Linear Regression with Categorical Covariates

This example shows how to perform a regression with categorical covariates using categorical arrays andfitlm.

Load sample data.

loadcarsmall

The variableMPGcontains measurements on the miles per gallon of 100 sample cars. The model year of each car is in the variableModel_Year, andWeightcontains the weight of each car.

Plot grouped data.

Draw a scatter plot ofMPGagainstWeight, 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 unique values,70,76, and82, corresponding to model years 1970, 1976, and 1982.

Create table and categorical array.

Create a table that contains the variablesMPG,Weight, andModel_Year. Convert the variableModel_Yearto a categorical array.

cars = table(MPG,Weight,Model_Year); cars.Model_Year = categorical(cars.Model_Year);

Fit a regression model.

Fit a regression model usingfitlmwithMPGas the dependent variable, andWeightandModel_Yearas the independent variables. BecauseModel_Yearis a categorical covariate with three levels, it should enter the model as two indicator variables.

The scatter plot suggests that the slope ofMPGagainstWeightmight differ for each model year. To assess this, include weight-year interaction terms.

The proposed model is

$E (M P G) = β_{0} + β_{1} W e i g h t + β_{2} I [1976] + β_{3} I [1982] + β_{4} W e i g h t \times I [1976] + β_{5} W e i g h t \times I [1982],$

whereI[1976] andI[1982]是虚拟变量表示模型ars 1976 and 1982, respectively.I[1976] takes the value 1 if model year is 1976 and takes the value 0 if it is not.I[1982] 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:

fitlmrecognizesModel_Yearas a categorical variable, and constructs the required indicator (dummy) variables. By default, the first level,70, is the reference group (usereordercatsto change the reference group).
The model specification,MPG~Weight*Model_Year, specifies the first-order terms forWeightandModel_Year, and all interactions.
The modelR²= 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

$M \hat{P} G = 37.4 - 0.006 W e i g h t + 4.7 I [1976] + 21.1 I [1982] - 0.0008 W e i g h t \times I [1976] - 0.005 W e i g h t \times I [1982] .$

Thus, the estimated regression equations for the model years are as follows.

Model Year	Predicted MPG Against Weight
1970	$M \hat{P} G = 37.4 - 0.006 W e i g h t$
1976	$M \hat{P} G = (37.4 + 4.7) - (0.006 + 0.0008) W e i g h t$
1982	$M \hat{P} G = (37.4 + 21.1) - (0.006 + 0.005) W e i g h t$

Model Year

Predicted MPG Against Weight

1970

$M \hat{P} G = 37.4 - 0.006 W e i g h t$

1976

$M \hat{P} G = (37.4 + 4.7) - (0.006 + 0.0008) W e i g h t$

1982

$M \hat{P} G = (37.4 + 21.1) - (0.006 + 0.005) W e i g h t$

The relationship betweenMPGandWeighthas an increasingly negative slope as the model year increases.

Plot fitted regression lines.

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 different slopes.

Test for significant differences between the slopes. This is equivalent to testing the hypothesis

$\begin{array}{l} H_{0} : β_{4} = β_{5} = 0 \\ H_{A} : β_{i} \neq 0 for at least one i . \end{array}$

anova(fit)

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

This output shows that thep-value for the test is0.0072(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 is5.2266. The numerator degrees of freedom for the test is2, 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.