Week 7: LM multiple predictors 2

class: center, middle, inverse, title-slide

# <b>Week 7: LM multiple predictors 2 </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 how to extend a simple regression to multiple predictors.

2. Understand and interpret the coefficients in multiple linear regression models

3. Understand how to include and interpret models with categorical variables with 2+ levels.

---
# Topics for today
+ Categorical predictors with more than 2 levels

---
#  Including categorical predictors with >2 levels in a regression 
+ When we have a categorical variable with 2+ levels, we will typically assign integers
  + But recall, these are not meaningful numbers

+ For example: What city do you live in?
  + 1 = Edinburgh; 2 = Glasgow, 3 = Birmingham etc.

+ So in analysing a categorical predictor with `$k$` levels, we need to take an additional step.

+ This step involves applying a coding scheme, where by each regressor = a difference in means between levels, or sets of levels.

+ Two common coding schemes are:
  + Dummy coding
  + Effects coding

---
#  Dummy coding 
+ Dummy coding uses 0's and 1's to represent group membership
	+ One level is chosen as a baseline
	+ All other levels are compared against that baseline
	
+ Notice, this is identical to binary variables already discussed.

+ Dummy coding is simply the process of producing a set of binary coded variables

+ For any categorical variable, we will create `$k$`-1 dummy variables
  + `$k$` = number of levels

---
# Steps in dummy coding
1. Choose a baseline level

2. Assign everyone in the baseline  group `0` for all `$k$`-1 dummy variables

3. Assign everyone in the next group a `1` for the first dummy variable and a `0` for all the other dummy variables

4. Assign everyone in the next again group a `1` for the second dummy variable and a `0` for all the other dummy variables

5. Repeat step 5 until all `$k$`-1 dummy variables have had 0's and 1's assigned

6. Enter the `$k$`-1 dummy variables into your regression

---
#  Choosing a baseline? 
+ Each level of your categorical predictor will be compared against the baseline.

+ Good baseline levels could be:
	+ The control group in an experiment
	+ The group expected to have the lowest score on the outcome
	+ The largest group

+ It is best the baseline is not:
	+ A poorly defined level, e.g. an `Other` group
	+ Much smaller than the other groups

---
#  Dummy coding 
+ Imagine 100 students took an exam and were each assigned to use one of three `study methods`
	+ 1 = Notes re-reading 
	+ 2 = Notes summarising
	+ 3 = Self-testing ([see here](https://www.psychologicalscience.org/publications/journals/pspi/learning-techniques.html))

+ We could use dummy coding to convert our `study methods` variable into `$k$`-1 regressors:

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> Level </th>
   <th style="text-align:right;"> D1 </th>
   <th style="text-align:right;"> D2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Notes re-reading </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Notes summarising </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Self-testing </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
</tbody>
</table>

---
#  Dummy coding

.pull-left[
+ We start out with  a dataset that looks like:

```
## # A tibble: 10 x 3
##    ID     exam method
##    <chr> <dbl> <fct> 
##  1 ID101    54 2     
##  2 ID102    59 3     
##  3 ID103    50 2     
##  4 ID104    55 3     
##  5 ID105    51 2     
##  6 ID106    52 2     
##  7 ID107    50 1     
##  8 ID108    57 2     
##  9 ID109    52 2     
## 10 ID110    52 2
```

]

.pull-right[
+ And end up with one that looks like:

```
## # A tibble: 10 x 5
##    ID     exam method dummy1 dummy2
##    <chr> <dbl> <fct>   <dbl>  <dbl>
##  1 ID101    54 2           1      0
##  2 ID102    59 3           0      1
##  3 ID103    50 2           1      0
##  4 ID104    55 3           0      1
##  5 ID105    51 2           1      0
##  6 ID106    52 2           1      0
##  7 ID107    50 1           0      0
##  8 ID108    57 2           1      0
##  9 ID109    52 2           1      0
## 10 ID110    52 2           1      0
```

]

---
#  Dummy coding with `lm`

+ `lm` automatically applies dummy coding when you include a variable of class `factor` in a model.

