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.

### 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:

• 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.

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.

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.