Add terms to linear regression model

## Description

example

NewMdl = addTerms(mdl,terms) returns a linear regression model fitted using the input data and settings in mdl with the terms terms added.

## Examples

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Create a linear regression model of the carsmall data set without any interactions, and then add an interaction term.

Load the carsmall data set and create a model of the MPG as a function of weight and model year.

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
F-statistic vs. constant model: 172, p-value = 5.52e-41

The model includes five terms, Intercept, Weight, Year_76, Year_82, and Weight^2, where Year_76 and Year_82 are indicator variables for the categorical variable Year that has three distinct values.

Add an interaction term between the Year and Weight variables to mdl.

terms = 'Year*Weight';
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
F-statistic vs. constant model: 117, p-value = 1.88e-39

NewMdl includes two additional terms, Weight*Year_76 and Weight*Year_82.

## Input Arguments

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Linear regression model, specified as a LinearModel object created using fitlm or stepwiselm.

Terms to add to the regression model mdl, specified 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.

• Terms matrix 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 of variable j in term i.

For example, suppose mdl has three variables A, B, and C in that order. Each row of T represents one term:

• [0 0 0] — Constant term or intercept

• [0 1 0]B; equivalently, A^0 * B^1 * C^0

• [1 0 1]A*C

• [2 0 0]A^2

• [0 1 2]B*(C^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 variable.

## Output Arguments

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Linear regression model with additional terms, returned as a LinearModel object. NewMdl is a newly fitted model that uses the input data and settings in mdl with additional terms specified in terms.

To overwrite the input argument mdl, assign the newly fitted model to mdl:

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### Wilkinson Notation

Wilkinson notation describes the terms present in a model. The notation relates to the terms present in a model, not to the multipliers (coefficients) of those terms.

Wilkinson notation uses these symbols:

• + means include the next variable.

• means do not include the next variable.

• : defines an interaction, which is a product of terms.

• * defines an interaction and all lower-order terms.

• ^ raises the predictor to a power, exactly as in * repeated, so ^ includes lower-order terms as well.

• () groups terms.

This table shows typical examples of Wilkinson notation.

Wilkinson NotationTerms in Standard Notation
1Constant (intercept) term
x1^k, where k is a positive integerx1, x12, ..., x1k
x1 + x2x1, x2
x1*x2x1, x2, x1*x2
x1:x2x1*x2 only
–x2Do not include x2
x1*x2 + x3x1, x2, x3, x1*x2
x1 + x2 + x3 + x1:x2x1, x2, x3, x1*x2
x1*x2*x3 – x1:x2:x3x1, x2, x3, x1*x2, x1*x3, x2*x3
x1*(x2 + x3)x1, x2, x3, x1*x2, x1*x3

For more details, see Wilkinson Notation.

## Algorithms

• 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 by using categories and reorder the categories by using reordercats to customize the reference level. For more details about creating indicator variables, see Automatic Creation of Dummy Variables.

• 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.

## Alternative Functionality

• Use stepwiselm to specify terms in a starting model and continue improving the model until no single step of adding or removing a term is beneficial.

• Use removeTerms to remove specific terms from a model.

• Use step to optimally improve a model by adding or removing terms.

## Version History

Introduced in R2012a