# stepwiselm

Perform stepwise regression

## Syntax

``mdl = stepwiselm(tbl)``
``mdl = stepwiselm(X,y)``
``mdl = stepwiselm(___,modelspec)``
``mdl = stepwiselm(___,Name,Value)``

## Description

example

````mdl = stepwiselm(tbl)` creates a linear model for the variables in the table or dataset array `tbl` using stepwise regression to add or remove predictors, starting from a constant model. `stepwiselm` uses the last variable of `tbl` as the response variable. `stepwiselm` uses forward and backward stepwise regression to determine a final model. At each step, the function searches for terms to add the model to or remove from the model, based on the value of the `'Criterion'` argument.```

example

````mdl = stepwiselm(X,y)` creates a linear model of the responses `y` to the predictor variables in the data matrix `X`.```

example

````mdl = stepwiselm(___,modelspec)` specifies the starting model `modelspec` using any of the input argument combinations in previous syntaxes.```

example

````mdl = stepwiselm(___,Name,Value)` specifies additional options using one or more name-value pair arguments. For example, you can specify the categorical variables, the smallest or largest set of terms to use in the model, the maximum number of steps to take, or the criterion that `stepwiselm` uses to add or remove terms.```

## Examples

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Load the `hald` data set, which measures the effect of cement composition on its hardening heat.

`load hald`

This data set includes the variables `ingredients` and `heat`. The matrix `ingredients` contains the percent composition of four chemicals present in the cement. The vector `heat` contains the values for the heat hardening after 180 days for each cement sample.

Fit a stepwise linear regression model to the data. Specify 0.06 as the threshold for the criterion to add a term to the model.

` mdl = stepwiselm(ingredients,heat,'PEnter',0.06)`
```1. Adding x4, FStat = 22.7985, pValue = 0.000576232 2. Adding x1, FStat = 108.2239, pValue = 1.105281e-06 3. Adding x2, FStat = 5.0259, pValue = 0.051687 4. Removing x4, FStat = 1.8633, pValue = 0.2054 ```
```mdl = 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 ```

By default, the starting model is a constant model. `stepwiselm` performs forward selection and adds the `x4`, `x1`, and `x2` terms (in that order), because the corresponding p-values are less than the `PEnter` value of 0.06. `stepwiselm` then uses backward elimination and removes `x4` from the model because, once `x2` is in the model, the p-value of `x4` is greater than the default value of `PRemove`, 0.1.

Perform stepwise regression using variables stored in a dataset array. Specify the starting model using Wilkinson notation, and identify the response and predictor variables using optional arguments.

`load hospital`

The hospital dataset array includes the gender, age, weight, and smoking status of patients.

Fit a linear model with a starting model of a constant term and `Smoker` as the predictor variable. Specify the response variable, `Weight`, and categorical predictor variables, `Sex`, `Age`, and `Smoker`.

```mdl = stepwiselm(hospital,'Weight~1+Smoker',... 'ResponseVar','Weight','PredictorVars',{'Sex','Age','Smoker'},... 'CategoricalVar',{'Sex','Smoker'})```
```1. Adding Sex, FStat = 770.0158, pValue = 6.262758e-48 2. Removing Smoker, FStat = 0.21224, pValue = 0.64605 ```
```mdl = Linear regression model: Weight ~ 1 + Sex Estimated Coefficients: Estimate SE tStat pValue ________ ______ ______ ___________ (Intercept) 130.47 1.1995 108.77 5.2762e-104 Sex_Male 50.06 1.7496 28.612 2.2464e-49 Number of observations: 100, Error degrees of freedom: 98 Root Mean Squared Error: 8.73 R-squared: 0.893, Adjusted R-Squared: 0.892 F-statistic vs. constant model: 819, p-value = 2.25e-49 ```

At each step, `stepwiselm` searches for terms to add and remove. At first step, stepwise algorithm adds `Sex` to the model with a $p$-value of 6.26e-48. Then, removes Smoker from the model, since given `Sex` in the model, the variable `Smoker` becomes redundant. `stepwiselm` only includes `Sex` in the final linear model. The weight of the patients do not seem to differ significantly according to age or the status of smoking.

Load a sample data set and define the matrix of predictors.

