Tutorial 2 - Categorical Variables

Categorical Predictors

Regression is really flexible because it allows you to include both categorical and continuous predictors
Simplest case of categorical variable is dichotomous
- E.g., Treatment and Control
Often we have categorical variables with more than 2 levels which we can call multicategorical variables
- E.g., Race

Multicategorical Variables

When these variables aren’t ordinal, we need to use a coding scheme to include them in our regression model
Several different coding schemes to choose from depending on what you are trying to understand
Changing the coding scheme doesn’t influence the model (e.g., how well it fits the data), only the interpretation of the coefficients

Example

library(haven)

## Warning: package 'haven' was built under R version 4.1.2

milit <- read_sav("milit.sav")
str(milit)

## tibble [215 x 3] (S3: tbl_df/tbl/data.frame)
##  $ ID            : num [1:215] 1 2 3 4 5 6 7 8 9 10 ...
##   ..- attr(*, "format.spss")= chr "F8.2"
##  $ group         : dbl+lbl [1:215] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
##    ..@ format.spss: chr "F8.2"
##    ..@ labels     : Named num [1:4] 0 1 2 3
##    .. ..- attr(*, "names")= chr [1:4] "mental health crisis team" "patrol" "K9" "tactical team"
##  $ militarization: num [1:215] 14.5 14.1 10.1 10.8 12.8 ...
##   ..- attr(*, "format.spss")= chr "F8.2"

Prepare our Dataset

Convert our categorical variable into a real factor

milit$groupf <- factor(milit$group, 
                      levels = 0:3, 
                      labels = c("MH", "Patrol", "K9", 
                                  "Tac"))

Check to make sure everything worked

attributes(milit$groupf)

## $levels
## [1] "MH"     "Patrol" "K9"     "Tac"   
## 
## $class
## [1] "factor"

Examine the means for our dependent variable at different levels of our independent variable

tapply(milit$militarization, milit$groupf, mean)

##       MH   Patrol       K9      Tac 
## 11.54941 20.39585 33.54770 41.96147

Indicator (Dummy) Coding

We create g-1 variables to represent a categorical variable with g categories
The one category that is treated differently (i.e., does not have its own dummy variable and is 0 on all other dummy variables) is referred to as the reference category
You can change which variable acts as the reference category by recoding
Example for a categorical variable with 4 levels

contr.treatment(4)

When no other predictors present:
- B0 is the mean of the reference group
- Bj indicates the difference between the mean of group j and the mean of the reference category

Indicator (Dummy) Coding

contrasts(milit$groupf) = contr.treatment(4)
indicator <- lm(militarization ~ groupf, milit)
indicator

## 
## Call:
## lm(formula = militarization ~ groupf, data = milit)
## 
## Coefficients:
## (Intercept)      groupf2      groupf3      groupf4  
##      11.549        8.846       21.998       30.412

Check for Yourself Exercise

tapply(milit$militarization, milit$groupf, mean)

##       MH   Patrol       K9      Tac 
## 11.54941 20.39585 33.54770 41.96147

contr.treatment(4)

Check for Yourself Exercise Answers

c1

20.3959-11.5494

## [1] 8.8465

c2

33.5477-11.5494

## [1] 21.9983

c3

41.9615-11.5494

## [1] 30.4121

## 
## Call:
## lm(formula = militarization ~ groupf, data = milit)
## 
## Coefficients:
## (Intercept)      groupf2      groupf3      groupf4  
##      11.549        8.846       21.998       30.412

Effect Coding

Similar to indicator coding in that a set of g-1 variables is created
However, the reference category in effect coding receives -1 on all variables instead of 0 as we saw in indicator coding

contr.sum(4)

##   [,1] [,2] [,3]
## 1    1    0    0
## 2    0    1    0
## 3    0    0    1
## 4   -1   -1   -1

When no other predictor variables present
- B0 is the unweighted mean of all group means(i.e., doesn’t take into consideration sample of each group)
- Bj quantifies the difference in the mean of group j and the mean of all groups

Effect Coding

contrasts(milit$groupf) = contr.sum(4)
effect<-lm(militarization ~ groupf, milit)
effect

## 
## Call:
## lm(formula = militarization ~ groupf, data = milit)
## 
## Coefficients:
## (Intercept)      groupf1      groupf2      groupf3  
##      26.864      -15.314       -6.468        6.684

