Documentation

# GeneralizedLinearModel.fit

Class: GeneralizedLinearModel

(Not Recommended) Create generalized linear regression model

`GeneralizedLinearModel.fit` is not recommended. Use `fitglm` instead.

## Syntax

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

## Description

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

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

`mdl = GeneralizedLinearModel.fit(...,modelspec)` creates a generalized linear model as specified by `modelspec`.

`mdl = GeneralizedLinearModel.fit(...,Name,Value)` or ```mdl = GeneralizedLinearModel.fit(...,modelspec,Name,Value)``` creates a generalized linear model with additional options specified by one or more `Name,Value` pair arguments.

## Input Arguments

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Input data including predictor and response variables, specified as a table or dataset array. The predictor variables and response variable can be numeric, logical, categorical, character, or string. The response variable can have a data type other than numeric only if `'Distribution'` is `'binomial'`.

• By default, `GeneralizedLinearModel.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 `stepwiseglm` function with the name-value pair arguments `'Lower'` and `'Upper'`, respectively.

You can verify the variable names in `tbl` by using the `isvarname` function. The following code returns logical `1` (`true`) for each variable that has a valid variable name.

`cellfun(@isvarname,tbl.Properties.VariableNames)`
If the variable names in `tbl` are not valid, then convert them by using the `matlab.lang.makeValidName` function.
`tbl.Properties.VariableNames = matlab.lang.makeValidName(tbl.Properties.VariableNames);`

Predictor variables, specified as an n-by-p matrix, where n is the number of observations and p is the number of predictor variables. Each column of `X` represents one variable, and each row represents one observation.

By default, there is a constant term in the model, unless you explicitly remove it, so do not include a column of 1s in `X`.

Data Types: `single` | `double`

Response variable, specified as a vector or matrix.

• If `'Distribution'` is not `'binomial'`, then `y` must be 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`. The data type must be single or double.

• If `'Distribution'` is `'binomial'`, then `y` can be an n-by-1 vector or n-by-2 matrix with counts in column 1 and `BinomialSize` in column 2.

Data Types: `single` | `double` | `logical` | `categorical`

Model specification, specified as one of the following:

• Character vector or string scalar specifying the type of model.

ValueModel Type
`'constant'`Model contains only a constant (intercept) term.
`'linear'`Model contains an intercept and linear term for each predictor.
`'interactions'`Model contains an intercept, linear term for each predictor, and all products of pairs of distinct predictors (no squared terms).
`'purequadratic'`Model contains an intercept term and linear and squared terms for each predictor.
`'quadratic'`Model contains an intercept term, linear and squared terms for each predictor, and all products of pairs of distinct predictors.
`'polyijk'`Model is a polynomial with all terms up to degree `i` in the first predictor, degree `j` in the second predictor, and so on. Specify the maximum degree for each predictor by using numerals 0 though 9. The model contains interaction terms, but the degree of each interaction term does not exceed the maximum value of the specified degrees. For example, `'poly13'` has an intercept and x1, x2, x22, x23, x1*x2, and x1*x22 terms, where x1 and x2 are the first and second predictors, respectively.
• t-by-(p+1) matrix, namely terms matrix, specifying terms to include in 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 or string scalar 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 quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Number of trials for binomial distribution, that is the sample size, specified as the comma-separated pair consisting of `'BinomialSize'` and the variable name in `tbl`, a numeric scalar, or a numeric 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 matrix with counts in column 1 and `BinomialSize` in column 2.

Data Types: `single` | `double` | `char` | `string`

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, `GeneralizedLinearModel.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`

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 or string scalar, 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.

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

f(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` | `string`

Predictor variables to use in the fit, specified as the comma-separated pair consisting of `'PredictorVars'` and either a string array or cell array of character vectors of the variable names in the table or dataset array `tbl`, or a logical or numeric index vector indicating which columns are predictor variables.

The string values or character vectors should be among the names in `tbl`, or the names you specify using the `'VarNames'` name-value pair argument.

The default is all variables in `X`, or all variables in `tbl` except for `ResponseVar`.

For example, you can specify the second and third variables as the predictor variables using either of the following examples.

Example: `'PredictorVars',[2,3]`

Example: `'PredictorVars',logical([0 1 1 0 0 0])`

Data Types: `single` | `double` | `logical` | `string` | `cell`

Response variable to use in the fit, specified as the comma-separated pair consisting of `'ResponseVar'` and either a character vector or string scalar containing the variable name in the table or dataset array `tbl`, or a logical or numeric index vector indicating which column is the response variable. You typically need to use `'ResponseVar'` when fitting a table or dataset array `tbl`.

For example, you can specify the fourth variable, say `yield`, as the response out of six variables, in one of the following ways.

Example: `'ResponseVar','yield'`

Example: `'ResponseVar',[4]`

Example: `'ResponseVar',logical([0 0 0 1 0 0])`

Data Types: `single` | `double` | `logical` | `char` | `string`

Names of variables, specified as the comma-separated pair consisting of `'VarNames'` and a string array or cell array of character vectors including the names for the columns of `X` first, and the name for the response variable `y` last.

`'VarNames'` is not applicable to variables in a table or dataset array, because those variables already have names.

The variable names do not have to be valid MATLAB identifiers. However, if the names are not valid, you cannot use a formula when you fit or adjust a model; for example:

Before specifying `'VarNames',varNames`, you can verify the variable names in `varNames` by using the `isvarname` function. The following code returns logical `1` (`true`) for each variable that has a valid variable name.

`cellfun(@isvarname,varNames)`
If the variable names in `varNames` are not valid, then convert them by using the `matlab.lang.makeValidName` function.
`varNames = matlab.lang.makeValidName(varNames);`

Example: `'VarNames',{'Horsepower','Acceleration','Model_Year','MPG'}`

Data Types: `string` | `cell`

Observation weights, specified as the comma-separated pair consisting of `'Weights'` and an n-by-1 vector of nonnegative scalar values, where n is the number of observations.

Data Types: `single` | `double`

## Output Arguments

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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.

## Examples

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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 $p$-value indicates the model might not differ statistically from a constant.

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## Tips

• 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`.

## Alternatives

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.

## 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.