Interactions 3

class: center, middle, inverse, title-slide

.title[
# Interactions 3 
]
.subtitle[
## Data Analysis for Psychology in R 2 
]
.author[
### dapR2 Team
]
.institute[
### Department of Psychology The University of Edinburgh
]

---

# Course Overview

.pull-left[

<table style="border: 1px solid black;>
 <tr style="padding: 0 1em 0 1em;">
 <td rowspan="5" style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1;text-align:center;vertical-align: middle">
 Introduction to Linear Models</td>
 <td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 Intro to Linear Regression</td>
 </tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 Interpreting Linear Models</td></tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 Testing Individual Predictors</td></tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 Model Testing & Comparison</td></tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 Linear Model Analysis</td></tr>

<tr style="padding: 0 1em 0 1em;">
 <td rowspan="5" style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1;text-align:center;vertical-align: middle">
 Analysing Experimental Studies</td>
 <td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 Categorical Predictors & Dummy Coding</td>
 </tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 	Effects Coding & Coding Specific Contrasts</td></tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 Assumptions & Diagnostics</td></tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 Bootstrapping</td></tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 Categorical Predictor Analysis</td></tr>
</table>

]

.pull-right[

<table style="border: 1px solid black;>
 <tr style="padding: 0 1em 0 1em;">
 <td rowspan="5" style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1;text-align:center;vertical-align: middle">
 Interactions</td>
 <td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 Interactions I</td>
 </tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 Interactions II</td></tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:1">
 Interactions III</td></tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:0.4">
 Analysing Experiments</td></tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:0.4">
 Interaction Analysis</td></tr>

<tr style="padding: 0 1em 0 1em;">
 <td rowspan="5" style="border: 1px solid black;padding: 0 1em 0 1em;opacity:0.4;text-align:center;vertical-align: middle">
 Advanced Topics</td>
 <td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:0.4">
 Power Analysis</td>
 </tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:0.4">
 Binary Logistic Regression I</td></tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:0.4">
 Binary Logistic Regression II</td></tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:0.4">
 Logistic Regresison Analysis</td></tr>
 <tr><td style="border: 1px solid black;padding: 0 1em 0 1em;opacity:0.4">
 	Exam Prep and Course Q&A</td></tr>
</table>

]

---

# Week's Learning Objectives

1. Interpret interactions between two categorical variables (with dummy coding)
  + Begin with 2x2 (two binary variables)
  + Move on to look at examples with more levels

2. Visualise and probe interactions

3. Be able to read interaction plots

---
#  General definition

+ When the effects of one predictor on the outcome differ across levels of another predictor

+ Categorical `$\times$` categorical interaction:
	+ There is a difference in the differences between groups across levels of a second factor

+ This idea of a difference in differences can be quite tricky to think about
  + So we will start with some visualisation, and then look at two examples

---
# Visualising interactions
+ In our class example we have looked at predicting salary from years of service (a continous variable) and department (a categorical variable)

+ Suppose our company also had two sites, Edinburgh and Dundee, and we wanted to see if the salaries across departments differed depending on location

+ The table below contains hypothetical average salaries in thousands of pounds for each group

|         | Edinburgh| Dundee|
|:--------|---------:|------:|
|Accounts |        50|     40|
|Manager  |        30|     20|

---
# Basic plot

.pull-left[

|         | Edinburgh| Dundee|
|:--------|---------:|------:|
|Accounts |        50|     40|
|Manager  |        30|     20|

+ Let's look at the plot:
  + x-axis shows locations
  + y-axis is our salaries
  + The colours represent departments
  
]

.pull-right[

![](dapr2_13_interactions3_files/figure-html/unnamed-chunk-6-1.png)
]

---
# Difference in differences (1)

.pull-left[

|         | Edinburgh| Dundee|
|:--------|---------:|------:|
|Accounts |        50|     40|
|Manager  |        30|     20|

+ In each plot we look at, think about subtracting the average store managers salary (blue triangle) from the average accounts salary (red circle)

