GeneralizedLinearModel

Generalized linear regression model class

Description

GeneralizedLinearModel is a fitted generalized linear regression model. A generalized linear regression model is a special class of nonlinear models that describe a nonlinear relationship between a response and predictors. A generalized linear regression model has generalized characteristics of a linear regression model. The response variable follows a normal, binomial, Poisson, gamma, or inverse Gaussian distribution with parameters including the mean response μ. A link function f defines the relationship between μ and the linear combination of predictors.

Use the properties of a GeneralizedLinearModel object to investigate a fitted generalized linear regression model. The object properties include information about coefficient estimates, summary statistics, fitting method, and input data. Use the object functions to predict responses and to modify, evaluate, and visualize the model.

Creation

Create a GeneralizedLinearModel object by using fitglm or stepwiseglm.

fitglm fits a generalized linear regression model to data using a fixed model specification. Use addTerms, removeTerms, or step to add or remove terms from the model. Alternatively, use stepwiseglm to fit a model using stepwise generalized linear regression.

Properties

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Coefficient Estimates

`CoefficientCovariance` — Covariance matrix of coefficient estimates
numeric matrix

This property is read-only.

Covariance matrix of coefficient estimates, specified as a p-by-p matrix of numeric values. p is the number of coefficients in the fitted model.

For details, see Coefficient Standard Errors and Confidence Intervals.

Data Types: single | double

`CoefficientNames` — Coefficient names
cell array of character vectors

This property is read-only.

Coefficient names, specified as a cell array of character vectors, each containing the name of the corresponding term.

Data Types: cell

`Coefficients` — Coefficient values
table

This property is read-only.

Coefficient values, specified as a table. Coefficients contains one row for each coefficient and these columns:

Estimate — Estimated coefficient value
SE — Standard error of the estimate
tStat — t-statistic for a test that the coefficient is zero
pValue — p-value for the t-statistic

Use coefTest to perform linear hypothesis tests on the coefficients. Use coefCI to find the confidence intervals of the coefficient estimates.

To obtain any of these columns as a vector, index into the property using dot notation. For example, obtain the estimated coefficient vector in the model mdl:

beta = mdl.Coefficients.Estimate

Data Types: table

`NumCoefficients` — Number of model coefficients
positive integer

This property is read-only.

Number of model coefficients, specified as a positive integer. NumCoefficients includes coefficients that are set to zero when the model terms are rank deficient.

Data Types: double

`NumEstimatedCoefficients` — Number of estimated coefficients
positive integer

This property is read-only.

Number of estimated coefficients in the model, specified as a positive integer. NumEstimatedCoefficients does not include coefficients that are set to zero when the model terms are rank deficient. NumEstimatedCoefficients is the degrees of freedom for regression.

Data Types: double

Summary Statistics

`Deviance` — Deviance of fit
numeric value

This property is read-only.

Deviance of the fit, specified as a numeric value. The deviance is useful for comparing two models when one model is a special case of the other model. The difference between the deviance of the two models has a chi-square distribution with degrees of freedom equal to the difference in the number of estimated parameters between the two models. For more information, see Deviance.

Data Types: single | double

`DFE` — Degrees of freedom for error
positive integer

This property is read-only.

Degrees of freedom for the error (residuals), equal to the number of observations minus the number of estimated coefficients, specified as a positive integer.

Data Types: double

`Diagnostics` — Observation diagnostics
table

This property is read-only.

Observation diagnostics, specified as a table that contains one row for each observation and the columns described in this table.

Column	Meaning	Description
`Leverage`	Diagonal elements of `HatMatrix`	`Leverage` for each observation indicates to what extent the fit is determined by the observed predictor values. A value close to `1` indicates that the fit is largely determined by that observation, with little contribution from the other observations. A value close to `0` indicates that the fit is largely determined by the other observations. For a model with `P` coefficients and `N` observations, the average value of `Leverage` is `P/N`. A `Leverage` value greater than `2*P/N` indicates high leverage.
`CooksDistance`	Cook's distance of scaled change in fitted values	`CooksDistance` is a measure of scaled change in fitted values. An observation with `CooksDistance` greater than three times the mean Cook's distance can be an outlier.
`HatMatrix`	Projection matrix to compute fitted from observed responses	`HatMatrix` is an `N`-by-`N` matrix such that `Fitted = HatMatrix*Y`, where `Y` is the response vector and `Fitted` is the vector of fitted response values.