```load carsmall X = [Acceleration,Weight];```

Define the starting model and the upper model using terms matrices.

`T_starting = [0 0 0] % a constant model`
```T_starting = 1×3 0 0 0 ```
`T_upper = [0 0 0;1 0 0;0 1 0;1 1 0] % a linear model with interactions`
```T_upper = 4×3 0 0 0 1 0 0 0 1 0 1 1 0 ```

Create a linear regression model using stepwise regression. Specify the starting model and the upper bound of the model using the terms matrices, and specify `'Verbose'` as 2 to display the evaluation process and the decision taken at each step.

`mdl = stepwiselm(X,MPG,T_starting,'upper',T_upper,'Verbose',2)`
``` pValue for adding x1 is 4.0973e-06 pValue for adding x2 is 1.6434e-28 1. Adding x2, FStat = 259.3087, pValue = 1.643351e-28 pValue for adding x1 is 0.18493 No candidate terms to remove ```
```mdl = Linear regression model: y ~ 1 + x2 Estimated Coefficients: Estimate SE tStat pValue __________ _________ _______ __________ (Intercept) 49.238 1.6411 30.002 2.7015e-49 x2 -0.0086119 0.0005348 -16.103 1.6434e-28 Number of observations: 94, Error degrees of freedom: 92 Root Mean Squared Error: 4.13 R-squared: 0.738, Adjusted R-Squared: 0.735 F-statistic vs. constant model: 259, p-value = 1.64e-28 ```

Fit a linear regression model with a categorical predictor using stepwise regression. `stepwiselm` adds or removes a group of indicator variables in one step to add or removes a categorical predictor. This example also shows how to create indicator variables manually and pass them to `stepwiselm` so that `stepwiselm` treats each indicator variable as a separate predictor.

Load the `carsmall` data set, and create a table using the `Weight`, `Model_Year`, and `MPG` variables.

```load carsmall Year = categorical(Model_Year); tbl1 = table(MPG,Weight,Year);```

Fit a linear regression model of `MPG` using stepwise regression. Specify the starting model as a function of `Weight`. Set the upper bound of the model to `'poly21'`, meaning the model can include (at most) a constant and the terms `Weight`, `Weight^2`, `Year`, and `Weight*Year`. Specify `'Verbose'` as 2 to display the evaluation process and the decision taken at each step.

`mdl1 = stepwiselm(tbl1,'MPG ~ Weight','Upper','poly21','Verbose',2)`
``` pValue for adding Year is 8.2284e-15 pValue for adding Weight^2 is 0.15454 1. Adding Year, FStat = 47.5136, pValue = 8.22836e-15 pValue for adding Weight^2 is 0.0022303 pValue for adding Weight:Year is 0.0071637 2. Adding Weight^2, FStat = 9.9164, pValue = 0.0022303 pValue for adding Weight:Year is 0.19519 pValue for removing Year is 2.9042e-16 ```
```mdl1 = 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 ```

`stepwiselm` creates two indicator variables, `Year_76` and `Year_82`, because `Year` includes three distinct values.

Because `'Verbose'` is 2, `stepwiselm` displays the evaluation process:

• `stepwiselm` creates a model as a function of `Weight`.

• `stepwiselm` computes the p-values for adding `Year` or `Weight^2`. The p-value for `Year` is less than both the p-value for `Weight^2` and the default threshold value of 0.05; therefore, `stepwiselm` adds `Year` to the model.

• `stepwiselm` computes the p-values for adding `Weight:Year` or `Weight^2`. Because the p-value for `Weight^2` is less than the p-value for `Weight:Year`, the `stepwiselm` function adds `Weight^2` to the model.

• After adding the quadratic term, `stepwiselm` computes the p-value for adding `Weight:Year` again, but the p-value is greater than the threshold value. Therefore, `stepwiselm` does not add the term to the model. `stepwiselm` does not examine adding `Weight^3` because of the upper bound specified by the `'Upper'` name-value pair argument.

• `stepwiselm` looks for terms to remove. `stepwiselm` already examined `Weight^2`, so it computes only the p-value for removing `Year`. Because the p-value is less than the default threshold value of 0.10, `stepwiselm` does not remove the term.

• Although the maximum allowed number of steps is 5, `stepwiselm` terminates the process after two steps because the model does not improve by adding or removing a term.

`stepwiselm` treats the two indicator variables as one predictor variable and adds `Year` in one step. To treat the two indicator variables as two distinct predictor variables, use `dummyvar` to create separate categorical variables.