Check for Yourself Exercise

tapply(milit$militarization, milit$groupf, mean)

##       MH   Patrol       K9      Tac 
## 11.54941 20.39585 33.54770 41.96147

contr.sum(4)

##   [,1] [,2] [,3]
## 1    1    0    0
## 2    0    1    0
## 3    0    0    1
## 4   -1   -1   -1

Check for Yourself

B0

(11.5494 + 20.39585 + 33.54770 + 41.96147) / 4

## [1] 26.8636

c1

11.5494-26.8636

## [1] -15.3142

c2

20.39585-26.8636

## [1] -6.46775

c3

33.54770-26.8636

## [1] 6.6841

## 
## Call:
## lm(formula = militarization ~ groupf, data = milit)
## 
## Coefficients:
## (Intercept)      groupf1      groupf2      groupf3  
##      26.864      -15.314       -6.468        6.684

Sequential Coding

Typically used when groups can be ordered and you are interested in examining how the mean of Y changes as the variable increases by one step
Can also be used for nominal variables if you strategically ordered your categories so that it allowed you to do the comparisons you were interested in

Sequential Coding

-With g groups, cj is set to 1 for cases that are members of a group ordinally higher than j on the variable defining groups; otherwise cj is set to 0

sequentialcodes = matrix(c(0, 1, 1, 1, 0, 0, 1, 1, 
                           0, 0, 0, 1), ncol = 3)
sequentialcodes

##      [,1] [,2] [,3]
## [1,]    0    0    0
## [2,]    1    0    0
## [3,]    1    1    0
## [4,]    1    1    1

When no other predictors present:
- B0 is mean of the ordinally lowest group
- Bj indicates the mean difference between group j and the group directly below it

Sequential Coding

contrasts(milit$groupf) = sequentialcodes
sequential <- lm(militarization ~ groupf, data = milit)
sequential

## 
## Call:
## lm(formula = militarization ~ groupf, data = milit)
## 
## Coefficients:
## (Intercept)      groupf1      groupf2      groupf3  
##      11.549        8.846       13.152        8.414

Check For Yourself

tapply(milit$militarization, milit$groupf, mean)

##       MH   Patrol       K9      Tac 
## 11.54941 20.39585 33.54770 41.96147

20.3959-11.5417

## [1] 8.8542

33.5477-20.3959

## [1] 13.1518

41.9615-33.5477

## [1] 8.4138

## 
## Call:
## lm(formula = militarization ~ groupf, data = milit)
## 
## Coefficients:
## (Intercept)      groupf1      groupf2      groupf3  
##      11.549        8.846       13.152        8.414

Helmert Coding

Useful for ordinal variables (but can also be used for nominal as well)
Allows you to compare one category of your variable with the mean of subsequent levels of the variable

Helmert Coding

my.helmert = matrix(c(3/4, -1/4, -1/4, -1/4, 0, 2/3, -1/3, -1/3, 0, 0, 1/
                        2, -1/2), ncol = 3)
my.helmert

##       [,1]       [,2] [,3]
## [1,]  0.75  0.0000000  0.0
## [2,] -0.25  0.6666667  0.0
## [3,] -0.25 -0.3333333  0.5
## [4,] -0.25 -0.3333333 -0.5

B0 is the unweighted mean of all groups
Bj is the difference between the mean of a j group and the average of the means of the groups above it

Helmert Coding

contrasts(milit$groupf) = my.helmert
helmert <- lm(militarization ~ groupf, milit)
helmert

## 
## Call:
## lm(formula = militarization ~ groupf, data = milit)
## 
## Coefficients:
## (Intercept)      groupf1      groupf2      groupf3  
##      26.864      -20.419      -17.359       -8.414

Check For Yourself

B0

(11.5494 + 20.39585 + 33.54770 + 41.96147) / 4

## [1] 26.8636

c1

(20.39585 + 33.54770 + 41.96147)/3

## [1] 31.96834

11.5494-31.9683

## [1] -20.4189

c2

(33.54770 + 41.96147)/2

## [1] 37.75458

20.3959-37.7546

## [1] -17.3587

c3

33.54770 - 41.9614

## [1] -8.4137

## 
## Call:
## lm(formula = militarization ~ groupf, data = milit)
## 
## Coefficients:
## (Intercept)      groupf1      groupf2      groupf3  
##      26.864      -20.419      -17.359       -8.414