Documentation

# LinearModel.fit

(Not Recommended) Create linear regression model

`LinearModel.fit` is not recommended. Use `fitlm` instead.

## Syntax

```mdl = LinearModel.fit(tbl) mdl = LinearModel.fit(X,y) mdl = LinearModel.fit(___,modelspec) mdl = LinearModel.fit(___,Name,Value) mdl = LinearModel.fit(___,modelspec,Name,Value) ```

## Description

`mdl = LinearModel.fit(tbl)` creates a linear model of a table or dataset array `tbl`.

`mdl = LinearModel.fit(X,y)` creates a linear model of the responses `y` to a data matrix `X`.

`mdl = LinearModel.fit(___,modelspec)` creates a linear model of the type specified by `modelspec`, using any of the previous syntaxes.

`mdl = LinearModel.fit(___,Name,Value)` or ```mdl = LinearModel.fit(___,modelspec,Name,Value)``` creates a linear model with additional options specified by one or more `Name,Value` pair arguments. For example, you can specify which predictor variables to include in the fit or include observation weights.

## 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, `LinearModel.fit` 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`

Model specification, 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.
• t-by-(p + 1) matrix, namely terms matrix, specifying terms to include in the model, where t is the number of terms and p is the number of predictor variables, and plus 1 is for the response variable.

• A character vector or string scalar representing a formula in the form

`'Y ~ terms'`,

where the `terms` are specified using Wilkinson Notation.

Example: `'quadratic'`

Example: `'y ~ X1 + X2^2 + X1:X2'`

### 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`.

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, `LinearModel.fit` 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`

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`

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`

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`

Indicator of the robust fitting type to use, specified as the comma-separated pair consisting of `'RobustOpts'` and one of these values.

• `'off'` — No robust fitting. `LinearModel.fit` uses ordinary least squares.

• `'on'` — Robust fitting using the `'bisquare'` weight function with the default tuning constant.

• Character vector or string scalar — Name of a robust fitting weight function from the following table. `LinearModel.fit` uses the corresponding default tuning constant specified in the table.

• Structure with the two fields `RobustWgtFun` and `Tune`.

• The `RobustWgtFun` field contains the name of a robust fitting weight function from the following table or a function handle of a custom weight function.

• The `Tune` field contains a tuning constant. If you do not set the `Tune` field, `LinearModel.fit` uses the corresponding default tuning constant.

Weight FunctionDescriptionDefault Tuning Constant
`'andrews'``w = (abs(r)<pi) .* sin(r) ./ r`1.339
`'bisquare'``w = (abs(r)<1) .* (1 - r.^2).^2` (also called biweight)4.685
`'cauchy'``w = 1 ./ (1 + r.^2)`2.385
`'fair'``w = 1 ./ (1 + abs(r))`1.400
`'huber'``w = 1 ./ max(1, abs(r))`1.345
`'logistic'``w = tanh(r) ./ r`1.205
`'ols'`Ordinary least squares (no weighting function)None
`'talwar'``w = 1 * (abs(r)<1)`2.795
`'welsch'``w = exp(-(r.^2))`2.985
function handleCustom weight function that accepts a vector `r` of scaled residuals, and returns a vector of weights the same size as `r`1
• The default tuning constants of built-in weight functions give coefficient estimates that are approximately 95% as statistically efficient as the ordinary least-squares estimates, provided the response has a normal distribution with no outliers. Decreasing the tuning constant increases the downweight assigned to large residuals; increasing the tuning constant decreases the downweight assigned to large residuals.

• The value r in the weight functions is

`r = resid/(tune*s*sqrt(1–h))`,

where `resid` is the vector of residuals from the previous iteration, `tune` is the tuning constant, `h` is the vector of leverage values from a least-squares fit, and `s` is an estimate of the standard deviation of the error term given by

`s = MAD/0.6745`.

`MAD` is the median absolute deviation of the residuals from their median. The constant 0.6745 makes the estimate unbiased for the normal distribution. If `X` has p columns, the software excludes the smallest p absolute deviations when computing the median.

For robust fitting, `LinearModel.fit` uses M-estimation to formulate estimating equations and solves them using the method of iterative reweighted least squares (IRLS).

Example: `'RobustOpts','andrews'`

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`

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.

If the value of the `'RobustOpts'` name-value pair is not `[]` or `'ols'`, the model is not a least-squares fit, but uses the robust fitting function.

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

## Examples

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Fit a linear regression model using a matrix input data set.

Load the `carsmall` data set, a matrix input data set.

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

Fit a linear regression model by using `fitlm`.

`mdl = fitlm(X,MPG)`
```mdl = Linear regression model: y ~ 1 + x1 + x2 + x3 Estimated Coefficients: Estimate SE tStat pValue __________ _________ _________ __________ (Intercept) 47.977 3.8785 12.37 4.8957e-21 x1 -0.0065416 0.0011274 -5.8023 9.8742e-08 x2 -0.042943 0.024313 -1.7663 0.08078 x3 -0.011583 0.19333 -0.059913 0.95236 Number of observations: 93, Error degrees of freedom: 89 Root Mean Squared Error: 4.09 R-squared: 0.752, Adjusted R-Squared: 0.744 F-statistic vs. constant model: 90, p-value = 7.38e-27 ```

The model display includes the model formula, estimated coefficients, and model summary statistics.