+ For both Edinburgh and Dundee, this difference is £20,000

+ Note, the lines are parallel
  + When the lines are parallel, there is no interaction
  + The effect of one variable (department) does not vary along values or levels of the other variable (location)

]

.pull-right[

![](dapr2_13_interactions3_files/figure-html/unnamed-chunk-9-1.png)

]

---
# Difference in differences (2)

.pull-left[

Let's imagine our group means look like this instead:

|         | Edinburgh| Dundee|
|:--------|---------:|------:|
|Accounts |        50|     40|
|Manager  |        40|     20|

+ This time we can see the difference differs
  + £20,000 in Dundee
  + £10,000 in Edinburgh
  
+ Note the lines are no longer parallel
  + Suggests interaction
  + But not crossing (so ordinal interaction)

]

.pull-right[

![](dapr2_13_interactions3_files/figure-html/unnamed-chunk-12-1.png)

]

---
# Difference in differences (3)

.pull-left[

Now consider the following group means:

|         | Edinburgh| Dundee|
|:--------|---------:|------:|
|Accounts |        40|     40|
|Manager  |        60|     20|

+ This time we can see the difference differs
  + postive difference of £20,000 in Dundee
  + negative difference of £20,000 in Edinburgh
  
+ Note the lines are no longer parallel
  + Suggests interaction
  + Now crossing (so disordinal interaction)

]

.pull-right[

![](dapr2_13_interactions3_files/figure-html/unnamed-chunk-15-1.png)

]

---
class: center, middle
# Questions?

---
#  Lecture notation

`$$y_i = \beta_0 + \beta_1 x_{i} + \beta_2 z_{i} + \beta_3 x_{i}z_{i} + \epsilon_i$$`

+ Lecture notation:
  + `$y$` is a continuous outcome
  
  + `$x$` is a categorical predictor ( `location` )
  
  + `$z$` is a categorical predictor (`department` )
  
  + `$xz$` is their product, or interaction predictor
  
---
# Product for dummy variables

+ Let's set Edinburgh as the baseline for `location` and Accounts as the baseline for `department`
+ The interaction is the product of the two variables
+ We can then figure out what to substitute for `$x$`, `$z$` and `$xz$` in our regression formula to get predictions for each group once we have estimated our `$\beta$` coefficients

|location  |department |  x|  z| xz|
|:---------|:----------|--:|--:|--:|
|Edinburgh |Accounts   |  0|  0|  0|
|Edinburgh |Manager    |  0|  1|  0|
|Dundee    |Accounts   |  1|  0|  0|
|Dundee    |Manager    |  1|  1|  1|

---
#  Interpretation: Categorical `$\times$` categorical interaction (dummy codes)

`$$y_i = \beta_0 + \beta_1 x_{i} + \beta_2 z_{i} + \beta_3 x_{i}z_{i} + \epsilon_i$$`

+ `$\beta_0$` = Value of `$y$` when `$x$` and `$z$` are 0

+ Expected salary for Accounts in Edinburgh

---
#  Interpretation: Categorical `$\times$` categorical interaction (dummy codes)

`$$y_i = \beta_0 + \beta_1 x_{i} + \beta_2 z_{i} + \beta_3 x_{i}z_{i} + \epsilon_i$$`

+ Remember `$x$` is our `location` variable and `$z$` is our `department` variable

+ `$\beta_1$` = Difference between levels of `$x$` when `$z$` = 0

+ The difference in salary between Accounts in Edinburgh and Dundee

---
#  Interpretation: Categorical `$\times$` categorical interaction (dummy codes)

`$$y_i = \beta_0 + \beta_1 x_{i} + \beta_2 z_{i} + \beta_3 x_{i}z_{i} + \epsilon_i$$`

+ Remember `$x$` is our `location` variable and `$z$` is our `department` variable

+ `$\beta_2$` = Difference between levels of `$z$` when `$x$` = 0

+ The difference in salary between Accounts and Store managers in Edinburgh

---
#  Interpretation: Categorical `$\times$` categorical interaction (dummy codes)