The software computes these values on the scale of the linear combination of the predictors, stored in the LinearPredictor field of the Fitted and Residuals properties. For example, the software computes the diagnostic values by using the fitted response and adjusted response values from the model mdl.

Yfit = mdl.Fitted.LinearPredictor
Yadjusted = mdl.Fitted.LinearPredictor + mdl.Residuals.LinearPredictor

Diagnostics contains information that is helpful in finding outliers and influential observations. For more details, see Leverage, Cook’s Distance, and Hat Matrix.

Use plotDiagnostics to plot observation diagnostics.

Rows not used in the fit because of missing values (in ObservationInfo.Missing) or excluded values (in ObservationInfo.Excluded) contain NaN values in the CooksDistance column and zeros in the Leverage and HatMatrix columns.

To obtain any of these columns as an array, index into the property using dot notation. For example, obtain the hat matrix in the model mdl:

HatMatrix = mdl.Diagnostics.HatMatrix;

Data Types: table

`Dispersion` — Scale factor of variance of response
numeric scalar

This property is read-only.

Scale factor of the variance of the response, specified as a numeric scalar.

If the 'DispersionFlag' name-value pair argument of fitglm or stepwiseglm is true, then the function estimates the Dispersion scale factor in computing the variance of the response. The variance of the response equals the theoretical variance multiplied by the scale factor. For example, the variance function for the binomial distribution is p(1–p)/n, where p is the probability parameter and n is the sample size parameter. If Dispersion is near 1, the variance of the data appears to agree with the theoretical variance of the binomial distribution. If Dispersion is larger than 1, the data set is “overdispersed” relative to the binomial distribution.

Data Types: double

`DispersionEstimated` — Flag to indicate use of dispersion scale factor
logical value

This property is read-only.

Flag to indicate whether fitglm used the Dispersion scale factor to compute standard errors for the coefficients in Coefficients.SE, specified as a logical value. If DispersionEstimated is false, fitglm used the theoretical value of the variance.

DispersionEstimated can be false only for the binomial and Poisson distributions.
Set DispersionEstimated by setting the 'DispersionFlag' name-value pair argument of fitglm or stepwiseglm.

Data Types: logical

`Fitted` — Fitted response values based on input data
table

This property is read-only.

Fitted (predicted) values based on the input data, specified as a table that contains one row for each observation and the columns described in this table.

Column	Description
`Response`	Predicted values on the scale of the response
`LinearPredictor`	Predicted values on the scale of the linear combination of the predictors (same as the link function applied to the `Response` fitted values)
`Probability`	Fitted probabilities (included only with the binomial distribution)

To obtain any of these columns as a vector, index into the property using dot notation. For example, obtain the vector f of fitted values on the response scale in the model mdl:

f = mdl.Fitted.Response

Use predict to compute predictions for other predictor values, or to compute confidence bounds on Fitted.

Data Types: table

`LogLikelihood` — Loglikelihood
numeric value

This property is read-only.

Loglikelihood of the model distribution at the response values, specified as a numeric value. The mean is fitted from the model, and other parameters are estimated as part of the model fit.

Data Types: single | double

`ModelCriterion` — Criterion for model comparison
structure

This property is read-only.

Criterion for model comparison, specified as a structure with these fields:

AIC — Akaike information criterion. AIC = –2*logL + 2*m, where logL is the loglikelihood and m is the number of estimated parameters.
AICc — Akaike information criterion corrected for the sample size. AICc = AIC + (2*m*(m + 1))/(n – m – 1), where n is the number of observations.
BIC — Bayesian information criterion. BIC = –2*logL + m*log(n).
CAIC — Consistent Akaike information criterion. CAIC = –2*logL + m*(log(n) + 1).

Information criteria are model selection tools that you can use to compare multiple models fit to the same data. These criteria are likelihood-based measures of model fit that include a penalty for complexity (specifically, the number of parameters). Different information criteria are distinguished by the form of the penalty.

When you compare multiple models, the model with the lowest information criterion value is the best-fitting model. The best-fitting model can vary depending on the criterion used for model comparison.

To obtain any of the criterion values as a scalar, index into the property using dot notation. For example, obtain the AIC value aic in the model mdl:

aic = mdl.ModelCriterion.AIC

Data Types: struct

`Residuals` — Residuals for fitted model
table

This property is read-only.

Residuals for the fitted model, specified as a table that contains one row for each observation and the columns described in this table.