The model formula in the display, `y ~ 1 + x1 + x2 + x3`, corresponds to $\mathit{y}={\beta }_{0}+{\beta }_{1}{\mathit{X}}_{1}+{\beta }_{2}{\mathit{X}}_{2}+{\beta }_{3}{\mathit{X}}_{3}+ϵ$.

The model display also shows the estimated coefficient information, which is stored in the `Coefficients` property. Display the `Coefficients` property.

`mdl.Coefficients`
```ans=4×4 table Estimate SE tStat pValue __________ _________ _________ __________ (Intercept) 47.977 3.8785 12.37 4.8957e-21 x1 -0.0065416 0.0011274 -5.8023 9.8742e-08 x2 -0.042943 0.024313 -1.7663 0.08078 x3 -0.011583 0.19333 -0.059913 0.95236 ```

The `Coefficient` property includes these columns:

• `Estimate` — Coefficient estimates for each corresponding term in the model. For example, the estimate for the constant term (`intercept`) is 47.977.

• `SE` — Standard error of the coefficients.

• `tStat`t-statistic for each coefficient to test the null hypothesis that the corresponding coefficient is zero against the alternative that it is different from zero, given the other predictors in the model. Note that `tStat = Estimate/SE`. For example, the t-statistic for the intercept is 47.977/3.8785 = 12.37.

• `pValue`p-value for the t-statistic of the hypothesis test that the corresponding coefficient is equal to zero or not. For example, the p-value of the t-statistic for `x2` is greater than 0.05, so this term is not significant at the 5% significance level given the other terms in the model.

The summary statistics of the model are:

• `Number of observations` — Number of rows without any `NaN` values. For example, `Number of observations` is 93 because the `MPG` data vector has six `NaN` values and the `Horsepower` data vector has one `NaN` value for a different observation, where the number of rows in `X` and `MPG` is 100.

• `Error degrees of freedom`n p, where n is the number of observations, and p is the number of coefficients in the model, including the intercept. For example, the model has four predictors, so the `Error degrees of freedom` is 93 – 4 = 89.

• `Root mean squared error` — Square root of the mean squared error, which estimates the standard deviation of the error distribution.

• `R-squared` and `Adjusted R-squared` — Coefficient of determination and adjusted coefficient of determination, respectively. For example, the `R-squared` value suggests that the model explains approximately 75% of the variability in the response variable `MPG`.

• `F-statistic vs. constant model` — Test statistic for the F-test on the regression model, which tests whether the model fits significantly better than a degenerate model consisting of only a constant term.

• `p-value`p-value for the F-test on the model. For example, the model is significant with a p-value of 7.3816e-27.

You can find these statistics in the model properties (`NumObservations`, `DFE`, `RMSE`, and `Rsquared`) and by using the `anova` function.

`anova(mdl,'summary')`
```ans=3×5 table SumSq DF MeanSq F pValue ______ __ ______ ______ __________ Total 6004.8 92 65.269 Model 4516 3 1505.3 89.987 7.3816e-27 Residual 1488.8 89 16.728 ```

Fit a linear regression model that contains a categorical predictor. Reorder the categories of the categorical predictor to control the reference level in the model. Then, use `anova` to test the significance of the categorical variable.

Model with Categorical Predictor

Load the `carsmall` data set and create a linear regression model of `MPG` as a function of `Model_Year`. To treat the numeric vector `Model_Year` as a categorical variable, identify the predictor using the `'CategoricalVars'` name-value pair argument.

```load carsmall mdl = fitlm(Model_Year,MPG,'CategoricalVars',1,'VarNames',{'Model_Year','MPG'})```
```mdl = Linear regression model: 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

$\mathrm{MPG}={\beta }_{0}+{\beta }_{1}{Ι}_{\mathrm{Year}=76}+{\beta }_{2}{Ι}_{\mathrm{Year}=82}+ϵ$,

where ${Ι}_{\mathrm{Year}=76}$ and ${Ι}_{\mathrm{Year}=82}$ are indicator variables whose value is one if the value of `Model_Year` is 76 and 82, respectively. The `Model_Year` variable includes three distinct values, which you can check by using the `unique` function.

