Interactions 3

class: center, middle, inverse, title-slide

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

---

# Weeks Learning Objectives
1. Interpret interactions for between two categorical variables (dummy coding).

2. Visualize and probe interactions.

3. Be able to read interaction plots.

---
# Topics for today

+ Categorical-categorical interaction.
  + Begin with 2x2 (two binary variables)
  + Move on to look at examples with more levels

+ Visualizing and understanding categorical interactions.

---
#  General definition

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

+ Categorical*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 visualization, and then look at two examples.

---
# Visualizing interactions
+ In our class example we have been looking at predicting salary from years of service and department.

+ Suppose our company also had two sites, London and Birmingham, and we wanted to see if the salaries across department differed depending on location.

+ The table below contains the average salary in thousands of pounds for each group.

<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> London </th>
   <th style="text-align:right;"> Birmingham </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Accounts </td>
   <td style="text-align:right;"> 50 </td>
   <td style="text-align:right;"> 40 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Manager </td>
   <td style="text-align:right;"> 30 </td>
   <td style="text-align:right;"> 20 </td>
  </tr>
</tbody>
</table>

---
# Basic plot

.pull-left[

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

.pull-right[

![](dapr2_08_interactions3_files/figure-html/unnamed-chunk-5-1.png)
]

---
# Difference in differences (1)

.pull-left[

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

+ In both cases it is £20,000.

+ Note, the lines are parallel
  + Remember what we have said about parallel lines...no interaction

]

.pull-right[

![](dapr2_08_interactions3_files/figure-html/unnamed-chunk-8-1.png)

]

---
# Difference in differences (2)

.pull-left[

<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> London </th>
   <th style="text-align:right;"> Birmingham </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Accounts </td>
   <td style="text-align:right;"> 50 </td>
   <td style="text-align:right;"> 40 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Manager </td>
   <td style="text-align:right;"> 40 </td>
   <td style="text-align:right;"> 20 </td>
  </tr>
</tbody>
</table>

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

]

.pull-right[

![](dapr2_08_interactions3_files/figure-html/unnamed-chunk-11-1.png)

]

---
# Difference in differences (3)

.pull-left[

<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> London </th>
   <th style="text-align:right;"> Birmingham </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Accounts </td>
   <td style="text-align:right;"> 40 </td>
   <td style="text-align:right;"> 40 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Manager </td>
   <td style="text-align:right;"> 60 </td>
   <td style="text-align:right;"> 20 </td>
  </tr>
</tbody>
</table>

+ This time we can see the difference differs.
  + £20,000 in Birmingham
  + £20,000 in London
  
+ Note the lines are no longer parallel.
  + Suggests interaction.
  + Now crossing (so disordinal interaction)

]

.pull-right[

![](dapr2_08_interactions3_files/figure-html/unnamed-chunk-14-1.png)

]

---
class: center, middle
# Question break ...

---
#  Lecture notation

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

+ 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

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> location </th>
   <th style="text-align:left;"> department </th>
   <th style="text-align:right;"> x </th>
   <th style="text-align:right;"> z </th>
   <th style="text-align:right;"> xz </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> London </td>
   <td style="text-align:left;"> Accounts </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> London </td>
   <td style="text-align:left;"> Manager </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Birmingham </td>
   <td style="text-align:left;"> Accounts </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Birmingham </td>
   <td style="text-align:left;"> Manager </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
</tbody>
</table>

---
#  Interpretation: Categorical*categorical interaction (dummy codes)

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

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

+ Expected salary for Accounts in London.

---
#  Interpretation: Categorical*categorical interaction (dummy codes)

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

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

+ The difference in salary between Accounts in London and Birmingham

---
#  Interpretation: Categorical*categorical interaction (dummy codes)

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

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

+ The difference in salary between Accounts and Store managers in London.

