Documentation

removeTerms

Remove terms from linear regression model

Description

example

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

Examples

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Create a linear regression model using the hald data set. Remove terms that have high p-values.

Load the data set.

X = ingredients; % predictor variables
y = heat; % response variable

Fit a linear regression model to the data.

mdl = fitlm(X,y)
mdl =
Linear regression model:
y ~ 1 + x1 + x2 + x3 + x4

Estimated Coefficients:
Estimate      SE        tStat       pValue
________    _______    ________    ________

(Intercept)      62.405     70.071      0.8906     0.39913
x1               1.5511    0.74477      2.0827    0.070822
x2              0.51017    0.72379     0.70486      0.5009
x3              0.10191    0.75471     0.13503     0.89592
x4             -0.14406    0.70905    -0.20317     0.84407

Number of observations: 13, Error degrees of freedom: 8
Root Mean Squared Error: 2.45
R-squared: 0.982,  Adjusted R-Squared: 0.974
F-statistic vs. constant model: 111, p-value = 4.76e-07

Remove the x3 and x4 terms because their p-values are high.

terms = 'x3 + x4'; % terms to remove
NewMdl = removeTerms(mdl,terms)
NewMdl =
Linear regression model:
y ~ 1 + x1 + x2

Estimated Coefficients:
Estimate       SE       tStat       pValue
________    ________    ______    __________

(Intercept)     52.577       2.2862    22.998    5.4566e-10
x1              1.4683       0.1213    12.105    2.6922e-07
x2             0.66225     0.045855    14.442     5.029e-08

Number of observations: 13, Error degrees of freedom: 10
Root Mean Squared Error: 2.41
R-squared: 0.979,  Adjusted R-Squared: 0.974
F-statistic vs. constant model: 230, p-value = 4.41e-09

NewMdl has the same adjusted R-squared value (0.974) as the previous model, meaning the fit is as good in the new model. All the terms in the new model have extremely low p-values.

Input Arguments

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

Terms to remove from the regression model mdl, specified as one of the following:

• A 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)

removeTerms treats a group of indicator variables for a categorical predictor as a single variable. Therefore, you cannot specify an indicator variable to remove from the model. If you specify a categorical predictor to remove from the model, removeTerms removes 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 fewer terms, returned as a LinearModel object. NewMdl is a newly fitted model that uses the input data and settings in mdl with the terms specified in terms removed from mdl.

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

mdl = removeTerms(mdl,terms);

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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 NotationTerm in Standard Notation
1Constant (intercept) term
A^k, where k is a positive integerA, A2, ..., Ak
A + BA, B
A*BA, B, A*B
A:BA*B only
–BDo not include B
A*B + CA, B, C, A*B
A + B + C + A:BA, B, C, A*B
A*B*C – A:B:CA, B, C, A*B, A*C, B*C
A*(B + C)A, B, C, A*B, A*C

Statistics and Machine Learning Toolbox™ notation always includes a constant term unless you explicitly remove the term using –1.

For more details, see Wilkinson Notation.

Algorithms

• removeTerms 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.

• removeTerms 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 addTerms to add specific terms to a model.

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