`unique(Model_Year)`
```ans = 3×1 70 76 82 ```

`fitlm` chooses the smallest value in `Model_Year` as a reference level (`'70'`) and creates two indicator variables ${Ι}_{\mathrm{Year}=76}$ and ${Ι}_{\mathrm{Year}=82}$. 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 of `mdl` as a model that has three indicator variables without an intercept term:

$\mathit{y}={\beta }_{0}{Ι}_{{\mathit{x}}_{1}=70}+\left({\beta }_{0}+{\beta }_{1}\right){Ι}_{{\mathit{x}}_{1}=76}+\left({{\beta }_{0}+\beta }_{2}\right){Ι}_{{\mathit{x}}_{2}=82}+ϵ$.

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 variable `Year`.

`Year = categorical(Model_Year);`

Check the order of categories by using the `categories` function.

`categories(Year)`
```ans = 3x1 cell array {'70'} {'76'} {'82'} ```

If you use `Year` as a predictor variable, then `fitlm` chooses the first category `'70'` as a reference level. Reorder `Year` by using the `reordercats` function.

```Year_reordered = reordercats(Year,{'76','70','82'}); categories(Year_reordered)```
```ans = 3x1 cell array {'76'} {'70'} {'82'} ```

The first category of `Year_reordered` is `'76'`. Create a linear regression model of `MPG` as a function of `Year_reordered`.

`mdl2 = fitlm(Year_reordered,MPG,'VarNames',{'Model_Year','MPG'})`
```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 ```

`mdl2` uses `'76'` as a reference level and includes two indicator variables ${Ι}_{\mathrm{Year}=70}$ and ${Ι}_{\mathrm{Year}=82}$.

Evaluate Categorical Predictor

The model display of `mdl2` includes a p-value of each term to test whether or not the corresponding coefficient is equal to zero. Each p-value examines each indicator variable. To examine the categorical variable `Model_Year` as a group of indicator variables, use `anova`. 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 table SumSq 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 the p-value of the `Model_Year` variable, which is smaller than the p-values of the indicator variables.

Fit a linear regression model to sample data. Specify the response and predictor variables, and include only pairwise interaction terms in the model.

`load hospital`

Fit a linear model with interaction terms to the data. Specify weight as the response variable, and sex, age, and smoking status as the predictor variables. Also, specify that sex and smoking status are categorical variables.

```mdl = fitlm(hospital,'interactions','ResponseVar','Weight',... 'PredictorVars',{'Sex','Age','Smoker'},... 'CategoricalVar',{'Sex','Smoker'})```
```mdl = Linear regression model: Weight ~ 1 + Sex*Age + Sex*Smoker + Age*Smoker Estimated Coefficients: Estimate SE tStat pValue ________ _______ ________ __________ (Intercept) 118.7 7.0718 16.785 6.821e-30 Sex_Male 68.336 9.7153 7.0339 3.3386e-10 Age 0.31068 0.18531 1.6765 0.096991 Smoker_1 3.0425 10.446 0.29127 0.77149 Sex_Male:Age -0.49094 0.24764 -1.9825 0.050377 Sex_Male:Smoker_1 0.9509 3.8031 0.25003 0.80312 Age:Smoker_1 -0.07288 0.26275 -0.27737 0.78211 Number of observations: 100, Error degrees of freedom: 93 Root Mean Squared Error: 8.75 R-squared: 0.898, Adjusted R-Squared: 0.892 F-statistic vs. constant model: 137, p-value = 6.91e-44 ```

The weight of the patients do not seem to differ significantly according to age, or the status of smoking, or interaction of these factors with patient sex at the 5% significance level.

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 robust linear regression model to the data.

`mdl = fitlm(ingredients,heat,'RobustOpts','on')`
```mdl = Linear regression model (robust fit): y ~ 1 + x1 + x2 + x3 + x4 Estimated Coefficients: Estimate SE tStat pValue ________ _______ ________ ________ (Intercept) 60.09 75.818 0.79256 0.4509 x1 1.5753 0.80585 1.9548 0.086346 x2 0.5322 0.78315 0.67957 0.51596 x3 0.13346 0.8166 0.16343 0.87424 x4 -0.12052 0.7672 -0.15709 0.87906 Number of observations: 13, Error degrees of freedom: 8 Root Mean Squared Error: 2.65 R-squared: 0.979, Adjusted R-Squared: 0.969 F-statistic vs. constant model: 94.6, p-value = 9.03e-07 ```

For more details, see the topic Robust Regression — Reduce Outlier Effects, which compares the results of a robust fit to a standard least-squares fit.

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## Tips

• Use robust fitting (`RobustOpts` name-value pair) to reduce the effect of outliers automatically.

• Do not use robust fitting when you want to subsequently adjust a model using `step`.

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

## Algorithms

The main fitting algorithm is QR decomposition. For robust fitting, the algorithm is `robustfit`.

## Alternatives

You can also construct a linear model using `fitlm`.

You can construct a model in a range of possible models using `stepwiselm`. However, you cannot use robust regression and stepwise regression together.