Course Outline
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segmentGetting Started (Don't Skip This Part)
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segmentStatistics and Data Science: A Modeling Approach
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segmentPART I: EXPLORING VARIATION
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segmentChapter 1 - Welcome to Statistics: A Modeling Approach
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segmentChapter 2 - Understanding Data
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segmentChapter 3 - Examining Distributions
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segmentChapter 4 - Explaining Variation
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segmentPART II: MODELING VARIATION
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segmentChapter 5 - A Simple Model
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segmentChapter 6 - Quantifying Error
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segmentChapter 7 - Adding an Explanatory Variable to the Model
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segmentChapter 8 - Digging Deeper into Group Models
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segmentChapter 9 - Models with a Quantitative Explanatory Variable
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9.4 Comparing Regression Models to Group Models
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segmentPART III: EVALUATING MODELS
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segmentChapter 10 - The Logic of Inference
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segmentChapter 11 - Model Comparison with F
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segmentChapter 12 - Parameter Estimation and Confidence Intervals
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segmentChapter 13 - What You Have Learned
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segmentFinishing Up (Don't Skip This Part!)
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segmentResources
list College / Introductory Statistics with R (ABC)
9.4 Comparing Regression Models to Group Models
Comparing the Height2Group Model and the Height Model
We now know how to specify and fit two different kinds of models: group models (e.g., Height2Group_model) and regression models (Height_model), let’s just think for a bit on what the similarities and differences are between these models.
| Symbol |
Group Model \(Y_i=b_0+b_1X_i+e_i\) \(\text{Thumb}_i=b_0+b_1\text{Height2Grouptall}_i+e_i\) |
Regression Model \(Y_i=b_0+b_1X_i+e_i\) \(\text{Thumb}_i=b_0+b_1\text{Height}_i+e_i\) |
|---|---|---|
| \(Y_i\) | Thumb length of a student i | Thumb length of a student i |
| \(b_0\) |
Predicted thumb length when \(\text{Height2Group}_i = 0\)) (mean thumb length for short group) |
Predicted thumb length when \(\text{Height}_i=0\) (y-intercept for regression line) |
| \(b_1\) |
Adjustment to predicted thumb length for a tall student (the mean difference between the two group means) |
Adjustment to predicted thumb length for a one-unit increase in height (the slope of the regression line) |
| \(X_i\) | Height2Group of a student i, coded as 0=not-tall, 1=tall | Height of a student i in inches |
| \(e_i\) | Error for student i | Error for student i |
| visualization of the model |
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Fitting a Regression Model By Accident When You Don’t Want One
Although R is pretty smart about knowing which model to fit, it won’t always do the right thing. If you code the grouping variable with character strings such as “female” and “male” or “short” and “tall,” R will make the right decision to fit a group model because it knows the variable must be categorical. But if you code the same grouping variable as 1 and 2 (maybe you forget to make it a factor), R may get confused and fit the model as though the explanatory variable is quantitative.
For example, we’ve added a new variable to our Fingers data called GenderNum. Here is what the data look like.
Thumb Gender GenderNum
1 66 male 2
2 64 female 1
3 56 female 1
4 70 male 2
5 52 female 1
6 62 male 2If you take a look at the variables Gender and GenderNum, they have the same information. Students 2, 3, and 5 are in one group and students 1, 4, and 6 are in another group. If we fit a model with Gender (and call it the Gender_model) or GenderNum (and call it the GenderNum_model), we would expect the same estimates. Let’s try it.
require(coursekata)
Fingers$GenderNum <- as.numeric(Fingers$Gender)
# fit a model of Thumb length based on Gender
Gender_model <- lm()
# fit a model of Thumb length based on GenderNum
GenderNum_model <- lm()
# this prints the parameter estimates from the two models
Gender_model
GenderNum_model
# fit a model of Thumb length based on Gender
Gender_model <- lm(Thumb ~ Gender, data=Fingers)
# fit a model of Thumb length based on GenderNum
GenderNum_model <- lm(Thumb ~ GenderNum, data=Fingers)
# this prints the parameter estimates from the two models
Gender_model
GenderNum_model
ex() %>% {
check_object(., "Gender_model") %>% check_equal()
check_object(., "GenderNum_model") %>% check_equal()
check_output_expr(., "Gender_model
GenderNum_model")
}Call:
lm(formula = Thumb ~ Gender, data = Fingers)
Coefficients:
(Intercept) Gendermale
58.256 6.447Call:
lm(formula = Thumb ~ GenderNum, data = Fingers)
Coefficients:
(Intercept) GenderNum
51.809 6.447Because Gender is a factor (i.e., a categorical variable), lm() fits a group model. But for GenderNum, lm() thinks the 1 or 2 coding refers to a quantitative variable. Because we did not tell R to treat GenderNum as a factor, it fits a regression line instead of a two-group model. If it does that, the meaning of the estimates will not be what you expect for the group model.
The \(b_1\) estimate will be the same as in the two-group model; because it represents the adjustment in thumb length for a one unit change in \(X_i\). For Gender, a 1-unit change is to go from not male (\(X_i=0\)) to male (\(X_i=1\)). For \(GenderNum\), a 1-unit change similarly goes from not male (\(X_i=1\)) to male (\(X_i=2\)).
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\(b_1\) of the \(Gender\) model, a group model |
\(b_1\) of the \(GenderNum\) model, an accidental regression model |
|---|---|
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But the \(b_0\) estimate will be different in the GenderNum model, where it represents the y-intercept of the regression line, or the predicted thumb length when \(X_i\) equals 0. This makes no sense, however, when there are only two groups and they are coded 1 and 2. This is an accidental regression model.
| \(b_0\) of the \(Gender\) model | \(b_0\) of the \(GenderNum\) model |
|---|---|
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