Column	Description
`Raw`	Observed minus fitted values
`LinearPredictor`	Residuals on the linear predictor scale, equal to the adjusted response value minus the fitted linear combination of the predictors
`Pearson`	Raw residuals divided by the estimated standard deviation of the response
`Anscombe`	Residuals defined on transformed data with the transformation selected to remove skewness
`Deviance`	Residuals based on the contribution of each observation to the deviance

Rows not used in the fit because of missing values (in ObservationInfo.Missing) contain NaN values.

To obtain any of these columns as a vector, index into the property using dot notation. For example, obtain the ordinary raw residual vector r in the model mdl:

r = mdl.Residuals.Raw

Data Types: table

`Rsquared` — R-squared value for model
structure

This property is read-only.

R-squared value for the model, specified as a structure with five fields:

Ordinary — Ordinary (unadjusted) R-squared
Adjusted — R-squared adjusted for the number of coefficients
LLR — Loglikelihood ratio
Deviance — Deviance
AdjGeneralized — Adjusted generalized R-squared

The R-squared value is the proportion of the total sum of squares explained by the model. The ordinary R-squared value relates to the SSR and SST properties:

Rsquared = SSR/SST

To obtain any of these values as a scalar, index into the property using dot notation. For example, obtain the adjusted R-squared value in the model mdl:

r2 = mdl.Rsquared.Adjusted

Data Types: struct

`SSE` — Sum of squared errors
numeric value

This property is read-only.

Sum of squared errors (residuals), specified as a numeric value.

Data Types: single | double

`SSR` — Regression sum of squares
numeric value

This property is read-only.

Regression sum of squares, specified as a numeric value. The regression sum of squares is equal to the sum of squared deviations of the fitted values from their mean.

Data Types: single | double

`SST` — Total sum of squares
numeric value

This property is read-only.

Total sum of squares, specified as a numeric value. The total sum of squares is equal to the sum of squared deviations of the response vector y from the mean(y).

Data Types: single | double

Fitting Information

`Steps` — Stepwise fitting information
structure

This property is read-only.

Stepwise fitting information, specified as a structure with the fields described in this table.

Field	Description
`Start`	Formula representing the starting model
`Lower`	Formula representing the lower bound model. The terms in `Lower` must remain in the model.
`Upper`	Formula representing the upper bound model. The model cannot contain more terms than `Upper`.
`Criterion`	Criterion used for the stepwise algorithm, such as `'sse'`
`PEnter`	Threshold for `Criterion` to add a term
`PRemove`	Threshold for `Criterion` to remove a term
`History`	Table representing the steps taken in the fit

The History table contains one row for each step, including the initial fit, and the columns described in this table.

Column	Description
`Action`	Action taken during the step: `'Start'` — First step `'Add'` — A term is added `'Remove'` — A term is removed
`TermName`	If `Action` is `'Start'`, `TermName` specifies the starting model specification. If `Action` is `'Add'` or `'Remove'`, `TermName` specifies the term added or removed in the step.
`Terms`	Model specification in a Terms Matrix
`DF`	Regression degrees of freedom after the step
`delDF`	Change in regression degrees of freedom from the previous step (negative for steps that remove a term)
`Deviance`	Deviance (residual sum of squares) at the step (only for a generalized linear regression model)
`FStat`	F-statistic that leads to the step
`PValue`	p-value of the F-statistic

The structure is empty unless you fit the model using stepwise regression.

Data Types: struct

Input Data

`Distribution` — Generalized distribution information
structure

This property is read-only.

Generalized distribution information, specified as a structure with the fields described in this table.

Field	Description
`Name`	Name of the distribution: `'normal'`, `'binomial'`, `'poisson'`, `'gamma'`, or `'inverse gaussian'`
`DevianceFunction`	Function that computes the components of the deviance as a function of the fitted parameter values and the response values
`VarianceFunction`	Function that computes the theoretical variance for the distribution as a function of the fitted parameter values. When `DispersionEstimated` is `true`, `Dispersion` multiplies the variance function in the computation of the coefficient standard errors.

Data Types: struct

`Formula` — Model information
`LinearFormula` object

This property is read-only.

Model information, specified as a LinearFormula object.

Display the formula of the fitted model mdl using dot notation:

mdl.Formula

`Link` — Link function
structure

This property is read-only.

Link function, specified as a structure with the fields described in this table.

