Add terms to linear regression model
Create a linear regression model of the
carsmall data set without any interactions, and then add an interaction term.
carsmall data set and create a model of the MPG as a function of weight and model year.
load carsmall tbl = table(MPG,Weight); tbl.Year = categorical(Model_Year); mdl = fitlm(tbl,'MPG ~ Year + Weight^2')
mdl = Linear regression model: MPG ~ 1 + Weight + Year + Weight^2 Estimated Coefficients: Estimate SE tStat pValue __________ __________ _______ __________ (Intercept) 54.206 4.7117 11.505 2.6648e-19 Weight -0.016404 0.0031249 -5.2493 1.0283e-06 Year_76 2.0887 0.71491 2.9215 0.0044137 Year_82 8.1864 0.81531 10.041 2.6364e-16 Weight^2 1.5573e-06 4.9454e-07 3.149 0.0022303 Number of observations: 94, Error degrees of freedom: 89 Root Mean Squared Error: 2.78 R-squared: 0.885, Adjusted R-Squared: 0.88 F-statistic vs. constant model: 172, p-value = 5.52e-41
The model includes five terms,
Year_82 are indicator variables for the categorical variable
Year that has three distinct values.
Add an interaction term between the
Weight variables to
terms = 'Year*Weight'; NewMdl = addTerms(mdl,terms)
NewMdl = Linear regression model: MPG ~ 1 + Weight*Year + Weight^2 Estimated Coefficients: Estimate SE tStat pValue ___________ __________ ________ __________ (Intercept) 48.045 6.779 7.0874 3.3967e-10 Weight -0.012624 0.0041455 -3.0454 0.0030751 Year_76 2.7768 3.0538 0.90931 0.3657 Year_82 16.416 4.9802 3.2962 0.0014196 Weight:Year_76 -0.00020693 0.00092403 -0.22394 0.82333 Weight:Year_82 -0.0032574 0.0018919 -1.7217 0.088673 Weight^2 1.0121e-06 6.12e-07 1.6538 0.10177 Number of observations: 94, Error degrees of freedom: 87 Root Mean Squared Error: 2.76 R-squared: 0.89, Adjusted R-Squared: 0.882 F-statistic vs. constant model: 117, p-value = 1.88e-39
NewMdl includes two additional terms,
terms— Terms to add to regression model
Terms to add to the regression model
as one of the following:
Character vector or string scalar formula in Wilkinson Notation representing one or more terms. The variable names in the formula must be valid MATLAB® identifiers.
T of size
t-by-p, where t is the
number of terms and p is the number of predictor variables in
mdl. The value of
T(i,j) is the exponent
j in term
For example, suppose
mdl has three variables
C in that
order. Each row of
T represents one term:
[0 0 0] — Constant term or intercept
[0 1 0] —
A^0 * B^1 * C^0
[1 0 1] —
[2 0 0] —
[0 1 2] —
addTerms treats a group of indicator variables for a
categorical predictor as a single variable. Therefore, you cannot specify an
indicator variable to add to the model. If you specify a categorical
predictor to add to the model,
addTerms adds a group of
indicator variables for the predictor in one step. See Modify Linear Regression Model Using step for an example that describes how to
create indicator variables manually and treat each one as a separate
NewMdl— Linear regression model with additional terms
addTerms treats a categorical predictor as follows:
A model with a categorical predictor that has L levels
(categories) includes L – 1 indicator variables. The model uses the first category as a
reference level, so it does not include the indicator variable for the reference
level. If the data type of the categorical predictor is
categorical, then you can check the order of categories
categories and reorder the
categories by using
reordercats to customize the
addTerms treats the group of L – 1 indicator variables as a single variable. If you want to treat
the indicator variables as distinct predictor variables, create indicator
variables manually by using
dummyvar. Then use the
indicator variables, except the one corresponding to the reference level of the
categorical variable, when you fit a model. For the categorical predictor
X, if you specify all columns of
dummyvar(X) and an intercept term as predictors, then the
design matrix becomes rank deficient.
Interaction terms between a continuous predictor and a categorical predictor with L levels consist of the element-wise product of the L – 1 indicator variables with the continuous predictor.
Interaction terms between two categorical predictors with L and M levels consist of the (L – 1)*(M – 1) indicator variables to include all possible combinations of the two categorical predictor levels.
You cannot specify higher-order terms for a categorical predictor because the square of an indicator is equal to itself.