Binary Predictors

class: center, middle, inverse, title-slide

# <b>Binary Predictors </b>
## Data Analysis for Psychology in R 2<br><br>
### dapR2 Team
### Department of Psychology<br>The University of Edinburgh

---

# Weeks Learning Objectives
1. Understand the calculation and interpretation of the coefficient of determination.

2. Understand the calculation and interpretation of the F-test of model utility.

3. Understand how to standardize model coefficients and when this is appropriate to do.

4. Understand the relationship between the correlation coefficient and the regression slope.

5. Understand the meaning of model coefficients in the case of a binary predictor.

---
# 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_07_LMBinary_files/figure-html/unnamed-chunk-7-1.png" width="504" />

---
#  Visualize the model
<img src="dapr2_07_LMBinary_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

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