Field	Description
`Name`	Name of the link function, specified as a character vector. If you specify the link function using a function handle, then `Name` is `''`.
`LinkFunction`	Function f that defines the link function, specified as a function handle
`DevianceFunction`	Derivative of f, specified as a function handle
`VarianceFunction`	Inverse of f, specified as a function handle

The link function is a function f that links the distribution parameter μ to the fitted linear combination Xb of the predictors:

f(μ) = Xb.

Data Types: struct

`NumObservations` — Number of observations
positive integer

This property is read-only.

Number of observations the fitting function used in fitting, specified as a positive integer. NumObservations is the number of observations supplied in the original table, dataset, or matrix, minus any excluded rows (set with the 'Exclude' name-value pair argument) or rows with missing values.

Data Types: double

`NumPredictors` — Number of predictor variables
positive integer

This property is read-only.

Number of predictor variables used to fit the model, specified as a positive integer.

Data Types: double

`NumVariables` — Number of variables
positive integer

This property is read-only.

Number of variables in the input data, specified as a positive integer. NumVariables is the number of variables in the original table or dataset, or the total number of columns in the predictor matrix and response vector.

NumVariables also includes any variables that are not used to fit the model as predictors or as the response.

Data Types: double

`ObservationInfo` — Observation information
table

This property is read-only.

Observation information, specified as an n-by-4 table, where n is equal to the number of rows of input data. ObservationInfo contains the columns described in this table.

Column	Description
`Weights`	Observation weights, specified as a numeric value. The default value is `1`.
`Excluded`	Indicator of excluded observations, specified as a logical value. The value is `true` if you exclude the observation from the fit by using the `'Exclude'` name-value pair argument.
`Missing`	Indicator of missing observations, specified as a logical value. The value is `true` if the observation is missing.
`Subset`	Indicator of whether or not the fitting function uses the observation, specified as a logical value. The value is `true` if the observation is not excluded or missing, meaning the fitting function uses the observation.

To obtain any of these columns as a vector, index into the property using dot notation. For example, obtain the weight vector w of the model mdl:

w = mdl.ObservationInfo.Weights

Data Types: table

`ObservationNames` — Observation names
cell array of character vectors

This property is read-only.

Observation names, specified as a cell array of character vectors containing the names of the observations used in the fit.

If the fit is based on a table or dataset containing observation names, ObservationNames uses those names.
Otherwise, ObservationNames is an empty cell array.

Data Types: cell

`Offset` — Offset variable
numeric vector

This property is read-only.

Offset variable, specified as a numeric vector with the same length as the number of rows in the data. Offset is passed from fitglm or stepwiseglm in the 'Offset' name-value pair argument. The fitting functions use Offset as an additional predictor variable with a coefficient value fixed at 1. In other words, the formula for fitting is

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

where f is the link function. The Offset predictor has 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: double

`PredictorNames` — Names of predictors used to fit model
cell array of character vectors

This property is read-only.

Names of predictors used to fit the model, specified as a cell array of character vectors.

Data Types: cell

`ResponseName` — Response variable name
character vector

This property is read-only.

Response variable name, specified as a character vector.

Data Types: char

`VariableInfo` — Information about variables
table

This property is read-only.

Information about variables contained in Variables, specified as a table with one row for each variable and the columns described in this table.

Column	Description
`Class`	Variable class, specified as a cell array of character vectors, such as `'double'` and `'categorical'`
`Range`	Variable range, specified as a cell array of vectors Continuous variable — Two-element vector `[min,max]`, the minimum and maximum values Categorical variable — Vector of distinct variable values
`InModel`	Indicator of which variables are in the fitted model, specified as a logical vector. The value is `true` if the model includes the variable.
`IsCategorical`	Indicator of categorical variables, specified as a logical vector. The value is `true` if the variable is categorical.

VariableInfo also includes any variables that are not used to fit the model as predictors or as the response.

Data Types: table

`VariableNames` — Names of variables
cell array of character vectors

This property is read-only.

Names of variables, specified as a cell array of character vectors.

If the fit is based on a table or dataset, this property provides the names of the variables in the table or dataset.
If the fit is based on a predictor matrix and response vector, VariableNames contains the values specified by the 'VarNames' name-value pair argument of the fitting method. The default value of 'VarNames' is {'x1','x2',...,'xn','y'}.

VariableNames also includes any variables that are not used to fit the model as predictors or as the response.

