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**Class: **GeneralizedLinearModel

Create generalized linear regression model

`GeneralizedLinearModel.fit`

will be removed
in a future release. Use `fitglm`

instead.

`mdl = GeneralizedLinearModel.fit(tbl)`

mdl = GeneralizedLinearModel.fit(X,y)

mdl = GeneralizedLinearModel.fit(...,modelspec)

mdl = GeneralizedLinearModel.fit(...,Name,Value)

mdl =
GeneralizedLinearModel.fit(...,modelspec,Name,Value)

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

= GeneralizedLinearModel.fit(`tbl`

)`tbl`

.

creates
a generalized linear model of the responses `mdl`

= GeneralizedLinearModel.fit(`X`

,`y`

)`y`

to
a data matrix `X`

.

creates
a generalized linear model as specified by `mdl`

= GeneralizedLinearModel.fit(...,`modelspec`

)`modelspec`

.

or `mdl`

= GeneralizedLinearModel.fit(...,`Name,Value`

)

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

=
GeneralizedLinearModel.fit(...,`modelspec`

,`Name,Value`

)`Name,Value`

pair arguments.

`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 and response variables can be numeric, or any grouping variable type, such as logical or categorical (see Grouping 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.

**Data Types: **`single`

| `double`

| `logical`

`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`

.

`modelspec`

— Model specificationcharacter vector specifying the model |

`'Y ~ terms'`

Model specification, which is the starting model for `stepwiseglm`

,
specified as one of the following:

Character vector specifying the type of model.

Character Vector 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 model, where*t*is the number of terms and*p*is the number of predictor variables, and plus one is for the response variable.Character vector representing a formula in the form

where the`'`

,~`Y`

'`terms`

`terms`

are in Wilkinson Notation.

**Example: **`'quadratic'`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside single quotes (`' '`

). You can
specify several name and value pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`'BinomialSize'`

— Number of trials for binomial distribution1 (default) | scalar value | vector

Number of trials for binomial distribution, that is the sample
size, specified as the comma-separated pair consisting of a scalar
value or a vector of the same length as the response. This is the
parameter `n`

for the fitted binomial distribution. `BinomialSize`

applies
only when the `Distribution`

parameter is `'binomial'`

.

If `BinomialSize`

is a scalar value, that means
all observations have the same number of trials.

As an alternative to `BinomialSize`

, you can
specify the response as a two-column vector with counts in column
1 and `BinomialSize`

in column 2.

**Data Types: **`single`

| `double`

`'CategoricalVars'`

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

Categorical variable list, specified as the comma-separated pair consisting of
`'CategoricalVars'`

and either a 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,`GeneralizedLinearModel.fit`

treats all categorical values, logical values, character 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`

| `cell`

`'DispersionFlag'`

— Indicator to compute dispersion parameter`false`

for `'binomial'`

and `'poisson'`

distributions (default) | `true`

Indicator to compute dispersion parameter for `'binomial'`

and `'poisson'`

distributions,
specified as the comma-separated pair consisting of `'DispersionFlag'`

and
one of the following.

`true` | Estimate a dispersion parameter when computing standard errors |

`false` | Default. Use the theoretical value when computing standard errors |

The fitting function always estimates the dispersion for other distributions.

**Example: **`'DispersionFlag',true`

`'Distribution'`

— Distribution of the response variable`'normal'`

(default) | `'binomial'`

| `'poisson'`

| `'gamma'`

| `'inverse gaussian'`

Distribution of the response variable, specified as the comma-separated
pair consisting of `'Distribution'`

and one of the
following.

`'normal'` | Normal distribution |

`'binomial'` | Binomial distribution |

`'poisson'` | Poisson distribution |

`'gamma'` | Gamma distribution |

`'inverse gaussian'` | Inverse Gaussian distribution |

**Example: **`'Distribution','gamma'`

`'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, not a formula or matrix.

**Example: **`'Intercept',false`

`'Link'`

— Link functionThe canonical link function (default) | scalar value | structure

Link function to use in place of the canonical link function,
specified as the comma-separated pair consisting of `'Link'`

and
one of the following.

Link Function Name | Link Function | Mean (Inverse) Function |
---|---|---|

`'identity'` | f(μ) = μ | μ = Xb |

`'log'` | f(μ) = log(μ) | μ = exp(Xb) |

`'logit'` | f(μ) = log(μ/(1–μ)) | μ = exp(Xb) / (1
+ exp(Xb)) |

`'probit'` | f(μ) = Φ^{–1}(μ) | μ = Φ(Xb) |

`'comploglog'` | f(μ) = log(–log(1
– μ)) | μ = 1 – exp(–exp(Xb)) |

`'reciprocal'` | f(μ) = 1/μ | μ = 1/(Xb) |

`p` (a number) | f(μ) = μ^{p} | μ = Xb^{1/p} |

`S` (a structure)with three fields. Each field holds a function handle that accepts a vector of inputs and returns a vector of the same size: `S.Link` — The link function`S.Inverse` — The inverse link function`S.Derivative` — The derivative of the link function
| f(μ) = `S.Link` (μ) | μ = `S.Inverse` (Xb) |

The link function defines the relationship *f*(*μ*)
= *X***b* between the mean response *μ* and
the linear combination of predictors *X***b*.

