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**Class: **LinearModel

Create linear regression model

`LinearModel.fit`

will be removed in a future
release. 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, specified as a table or dataset array. When `modelspec`

is
a `formula`

, it specifies the variables to be used
as the predictors and response. Otherwise, if you do not specify the
predictor and response variables, the last variable is the response
variable and the others are the predictor variables by default.

Predictor variables can be numeric, or any grouping variable type, such as logical or categorical (see Grouping Variables). The response must be numeric or logical.

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.

`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`

| `logical`

`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 specificationcharacter 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 terms for each predictor. `'interactions'`

Model contains an intercept, linear terms, and all products of pairs of distinct predictors (no squared terms). `'purequadratic'`

Model contains an intercept, linear terms, and squared terms. `'quadratic'`

Model contains an intercept, linear terms, interactions, and squared terms. `'poly`

'`ijk`

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

in the second predictor, etc. Use numerals`j`

`0`

through`9`

. For example,`'poly2111'`

has a constant plus all linear and product terms, and also contains terms with predictor 1 squared.*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 | structure with character vector, string scalar, or function handleIndicator of the robust fitting type to use, specified as the
comma-separated pair consisting of `'RobustOpts'`

and
one of the following.

`'off'`

— No robust fitting.`fitlm`

uses ordinary least squares.`'on'`

— Robust fitting. When you use robust fitting,`'bisquare'`

weight function is the default.Character vector or string scalar — Name of the robust fitting weight function from the following table.

`fitlm`

uses the corresponding default tuning constant in the table.Structure with the character vector or string scalar

`RobustWgtFun`

containing the name of the robust fitting weight function from the following table and optional scalar`Tune`

fields —`fitlm`

uses the`RobustWgtFun`

weight function and`Tune`

tuning constant from the structure. You can choose the name of the robust fitting weight function from this table. If you do not supply a`Tune`

field, the fitting function uses the corresponding default tuning constant.Weight Function Equation Default Tuning Constant `'andrews'`

`w = (abs(r)<pi) .* sin(r) ./ r`

1.339 `'bisquare'`

(default)`w = (abs(r)<1) .* (1 - r.^2).^2`

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 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,`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 there are

*p*columns in`X`

, the smallest*p*absolute deviations are excluded when computing the median.Default tuning constants 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.

Structure with the function handle

`RobustWgtFun`

and optional scalar`Tune`

fields — You can specify a custom weight function.`fitlm`

uses the`RobustWgtFun`

weight function and`Tune`

tuning constant from the structure. Specify`RobustWgtFun`

as a function handle that accepts a vector of residuals, and returns a vector of weights the same size. The fitting function scales the residuals, dividing by the tuning constant (default`1`

) and by an estimate of the error standard deviation before it calls the weight function.

**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.

For example, if in your data, horsepower, acceleration, and model year of the cars are the predictor variables, and miles per gallon (MPG) is the response variable, then you can name the variables as follows.

**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 model of the Hald data.

Load the data.

load hald X = ingredients; % Predictor variables y = heat; % Response

Fit a default linear 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

Fit a model of a table that contains a categorical predictor.

Load the `carsmall`

data.

`load carsmall`

Construct a table containing continuous predictor variable `Weight`

, categorical predictor variable `Year`

, and response variable `MPG`

.

tbl = table(MPG,Weight); tbl.Year = categorical(Model_Year);

Create a fitted model of `MPG`

as a function of `Year`

, `Weight`

, and `Weight^2`

. You don't have to include `Weight`

explicitly in your formula because it is a lower-order term of `Weight^2`

. The `fitlm`

function includes `Weight`

automatically.

`mdl = fitlm(tbl,'MPG ~ Year + Weight^2')`

mdl = Linear regression model: MPG ~ 1 + Weight + Year + Weight^2 Estimated Coefficients: Estimate SE tStat pValue __________ __________ _______ __________ (Intercept) 54.206 4.7117 11.505 2.6648e-19 Weight -0.016404 0.0031249 -5.2493 1.0283e-06 Year_76 2.0887 0.71491 2.9215 0.0044137 Year_82 8.1864 0.81531 10.041 2.6364e-16 Weight^2 1.5573e-06 4.9454e-07 3.149 0.0022303 Number of observations: 94, Error degrees of freedom: 89 Root Mean Squared Error: 2.78 R-squared: 0.885, Adjusted R-Squared 0.88 F-statistic vs. constant model: 172, p-value = 5.52e-41

`fitlm`

creates two dummy (indicator) variables for the categorical variate, `Year`

. The dummy variable `Year_76`

takes the value 1 if model year is 1976 and takes the value 0 if it is not. The dummy variable `Year_82`

takes the value 1 if model year is 1982 and takes the value 0 if it is not. And the year 1970 is the reference year. The corresponding model is

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.

Fit a linear regression model using a robust fitting method.

Load the sample data.