```temp_Year = dummyvar(Year); Year_76 = logical(temp_Year(:,2)); Year_82 = logical(temp_Year(:,3));```

Create a table containing `MPG`, `Weight`, `Year_76`, and `Year_82`.

`tbl2 = table(MPG,Weight,Year_76,Year_82);`

Create a stepwise linear regression model from the same starting model used for `mdl1`.

`mdl2 = stepwiselm(tbl2,'MPG ~ Weight','Upper','poly211')`
```1. Adding Year_82, FStat = 83.1956, pValue = 1.76163e-14 2. Adding Weight:Year_82, FStat = 8.0641, pValue = 0.0055818 3. Adding Year_76, FStat = 8.1284, pValue = 0.0054157 ```
```mdl2 = Linear regression model: MPG ~ 1 + Year_76 + Weight*Year_82 Estimated Coefficients: Estimate SE tStat pValue __________ __________ _______ __________ (Intercept) 38.844 1.5294 25.397 1.503e-42 Weight -0.006272 0.00042673 -14.698 1.5622e-25 Year_76_1 2.0395 0.71537 2.851 0.0054157 Year_82_1 19.607 3.8731 5.0623 2.2163e-06 Weight:Year_82_1 -0.0046268 0.0014979 -3.0888 0.0026806 Number of observations: 94, Error degrees of freedom: 89 Root Mean Squared Error: 2.79 R-squared: 0.885, Adjusted R-Squared: 0.88 F-statistic vs. constant model: 171, p-value = 6.54e-41 ```

The model `mdl2` includes the interaction term `Weight:Year_82_1` instead of `Weight^2`, the term included in `mdl1`.

## Input Arguments

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Input data including predictor and response variables, specified as a table or dataset array. The predictor variables can be numeric, logical, categorical, character, or string. The response variable must be numeric or logical.

• By default, `stepwiselm` takes the last variable as the response variable and the others as the predictor variables.

• To set a different column as the response variable, use the `ResponseVar` name-value pair argument.

• To use a subset of the columns as predictors, use the `PredictorVars` name-value pair argument.

• To define a model specification, set the `modelspec` argument using a formula or terms matrix. The formula or terms matrix specifies which columns to use as the predictor or response variables.

The variable names in a table do not have to be valid MATLAB® identifiers. However, if the names are not valid, you cannot use a formula when you fit or adjust a model; for example:

• You cannot specify `modelspec` using a formula.

• You cannot use a formula to specify the terms to add or remove when you use the `addTerms` function or the `removeTerms` function, respectively.

• You cannot use a formula to specify the lower and upper bounds of the model when you use the `step` or `stepwiselm` function with the name-value pair arguments `'Lower'` and `'Upper'`, respectively.

You can verify the variable names in `tbl` by using the `isvarname` function. The following code returns logical `1` (`true`) for each variable that has a valid variable name.

`cellfun(@isvarname,tbl.Properties.VariableNames)`
If the variable names in `tbl` are not valid, then convert them by using the `matlab.lang.makeValidName` function.
`tbl.Properties.VariableNames = matlab.lang.makeValidName(tbl.Properties.VariableNames);`

Predictor variables, specified as an n-by-p matrix, where n is the number of observations and p is the number of predictor variables. Each column of `X` represents one variable, and each row represents one observation.

