Chapter 8 - Digging Deeper into Group Models
8.1 Extending to a Three-Group Model
You have now learned how to specify a model with a single categorical explanatory variable consisting of two groups. It’s actually pretty simple to extend this idea to a categorical variable with three groups.
First, a New Two-Group Model
Let’s use a new explanatory variable to explain variation in thumb
length: Height. Height, in our dataset, is a
quantitative variable measured in inches. But we can make a new variable
that turns Height into a categorical variable with two
categories: short and tall.
We can do this using the ntile() function in R. The code
below will cut the sample up into two equal-sized groups based on
Height and save the result into a new variable called
Height2Group.
Fingers$Height2Group <- ntile(Fingers$Height, 2)
head(select(Fingers, Thumb, Height, Height2Group), 10)We used head() and select() to look at the
first 10 rows of the relevant variables – Thumb,
Height, and Height2Group:
| row | Thumb | Height | Height2Group |
|---|---|---|---|
| 1 | 66.00 | 70.5 | 2 |
| 2 | 64.00 | 64.8 | 1 |
| 3 | 56.00 | 64.0 | 1 |
| 4 | 58.42 | 70.0 | 2 |
| 5 | 74.00 | 68.0 | 2 |
| 6 | 60.00 | 68.0 | 2 |
| 7 | 70.00 | 69.0 | 2 |
| 8 | 55.00 | 65.7 | 2 |
| 9 | 60.00 | 62.5 | 1 |
| 10 | 52.00 | 63.4 | 1 |
In the code window below, use the factor() function to
add labels to Height2Group so that the 1s are labeled as
short and the 2s are labeled as tall.
require(coursekata)
Fingers <- Fingers %>% mutate(
Height2Group = ntile(Height, 2)
)
# this creates Height2Group, a numeric variable
Fingers$Height2Group <- ntile(Fingers$Height, 2)
# this is how we used factor() before:
Fingers$Gender <- factor(Fingers$Gender, levels = c(1,2), labels = c("female", "male"))
# modify this line so that 1s are labeled as "short" and 2s are labeled as "tall"
Fingers$Height2Group <- factor()
# this prints out 10 rows of Fingers for the selected columns
head(select(Fingers, Thumb, Height, Height2Group), 10)
Fingers$Height2Group <- factor(Fingers$Height2Group, levels = 1:2, labels = c("short", "tall"))
head(select(Fingers, Thumb, Height, Height2Group), 10)
ex() %>% {
check_object(., "Fingers") %>% check_column("Height2Group") %>% check_equal()
check_output_expr(., "head(select(Fingers, Thumb, Height, Height2Group), 10)")
}| row | Thumb | Height | Height2Group |
|---|---|---|---|
| 1 | 66.00 | 70.5 | tall |
| 2 | 64.00 | 64.8 | short |
| 3 | 56.00 | 64.0 | short |
| 4 | 58.42 | 70.0 | tall |
| 5 | 74.00 | 68.0 | tall |
| 6 | 60.00 | 68.0 | tall |
| 7 | 70.00 | 69.0 | tall |
| 8 | 55.00 | 65.7 | tall |
| 9 | 60.00 | 62.5 | short |
| 10 | 52.00 | 63.4 | short |
Using the same approach we used for gender, we can write the model
for Height2Group like this:
\[\text{Thumb}_i=b_0+b_1\text{Height2Group}_i+e_i\]
Go ahead and fit the Height2Group model, and print out
the parameter estimates and ANOVA table for the model.
