Accelerating the pace of engineering and science

# GeneralizedLinearModel class

Generalized linear regression model class

## Description

An object comprising training data, model description, diagnostic information, and fitted coefficients for a generalized linear regression. Predict model responses with the predict or feval methods.

## Construction

mdl = fitglm(tbl) or mdl = fitglm(X,y) creates a generalized linear model of a table or dataset array tbl, or of the responses y to a data matrix X. For details, see fitglm.

mdl = stepwiseglm(tbl) or mdl = stepwiseglm(X,y) creates a generalized linear model of a table or dataset array tbl, or of the responses y to a data matrix X, with unimportant predictors excluded. For details, see stepwiseglm.

expand all

### 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 can be numeric, or any grouping variable type, such as logical or categorical (see Grouping Variables). The response must be numeric or logical.

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.

Data Types: single | double

## Properties

CoefficientCovariance

Covariance matrix of coefficient estimates.

CoefficientNames

Cell array of strings containing a label for each coefficient.

Coefficients

Coefficient values stored as a table. Coefficients has one row for each coefficient and these columns:

• Estimate — Estimated coefficient value

• SE — Standard error of the estimate

• tStatt statistic for a test that the coefficient is zero

• pValuep-value for the t statistic

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

`beta = mdl.Coefficients.Estimate`

Use coefTest to perform other tests on the coefficients.

Deviance

Deviance of the fit. It is useful for comparing two models when one is a special case of the other. 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 on deviance, see Deviance.

DFE

Degrees of freedom for error (residuals), equal to the number of observations minus the number of estimated coefficients.

Diagnostics

Table with diagnostics helpful in finding outliers and influential observations. The table contains the following fields:

FieldMeaningUtility
LeverageDiagonal elements of HatMatrixLeverage indicates to what extent the predicted value for an observation is determined by the observed value for that observation. A value close to 1 indicates that the prediction is largely determined by that observation, with little contribution from the other observations. A value close to 0 indicates 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. An observation with Leverage larger than 2*p/n can be an outlier.
CooksDistanceCook's measure of scaled change in fitted valuesCooksDistance is a measure of scaled change in fitted values. An observation with CooksDistance larger than three times the mean Cook's distance can be an outlier.
HatMatrixProjection matrix to compute fitted from observed responsesHatMatrix 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.