By default, there is a constant term in the model, unless you explicitly remove it, so do not include a column of 1s in `X`.

Data Types: `single` | `double`

Response variable, specified as an n-by-1 vector, where n is the number of observations. Each entry in `y` is the response for the corresponding row of `X`.

Data Types: `single` | `double` | `logical`

Starting model for the stepwise regression, specified as one of the following:

• A character vector or string scalar naming the model.

ValueModel Type
`'constant'`Model contains only a constant (intercept) term.
`'linear'`Model contains an intercept and linear term for each predictor.
`'interactions'`Model contains an intercept, linear term for each predictor, and all products of pairs of distinct predictors (no squared terms).
`'purequadratic'`Model contains an intercept term and linear and squared terms for each predictor.
`'quadratic'`Model contains an intercept term, linear and squared terms for each predictor, and all products of pairs of distinct predictors.
`'polyijk'`Model is a polynomial with all terms up to degree `i` in the first predictor, degree `j` in the second predictor, and so on. Specify the maximum degree for each predictor by using numerals 0 though 9. The model contains interaction terms, but the degree of each interaction term does not exceed the maximum value of the specified degrees. For example, `'poly13'` has an intercept and x1, x2, x22, x23, x1*x2, and x1*x22 terms, where x1 and x2 are the first and second predictors, respectively.
• A t-by-(p + 1) matrix, or a Terms Matrix, specifying terms in the model, where t is the number of terms and p is the number of predictor variables, and +1 accounts for the response variable. A terms matrix is convenient when the number of predictors is large and you want to generate the terms programmatically.

• A character vector or string scalar Formula in the form

`'y ~ terms'`,

where the `terms` are in Wilkinson Notation. The variable names in the formula must be variable names in `tbl` or variable names specified by `Varnames`. Also, the variable names must be valid MATLAB identifiers.

The software determines the order of terms in a fitted model by using the order of terms in `tbl` or `X`. Therefore, the order of terms in the model can be different from the order of terms in the specified formula.

If you want to specify the smallest or largest set of terms in the model that `stepwiselm` fits, use the `Lower` and `Upper` name-value pair arguments.

Data Types: `char` | `string` | `single` | `double`

### Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: `'Criterion','aic','Upper','interactions','Verbose',1` instructs `stepwiselm` to use the Akaike information criterion, display the action it takes at each step, and include at most the interaction terms in the model.

Categorical variable list, specified as the comma-separated pair consisting of `'CategoricalVars'` and either a string array or cell array of character vectors containing categorical variable names in the table or dataset array `tbl`, or a logical or numeric index vector indicating which columns are categorical.

• If data is in a table or dataset array `tbl`, then, by default, `stepwiselm` treats all categorical values, logical values, character arrays, string arrays, and cell arrays of character vectors as categorical variables.

• If data is in matrix `X`, then the default value of `'CategoricalVars'` is an empty matrix `[]`. That is, no variable is categorical unless you specify it as categorical.

For example, you can specify the observations 2 and 3 out of 6 as categorical using either of the following examples.

Example: `'CategoricalVars',[2,3]`

Example: `'CategoricalVars',logical([0 1 1 0 0 0])`

Data Types: `single` | `double` | `logical` | `string` | `cell`

Criterion to add or remove terms, specified as the comma-separated pair consisting of `'Criterion'` and one of these values:

• `'sse'`p-value for an F-test of the change in the sum of squared error that results from adding or removing the term

• `'aic'` — Change in the value of Akaike information criterion (AIC)

• `'bic'` — Change in the value of Bayesian information criterion (BIC)

• `'rsquared'` — Increase in the value of R2

• `'adjrsquared'` — Increase in the value of adjusted R2

Example: `'Criterion','bic'`

Observations to exclude from the fit, specified as the comma-separated pair consisting of `'Exclude'` and a logical or numeric index vector indicating which observations to exclude from the fit.

For example, you can exclude observations 2 and 3 out of 6 using either of the following examples.