`load hald`

The `hald`

data measures the effect of cement composition on its hardening heat. The matrix `ingredients`

contains the percent composition of four chemicals present in the cement. The array `heat`

contains the heat of hardening after 180 days for each cement sample.

Fit a robust linear model to the data.

mdl = fitlm(ingredients,heat,'linear','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

A terms matrix 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 plus one is for the response variable.

The value of `T(i,j)`

is the exponent of variable `j`

in
term `i`

. Suppose there are three predictor variables `A`

, `B`

,
and `C`

:

[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)

`0`

at
the end of each term represents the response variable. In general,
If you have the variables in a table or dataset array, then

`0`

must represent the response variable depending on the position of the response variable. The following example illustrates this using a table.Load the sample data and define a table.

load hospital dsa = table(hospital.Sex,hospital.BloodPressure(:,1), ... hospital.Age,hospital.Smoker,'VariableNames', ... {'Sex','BloodPressure','Age','Smoker'});

Represent the linear model

`'BloodPressure ~ 1 + Sex + Age + Smoker'`

in a terms matrix. The response variable is in the second column of the table, so the second column of the terms matrix must be a column of 0s for the response variable.T = [0 0 0 0;1 0 0 0;0 0 1 0;0 0 0 1]

T = 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1

Redefine the table.

dsa = table(hospital.BloodPressure(:,1),hospital.Sex, ... hospital.Age,hospital.Smoker,'VariableNames', ... {'BloodPressure','Sex','Age','Smoker'});

Now, the response variable is the first term in the table. Specify the same linear model,

`'BloodPressure ~ 1 + Sex + Age + Smoker'`

, using a terms matrix.T = [0 0 0 0;0 1 0 0;0 0 1 0;0 0 0 1]

T = 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

If you have the predictor and response variables in a matrix and column vector, then you must include

`0`

for the response variable at the end of each term. The following example illustrates this.Load the sample data and define the matrix of predictors.

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

Specify the model

`'MPG ~ Acceleration + Weight + Acceleration:Weight + Weight^2'`

using a term matrix and fit the model to the data. This model includes the main effect and two-way interaction terms for the variables,`Acceleration`

and`Weight`

, and a second-order term for the variable,`Weight`

.T = [0 0 0;1 0 0;0 1 0;1 1 0;0 2 0]

T = 0 0 0 1 0 0 0 1 0 1 1 0 0 2 0

Fit a linear model.

mdl = fitlm(X,MPG,T)

mdl = Linear regression model: y ~ 1 + x1*x2 + x2^2 Estimated Coefficients: Estimate SE tStat pValue (Intercept) 48.906 12.589 3.8847 0.00019665 x1 0.54418 0.57125 0.95261 0.34337 x2 -0.012781 0.0060312 -2.1192 0.036857 x1:x2 -0.00010892 0.00017925 -0.6076 0.545 x2^2 9.7518e-07 7.5389e-07 1.2935 0.19917 Number of observations: 94, Error degrees of freedom: 89 Root Mean Squared Error: 4.1 R-squared: 0.751, Adjusted R-Squared 0.739 F-statistic vs. constant model: 67, p-value = 4.99e-26

Only the intercept and

`x2`

term, which correspond to the`Weight`

variable, are significant at the 5% significance level.Now, perform a stepwise regression with a constant model as the starting model and a linear model with interactions as the upper model.

`T = [0 0 0;1 0 0;0 1 0;1 1 0]; mdl = stepwiselm(X,MPG,[0 0 0],'upper',T)`

1. Adding x2, FStat = 259.3087, pValue = 1.643351e-28 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

The results of the stepwise regression are consistent with the results of

`fitlm`

in the previous step.

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

* Y* ~

`terms`

where

is the response name.`Y`

contains`terms`

Variable names

`+`

means include the next variable`-`

means do not include the next variable`:`

defines an interaction, 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

Formulas include a constant (intercept) term by default. To
exclude a constant term from the model, include `-1`

in
the formula.

For example,

`'Y ~ A + B + C'`

means a three-variable
linear model with intercept.

```
'Y ~ A + B +
C - 1'
```

is a three-variable linear model without intercept.

`'Y ~ A + B + C + B^2'`

is a three-variable
model with intercept and a `B^2`

term.

```
'Y
~ A + B^2 + C'
```

is the same as the previous example because `B^2`

includes
a `B`

term.

```
'Y ~ A + B +
C + A:B'
```

includes an `A*B`

term.

```
'Y
~ A*B + C'
```

is the same as the previous example because ```
A*B
= A + B + A:B
```

.

`'Y ~ A*B*C - A:B:C'`

has
all interactions among `A`

, `B`

,
and `C`

, except the three-way interaction.

```
'Y
~ A*(B + C + D)'
```

has all linear terms, plus products of `A`

with
each of the other variables.

Wilkinson notation describes the factors present in models. The notation relates to factors present in models, not to the multipliers (coefficients) of those factors.

Wilkinson Notation | Factors 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`

.

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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