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# fitglm

Create generalized linear regression model

## Syntax

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

## Description

example

````mdl = fitglm(tbl)` returns a generalized linear model fit to variables in the table or dataset array `tbl`. By default, `fitglm` takes the last variable as the response variable.```

example

````mdl = fitglm(X,y)` returns a generalized linear model of the responses `y`, fit to the data matrix `X`.```

example

````mdl = fitglm(___,modelspec)` returns a generalized linear model of the type you specify in `modelspec`.```

example

````mdl = fitglm(___,Name,Value)` returns a generalized linear model with additional options specified by one or more `Name,Value` pair arguments.For example, you can specify which variables are categorical, the distribution of the response variable, and the link function to use.```

## Examples

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Make a logistic binomial model of the probability of smoking as a function of age, weight, and sex, using a two-way interactions model.

Load the `hospital` dataset array.

```load hospital dsa = hospital; ```

Specify the model using a formula that allows up to two-way interactions between the variables age, weight, and sex. Smoker is the response variable.

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

Fit a logistic binomial model.

```mdl = fitglm(dsa,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 ```

All of the p-values (under pValue) are large. This means none of the coefficients are significant. The large -value for the test of the model, 0.535, indicates that this model might not differ statistically from a constant model.

Create sample data with 20 predictors, and Poisson response using just three of the predictors, plus a constant.

```rng('default') % for reproducibility X = randn(100,7); mu = exp(X(:,[1 3 6])*[.4;.2;.3] + 1); y = poissrnd(mu); ```

Fit a generalized linear model using the Poisson distribution.

```mdl = fitglm(X,y,'linear','Distribution','poisson') ```
```mdl = Generalized linear regression model: log(y) ~ 1 + x1 + x2 + x3 + x4 + x5 + x6 + x7 Distribution = Poisson Estimated Coefficients: Estimate SE tStat pValue _________ ________ ________ __________ (Intercept) 0.88723 0.070969 12.502 7.3149e-36 x1 0.44413 0.052337 8.4858 2.1416e-17 x2 0.0083388 0.056527 0.14752 0.88272 x3 0.21518 0.063416 3.3932 0.00069087 x4 -0.058386 0.065503 -0.89135 0.37274 x5 -0.060824 0.073441 -0.8282 0.40756 x6 0.34267 0.056778 6.0352 1.5878e-09 x7 0.04316 0.06146 0.70225 0.48252 100 observations, 92 error degrees of freedom Dispersion: 1 Chi^2-statistic vs. constant model: 119, p-value = 1.55e-22 ```

The `p`-values of 2.14e-17, 0.00069, and 1.58e-09 indicate that the coefficients of the variables `x1`, `x3`, and |x6|are statistically significant.

## Input Arguments

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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`

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`

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`.

Model specification, which is the starting model for `stepwiseglm`, specified as one of the following:

• Character vector specifying the type of model.

Character VectorModel 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.
`'polyijk'`Model is a polynomial with all terms up to degree `i` in the first predictor, degree `j` in the second predictor, etc. Use numerals `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

`'Y ~ terms'`,

where the `terms` are in Wilkinson Notation.

Example: `'quadratic'`

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

Example: `'Distribution','normal','link','probit','Exclude',[23,59]` specifies that the distribution of the response is normal, and instructs `fitglm` to use the probit link function and exclude the 23rd and 59th observations from the fit.

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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`

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, `fitglm` 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`

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 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'`

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, not a formula or matrix.

Example: `'Intercept',false`

Offset 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`

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`

Response 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`

Names 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`

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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Generalized 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.

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### Terms Matrix

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)```
The `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.

### Formula

A formula for model specification is a character vector of the form `'Y ~ terms'`

where

• `Y` is the response name.

• `terms` contains

• 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

### Note

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

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 NotationFactors in Standard Notation
`1`Constant (intercept) term
`A^k`, where `k` is a positive integer`A`, `A2`, ..., `Ak`
`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

`'normal'``'identity'`f(μ) = μμ = 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/μμ = 1/(Xb)
`'inverse gaussian'``-2`f(μ) = 1/μ2μ = (Xb)–1/2

## Tips

• The generalized linear model `mdl` is a standard linear model unless you specify otherwise with the `Distribution` name-value pair.

• For methods such as `plotResiduals` or `devianceTest`, or properties of the `GeneralizedLinearModel` object, see `GeneralizedLinearModel`.

• After training a `GeneralizedLinearModel` model object by using `fitglm`, use the function `predict` and the trained model object to generate C code that predicts responses for new data. For details, see Code Generation.

## Alternative Functionality

Use `stepwiseglm` to select a model specification automatically. Use `step`, `addTerms`, or `removeTerms` to adjust a fitted model.

## References

[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.