For more information on the canonical link functions, see `Definitions`

.

**Example: **`'Link','probit'`

`'Offset'`

— Offset variable[ ] (default) |

`vector`

| character vectorOffset variable in the fit, specified as the comma-separated
pair consisting of `'Offset'`

and a vector or name
of a variable with the same length as the response.

`fitglm`

and `stepwiseglm`

use `Offset`

as
an additional predictor, with a coefficient value fixed at 1.0. In
other words, the formula for fitting is

μ` ~ Offset + (terms involving real predictors)`

with the `Offset`

predictor having coefficient `1`

.

For example, consider a Poisson regression model. Suppose the
number of counts is known for theoretical reasons to be proportional
to a predictor `A`

. By using the log link function
and by specifying `log(A)`

as an offset, you can
force the model to satisfy this theoretical constraint.

**Data Types: **`single`

| `double`

| `char`

`'PredictorVars'`

— Predictor variablescell 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
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 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`

| `cell`

`'ResponseVar'`

— Response variablelast column in

`tbl`

(default) | character vector 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 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`

`'VarNames'`

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

(default) | cell array of character vectorsNames of variables in fit, specified as the comma-separated
pair consisting of `'VarNames'`

and a 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: **`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`

— Generalized linear model`GeneralizedLinearModel`

objectGeneralized linear model representing a least-squares fit of
the link of the response to the data, returned as a `GeneralizedLinearModel`

object.

For properties and methods of the generalized linear model object, `mdl`

,
see the `GeneralizedLinearModel`

class
page.

Fit a logistic regression model of probability of smoking as a function of age, weight, and sex, using a two-way interactions model.

Load the `hospital`

dataset array.

load hospital ds = hospital; % just to use the ds name

Specify the model using a formula that allows up to two-way interactions.

```
modelspec = 'Smoker ~ Age*Weight*Sex - Age:Weight:Sex';
```

Create the generalized linear model.

mdl = fitglm(ds,modelspec,'Distribution','binomial')

mdl = Generalized linear regression model: logit(Smoker) ~ 1 + Sex*Age + Sex*Weight + Age*Weight Distribution = Binomial Estimated Coefficients: Estimate SE tStat pValue ___________ _________ ________ _______ (Intercept) -6.0492 19.749 -0.3063 0.75938 Sex_Male -2.2859 12.424 -0.18399 0.85402 Age 0.11691 0.50977 0.22934 0.81861 Weight 0.031109 0.15208 0.20455 0.83792 Sex_Male:Age 0.020734 0.20681 0.10025 0.92014 Sex_Male:Weight 0.01216 0.053168 0.22871 0.8191 Age:Weight -0.00071959 0.0038964 -0.18468 0.85348 100 observations, 93 error degrees of freedom Dispersion: 1 Chi^2-statistic vs. constant model: 5.07, p-value = 0.535

The large -value indicates the model might not differ statistically from a constant.

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.Load the sample data and define the dataset array.

load hospital dsa = dataset(hospital.Sex,hospital.BloodPressure(:,1),hospital.Age,... hospital.Smoker,'VarNames',{'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 dataset array, so there must be a column of 0s for the response variable in the second column of the terms matrix.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 dataset array.

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

Now, the response variable is the first term in the dataset array. 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 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`

.

The default link function for a generalized linear model is
the *canonical link function*.

**Canonical Link Functions for Generalized Linear Models**

Distribution | Link Function Name | Link Function | Mean (Inverse) Function |
---|---|---|---|

`'normal'` | `'identity'` | f(μ) = μ | μ = Xb |

`'binomial'` | `'logit'` | f(μ) = log(μ/(1–μ)) | μ = exp(Xb) / (1
+ exp(Xb)) |

`'poisson'` | `'log'` | f(μ) = log(μ) | μ = exp(Xb) |

`'gamma'` | `-1` | f(μ) = 1/μ | μ = 1/(Xb) |

`'inverse gaussian'` | `-2` | f(μ) = 1/μ^{2} | μ = (Xb)^{–1/2} |

The generalized linear model

`mdl`

is a standard linear model unless you specify otherwise with the`Distribution`

name-value pair.For other methods such as

`devianceTest`

, or properties of the`GeneralizedLinearModel`

object, see`GeneralizedLinearModel`

.

You can also construct a generalized linear model using `fitglm`

.

Use `stepwiseglm`

to select
a model specification automatically. Use `step`

, `addTerms`

,
or `removeTerms`

to adjust a fitted model.

[1] Collett, D. *Modeling Binary
Data*. New York: Chapman & Hall, 2002.

[2] Dobson, A. J. *An Introduction
to Generalized Linear Models*. New York: Chapman &
Hall, 1990.

[3] McCullagh, P., and J. A. Nelder. *Generalized
Linear Models*. New York: Chapman & Hall, 1990.

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