---
#  Interpretation: Categorical*categorical interaction (dummy codes)

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

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

+ The difference between salary in Accounts and Store managers between London and Birmingham

---
#  Example: Categorical*categorical

.pull-left[

```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 London     Accounts     50.7
## 2 London     Manager      47.2
## 3 Birmingham Accounts     48.7
## 4 Birmingham Manager      36.8
```
]

---
#  Example: Categorical*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 ***
## locationBirmingham                    -1.9796     1.4872  -1.331 0.186297    
## departmentManager                     -3.4625     1.3365  -2.591 0.011072 *  
## locationBirmingham: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*categorical

.pull-left[

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

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

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

.pull-right[
![](dapr2_08_interactions3_files/figure-html/unnamed-chunk-21-1.png)

]

---
#  Example: Categorical*categorical

```
##                                      Estimate Std. Error t value Pr(>|t|)
## (Intercept)                             50.67       0.71   70.93     0.00
## locationBirmingham                      -1.98       1.49   -1.33     0.19
## departmentManager                       -3.46       1.34   -2.59     0.01
## locationBirmingham: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 London is £50,670.
]

.pull-right[
<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> London </th>
   <th style="text-align:right;"> Birmingham </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Accounts </td>
   <td style="text-align:right;"> 50.67 </td>
   <td style="text-align:right;"> 48.69 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Manager </td>
   <td style="text-align:right;"> 47.21 </td>
   <td style="text-align:right;"> 36.81 </td>
  </tr>
</tbody>
</table>
]

---
#  Example: Categorical*categorical

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

+ The difference in salary between Accounts in London and Birmingham is £1,980. The salary is lower in Birmingham.

+ These are calculated as:

`$$\text{Group} - \text{Reference group (intercept)}$$`

]

---
#  Example: Categorical*categorical

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

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

+ These are calculated as:

`$$\text{Group} - \text{Reference group (intercept)}$$`

]

---
#  Example: Categorical*categorical

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

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

+ These are fiddly (handout online that goes through this in detail)

]

---
class: center, middle
# Good time to pause

---
# Extending past 2x2
+ When fitting an interaction to categorical variables with more than 2 levels, we need additional interaction terms.

+ Remember, I categorical variable with 3 levels is represented by 2 dummy coded variables.
  + So that means we have two variables to create products with.

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

`$$(r-1)*(c-1)$$`

+ 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 a 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.

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

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

---
# The data

```r
hosp_tbl <- read_csv("hospital.csv", col_types = "dff")
hosp_tbl %>%
  slice(1:10)
```

```
## # 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

<table>
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:left;"> Hosp1 </th>
   <th style="text-align:left;"> Hosp2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> TreatA </td>
   <td style="text-align:left;"> 10.80 </td>
   <td style="text-align:left;"> 7.85 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> TreatB </td>
   <td style="text-align:left;"> 9.43 </td>
   <td style="text-align:left;"> 13.11 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> TreatC </td>
   <td style="text-align:left;"> 10.10 </td>
   <td style="text-align:left;"> 7.98 </td>
  </tr>
</tbody>
</table>

---
# 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
```

---
# Interpretation with dummy coding

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

+ `$b_0$` = 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[
<table>
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:left;"> Hosp1 </th>
   <th style="text-align:left;"> Hosp2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> TreatA </td>
   <td style="text-align:left;"> 10.80 </td>
   <td style="text-align:left;"> 7.85 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> TreatB </td>
   <td style="text-align:left;"> 9.43 </td>
   <td style="text-align:left;"> 13.11 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> TreatC </td>
   <td style="text-align:left;"> 10.10 </td>
   <td style="text-align:left;"> 7.98 </td>
  </tr>
</tbody>
</table>

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

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

+ Consider our salary example:
  + We could ask does the change in difference in the effect of years of service across departments, differ dependent on whether you are based in London or Birmingham.
  
+ Note we have three predictors here:
  + Years of service (continuous)
  + department (binary; Store Manager, Accounts)
  + location (binary; London, Birmingham)

+ 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 focussed on dummy coded (0,1) variables.

+ Next semester, we will return to this topic and consider other ways of treating categorical data when we have specific hypotheses we want to test.

---
class: center, middle
# Thanks for listening!