+ It selects the first group as the baseline group

+ We write:

```r
mod1 <- lm(exam ~ method, data = dum_dat)
```

+ And `lm` does all the dummy coding work for us

---
#  Dummy coding with `lm`

.pull-left[
+ The intercept is the mean of the baseline group (notes re-reading)

+ The coefficient for `method2` is the mean difference between the notes summarising group and the baseline group

+ The coefficient for `method3` is the mean difference between the self-test group and the baseline group

]

.pull-right[

```r
mod1 <- lm(exam ~ method, data = dum_dat)
mod1
```

```
## 
## Call:
## lm(formula = exam ~ method, data = dum_dat)
## 
## Coefficients:
## (Intercept)      method2      method3  
##      51.696        1.878        4.348
```

]

---
#  Dummy coding with `lm` (full results)

```r
summary(mod1)
```

```
## 
## Call:
## lm(formula = exam ~ method, data = dum_dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5741 -1.5741  0.3651  1.4259  5.3043 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  51.6957     0.4261 121.328  < 2e-16 ***
## method2       1.8784     0.5088   3.692 0.000368 ***
## method3       4.3478     0.6026   7.215  1.2e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.043 on 97 degrees of freedom
## Multiple R-squared:  0.3515,	Adjusted R-squared:  0.3382 
## F-statistic: 26.29 on 2 and 97 DF,  p-value: 7.529e-10
```

---
#  Changing the baseline group

+ The level that `lm` chooses as it's baseline may not always be the best choice
	+ You can change it using:

```r
contrasts(dum_dat$method) <- contr.treatment(3, base = 2)
```

+ `contrasts` updates the variable with the new coding scheme

+ `contr.treatment` Specifies that you want dummy coding

+ `3` is No. of levels of your predictor

+ `base=2` is the level number of your new baseline

---
#  Results using the new baseline

.pull-left[
+ The intercept is the now the mean of the second group (Notes summarising)

+ `method1` is now the difference between Notes re-reading and Notes summarising

+ `method3` is now the difference between Self-testing and Notes summarising

]

.pull-right[

```r
contrasts(dum_dat$method) <- contr.treatment(3, base = 2)
mod2 <- lm(exam ~ method, data = dum_dat)
mod2
```

```
## 
## Call:
## lm(formula = exam ~ method, data = dum_dat)
## 
## Coefficients:
## (Intercept)      method1      method3  
##      53.574       -1.878        2.469
```

]

---
#  New baseline (full results)

```r
summary(mod2)
```

```
## 
## Call:
## lm(formula = exam ~ method, data = dum_dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5741 -1.5741  0.3651  1.4259  5.3043 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  53.5741     0.2781 192.661  < 2e-16 ***
## method1      -1.8784     0.5088  -3.692 0.000368 ***
## method3       2.4694     0.5088   4.853 4.64e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.043 on 97 degrees of freedom
## Multiple R-squared:  0.3515,	Adjusted R-squared:  0.3382 
## F-statistic: 26.29 on 2 and 97 DF,  p-value: 7.529e-10
```

???
+ Note that the choice of baseline does not affect the R^2 or F-ratio

---
# Exercise in understanding

+ Once you are finished watching this recording, please do the following:

+ Download the data used in this lecture from LEARN (`dummy_code_data.csv`)
  + Read into R
  + Run the code for `mod1`, creating level 2 as baseline, and `mod2`
  + Calculate the group means
  + Try to guess the value of both `method1` and `method2` if you made the third level the baseline
  + Make it the baseline
  + Create `mod3` and check your estimate
  
+ The answer will appear on LEARN at the end of the week.

---
# Summary of today

+ Categorical variables with 2+ levels require a coding scheme.

+ Dummy coding is one of the most common

+ Dummy coding creates a set of `$k$`-1 0-1 binary variables

+ These compare each of the other levels to a baseline level.

+ Each dummy variable is interpreted as a group difference.

---
# Next tasks
+ This week:
  + Complete your lab
  + Come to office hours
  + Weekly quiz - content from week 5
      + Open Monday 09:00
      + Closes Sunday 17:00