(Not Recommended) Create linear regression model

`LinearModel.fit`

is not recommended. Use `fitlm`

instead.

`mdl = LinearModel.fit(tbl)`

mdl = LinearModel.fit(X,y)

mdl = LinearModel.fit(___,modelspec)

mdl = LinearModel.fit(___,Name,Value)

mdl =
LinearModel.fit(___,modelspec,Name,Value)

creates a linear model of a table or dataset array `mdl`

= LinearModel.fit(`tbl`

)`tbl`

.

creates a linear model of the responses `mdl`

= LinearModel.fit(`X`

,`y`

)`y`

to a data matrix
`X`

.

creates a linear model of the type specified by `mdl`

= LinearModel.fit(___,`modelspec`

)`modelspec`

, using
any of the previous syntaxes.

or
`mdl`

= LinearModel.fit(___,`Name,Value`

)

creates a linear model with additional options specified by one or more
`mdl`

=
LinearModel.fit(___,`modelspec`

,`Name,Value`

)`Name,Value`

pair arguments. For example, you can specify which
predictor variables to include in the fit or include observation weights.

`tbl`

— Input datatable | dataset array

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)

`tbl`

are not valid, then convert them by using the `matlab.lang.makeValidName`

function.tbl.Properties.VariableNames = matlab.lang.makeValidName(tbl.Properties.VariableNames);

`X`

— Predictor variablesmatrix

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`

`y`

— Response variablevector

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`

`modelspec`

— Model specification`'linear'`

(default) | character vector or string scalar naming the model | ```
'Y ~
terms'
```

Model specification, specified as one of the following.

A character vector or string scalar naming the model.

Value Model 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. `'poly`

'`ijk`

Model is a polynomial with all terms up to degree in the first predictor, degree`i`

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,`j`

`'poly13'`

has an intercept and*x*_{1},*x*_{2},*x*_{2}^{2},*x*_{2}^{3},*x*_{1}**x*_{2}, and*x*_{1}**x*_{2}^{2}terms, where*x*_{1}and*x*_{2}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

where the`'Y ~ terms'`

,`terms`

are specified using Wilkinson Notation.

**Example: **`'quadratic'`

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

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`

.

`'CategoricalVars'`

— Categorical variable liststring array | cell array of character vectors | logical or numeric index vector

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`

`'Exclude'`

— Observations to excludelogical or numeric index vector

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`

`'Intercept'`

— Indicator for constant term`true`

(default) | `false`

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`

`'PredictorVars'`

— Predictor variablesstring array | cell array of character vectors | logical or numeric index vector

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`

`'ResponseVar'`

— Response variablelast column in

`tbl`

(default) | character vector or string scalar containing variable name | logical or numeric index vectorResponse 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`

`'RobustOpts'`

— Indicator of robust fitting type`'off'`

(default) | `'on'`

| character vector | string scalar | structureIndicator 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 Function Description Default 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 handle Custom 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'`

`'VarNames'`

— Names of variables`{'x1','x2',...,'xn','y'}`

(default) | string array | cell array of character vectorsNames 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:

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.

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)

`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`

`'Weights'`

— Observation weights`ones(n,1)`

(default) | 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`

`mdl`

— Linear model`LinearModel`

objectLinear 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.

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 $\mathit{y}={\beta}_{0}+{\beta}_{1}{\mathit{X}}_{1}+{\beta}_{2}{\mathit{X}}_{2}+{\beta}_{3}{\mathit{X}}_{3}+\u03f5$.

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

$\mathrm{MPG}={\beta}_{0}+{\beta}_{1}{{\rm I}}_{\mathrm{Year}=76}+{\beta}_{2}{{\rm I}}_{\mathrm{Year}=82}+\u03f5$,

where ${{\rm I}}_{\mathrm{Year}=76}$ and ${{\rm I}}_{\mathrm{Year}=82}$ 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 ${{\rm I}}_{\mathrm{Year}=76}$ and ${{\rm I}}_{\mathrm{Year}=82}$. 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:

$\mathit{y}={\beta}_{0}{{\rm I}}_{{\mathit{x}}_{1}=70}+\left({\beta}_{0}+{\beta}_{1}\right){{\rm I}}_{{\mathit{x}}_{1}=76}+\left({{\beta}_{0}+\beta}_{2}\right){{\rm I}}_{{\mathit{x}}_{2}=82}+\u03f5$.

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*
{'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*
{'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 ${{\rm I}}_{\mathrm{Year}=70}$ and ${{\rm I}}_{\mathrm{Year}=82}$.

**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 sample data.

`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.

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 `A`

,
`B`

, and `C`

and the response variable
`Y`

in the order `A`

, `B`

,
`C`

, and `Y`

. Each row of `T`

represents one term:

`[0 0 0 0]`

— Constant term or intercept`[0 1 0 0]`

—`B`

; equivalently,`A^0 * B^1 * C^0`

`[1 0 1 0]`

—`A*C`

`[2 0 0 0]`

—`A^2`

`[0 1 2 0]`

—`B*(C^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.

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

.* Y* ~

`terms`

is the response name.`Y`

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

For example:

`'Y ~ A + B + C'`

specifies a three-variable linear model with intercept.`'Y ~ A + B + C – 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.

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 Notation | Term in Standard Notation |
---|---|

`1` | Constant (intercept) term |

`A^k` , where `k` is a positive
integer | `A` ,
`A` , ...,
`A` |

`A + B` | `A` , `B` |

`A*B` | `A` , `B` ,
`A*B` |

`A:B` | `A*B` only |

`–B` | Do not include `B` |

`A*B + C` | `A` , `B` , `C` ,
`A*B` |

`A + B + C + A:B` | `A` , `B` , `C` ,
`A*B` |

`A*B*C – A:B:C` | `A` , `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.

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`

.

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

.

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.

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