Data Types: cell

`Variables` — Input data
table

This property is read-only.

Input data, specified as a table. Variables contains both predictor and response values. If the fit is based on a table or dataset array, Variables contains all the data from the table or dataset array. Otherwise, Variables is a table created from the input data matrix X and the response vector y.

Variables also includes any variables that are not used to fit the model as predictors or as the response.

Data Types: table

Object Functions

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Create `CompactGeneralizedLinearModel`

compact Compact generalized linear regression model

Add or Remove Terms from Generalized Linear Model

`addTerms`	Add terms to generalized linear regression model
`removeTerms`	Remove terms from generalized linear regression model
`step`	Improve generalized linear regression model by adding or removing terms

Predict Responses

`feval`	Predict responses of generalized linear regression model using one input for each predictor
`predict`	Predict responses of generalized linear regression model
`random`	Simulate responses with random noise for generalized linear regression model

Evaluate Generalized Linear Model

`coefCI`	Confidence intervals of coefficient estimates of generalized linear regression model
`coefTest`	Linear hypothesis test on generalized linear regression model coefficients
`devianceTest`	Analysis of deviance for generalized linear regression model
`partialDependence`	Compute partial dependence

Visualize Generalized Linear Model and Summary Statistics

`plotDiagnostics`	Plot observation diagnostics of generalized linear regression model
`plotPartialDependence`	Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
`plotResiduals`	Plot residuals of generalized linear regression model
`plotSlice`	Plot of slices through fitted generalized linear regression surface

Gather Properties of Generalized Linear Model

gather Gather properties of linear or generalized linear regression model

Examples

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Create Generalized Linear Regression Model

Open Live Script

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

Load the hospital data set.

load hospital

Convert the dataset array to a table.

tbl = dataset2table(hospital);

Specify the model using a formula that includes two-way interactions and lower-order terms.

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

Create the generalized linear model.

mdl = fitglm(tbl,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 that the model might not differ statistically from a constant.

Create Generalized Linear Regression Model Using Stepwise Regression

Open Live Script

Create response data using three of 20 predictor variables, and create a generalized linear model using stepwise regression from a constant model to see if stepwiseglm finds the correct predictors.

Generate sample data that has 20 predictor variables. Use three of the predictors to generate the Poisson response variable.

rng default % for reproducibility
X = randn(100,20);
mu = exp(X(:,[5 10 15])*[.4;.2;.3] + 1);
y = poissrnd(mu);

Fit a generalized linear regression model using the Poisson distribution. Specify the starting model as a model that contains only a constant (intercept) term. Also, specify a model with an intercept and linear term for each predictor as the largest model to consider as the fit by using the 'Upper' name-value pair argument.

mdl =  stepwiseglm(X,y,'constant','Upper','linear','Distribution','poisson')

1. Adding x5, Deviance = 134.439, Chi2Stat = 52.24814, PValue = 4.891229e-13
2. Adding x15, Deviance = 106.285, Chi2Stat = 28.15393, PValue = 1.1204e-07
3. Adding x10, Deviance = 95.0207, Chi2Stat = 11.2644, PValue = 0.000790094

mdl = 
Generalized linear regression model:
    log(y) ~ 1 + x5 + x10 + x15
    Distribution = Poisson

Estimated Coefficients:
                   Estimate       SE       tStat       pValue  
                   ________    ________    ______    __________

    (Intercept)     1.0115     0.064275    15.737    8.4217e-56
    x5             0.39508     0.066665    5.9263    3.0977e-09
    x10            0.18863      0.05534    3.4085     0.0006532
    x15            0.29295     0.053269    5.4995    3.8089e-08


100 observations, 96 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 91.7, p-value = 9.61e-20

stepwiseglm finds the three correct predictors: x5, x10, and x15.

More About

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Canonical Link Function

The default link function for a generalized linear model is the canonical link function. You can specify a link function when you fit a model with fitglm or stepwiseglm by using the 'Link' name-value pair argument.

Distribution	Canonical 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/μ²	μ = (Xb)^–1/2

Cook’s Distance

The Cook’s distance D_i of observation i is

$D_{i} = w_{i} \frac{e_{i}^{2}}{p \hat{φ}} \frac{h_{i i}}{{(1 - h_{i i})}^{2}},$

where

$\hat{φ}$ is the dispersion parameter (estimated or theoretical).
e_i is the linear predictor residual, $g (y_{i}) - x_{i} \hat{β}$ , where
- g is the link function.
- y_i is the observed response.
- x_i is the observation.
- $\hat{β}$ is the estimated coefficient vector.
p is the number of coefficients in the regression model.
h_ii is the ith diagonal element of the Hat Matrix H.

