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This example shows how to perform a regression with categorical
covariates using categorical arrays and
MPG contains measurements on the miles per
gallon of 100 sample cars. The model year of each car is in the variable
Weight contains the
weight of each car.
Draw a scatter plot of
Weight, grouped by model year.
figure() gscatter(Weight,MPG,Model_Year,'bgr','x.o') title('MPG vs. Weight, Grouped by Model Year')
The grouping variable,
Model_Year, has three
82, corresponding to model years 1970, 1976, and
Create a table that contains the variables
Model_Year. Convert the
Model_Year to a categorical array.
cars = table(MPG,Weight,Model_Year); cars.Model_Year = categorical(cars.Model_Year);
Fit a regression model using
MPG as the dependent variable, and
Model_Year as the
independent variables. Because
Model_Year is a categorical
covariate with three levels, it should enter the model as two indicator
The scatter plot suggests that the slope of
Weight might differ for each model year. To assess this,
include weight-year interaction terms.
The proposed model is
where I and I are dummy variables indicating the model years 1976 and 1982, respectively. I takes the value 1 if model year is 1976 and takes the value 0 if it is not. I takes the value 1 if model year is 1982 and takes the value 0 if it is not. In this model, 1970 is the reference year.
fit = fitlm(cars,'MPG~Weight*Model_Year')
fit = Linear regression model: MPG ~ 1 + Weight*Model_Year Estimated Coefficients: Estimate SE ___________ __________ (Intercept) 37.399 2.1466 Weight -0.0058437 0.00061765 Model_Year_76 4.6903 2.8538 Model_Year_82 21.051 4.157 Weight:Model_Year_76 -0.00082009 0.00085468 Weight:Model_Year_82 -0.0050551 0.0015636 tStat pValue ________ __________ (Intercept) 17.423 2.8607e-30 Weight -9.4612 4.6077e-15 Model_Year_76 1.6435 0.10384 Model_Year_82 5.0641 2.2364e-06 Weight:Model_Year_76 -0.95953 0.33992 Weight:Model_Year_82 -3.2329 0.0017256 Number of observations: 94, Error degrees of freedom: 88 Root Mean Squared Error: 2.79 R-squared: 0.886, Adjusted R-Squared 0.88 F-statistic vs. constant model: 137, p-value = 5.79e-40
The regression output shows:
Model_Year as a categorical variable, and
constructs the required indicator (dummy) variables. By default, the
70, is the reference group (use
reordercats to change the reference
The model specification,
specifies the first-order terms for
Model_Year, and all interactions.
The model R2 = 0.886, meaning the variation in miles per gallon is reduced by 88.6% when you consider weight, model year, and their interactions.
The fitted model is
Thus, the estimated regression equations for the model years are as follows.
|Model Year||Predicted MPG Against Weight|
The relationship between
has an increasingly negative slope as the model year increases.
Plot the data and fitted regression lines.
w = linspace(min(Weight),max(Weight)); figure() gscatter(Weight,MPG,Model_Year,'bgr','x.o') line(w,feval(fit,w,'70'),'Color','b','LineWidth',2) line(w,feval(fit,w,'76'),'Color','g','LineWidth',2) line(w,feval(fit,w,'82'),'Color','r','LineWidth',2) title('Fitted Regression Lines by Model Year')
Test for significant differences between the slopes. This is equivalent to testing the hypothesis
ans = SumSq DF MeanSq F pValue Weight 2050.2 1 2050.2 263.87 3.2055e-28 Model_Year 807.69 2 403.84 51.976 1.2494e-15 Weight:Model_Year 81.219 2 40.609 5.2266 0.0071637 Error 683.74 88 7.7698
0.0072(from the interaction row,
Weight:Model_Year), so the null hypothesis is rejected at the 0.05 significance level. The value of the test statistic is
5.2266. The numerator degrees of freedom for the test is
2, which is the number of coefficients in the null hypothesis.
There is sufficient evidence that the slopes are not equal for all three model years.