Week 4: Binary Predictors

class: center, middle, inverse, title-slide

# <b>Week 4: Binary Predictors </b>
## Data Analysis for Psychology in R 2<br><br>
### TOM BOOTH & ALEX DOUMAS
### Department of Psychology<br>The University of Edinburgh
### AY 2020-2021

---

# Weeks Learning Objectives
1. Understand the meaning of model coefficients in the case of a binary predictor.

2. Be able to state the assumptions underlying a linear model.

3. Understand how to assess if a fitted model satisfies the linear model assumptions.

4. Understand how to use transformations when the model violates assumptions.

---
# Topics for today
+ Today will will focus on the linear model with a binary predictor.

+ Recap categorical and binary variables

+ Extend our data example

+ Run through the steps and interpretation for the linear model

---
#  Recap: Categorical variables 
+ Categorical variables can only take discrete values
	+ E.g., animal type: 1= duck, 2= cow, 3= wasp

+ They are mutually exclusive
  + No duck-wasps or cow-ducks!

+ In R, categorical variables should be of class `factor`
  + The discrete values are `levels`
  + Levels can have numeric values (1, 2 3) and labels (e.g. "duck", "cow", "wasp")

---
# Recap: Binary variable
+ Binary variable is a categorical variable with two levels.

+ Traditionally coded with a 0 and 1
  + Referred to as dummy coding
  + We will come back to this for categorical variables with 2+ levels

+ Why 0 and 1?
  + Quick version: It has some nice properties when it comes to interpretation.

---
# Extending our example

.pull-left[
+ Our in class example so far has used test scores and revision time for 10 students.

+ Let's say we collect this data on 150 students.

+ We also collected data on who they studied with;
  + 0 = alone
  + 1 = with others
  
+ So our variable `study` is a binary
]

.pull-right[

+ This data set is available on LEARN

```r
*df <- read_csv("./dapr2_lec07.csv")
```

<div style="border: 1px solid #ddd; padding: 0px; overflow-y: scroll; height:300px; "><table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;position: sticky; top:0; background-color: #FFFFFF;"> ID </th>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> score </th>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> hours </th>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> study </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> ID1 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.3 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID2 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 2.6 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID3 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.7 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID4 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.6 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID5 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.7 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID6 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 4.4 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID7 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.6 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID8 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.1 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID9 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.6 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID10 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.9 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID11 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 2.2 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID12 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 5.7 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID13 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.8 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID14 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.5 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID15 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.8 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID16 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.6 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID17 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID18 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 2.9 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID19 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.8 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID20 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.4 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID21 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.7 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID22 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.7 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID23 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.6 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID24 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.5 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID25 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID26 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 1.8 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID27 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 2.5 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID28 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 1.9 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID29 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.3 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID30 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.5 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID31 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.3 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID32 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.6 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID33 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.5 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID34 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID35 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.2 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID36 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 4.8 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID37 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 5.2 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID38 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 3.9 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID39 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.6 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID40 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.9 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID41 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.3 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID42 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 4.8 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID43 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.3 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID44 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 4.0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID45 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 2.0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID46 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 3.3 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID47 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.1 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID48 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 2.8 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID49 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 5.9 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID50 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.4 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID51 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID52 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.8 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID53 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.8 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID54 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 5.7 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID55 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.9 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID56 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID57 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 1.6 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID58 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.6 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID59 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.5 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID60 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 0.8 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID61 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.7 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID62 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.1 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID63 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.1 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID64 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.2 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID65 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.5 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID66 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.4 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID67 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.7 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID68 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 4.1 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID69 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.8 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID70 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.4 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID71 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 5.4 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID72 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.8 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID73 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 4.2 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID74 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 5.2 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID75 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.2 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID76 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.1 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID77 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.1 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID78 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.3 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID79 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.1 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID80 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 5.3 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID81 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID82 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 2.9 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID83 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.7 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID84 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.1 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID85 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 4.2 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID86 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.8 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID87 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.9 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID88 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.6 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID89 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.2 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID90 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 2.8 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID91 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID92 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.9 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID93 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 2.6 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID94 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.2 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID95 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 2.6 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID96 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 2.8 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID97 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 4.3 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID98 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 3.1 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID99 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 2.3 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID100 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.7 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID101 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.2 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID102 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.5 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID103 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.4 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID104 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 2.0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID105 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 2.8 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID106 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 2.3 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID107 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.3 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID108 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 2.8 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID109 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.2 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID110 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 4.5 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID111 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 4.6 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID112 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID113 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.7 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID114 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 4.1 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID115 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 4.1 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID116 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.4 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID117 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.6 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID118 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 1.9 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID119 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.8 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID120 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 3.2 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID121 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.8 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID122 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.3 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID123 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.9 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID124 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.6 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID125 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID126 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.6 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID127 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.6 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID128 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 3.7 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID129 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.3 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID130 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.7 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID131 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 1.8 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID132 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 5.1 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID133 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.8 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID134 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.6 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID135 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.5 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID136 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.7 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID137 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.8 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID138 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.7 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID139 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.4 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID140 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 2.9 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID141 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 6.5 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID142 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.3 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID143 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.6 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID144 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4.0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID145 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.6 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID146 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 3.0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID147 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 2.4 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID148 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 4.2 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID149 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 2.9 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ID150 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 3.6 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table></div>