Leverage

Leverage is a measure of the effect of a particular observation on the regression predictions due to the position of that observation in the space of the inputs.

The leverage of observation i is the value of the ith diagonal term h_ii of the hat matrix H. Because the sum of the leverage values is p (the number of coefficients in the regression model), an observation i can be considered an outlier if its leverage substantially exceeds p/n, where n is the number of observations.

Hat Matrix

The hat matrix H is defined in terms of the data matrix X and a diagonal weight matrix W:

H = X(X^TWX)^–1X^TW^T.

W has diagonal elements w_i:

$w_{i} = \frac{g^{'} (μ_{i})}{\sqrt{V (μ_{i})}},$

where

g is the link function mapping y_i to x_ib.
$g^{'}$ is the derivative of the link function g.
V is the variance function.
μ_i is the ith mean.

The diagonal elements H_ii satisfy

$\begin{array}{l} 0 \leq h_{i i} \leq 1 \\ \sum_{i = 1}^{n} h_{i i} = p, \end{array}$

where n is the number of observations (rows of X), and p is the number of coefficients in the regression model.

Deviance

Deviance of a model M₁ is twice the difference between the loglikelihood of the model M₁ and the saturated model M_s. A saturated model is the model with the maximum number of parameters that you can estimate.

For example, if you have n observations (y_i, i = 1, 2, ..., n) with potentially different values for X_i^Tβ, then you can define a saturated model with n parameters. Let L(b,y) denote the maximum value of the likelihood function for a model with the parameters b. Then the deviance of the model M₁ is

$- 2 (\log L (b_{1}, y) - \log L (b_{S}, y)),$

where b₁ and b_s contain the estimated parameters for the model M₁ and the saturated model, respectively. The deviance has a chi-square distribution with n – p degrees of freedom, where n is the number of parameters in the saturated model and p is the number of parameters in the model M₁.

Assume you have two different generalized linear regression models M₁ and M₂, and M₁ has a subset of the terms in M₂. You can assess the fit of the models by comparing the deviances D₁ and D₂ of the two models. The difference of the deviances is

$\begin{array}{l} D = D_{2} - D_{1} = - 2 (\log L (b_{2}, y) - \log L (b_{S}, y)) + 2 (\log L (b_{1}, y) - \log L (b_{S}, y)) \\ = - 2 (\log L (b_{2}, y) - \log L (b_{1}, y)) . \end{array}$

Asymptotically, the difference D has a chi-square distribution with degrees of freedom v equal to the difference in the number of parameters estimated in M₁ and M₂. You can obtain the p-value for this test by using 1 – chi2cdf(D,v).

Typically, you examine D using a model M₂ with a constant term and no predictors. Therefore, D has a chi-square distribution with p – 1 degrees of freedom. If the dispersion is estimated, the difference divided by the estimated dispersion has an F distribution with p – 1 numerator degrees of freedom and n – p denominator degrees of freedom.

Terms Matrix

A terms matrix T 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 +1 accounts for the response variable. The value of T(i,j) is the exponent of variable j in term i.

For example, suppose that an input includes three predictor variables x1, x2, and x3 and the response variable y in the order x1, x2, x3, and y. Each row of T represents one term:

[0 0 0 0] — Constant term or intercept
[0 1 0 0] — x2; equivalently, x1^0 * x2^1 * x3^0
[1 0 1 0] — x1*x3
[2 0 0 0] — x1^2
[0 1 2 0] — x2*(x3^2)

The 0 at the end of each term represents the response variable. In general, a column vector of zeros in a terms matrix represents the position of the response variable. If you have the predictor and response variables in a matrix and column vector, then you must include 0 for the response variable in the last column of each row.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

The predict and random functions support code generation.
When you fit a model by using fitglm or stepwiseglm, the following restrictions apply.
- Code generation does not support categorical predictors. You cannot supply training data in a table that contains a logical vector, character array, categorical array, string array, or cell array of character vectors. Also, you cannot use the 'CategoricalVars' name-value pair argument. To include categorical predictors in a model, preprocess the categorical predictors by using dummyvar before fitting the model.
- The Link, Derivative, and Inverse fields of the 'Link' name-value pair argument cannot be anonymous functions. That is, you cannot generate code using a generalized linear model that was created using anonymous functions for links. Instead, define functions for link components.