All of these quantities are computed on the scale of the linear predictor. So, for example, in the equation that defines the hat matrix,

```Yfit = glm.Fitted.LinearPredictor
Y = glm.Fitted.LinearPredictor + glm.Residuals.LinearPredictor```

Dispersion

Scale factor of the variance of the response. Dispersion multiplies the variance function for the distribution.

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 are "overdispersed" relative to the binomial distribution.

DispersionEstimated

Logical value indicating whether fitglm used the Dispersion property to compute standard errors for the coefficients in Coefficients.SE. If DispersionEstimated is false, fitglm used the theoretical value of the variance.

• DispersionEstimated can be false only for 'binomial' or 'poisson' distributions.

• Set DispersionEstimated by setting the DispersionFlag name-value pair in fitglm.

Distribution

Structure with the following fields relating to the generalized distribution:

FieldDescription
NameName of the distribution, one of 'normal', 'binomial', 'poisson', 'gamma', or 'inverse gamma'.
DevianceFunctionFunction that computes the components of the deviance as a function of the fitted parameter values and the response values.
VarianceFunctionFunction 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.

Fitted

Table of predicted (fitted) values based on the training data, a table with one row for each observation and the following columns.

FieldDescription
ResponsePredicted values on the scale of the response.
LinearPredictorPredicted values on the scale of the linear predictor. These are the same as the link function applied to the Response fitted values.
ProbabilityFitted probabilities (this column is included only with the binomial distribution).

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

`f = mdl.Fitted.Response`

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

Formula

Object containing information about the model.

Link

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

f(μ) = Xb.

The structure has the following fields.

FieldDescription
NameName of the link function, or '' if you specified the link as a function handle rather than a string.
LinkFunctionThe function that defines f, a function handle.
DevianceFunctionDerivative of f, a function handle.
VarianceFunctionInverse of f, a function handle.

LogLikelihood

Log likelihood of the model distribution at the response values, with mean fitted from the model, and other parameters estimated as part of the model fit.

ModelCriterion

AIC and other information criteria for comparing models. A structure with fields:

• AIC — Akaike information criterion

• AICc — Akaike information criterion corrected for sample size

• BIC — Bayesian information criterion

• CAIC — Consistent Akaike information criterion

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

`aic = mdl.ModelCriterion.AIC`

NumCoefficients

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

NumEstimatedCoefficients

Number of estimated coefficients in the model, 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.

NumObservations

Number of observations the fitting function used in fitting. This is the number of observations supplied in the original table, dataset, or matrix, minus any excluded rows (set with the Excluded name-value pair) or rows with missing values.

NumPredictors

Number of variables fitlm used as predictors for fitting.

NumVariables

Number of variables in the data. 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 when the fit is based on those arrays. It includes variables, if any, that are not used as predictors or as the response.

ObservationInfo

Table with the same number of rows as the input data (tbl or X).

FieldDescription
WeightsObservation weights. Default is all 1.
ExcludedLogical value, 1 indicates an observation that you excluded from the fit with the Exclude name-value pair.
MissingLogical value, 1 indicates a missing value in the input. Missing values are not used in the fit.
SubsetLogical value, 1 indicates the observation is not excluded or missing, so is used in the fit.

ObservationNames

Cell array of strings 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

Offset

Vector with the same length as the number of rows in the data, passed from fitglm or stepwiseglm in the Offset name-value pair. The fitting function used Offset as a predictor variable, but with the coefficient set to exactly 1. In other words, the formula for fitting was

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.

PredictorNames

Cell array of strings, the names of the predictors used in fitting the model.

Residuals

Table containing residuals, with one row for each observation and these variables.

FieldDescription
RawObserved minus fitted values.
LinearPredictorResiduals on the linear predictor scale, equal to the adjusted response value minus the fitted linear combination of the predictors.
PearsonRaw residuals divided by the estimated standard deviation of the response.
AnscombeResiduals defined on transformed data with the transformation chosen to remove skewness.
DevianceResiduals based on the contribution of each observation to the deviance.

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

`r = mdl.Residuals.Raw`

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

Rows not used in the fit because of excluded values (in ObservationInfo.Excluded) contain NaN values, with the following exceptions:

• raw contains the difference between the observed and predicted values.

• standardized is the residual, standardized in the usual way.

• studentized matches the standardized values because this residual is not used in the estimate of the residual standard deviation.

ResponseName

String giving naming the response variable.

Rsquared

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

Rsquared = SSR/SST = 1 - SSE/SST.

For a linear or nonlinear model, Rsquared is a structure with two fields:

• Ordinary — Ordinary (unadjusted) R-squared

• Adjusted — R-squared adjusted for the number of coefficients

For a generalized linear model, Rsquared is a structure with five fields:

• Ordinary — Ordinary (unadjusted) R-squared

• Adjusted — R-squared adjusted for the number of coefficients

• LLR — Log-likelihood ratio

• Deviance — Deviance

• AdjGeneralized — Adjusted generalized R-squared

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

`r2 = mdl.Rsquared.Adjusted`

SSE

Sum of squared errors (residuals).

The Pythagorean theorem implies

SST = SSE + SSR.

SSR

Regression sum of squares, the sum of squared deviations of the fitted values from their mean.

The Pythagorean theorem implies

SST = SSE + SSR.

SST

Total sum of squares, the sum of squared deviations of y from mean(y).

The Pythagorean theorem implies

SST = SSE + SSR.

Steps

Structure that is empty unless stepwiselm constructed the model.

FieldDescription
StartFormula representing the starting model
LowerFormula representing the lower bound model, these terms that must remain in the model