`$$y_i = \beta_0 + \beta_1 x_{i} + \beta_2 z_{i} + \beta_3 x_{i}z_{i} + \epsilon_i$$`

+  `$\beta_3$` = Difference between levels of `$x$` across levels of `$z$`

+ The difference between salary in Accounts and Store managers between Edinburgh and Dundee

+ "Difference in differences"

---
#  Example: Categorical `$\times$` categorical

.pull-left[

Let's examine the actual group means in our data set

``` r
salary3 %>%
  group_by(location, department) %>%
  summarise(
    Salary = mean(salary)
  )
```
]

.pull-right[

```
## # A tibble: 4 × 3
## # Groups: location [2]
## location department Salary
## <fct> <fct> <dbl>
## 1 Edinburgh Accounts 50.7
## 2 Edinburgh Manager 47.2
## 3 Dundee Accounts 48.7
## 4 Dundee Manager 36.8
```
]

---
#  Example: Categorical `$\times$` categorical

``` r
m1 <- lm(salary ~ location*department, salary3)
```

```
## 
## Call:
## lm(formula = salary ~ location * department, data = salary3)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -10.551 -3.389 0.579 2.937 14.325 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 50.6704 0.7144 70.927 < 2e-16 ***
## locationDundee -1.9796 1.4872 -1.331 0.186297 
## departmentManager -3.4625 1.3365 -2.591 0.011072 * 
## locationDundee:departmentManager -8.4153 2.2779 -3.694 0.000367 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.052 on 96 degrees of freedom
## Multiple R-squared: 0.4774,	Adjusted R-squared: 0.461 
## F-statistic: 29.23 on 3 and 96 DF, p-value: 1.636e-13
```

---
#  Example: Categorical `$\times$` categorical

.pull-left[

+ We can visualise categorical interactions using `cat_plot()` from the `interactions` package

+ `probe_interaction()` does not work with two categorical predictors

``` r
cat_plot(m1,
         pred = location,
         modx = department)
```

+ Plot shows group means and 95% confidence intervals based on `m1`

]

.pull-right[
![](dapr2_13_interactions3_files/figure-html/unnamed-chunk-22-1.png)

]

---
#  Example: Categorical `$\times$` categorical

```
##                                  Estimate Std. Error t value Pr(>|t|)
## (Intercept)                         50.67       0.71   70.93     0.00
## locationDundee                      -1.98       1.49   -1.33     0.19
## departmentManager                   -3.46       1.34   -2.59     0.01
## locationDundee:departmentManager    -8.42       2.28   -3.69     0.00
```

.pull-left[
+ `$\beta_0$` = Value of `$y$` when `$x$` and `$z$` are 0

+ Expected salary for Accounts in Edinburgh is £50,670
]

.pull-right[

|         | Edinburgh| Dundee|
|:--------|---------:|------:|
|Accounts |     50.67|  48.69|
|Manager  |     47.21|  36.81|
]

---
#  Example: Categorical `$\times$` categorical

.pull-left[
+ `$\beta_1$` = Difference between levels of `$x$` when `$z$` = 0

+ The difference in salary between Accounts in Edinburgh and Dundee is £1,980. The salary is lower in Dundee. (But note this is not statistically significant)

+ With dummy coding, slopes for marginal effects refer to `$\text{Group} - \text{Reference group (intercept)}$`

]

.pull-right[

|         | Edinburgh| Dundee|
|:--------|---------:|------:|
|Accounts |     50.67|  48.69|
|Manager  |     47.21|  36.81|
]

---
#  Example: Categorical `$\times$` categorical

.pull-left[
+ `$\beta_2$` = Difference between levels of `$z$` when `$x$` = 0

+ The difference in salary between Accounts and Store managers in Edinburgh is £3,460. The salary is lower for Store Managers.