Example: `'Exclude',[2,3]`

Example: `'Exclude',logical([0 1 1 0 0 0])`

Data Types: `single` | `double` | `logical`

Indicator for the constant term (intercept) in the fit, specified as the comma-separated pair consisting of `'Intercept'` and either `true` to include or `false` to remove the constant term from the model.

Use `'Intercept'` only when specifying the model using a character vector or string scalar, not a formula or matrix.

Example: `'Intercept',false`

Model specification describing terms that cannot be removed from the model, specified as the comma-separated pair consisting of `'Lower'` and one of the options for `modelspec` naming the model.

Example: `'Lower','linear'`

Maximum number of steps to take, specified as the comma-separated pair consisting of `'NSteps'` and a positive integer.

Example: `'NSteps',5`

Data Types: `single` | `double`

Threshold for the criterion to add a term, specified as the comma-separated pair consisting of `'PEnter'` and a scalar value, as described in this table.

CriterionDefault ValueDecision
`'SSE'`0.05If the p-value of the F-statistic is less than `PEnter` (p-value to enter), add the term to the model.
`'AIC'`0If the change in the AIC of the model is less than `PEnter`, add the term to the model.
`'BIC'`0If the change in the BIC of the model is less than `PEnter`, add the term to the model.
`'Rsquared'`0.1If the increase in the R-squared value of the model is greater than `PEnter`, add the term to the model.
`'AdjRsquared'`0If the increase in the adjusted R-squared value of the model is greater than `PEnter`, add the term to the model.

For more information, see the `Criterion` name-value pair argument.

Example: `'PEnter',0.075`

Predictor variables to use in the fit, specified as the comma-separated pair consisting of `'PredictorVars'` and either a string array or cell array of character vectors of the variable names in the table or dataset array `tbl`, or a logical or numeric index vector indicating which columns are predictor variables.

The string values or character vectors should be among the names in `tbl`, or the names you specify using the `'VarNames'` name-value pair argument.

The default is all variables in `X`, or all variables in `tbl` except for `ResponseVar`.

For example, you can specify the second and third variables as the predictor variables using either of the following examples.

Example: `'PredictorVars',[2,3]`

Example: `'PredictorVars',logical([0 1 1 0 0 0])`

Data Types: `single` | `double` | `logical` | `string` | `cell`

Threshold for the criterion to remove a term, specified as the comma-separated pair consisting of `'PRemove'` and a scalar value, as described in this table.

CriterionDefault ValueDecision
`'SSE'`0.10If the p-value of the F-statistic is greater than `PRemove` (p-value to remove), remove the term from the model.
`'AIC'`0.01If the change in the AIC of the model is greater than `PRemove`, remove the term from the model.
`'BIC'`0.01If the change in the BIC of the model is greater than `PRemove`, remove the term from the model.
`'Rsquared'`0.05If the increase in the R-squared value of the model is less than `PRemove`, remove the term from the model.
`'AdjRsquared'`-0.05If the increase in the adjusted R-squared value of the model is less than `PRemove`, remove the term from the model.

At each step, the `stepwiselm` function also checks whether a term is redundant (linearly dependent) with other terms in the current model. When any term is linearly dependent with other terms in the current model, the `stepwiselm` function removes the redundant term, regardless of the criterion value.

For more information, see the `Criterion` name-value pair argument.

Example: `'PRemove',0.05`

Response variable to use in the fit, specified as the comma-separated pair consisting of `'ResponseVar'` and either a character vector or string scalar containing the variable name in the table or dataset array `tbl`, or a logical or numeric index vector indicating which column is the response variable. You typically need to use `'ResponseVar'` when fitting a table or dataset array `tbl`.

For example, you can specify the fourth variable, say `yield`, as the response out of six variables, in one of the following ways.