]

---
#  LM with binary predictors 
+ In previous lectures we asked the question:

+ **Do students who study more get higher scores on the test?**

+ And we specified a linear model:

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

+ Or

`$$score_i = \beta_0 + \beta_1 hours_{i} + \epsilon_i$$`

+ And nothing changes with our binary variable. We can ask the question:

+ **Do students who study with others score better than students who study alone?**

`$$score_i = \beta_0 + \beta_1 study_{i} + \epsilon_i$$`

---
# In `R`

```r
res <- lm(score ~ study, data = df)
summary(res)
```

```
## 
## Call:
## lm(formula = score ~ study, data = df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.8333 -0.8333  0.1667  0.7778  2.1667 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   5.2222     0.1076  48.552  < 2e-16 ***
## study         0.6111     0.1492   4.097 6.87e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9127 on 148 degrees of freedom
## Multiple R-squared:  0.1019,	Adjusted R-squared:  0.0958 
## F-statistic: 16.79 on 1 and 148 DF,  p-value: 6.866e-05
```

---
# Interpretation

.pull-left[
+ As before, the intercept `$\hat \beta_0$` is the expected value of `$y$` when `$x=0$`

+ What is `$x=0$` here?
  + It is the students who study alone.

+ So what about `$\hat \beta_1$`?

+ **Look at the output on the right hand side.** 
  + What do you notice about the difference in averages?

]

.pull-right[

```r
df %>%
* group_by(., study) %>%
  summarise(
*   Average = round(mean(score),4)
  )
```

```
## # A tibble: 2 x 2
##   study Average
##   <dbl>   <dbl>
## 1     0    5.22
## 2     1    5.83
```

]

---
# Interpretation
+ `$\hat \beta_0$` = predicted expected value of `$y$` when `$x = 0$`
  + Or, the mean of group coded 0 (those who study alone)
  
+ `$\hat \beta_1$` = predicted difference between the means of the two groups.
  + Group 1 - Group 0 (Mean `score` for those who study with others - mean `score` of those who study alone)
  
+ Notice how this maps to our question. 
  + Do students who study with others score better than students who study alone?

---
class: center, middle
# Time for a break

**Have a go at the binary variable quiz**

---
class: center, middle
# Welcome Back!

**Where we left off... **

Let's think about the interpretation by group a little more.

---
#  Equations for each group 
+ What would our linear model look like if we added the values for `$x$`.

`$$\widehat{score} = \hat \beta_0 + \hat \beta_1 study$$`

+ For those who study alone ( `$study = 0$` ):

`$$\widehat{score}_{alone} = \hat \beta_0 + \hat \beta_1 \times 0$$`

+ So;

`$$\widehat{score}_{alone} = \hat \beta_0$$`

---
#  Equations for each group 
+ For those who study with others ( `$study = 1$` ):

`$$\widehat{score}_{others} = \hat \beta_0 + \hat \beta_1 \times 1$$`

+ So;

`$$\widehat{score}_{others} = \hat \beta_0 + \hat \beta_1$$`

+ And if we re-arrange;

`$$\hat \beta_1 = \widehat{score}_{others} - \hat \beta_0$$`

+ Remembering that `$\widehat{score}_{alone} = \hat \beta_0$`, we finally obtain:

`$$\hat \beta_1 = \widehat{score}_{others} - \widehat{score}_{alone}$$`

---
#  Visualize the model

---
#  Visualize the model
<img src="dapR2_lec07_LMcategorical_files/figure-html/unnamed-chunk-7-1.png" width="504" />

---
#  Visualize the model
<img src="dapR2_lec07_LMcategorical_files/figure-html/unnamed-chunk-8-1.png" width="504" />

---
#  Visualize the model

---
#  Evaluation of model and significance of `$\beta_1$`

+ `$R^2$` and `$F$`-ratio interpretation are identical to their interpretation in models with only continuous predictors.

+ And we assess the significance of predictors in the same way

+ We use the standard error of the coefficient to construct:
  + We calculate the `$\hat \beta_1$` = difference between groups
	+ `$t$`-value and associated `$p$`-value for the coefficient
	+ Or a confidence interval around the coefficient

---
# Hold on... it's a t-test

```r
df %>%
  t.test(score ~ study, .)
```

```
## 
## 	Welch Two Sample t-test
## 
## data:  score by study
## t = -4.0883, df = 145.39, p-value = 7.163e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.9065440 -0.3156782
## sample estimates:
## mean in group 0 mean in group 1 
##        5.222222        5.833333
```

???
Yup!

---
# Standardizing `$\hat \beta_1$`

+ When discussing continuous predictors we discussed standardized `$b_1$` and unstandardized `$\hat \beta_1$`

+ Recall, when we calculated `$b_1$` we used the SD of x, `$s_x$`.

+ For a binary categorical variable, the SD is not appropriate.

+ 1 unit also has meaning - it is the membership of a different group.

+ As such, we do not standardize

---
# Summary of today

+ Recapped categorical and binary variables

+ Introduced the linear model with a single binary variable

+ Considered the interpretation of the coefficients
  + And saw it is an independent sample `$t$`-test

---
# Next tasks

+ This week:
  + Complete your lab
  + Come to office hours
  + Weekly quiz: Assessed quiz - Week 3 content.
      + Open Monday 09:00
      + Closes Sunday 17:00