Interactions (II)

class: center, middle, inverse, title-slide

# Interactions (II)

---

## Last time...

Introduction to interactions with two continuous predictors

---

### Recap

We use interaction terms to test the hypothesis that the relationship between X and Y changes as a function of Z. 
  - social support buffers the effect of anxiety and stress
  - conscientiousness predicts better health for affluent individuals and worse health for non-affluent individuals
  
The interaction term represents how much the slope of X changes as you increase on Z, and also how much the slope of Z changes as you increase on X. Interactions are symmetric.

---
### Recap: Output

```r
cars_model = lm(mpg ~ disp*hp, data = mtcars)
summary(cars_model)
```

```
## 
## Call:
## lm(formula = mpg ~ disp * hp, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.5153 -1.6315 -0.6346  0.9038  5.7030 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.967e+01  2.914e+00  13.614 7.18e-14 ***
## disp        -7.337e-02  1.439e-02  -5.100 2.11e-05 ***
## hp          -9.789e-02  2.474e-02  -3.956 0.000473 ***
## disp:hp      2.900e-04  8.694e-05   3.336 0.002407 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.692 on 28 degrees of freedom
## Multiple R-squared:  0.8198,	Adjusted R-squared:  0.8005 
## F-statistic: 42.48 on 3 and 28 DF,  p-value: 1.499e-10
```

---
### Recap: Simple slopes

```r
library(reghelper)
simple_slopes(cars_model, levels = list(hp = c(78, 147, 215)))
```

```
##     disp  hp Test Estimate Std. Error t value df  Pr(>|t|) Sig.
## 1 sstest  78       -0.0507     0.0088 -5.7448 28 3.645e-06  ***
## 2 sstest 147       -0.0307     0.0064 -4.8207 28 4.528e-05  ***
## 3 sstest 215       -0.0110     0.0086 -1.2782 28    0.2117
```
---
### Recap: Plot simple slopes

```r
library(sjPlot)
plot_model(cars_model, type = "int", mdrt.values = "meansd")
```

![](15-interactions_files/figure-html/unnamed-chunk-3-1.png)

---

## Today

Mixing categorical and continuous predictors

Two categorical predictors

Start discussing Factorial ANOVA

---

## Mixing categorical and continuous

Consider the case where D is a variable representing two groups. In a univariate regression, how do we interpret the coefficient for D?

`$$\hat{Y} = b_{0} + b_{1}D$$`

`$b_0$` is the mean of the reference group, and D represents the difference in means between the two groups.

---

### Interpreting slopes

Extending this to the multivariate case, where X is continuous and D is a dummy code representing two groups.

`$$\hat{Y} = b_{0} + b_{1}D + b_2X$$`

How do we interpret `$b_1?$`

`$b_1$` is the difference in means between the two groups *if the two groups have the same average level of X* or holding X constant.

This, by the way, is ANCOVA.

---

### Visualizing

![](15-interactions_files/figure-html/unnamed-chunk-4-1.png)

---
### Visualizing
![](15-interactions_files/figure-html/unnamed-chunk-5-1.png)

---
### Visualizing
![](15-interactions_files/figure-html/unnamed-chunk-6-1.png)
---

### 3 or more groups

We might be interested in the relative contributions of our two variables, but we have to remember that they're on different scales, so we cannot compare them using the unstandardized regression coefficient.

Standardized coefficients can be used if we only have two groups, but what if we have 3 or more?

Just like we use `$R^2$` to report how much variance in Y is explained by the model, we can break this down into the unique contributions of each variable in the model, including factors with 3+ levels.

`$$\large \eta^2 = \frac{SS_{\text{Variable}}}{SS_{Y}}== \frac{SS_{\text{Variable}}}{SS_{\text{Total}}}$$`