require(coursekata)
Fingers <- Fingers %>% mutate(
Height2Group = factor(ntile(Height, 2), 1:2, c("short", "tall"))
)
# fit a model for Thumb ~ Height2Group
Height2Group_model <-
# this prints out the estimates
Height2Group_model
# fit a model for Thumb ~ Height2Group
Height2Group_model <- lm(formula = Thumb ~ Height2Group, data = Fingers)
# this prints out the estimates
Height2Group_model
ex() %>% {
check_function(., "lm") %>% check_arg("formula") %>% check_equal()
check_object(., "Height2Group_model") %>% check_equal()
check_output_expr(., "Height2Group_model")
}| (Intercept) | Height2Grouptall |
|---|---|
| 57.818 | 4.601 |
| Source of variation | SS | df | MS | F | PRE | p |
|---|---|---|---|---|---|---|
| Model (error reduced) | 830.880 | 1 | 830.880 | 11.656 | 0.0699 | .0008 |
| Error (from model) | 11049.331 | 155 | 71.286 | |||
| Total (empty model) | 11880.211 | 156 | 76.155 |
A Three-Group Model
Now let’s try this same approach with three height groups: short, medium, and tall.
Revise the code below to make a new variable called
Height3Group that divides the sample into three categories
based on Height, each with an equal number of students.
Label the levels (1,2,3) as short, medium, and
tall.
require(coursekata)
Fingers <- Fingers %>% mutate(
Height2Group = factor(ntile(Height, 2), 1:2, c("short", "tall"))
)
Height2Group.model <- lm(Thumb ~ Height2Group, data = Fingers)
# modify these two lines of code to create 3 Height groups with the labels "short", "medium", and "tall"
# make sure you save to a new variable in Fingers called Height3Group
Fingers$Height2Group <- ntile(Fingers$Height, 2)
Fingers$Height2Group <- factor(Fingers$Height2Group, levels = c(1,2), labels = c("short", "tall"))
# this prints out 10 rows of Fingers for selected columns
head(select(Fingers, Thumb, Height, Height3Group), 10)
# modify these two lines of code to create 3 Height groups with the labels "short", "medium", and "tall"
# make sure you save to a new variable in Fingers called Height3Group
Fingers$Height3Group <- ntile(Fingers$Height, 3)
Fingers$Height3Group <- factor(Fingers$Height3Group, levels = c(1,2,3), labels = c("short", "medium", "tall"))
# this prints out 10 rows of Fingers for selected columns
head(select(Fingers, Thumb, Height, Height3Group), 10)
ex() %>% {
check_object(., "Fingers") %>% check_column("Height3Group") %>% check_equal()
check_output_expr(., "head(select(Fingers, Thumb, Height, Height3Group),10)")
}| row | Thumb | Height | Height3Group |
|---|---|---|---|
| 1 | 66.00 | 70.5 | tall |
| 2 | 64.00 | 64.8 | medium |
| 3 | 56.00 | 64.0 | short |
| 4 | 58.42 | 70.0 | tall |
| 5 | 74.00 | 68.0 | tall |
| 6 | 60.00 | 68.0 | tall |
| 7 | 70.00 | 69.0 | tall |
| 8 | 55.00 | 65.7 | medium |
| 9 | 60.00 | 62.5 | short |
| 10 | 52.00 | 63.4 | short |
Calculate and print out the group means of Thumb for the
three height groups.
require(coursekata)
Fingers <- Fingers %>% mutate(
Height2Group = factor(ntile(Height, 2), 1:2, c("short", "tall")),
Height3Group = factor(ntile(Height, 3), 1:3, c("short", "medium", "tall"))
)
# use favstats() to print the group means of Thumb length for the three height groups you created earlier
favstats()
# use favstats() to print the group means of Thumb length for the three height groups you created earlier
favstats(Thumb ~ Height3Group, data = Fingers)
ex() %>% check_function("favstats") %>% check_result() %>% check_equal()| row | Height3Group | min | Q1 | median | Q3 | max | mean | sd | n | missing |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | short | 39.00 | 51.00 | 55 | 58.42 | 79.00 | 56.07113 | 7.499937 | 53 | 0 |
| 2 | medium | 45.00 | 55.00 | 60 | 64.00 | 86.36 | 60.22375 | 8.490406 | 52 | 0 |
| 3 | tall | 44.45 | 59.75 | 64 | 68.25 | 90.00 | 64.09365 | 8.388113 | 52 | 0 |
Here is a jitter plot that shows the distribution of thumb lengths for each of the three height groups and the mean of each group. On the next page, we'll learn how to create a model of thumb length based on the three height groups.
