Health",
                             n=3,modxVals="std.dev")
```

- Lets plot just the main effect (and think about what it means relative the power)

```{r, echo=TRUE, warning=FALSE}
plotCurves(Smoke.Model.1, plotx = "Health")
```

- Does not look very quadratic, does it? In other words, you cannot see the higher order effects as they can be hidden by the moderating variable
- How to find these hidden things? 
- You have to test for interactions and changes in $R^2$
- You have to try to add higher order terms, but they should be motivated by some theory
- What if you did not know the second order term was in the model above. 

```{r, echo=TRUE, warning=FALSE}
Smoke.Model.1.L<-lm(Smoking~Health+SelfEfficacy,Smoke.Data)
Smoke.Model.2.L<-lm(Smoking~Health*SelfEfficacy,Smoke.Data)
```

```{r, echo=FALSE, results='asis'}
stargazer(Smoke.Model.1.L,Smoke.Model.2.L,type="html",
          column.labels = c("Main Effects", "Interaction"),
          intercept.bottom = FALSE,
          single.row=TRUE, 
          notes.append = FALSE,
          header=FALSE)

```

```{r}
ChangeInR.Smoke<-anova(Smoke.Model.1.L,Smoke.Model.2.L)
knitr::kable(ChangeInR.Smoke, digits=4)
```

- There is no significant interaction, but let's plot it anyway.
- Replotting as linear interaction

```{r}
SmokeInterPlot.Linear <- plotSlopes(Smoke.Model.2.L, modx = "SelfEfficacy", plotx = "Health", n=3, modxVals="std.dev")
```


- Let's check the change in $R^2$ from Linear interaction model to poly interaction model now

```{r, echo=TRUE, results='asis'}
stargazer(Smoke.Model.2.L,Smoke.Model.2,type="html",
          column.labels = c("Linear", "Poly"),
          intercept.bottom = FALSE,
          single.row=TRUE, 
          notes.append = FALSE,
          header=FALSE)
```

```{r}
ChangeInR.Smoke.2<-anova(Smoke.Model.2,Smoke.Model.2.L)
knitr::kable(ChangeInR.Smoke.2, digits=4)
```

## Notes
- So, model with poly is a significantly better fit
- Would it change the story by much? In this case YES, but not always! 
    - Sometimes the improvement in fit is minor and does not change the story
        - Often ignoring the higher order term can make it easier to test simple slopes and tell a story
    - However, ignoring the higher order terms can violate your assumptions when the results, let's compare our linear vs poly models residuals

- Linear Model
```{r, fig.width=6}
autoplot(Smoke.Model.2, which = 1:2, label.size = 1) + theme_bw()
```

- Quadtradic Model
```{r, fig.width=6}
autoplot(Smoke.Model.2.L, which = 1:2, label.size = 1) + theme_bw()
```

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