+ With dummy coding, slopes for marginal effects refer to `$\text{Group} - \text{Reference group (intercept)}$`

]

.pull-right[

|         | Edinburgh| Dundee|
|:--------|---------:|------:|
|Accounts |     50.67|  48.69|
|Manager  |     47.21|  36.81|
]

---
#  Example: Categorical `$\times$` categorical

.pull-left[
+  `$\beta_3$` = Difference between levels of `$x$` across levels of `$z$`

+ The difference between salary for Accounts and Store managers differs by £8,420 between Edinburgh and Dundee. The difference is greater in Dundee than in Edinburgh.

]

.pull-right[

|         | Edinburgh| Dundee|
|:--------|---------:|------:|
|Accounts |     50.67|  48.69|
|Manager  |     47.21|  36.81|
]

---
# Predictions from the regression formula

+ Let's plug our `$\beta$` coefficients back into our regression formula:

`$$\hat{y} = \beta_0 + \beta_1 x + \beta_2 z + \beta_3 xz$$`

`$$\hat{y} = 50.67 -1.98 x -3.46 z -8.42 xz$$`

> **Test yourself: What is the estimated mean salary for each group?**

---
class: center, middle
# Questions?

---
# Extending past 2x2

+ When fitting an interaction to categorical variables with more than 2 levels, we need additional interaction terms

+ Remember, a categorical variable with 3 levels is represented by the model as 2 binary variables
  + So that means we have two variables to create products with

+ The general rule on the number of interaction terms is:

`$$(r-1) \times (c-1)$$`

+ Where 
  + `$r$` (row) = number of levels of the first categorical variable
  + `$c$` (column) = number of levels of the second categorical variable

---
# Example

+ The data comes from a study into patient care in paediatric wards

+ A researcher was interested in whether the subjective well-being of patients differed dependent on the post-operation treatment schedule they were given, and the hospital in which they were staying

+ **Factor 1**: `Treatment` (Levels: TreatA, TreatB, TreatC).
  
+ **Factor 2**: `Hosp` (Levels: Hosp1, Hosp2). 
  
+ Total sample n = 180 (30 patients in each of 6 groups)
  + Between-person design

+ **Outcome variable**: Subjective well-being (SWB)
  + An average of multiple raters (the patient, a member of their family, and a friend)
  + SWB scores ranged from 0 to 20

---
# The data

``` r
head(hosp_tbl, 10)
```

```
## # A tibble: 10 × 3
## SWB Treatment Hospital
## <dbl> <fct> <fct> 
## 1 6.2 TreatA Hosp1 
## 2 15.9 TreatA Hosp1 
## 3 7.2 TreatA Hosp1 
## 4 11.3 TreatA Hosp1 
## 5 11.2 TreatA Hosp1 
## 6 9 TreatA Hosp1 
## 7 14.5 TreatA Hosp1 
## 8 7.3 TreatA Hosp1 
## 9 13.7 TreatA Hosp1 
## 10 12.6 TreatA Hosp1
```

---
# The group means

|       |Hosp1 |Hosp2 |
|:------|:-----|:-----|
|TreatA |10.80 |7.85  |
|TreatB |9.43  |13.11 |
|TreatC |10.10 |7.98  |

Some ways of phrasing our research question:

+ Is there a difference in how subjective wellbeing varies for the three different treatment options between the two hospitals?
+ Does the hospital a patient stays at influence how the type of treatment influences their subjective wellbeing?