UpperFormula representing the upper bound model, model cannot contain more terms than Upper
CriterionCriterion used for the stepwise algorithm, such as 'sse'
PEnterValue of the parameter, such as 0.05
PRemoveValue of the parameter, such as 0.10
HistoryTable representing the steps taken in the fit

The History table has one row for each step including the initial fit, and the following variables (columns).

FieldDescription
ActionAction taken during this step, one of:
• 'Start' — First step

• 'Add' — A term is added

• 'Remove' — A term is removed

TermName
• 'Start' step: The starting model specification

• 'Add' or 'Remove' steps: The term moved in that step

TermsTerms matrix (see modelspec of fitlm)
DFRegression degrees of freedom after this step
delDFChange in regression degrees of freedom from previous step (negative for steps that remove a term)
DevianceDeviance (residual sum of squares) at that step
FStatF statistic that led to this step
PValuep-value of the F statistic

VariableInfo

Table containing metadata about Variables. There is one row for each term in the model, and the following columns.

FieldDescription
ClassString giving variable class, such as 'double'
RangeCell array giving variable range:
• Continuous variable — Two-element vector [min,max], the minimum and maximum values

• Categorical variable — Cell array of distinct variable values

InModelLogical vector, where true indicates the variable is in the model
IsCategoricalLogical vector, where true indicates a categorical variable

VariableNames

Cell array of strings containing names of the variables in the fit.

• If the fit is based on a table or dataset, this property provides the names of the variables in that table or dataset.

• If the fit is based on a predictor matrix and response vector, VariableNames is the values in the VarNames name-value pair of the fitting method.

• Otherwise the variables have the default fitting names.

Variables

Table containing the data, both observations and responses, that the fitting function used to construct the fit. If the fit is based on a table or dataset array, Variables contains all of the data from that table or dataset array. Otherwise, Variables is a table created from the input data matrix X and response vector y.

## Methods

 addTerms Add terms to generalized linear model coefCI Confidence intervals of coefficient estimates of generalized linear model coefTest Linear hypothesis test on generalized linear regression model coefficients devianceTest Analysis of deviance disp Display generalized linear regression model feval Evaluate generalized linear regression model prediction fit Create generalized linear regression model plotDiagnostics Plot diagnostics of generalized linear regression model plotResiduals Plot residuals of generalized linear regression model plotSlice Plot of slices through fitted generalized linear regression surface predict Predict response of generalized linear regression model random Simulate responses for generalized linear regression model removeTerms Remove terms from generalized linear model step Improve generalized linear regression model by adding or removing terms stepwise Create generalized linear regression model by stepwise regression

## Definitions

### Canonical Link Function

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

Canonical Link Functions for Generalized Linear Models

DistributionLink Function NameLink FunctionMean (Inverse) Function
'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'-1f(μ) = 1/μμ = 1/(Xb)
'inverse gaussian'-2f(μ) = 1/μ2μ = (Xb)–1/2

### Hat Matrix

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

H = X(XTWX)–1XTWT.

W has diagonal elements wi:

${w}_{i}=\frac{{g}^{\prime }\left({\mu }_{i}\right)}{\sqrt{V\left({\mu }_{i}\right)}},$

where

• g is the link function mapping yi to xib.

• ${g}^{\prime }$ is the derivative of the link function g.

• V is the variance function.

• μi is the ith mean.

The diagonal elements Hii satisfy

$\begin{array}{l}0\le {h}_{ii}\le 1\\ \sum _{i=1}^{n}{h}_{ii}=p,\end{array}$

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

### Leverage

The leverage of observation i is the value of the ith diagonal term, hii, 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 to be an outlier if its leverage substantially exceeds p/n, where n is the number of observations.

### Cook's Distance

The Cook's distance Di of observation i is

${D}_{i}={w}_{i}\frac{{e}_{i}^{2}}{p\stackrel{^}{\phi }}\frac{{h}_{ii}}{{\left(1-{h}_{ii}\right)}^{2}},$

where

• $\stackrel{^}{\phi }$ is the dispersion parameter (estimated or theoretical).

• ei is the linear predictor residual, $g\left({y}_{i}\right)-{x}_{i}\stackrel{^}{\beta }$, where

• g is the link function.

• yi is the observed response.

• xi is the observation.

• $\stackrel{^}{\beta }$ is the estimated coefficient vector.

• p is the number of coefficients in the regression model.

• hii is the ith diagonal element of the Hat Matrix H.

### Deviance

Deviance of a model M1 is twice the difference between the loglikelihood of that model and the saturated model, MS. The saturated model is the model with the maximum number of parameters that can be estimated. For example, if there are n observations yi, i = 1, 2, ..., n, with potentially different values for XiTβ, 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. Then the deviance of model M1 is

$-2\left(\mathrm{log}L\left({b}_{1},y\right)-\mathrm{log}L\left({b}_{S},y\right)\right),$

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

If M1 and M2 are two different generalized linear models, then the fit of the models can be assessed by comparing the deviances D1 and D2 of these models. The difference of the deviances is

$\begin{array}{l}D={D}_{2}-{D}_{1}=-2\left(\mathrm{log}L\left({b}_{2},y\right)-\mathrm{log}L\left({b}_{S},y\right)\right)+2\left(\mathrm{log}L\left({b}_{1},y\right)-\mathrm{log}L\left({b}_{S},y\right)\right)\\ \text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}=-2\left(\mathrm{log}L\left({b}_{2},y\right)-\mathrm{log}L\left({b}_{1},y\right)\right).\end{array}$

Asymptotically, this difference has a chi-square distribution with degrees of freedom v equal to the number of parameters that are estimated in one model but fixed (typically at 0) in the other. That is, it is equal to the difference in the number of parameters estimated in M1 and M2. You can get the p-value for this test using 1 - chi2cdf(D,V), where D = D2D1.

## Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB® documentation.

## Examples

expand all

### Fit a Generalized Linear Model

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.

### Create a Generalized Linear Model Stepwise

Create response data using just three of 20 predictors, and create a generalized linear model stepwise to see if it uses just the correct predictors.

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

```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 model using the Poisson distribution.

```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```

## More About

Was this topic helpful?