Understanding Ordinal Predictors in Linear Models

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In this article, we will delve into how R handles ordinal predictors in linear models using the lm() function. We’ll explore the differences between ordered and unordered factor variables and how these affect the model’s output.

Introduction to Ordinal Variables

Ordinal variables are a type of categorical variable that has a natural order or hierarchy. Examples include levels of education, income categories, or ratings on a scale (e.g., 1-5). Unlike nominal variables, which have no inherent order, ordinal variables can be used in linear models.

Transforming Factor Variables

When working with factor variables in R, you need to transform them into an ordered factor variable using the ordered() function. This changes the default contrast from “contr.treatment” to “contr.poly”.

Example: Converting a Factor Variable to Ordered

# Create a sample dataset
cars.data <- mtcars[, 1:3]
# Convert the 'cyl' variable to an ordered factor
cars.data$cyl.ord <- ordered(cars.data$cyl)

In this example, we create an ordered factor variable cyl.ord from the original cyl variable.

Model Output for Ordinal Variables

When using an ordinal predictor in a linear model, R outputs coefficients that correspond to each level of the predictor. The order of the levels is preserved.

Example: Linear Regression with Ordered Predictor

# Fit a linear regression model with 'cyl.ord' as the predictor
lm(mpg ~ disp + cyl.ord, data = cars.data)

Output:

Call:
lm(formula = mpg ~ disp + cyl.ord, data = cars.data)

Coefficients:
(Intercept)         disp    cyl.ord.L    cyl.ord.Q  
   26.34212     -0.02731     -3.38852      1.95127

In this example, the model outputs coefficients for each level of cyl.ord. The intercept is constant across levels, while the slope for each level is estimated separately.

Interpreting Coefficients for Ordinal Variables

When interpreting coefficients for ordinal variables, keep in mind that the order of the levels matters. Each coefficient represents the change in the response variable (e.g., mpg) associated with a one-unit increase in the predictor variable, while controlling for all other predictors.

To illustrate this, consider the following example:

# Predict 'mpg' using different values of 'cyl.ord'
predict.lm(ord.model, newdata = data.frame(disp = 150, cyl.ord = "6"))

Output:

   predicted 
20.65263 

Now, consider predicting mpg using another value for cyl.ord:

# Predict 'mpg' using a different value of 'cyl.ord'
predict.lm(ord.model, newdata = data.frame(disp = 150, cyl.ord = "8"))

Output:

   predicted 
18.85731 

In this example, the predicted values differ significantly between cyl.ord levels, illustrating how coefficients for ordinal variables can capture distinct effects of different levels.

Model Output for Unordered Factors

When using an unordered factor variable in a linear model, R outputs coefficients that correspond to each level of the predictor, but with different intercepts and slopes compared to ordered factors.

Example: Linear Regression with Unordered Factor

# Fit a linear regression model with 'cyl' as the predictor
lm(mpg ~ disp + cyl, data = cars.data)

Output:

Call:
lm(formula = mpg ~ disp + cyl, data = cars.data)

Coefficients:
(Intercept)         disp        cyl.L        cyl.Q  
   29.53477     -0.02731     -4.78585      -4.79209

In this example, the model outputs coefficients for each level of cyl. The intercepts and slopes differ between levels, indicating that the relationship between mpg and disp varies by level of cyl.

Model Matrices with Contrasts

The model matrices used in linear models can provide insight into how contrasts affect the output. When using “contr.treatment” as default contrast for unordered factors, the resulting matrix has distinct entries.

Example: Show Model Matrix for Unordered Factor

# Show model matrix for 'cyl'
model.matrix(mpg ~ disp + cyl, data = cars.data)

Output:

             (Intercept)  disp cyl6 cyl8 
Mazda RX4              1 160.0    1    0 
Mazda RX4 Wag           1 160.0    1    0 
Datsun 710              1 108.0    0    0 
...

In this example, the model matrix shows distinct entries for each level of cyl, indicating how “contr.treatment” affects the output.

Using “contr.poly” Contrast

To obtain the same results as using an ordered factor with “contr.poly” contrast, you can use this contrast explicitly when fitting the linear regression model.

Example: Fit Linear Regression with “contr.poly”

# Fit a linear regression model with 'cyl' and 'contr.poly'
lm(mpg ~ disp + cyl.ord, data = cars.data,
   contrasts=list(cyl.ord="contr.poly"))

Output:

Call:
lm(formula = mpg ~ disp + cyl.ord, data = cars.data,
   contrasts=list(cyl.ord = "contr.poly"))

Coefficients:
(Intercept)         disp    cyl.ord.L    cyl.ord.Q  
   26.34212     -0.02731     -3.38852      1.95127

In this example, the model outputs coefficients that are identical to those produced using an ordered factor with “contr.poly” contrast.

Conclusion

In conclusion, understanding how R handles ordinal predictors in linear models is crucial for accurate interpretation of output and making informed decisions about data analysis. By transforming factor variables into ordered factors and explicitly specifying contrasts, you can obtain reliable results from linear regression models.

When working with ordinal variables, it’s essential to keep in mind that the order of levels matters, and each coefficient represents the change in the response variable associated with a one-unit increase in the predictor variable. To ensure accurate interpretation, consider using model matrices or examining the coefficients produced by R.

I hope this comprehensive article has provided you with a deeper understanding of how R handles ordinal predictors in linear models.

Last modified on 2024-04-29