Example: `'ResponseVar','yield'`

Example: `'ResponseVar',[4]`

Example: `'ResponseVar',logical([0 0 0 1 0 0])`

Data Types: `single` | `double` | `logical` | `char` | `string`

Model specification describing the largest set of terms in the fit, specified as the comma-separated pair consisting of `'Upper'` and one of the options for `modelspec` naming the model.

Example: `'Upper','quadratic'`

Names of variables, specified as the comma-separated pair consisting of `'VarNames'` and a string array or cell array of character vectors including the names for the columns of `X` first, and the name for the response variable `y` last.

`'VarNames'` is not applicable to variables in a table or dataset array, because those variables already have names.

The variable names do not have to be valid MATLAB identifiers. However, if the names are not valid, you cannot use a formula when you fit or adjust a model; for example:

Before specifying `'VarNames',varNames`, you can verify the variable names in `varNames` by using the `isvarname` function. The following code returns logical `1` (`true`) for each variable that has a valid variable name.

`cellfun(@isvarname,varNames)`
If the variable names in `varNames` are not valid, then convert them by using the `matlab.lang.makeValidName` function.
`varNames = matlab.lang.makeValidName(varNames);`

Example: `'VarNames',{'Horsepower','Acceleration','Model_Year','MPG'}`

Data Types: `string` | `cell`

Control for the display of information, specified as the comma-separated pair consisting of `'Verbose'` and one of these values:

• `0` — Suppress all display.

• `1` — Display the action taken at each step.

• `2` — Display the evaluation process and the action taken at each step.

Example: `'Verbose',2`

Observation weights, specified as the comma-separated pair consisting of `'Weights'` and an n-by-1 vector of nonnegative scalar values, where n is the number of observations.

Data Types: `single` | `double`

## Output Arguments

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Linear model representing a least-squares fit of the response to the data, returned as a `LinearModel` object.

For the properties and methods of the linear model object, `mdl`, see the `LinearModel` class page.

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### Terms Matrix

A terms matrix `T` is a t-by-(p + 1) matrix specifying terms in a model, where t is the number of terms, p is the number of predictor variables, and +1 accounts for the response variable. The value of `T(i,j)` is the exponent of variable `j` in term `i`.

For example, suppose that an input includes three predictor variables `x1`, `x2`, and `x3` and the response variable `y` in the order `x1`, `x2`, `x3`, and `y`. Each row of `T` represents one term:

• `[0 0 0 0]` — Constant term or intercept

• `[0 1 0 0]``x2`; equivalently, `x1^0 * x2^1 * x3^0`

• `[1 0 1 0]``x1*x3`

• `[2 0 0 0]``x1^2`

• `[0 1 2 0]``x2*(x3^2)`

The `0` at the end of each term represents the response variable. In general, a column vector of zeros in a terms matrix represents the position of the response variable. If you have the predictor and response variables in a matrix and column vector, then you must include `0` for the response variable in the last column of each row.

### Formula

A formula for model specification is a character vector or string scalar of the form ```'y ~ terms'```.

• `y` is the response name.

• `terms` represents the predictor terms in a model using Wilkinson notation.

To represent predictor and response variables, use the variable names of the table input `tbl` or the variable names specified by using `VarNames`. The default value of `VarNames` is `{'x1','x2',...,'xn','y'}`.

For example:

• `'y ~ x1 + x2 + x3'` specifies a three-variable linear model with intercept.

• `'y ~ x1 + x2 + x3 – 1'` specifies a three-variable linear model without intercept. Note that formulas include a constant (intercept) term by default. To exclude a constant term from the model, you must include `–1` in the formula.

A formula includes a constant term unless you explicitly remove the term using `–1`.

### 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
`1`Constant (intercept) term
`x1^k`, where `k` is a positive integer`x1`, `x12`, ..., `x1k`
`x1 + x2``x1`, `x2`
`x1*x2``x1`, `x2`, `x1*x2`
`x1:x2``x1*x2` only
`–x2`Do not include `x2`
`x1*x2 + x3``x1`, `x2`, `x3`, `x1*x2`
`x1 + x2 + x3 + x1:x2``x1`, `x2`, `x3`, `x1*x2`
`x1*x2*x3 – x1:x2:x3``x1`, `x2`, `x3`, `x1*x2`, `x1*x3`, `x2*x3`
`x1*(x2 + x3)``x1`, `x2`, `x3`, `x1*x2`, `x1*x3`

For more details, see Wilkinson Notation.