---
# Model equations and coding

`$$y_{ijk} = b_0 + \underbrace{(b_1D_1 + b_2D_2)}_{\text{Treatment}} + \underbrace{b_3D_3}_{\text{Hospital}} + \underbrace{b_4D_{13} + b_5D_{23}}_{\text{Interactions}} + \epsilon_{i}$$`

```
## # A tibble: 6 × 7
## Treatment Hospital D1 D2 D3 D13 D23
## <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 A Hosp1 0 0 0 0 0
## 2 A Hosp2 0 0 1 0 0
## 3 B Hosp1 1 0 0 0 0
## 4 B Hosp2 1 0 1 1 0
## 5 C Hosp1 0 1 0 0 0
## 6 C Hosp2 0 1 1 0 1
```

+ Note that "D" stands for dummy-coded variables

---
# Interpretation with dummy coding

`$$y_{ijk} = b_0 + \underbrace{(b_1D_1 + b_2D_2)}_{\text{Treatment}} + \underbrace{b_3D_3}_{\text{Hospital}} + \underbrace{b_4D_{13} + b_5D_{23}}_{\text{Interactions}} + \epsilon_{i}$$`

+ Treatment A and Hospital 1 as reference levels

+ `$b_0$` = Mean of treatment A in hospital 1
+ `$b_1$` = Difference between Treatment B and Treatment A in Hospital 1
+ `$b_2$` = Difference between Treatment C and Treatment A in Hospital 1
+ `$b_3$` = Difference between Treatment A in Hospital 1 and Hospital 2
+ `$b_4$` = Difference between Treatment A and Treatment B between Hospital 1 and Hospital 2
+ `$b_5$` = Difference between Treatment A and Treatment C between Hospital 1 and Hospital 2

---
# Our results

``` r
m4 <- lm(SWB ~ Treatment + Hospital + Treatment*Hospital, data = hosp_tbl)
m4sum <- summary(m4)
m4res <- round(m4sum$coefficients,2)
m4res
```

```
##                               Estimate Std. Error t value Pr(>|t|)
## (Intercept)                      10.80       0.37   29.19     0.00
## TreatmentTreatB                  -1.37       0.52   -2.62     0.01
## TreatmentTreatC                  -0.70       0.52   -1.33     0.18
## HospitalHosp2                    -2.95       0.52   -5.63     0.00
## TreatmentTreatB:HospitalHosp2     6.63       0.74    8.97     0.00
## TreatmentTreatC:HospitalHosp2     0.82       0.74    1.11     0.27
```

---
# Interpretation with dummy coding

.pull-left[
+ `$b_0$` = Mean of treatment A in hospital 1
+ `$b_1$` = Difference between Treatment B and Treatment A in Hospital 1
+ `$b_2$` = Difference between Treatment C and Treatment A in Hospital 1
+ `$b_3$` = Difference between Treatment A in Hospital 1 and Hospital 2
+ `$b_4$` = Difference between Treatment A and Treatment B between Hospital 1 and Hospital 2
+ `$b_5$` = Difference between Treatment A and Treatment C between Hospital 1 and Hospital 2
]

.pull-right[

|       |Hosp1 |Hosp2 |
|:------|:-----|:-----|
|TreatA |10.80 |7.85  |
|TreatB |9.43  |13.11 |
|TreatC |10.10 |7.98  |

]
---
# Brief comment: 3-way interactions

+ **In principle** we can extend interactions to more than two variables

+ Consider our salary example:
  + We could ask: Are there location-based differences in how years of service contribute to salary differently across departments? 
  
+ Note that we'd then have three interacting predictors:
  + Years of service (continuous)
  + Department (binary; Store Manager, Accounts)
  + Location (binary; Edinburgh, Dundee)

+ This is a plausible question concerning organisatonal fairness

---
# Brief comment: 3-way interactions

+ In general, three-way interactions are tricky to interpret

+ Require lots of data to test as the effects are often very small
  + Extends the issues of power already discussed

+ **Only test a three-way interaction if you have strong theoretical reason for doing so**

---
# Summary

+ We have considered how we fit and interpret linear models with categorical interactions

+ We have focused on dummy coded variables (we'll return to effects coding next week)

+ We saw an example of an interaction involving categorical variables with more than two levels (multiple interaction terms)

---

## This week

.pull-left[

### Tasks

**Attend your lab and work together on the exercises**

**Complete the weekly quiz**

]

.pull-right[

### Support

**Help each other on the Piazza forum**

**Attend office hours (see Learn page for details)**

]