---

```r
mod = lm(Y ~ X + D, data = df)
anova(mod)
```

```
## Analysis of Variance Table
## 
## Response: Y
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## X          1 64.045  64.045  61.489 4.788e-07 ***
## D          1 20.071  20.071  19.270 0.0003998 ***
## Residuals 17 17.707   1.042                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

`$$\large \eta^2_{X} = \frac{64.045}{64.045+20.071+17.707} = .62899 = 63\%$$`
`$$\large \eta^2_{D} = \frac{20.071}{64.045+20.071+17.707} = .19712 = 20\%$$`

---

## Interactions

Now extend this example to include joint effects, not just additive effects:

`$$\hat{Y} = b_{0} + b_{1}D + b_2X + b_3DX$$`

How do we interpret `$b_1?$`

`$b_1$` is the difference in means between the two groups *when X is 0*.

What is the interpretation of `$b_2$`?

`$b_2$` is the slope of X among the reference group.

What is the interpretation of `$b_3?$`

`$b_3$` is the difference in slopes between the reference group and the other group.

---

### Visualizing

![](15-interactions_files/figure-html/unnamed-chunk-8-1.png)

Where should we draw the segment to compare means?

???

Where you draw the segment changes the difference in means. That's why `$b_1$` can only be interpreted as the difference in means when X = 0.

---

## Example

The University of Oregon is interested in understanding how undergraduates' academic performance and choice of major impacts their career success. They contact 150 alumni between the ages of 25 and 35 and collect their current salary (in thousands of dollars), their primary undergarduate major, and their GPA upon graduating.

---

### `R` code

```r
library(psych)
table(inc_data$major)
```

```
## 
##    Econ English   Psych 
##      50      50      50
```

```r
describe(inc_data[,c("gpa", "income")], fast = T)
```

```
##        vars   n  mean   sd   min    max  range   se
## gpa       1 150  3.36  0.4  2.44   4.19   1.74 0.03
## income    2 150 84.35 34.0 24.67 160.27 135.60 2.78
```
---
### Model summary

```r
career.mod = lm(income ~ gpa*major, data = inc_data)
summary(career.mod)
```

```
## 
## Call:
## lm(formula = income ~ gpa * major, data = inc_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -42.625 -11.869   0.376   9.301  40.942 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       -59.181     22.902  -2.584   0.0108 *  
## gpa                59.660      7.705   7.743 1.58e-12 ***
## majorEnglish      -81.747     37.149  -2.201   0.0294 *  
## majorPsych       -175.314     35.462  -4.944 2.10e-06 ***
## gpa:majorEnglish   -4.562     11.089  -0.411   0.6814    
## gpa:majorPsych     29.545     10.949   2.698   0.0078 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.91 on 144 degrees of freedom
## Multiple R-squared:  0.8142,	Adjusted R-squared:  0.8077 
## F-statistic: 126.2 on 5 and 144 DF,  p-value: < 2.2e-16
```

---
### Model summary: centering predictors

```r
inc_data$gpa_c = inc_data$gpa - mean(inc_data$gpa)
career.mod_c = lm(income ~ gpa_c*major, data = inc_data)
summary(career.mod_c)
```

```
## 
## Call:
## lm(formula = income ~ gpa_c * major, data = inc_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -42.625 -11.869   0.376   9.301  40.942 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         141.428      3.752  37.691  < 2e-16 ***
## gpa_c                59.660      7.705   7.743 1.58e-12 ***
## majorEnglish        -97.086      4.907 -19.783  < 2e-16 ***
## majorPsych          -75.965      4.384 -17.327  < 2e-16 ***
## gpa_c:majorEnglish   -4.562     11.089  -0.411   0.6814    
## gpa_c:majorPsych     29.545     10.949   2.698   0.0078 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.91 on 144 degrees of freedom
## Multiple R-squared:  0.8142,	Adjusted R-squared:  0.8077 
## F-statistic: 126.2 on 5 and 144 DF,  p-value: < 2.2e-16
```

---

### Plotting results

```r
library(sjPlot)
plot_model(career.mod, type = "int", show.data = T, axis.title = c("Grade Point Average", "Income (in thousands of dollars)"), legend.title = "Undergraduate Major", title = "I'm in the wrong profession", wrap.title = T)
```

![](15-interactions_files/figure-html/unnamed-chunk-18-1.png)
---

## Two categorical predictors

If both X and M are categorical variables, the interpretation of coefficients is no longer the value of means and slopes, but means and differences in means.

Recall our Solomon's paradox example from a few weeks ago:

```r
library(here)
```

```
## here() starts at /Users/sweston2/Google Drive/Work/Teaching/Courses/Graduate/Statistics and Methods Sequence/psy612
```

```r
solomon = read.csv(here("data/solomon.csv"))
```

```r
head(solomon[,c("PERSPECTIVE", "DISTANCE", "WISDOM")])
```

```
##   PERSPECTIVE  DISTANCE      WISDOM
## 1       other  immersed -0.27589395
## 2       other distanced  0.42949213
## 3       other distanced -0.02785874
## 4       other distanced  0.53271500
## 5        self distanced  0.62299793
## 6        self distanced -1.99578129
```
---

### Model summary