Model with Categorical Predictor

Load thecarsmalldata set and create a linear regression model ofMPGas a function ofModel_Year. To treat the numeric vectorModel_Yearas a categorical variable, identify the predictor using the'CategoricalVars'name-value pair argument.

loadcarsmallmdl = fitlm(Model_Year,MPG,'CategoricalVars',1,'VarNames',{'Model_Year',“英里”})

mdl =线性回归模型:MPG ~ 1 + Model_Year Estimated Coefficients: Estimate SE tStat pValue ________ ______ ______ __________ (Intercept) 17.69 1.0328 17.127 3.2371e-30 Model_Year_76 3.8839 1.4059 2.7625 0.0069402 Model_Year_82 14.02 1.4369 9.7571 8.2164e-16 Number of observations: 94, Error degrees of freedom: 91 Root Mean Squared Error: 5.56 R-squared: 0.531, Adjusted R-Squared: 0.521 F-statistic vs. constant model: 51.6, p-value = 1.07e-15

The model formula in the display,MPG ~ 1 + Model_Year, corresponds to

$\mathrm{MPG}={\beta }_0+{\beta }_1{{\rm I}}_{\mathrm{Year}=76}+{\beta }_2{{\rm I}}_{\mathrm{Year}=82}+ϵ$,

where ${{\rm I}}_{\mathrm{Year}=76}$and ${{\rm I}}_{\mathrm{Year}=82}$are indicator variables whose value is one if the value ofModel_Yearis 76 and 82, respectively. TheModel_Yearvariable includes three distinct values, which you can check by using theuniquefunction.

unique(Model_Year)

ans =3×170 76 82

fitlmchooses the smallest value inModel_Yearas a reference level ('70') and creates two indicator variables ${{\rm I}}_{\mathrm{Year}=76}$and ${{\rm I}}_{\mathrm{Year}=82}$. The model includes only two indicator variables because the design matrix becomes rank deficient if the model includes three indicator variables (one for each level) and an intercept term.

Model with Full Indicator Variables

You can interpret the model formula ofmdlas a model that has three indicator variables without an intercept term:

$\mathit{y}={\beta }_0{{\rm I}}_{{\mathit{x}}_1=70}+\left({\beta }_0+{\beta }_1\right){{\rm I}}_{{\mathit{x}}_1=76}+\left({{\beta }_0+\beta }_2\right){{\rm I}}_{{\mathit{x}}_2=82}+ϵ$.

Alternatively, you can create a model that has three indicator variables without an intercept term by manually creating indicator variables and specifying the model formula.

temp_Year = dummyvar(categorical(Model_Year)); Model_Year_70 = temp_Year(:,1); Model_Year_76 = temp_Year(:,2); Model_Year_82 = temp_Year(:,3); tbl = table(Model_Year_70,Model_Year_76,Model_Year_82,MPG); mdl = fitlm(tbl,'MPG ~ Model_Year_70 + Model_Year_76 + Model_Year_82 - 1')

mdl = Linear regression model: MPG ~ Model_Year_70 + Model_Year_76 + Model_Year_82 Estimated Coefficients: Estimate SE tStat pValue ________ _______ ______ __________ Model_Year_70 17.69 1.0328 17.127 3.2371e-30 Model_Year_76 21.574 0.95387 22.617 4.0156e-39 Model_Year_82 31.71 0.99896 31.743 5.2234e-51 Number of observations: 94, Error degrees of freedom: 91 Root Mean Squared Error: 5.56

Choose Reference Level in Model

You can choose a reference level by modifying the order of categories in a categorical variable. First, create a categorical variableYear.

Year = categorical(Model_Year);

Check the order of categories by using thecategoriesfunction.

categories(Year)

ans =3x1 cell{'70'} {'76'} {'82'}

If you useYearas a predictor variable, thenfitlmchooses the first category'70'as a reference level. ReorderYearby using thereordercatsfunction.

Year_reordered = reordercats(Year,{'76','70','82'}); categories(Year_reordered)

ans =3x1 cell{'76'} {'70'} {'82'}

The first category ofYear_reorderedis'76'. Create a linear regression model ofMPGas a function ofYear_reordered.

mdl2 = fitlm(Year_reordered,MPG,'VarNames',{'Model_Year',“英里”})

mdl2 = Linear regression model: MPG ~ 1 + Model_Year Estimated Coefficients: Estimate SE tStat pValue ________ _______ _______ __________ (Intercept) 21.574 0.95387 22.617 4.0156e-39 Model_Year_70 -3.8839 1.4059 -2.7625 0.0069402 Model_Year_82 10.136 1.3812 7.3385 8.7634e-11 Number of observations: 94, Error degrees of freedom: 91 Root Mean Squared Error: 5.56 R-squared: 0.531, Adjusted R-Squared: 0.521 F-statistic vs. constant model: 51.6, p-value = 1.07e-15

mdl2uses'76'as a reference level and includes two indicator variables ${{\rm I}}_{\mathrm{Year}=70}$and ${{\rm I}}_{\mathrm{Year}=82}$.

Evaluate Categorical Predictor

The model display ofmdl2includes ap-value of each term to test whether or not the corresponding coefficient is equal to zero. Eachp-value examines each indicator variable. To examine the categorical variableModel_Yearas a group of indicator variables, useanova. Use the'components'(default) option to return a component ANOVA table that includes ANOVA statistics for each variable in the model except the constant term.

anova(mdl2,'components')

ans=2×5 tableSumSq DF MeanSq F pValue ______ __ ______ _____ __________ Model_Year 3190.1 2 1595.1 51.56 1.0694e-15 Error 2815.2 91 30.936

The component ANOVA table includes thep-value of theModel_Yearvariable, which is smaller than thep-values of the indicator variables.