## Tips

• You cannot use robust regression with stepwise regression. Check your data for outliers before using `stepwiselm`.

• For other methods such as `anova`, or properties of the `LinearModel` object, see `LinearModel`.

• After training a model, you can generate C/C++ code that predicts responses for new data. Generating C/C++ code requires MATLAB Coder™. For details, see Introduction to Code Generation.

## Algorithms

• Stepwise regression is a systematic method for adding and removing terms from a linear or generalized linear model based on their statistical significance in explaining the response variable. The method begins with an initial model, specified using `modelspec`, and then compares the explanatory power of incrementally larger and smaller models.

The `stepwiselm` function uses forward and backward stepwise regression to determine a final model. At each step, the function searches for terms to add to the model or remove from the model based on the value of the `'Criterion'` name-value pair argument.

The default value of `'Criterion'` for a linear regression model is `'sse'`. In this case, `stepwiselm` and `step` of `LinearModel` use the p-value of an F-statistic to test models with and without a potential term at each step. If a term is not currently in the model, the null hypothesis is that the term would have a zero coefficient if added to the model. If there is sufficient evidence to reject the null hypothesis, the function adds the term to the model. Conversely, if a term is currently in the model, the null hypothesis is that the term has a zero coefficient. If there is insufficient evidence to reject the null hypothesis, the function removes the term from the model.

Stepwise regression takes these steps when `'Criterion'` is `'sse'`:

1. Fit the initial model.

2. Examine a set of available terms not in the model. If any of the terms have p-values less than an entrance tolerance (that is, if it is unlikely a term would have a zero coefficient if added to the model), add the term with the smallest p-value and repeat this step; otherwise, go to step 3.

3. If any of the available terms in the model have p-values greater than an exit tolerance (that is, the hypothesis of a zero coefficient cannot be rejected), remove the term with the largest p-value and return to step 2; otherwise, end the process.

At any stage, the function will not add a higher-order term if the model does not also include all lower-order terms that are subsets of the higher-order term. For example, the function will not try to add the term `X1:X2^2` unless both `X1` and `X2^2` are already in the model. Similarly, the function will not remove lower-order terms that are subsets of higher-order terms that remain in the model. For example, the function will not try to remove `X1` or `X2^2` if `X1:X2^2` remains in the model.

The default value of `'Criterion'` for a generalized linear model is `'Deviance'`. `stepwiseglm` and `step` of `GeneralizedLinearModel` follow a similar procedure for adding or removing terms.

You can specify other criteria by using the `'Criterion'` name-value pair argument. For example, you can specify the change in the value of the Akaike information criterion, Bayesian information criterion, R-squared, or adjusted R-squared as the criterion to add or remove terms.

Depending on the terms included in the initial model, and the order in which the function adds and removes terms, the function might build different models from the same set of potential terms. The function terminates when no single step improves the model. However, a different initial model or a different sequence of steps does not guarantee a better fit. In this sense, stepwise models are locally optimal, but might not be globally optimal.

• `stepwiselm` 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.

• `stepwiselm` 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.

Therefore, if `stepwiselm` adds or removes a categorical predictor, the function actually adds or removes the group of indicator variables in one step. Similarly, if `stepwiselm` adds or removes an interaction term with a categorical predictor, the function actually adds or removes the group of interaction terms including the categorical predictor.

• `stepwiselm` considers `NaN`, `''` (empty character vector), `""` (empty string), `<missing>`, and `<undefined>` values in `tbl`, `X`, and `Y` to be missing values. `stepwiselm` does not use observations with missing values in the fit. The `ObservationInfo` property of a fitted model indicates whether or not `stepwiselm` uses each observation in the fit.