For more information, see Introduction to Code Generation.

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

The following object functions fully support GPU arrays:
The following object functions support model objects fitted with GPU array input arguments:

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Documentation

GeneralizedLinearModel

Description

Creation

Properties

Coefficient Estimates

CoefficientCovariance — Covariance matrix of coefficient estimates numeric matrix

CoefficientNames — Coefficient names cell array of character vectors

Coefficients — Coefficient values table

NumCoefficients — Number of model coefficients positive integer

NumEstimatedCoefficients — Number of estimated coefficients positive integer

Summary Statistics

Deviance — Deviance of fit numeric value

DFE — Degrees of freedom for error positive integer

Diagnostics — Observation diagnostics table

Dispersion — Scale factor of variance of response numeric scalar

DispersionEstimated — Flag to indicate use of dispersion scale factor logical value

Fitted — Fitted response values based on input data table

LogLikelihood — Loglikelihood numeric value

ModelCriterion — Criterion for model comparison structure

Residuals — Residuals for fitted model table

Rsquared — R-squared value for model structure

SSE — Sum of squared errors numeric value

SSR — Regression sum of squares numeric value

SST — Total sum of squares numeric value

Fitting Information

Steps — Stepwise fitting information structure

Input Data

Distribution — Generalized distribution information structure

Formula — Model information LinearFormula object

Link — Link function structure

NumObservations — Number of observations positive integer

NumPredictors — Number of predictor variables positive integer

NumVariables — Number of variables positive integer

ObservationInfo — Observation information table

ObservationNames — Observation names cell array of character vectors

Offset — Offset variable numeric vector

PredictorNames — Names of predictors used to fit model cell array of character vectors

ResponseName — Response variable name character vector

VariableInfo — Information about variables table

VariableNames — Names of variables cell array of character vectors

Variables — Input data table

Object Functions

Create CompactGeneralizedLinearModel

Add or Remove Terms from Generalized Linear Model

Predict Responses

Evaluate Generalized Linear Model

Visualize Generalized Linear Model and Summary Statistics

Gather Properties of Generalized Linear Model

Examples

Create Generalized Linear Regression Model

Create Generalized Linear Regression Model Using Stepwise Regression

More About

Canonical Link Function

Cook’s Distance

Leverage

Hat Matrix

Deviance

Terms Matrix

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

GPU Arrays Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

See Also

Topics

Statistics and Machine Learning Toolbox Documentation

Support

`CoefficientCovariance` — Covariance matrix of coefficient estimates
numeric matrix

`CoefficientNames` — Coefficient names
cell array of character vectors

`Coefficients` — Coefficient values
table

`NumCoefficients` — Number of model coefficients
positive integer

`NumEstimatedCoefficients` — Number of estimated coefficients
positive integer

`Deviance` — Deviance of fit
numeric value

`DFE` — Degrees of freedom for error
positive integer

`Diagnostics` — Observation diagnostics
table

`Dispersion` — Scale factor of variance of response
numeric scalar

`DispersionEstimated` — Flag to indicate use of dispersion scale factor
logical value

`Fitted` — Fitted response values based on input data
table

`LogLikelihood` — Loglikelihood
numeric value

`ModelCriterion` — Criterion for model comparison
structure

`Residuals` — Residuals for fitted model
table

`Rsquared` — R-squared value for model
structure

`SSE` — Sum of squared errors
numeric value

`SSR` — Regression sum of squares
numeric value

`SST` — Total sum of squares
numeric value

`Steps` — Stepwise fitting information
structure

`Distribution` — Generalized distribution information
structure

`Formula` — Model information
`LinearFormula` object

`Link` — Link function
structure

`NumObservations` — Number of observations
positive integer

`NumPredictors` — Number of predictor variables
positive integer

`NumVariables` — Number of variables
positive integer

`ObservationInfo` — Observation information
table

`ObservationNames` — Observation names
cell array of character vectors

`Offset` — Offset variable
numeric vector

`PredictorNames` — Names of predictors used to fit model
cell array of character vectors

`ResponseName` — Response variable name
character vector

`VariableInfo` — Information about variables
table

`VariableNames` — Names of variables
cell array of character vectors

`Variables` — Input data
table

Create `CompactGeneralizedLinearModel`

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.