# 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

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, as given by NumCoefficients.

For details, see Coefficient Standard Errors and Confidence Intervals.

Data Types: single | double

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

Data Types: cell

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

• tStatt-statistic for a two-sided test with the null hypothesis that the coefficient is zero

• pValuep-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

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

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 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

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

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

ColumnMeaningDescription
LeverageDiagonal elements of HatMatrixLeverage 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.
CooksDistanceCook's distance of scaled change in fitted valuesCooksDistance 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.
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.

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

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

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

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 (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.

ColumnDescription
ResponsePredicted values on the scale of the response
LinearPredictorPredicted values on the scale of the linear combination of the predictors (same as the link function applied to the Response fitted values)
ProbabilityFitted 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 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

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

ColumnDescription
RawObserved minus fitted values
PearsonRaw residuals divided by the estimated standard deviation of the response
AnscombeResiduals defined on transformed data with the transformation selected to remove skewness
DevianceResiduals 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

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

FieldDescriptionEquation

${R}_{\text{Ordinary}}^{2}=1-\frac{\text{SSE}}{\text{SST}}$

SSE is the sum of squared errors, and SST is the total sum of squared deviations of the response vector from the mean of the response vector.

${R}_{\text{Adjusted}}^{2}=1-\frac{\text{SSE}}{\text{SST}}\cdot \frac{N-1}{\text{DFE}}$

N is the number of observations (NumObservations), and DFE is the degrees of freedom for the error (residuals).

LLRLoglikelihood ratio

${R}_{\text{LLR}}^{2}=1-\frac{L}{{L}_{0}}$

L is the loglikelihood of the fitted model (LogLikelihood), and L0 is the loglikelihood of a model that includes only a constant term. R2LLR is the McFadden pseudo R-squared value [1] for logistic regression models.

DevianceDeviance R-squared

${R}_{\text{Deviance}}^{2}=1-\frac{D}{{D}_{0}}$

D is the deviance of the fitted model (Deviance), and D0 is the deviance of a model that includes only a constant term.

${R}_{\text{AdjGeneralized}}^{2}=\frac{1-\mathrm{exp}\left(\frac{2\left({L}_{0}-L\right)}{N}\right)}{1-\mathrm{exp}\left(\frac{2{L}_{0}}{N}\right)}$

R2AdjGeneralized is the Nagelkerke adjustment [2] to a formula proposed by Maddala [3], Cox and Snell [4], and Magee [5] for logistic regression models.

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

Data Types: struct

Sum of squared errors (residuals), specified as a numeric value. If the model was trained with observation weights, the sum of squares in the SSE calculation is the weighted sum of squares.

Data Types: single | double

Regression sum of squares, specified as a numeric value. SSR is equal to the sum of the squared deviations between the fitted values and the mean of the response. If the model was trained with observation weights, the sum of squares in the SSR calculation is the weighted sum of squares.

Data Types: single | double

Total sum of squares, specified as a numeric value. SST is equal to the sum of squared deviations of the response vector y from the mean(y). If the model was trained with observation weights, the sum of squares in the SST calculation is the weighted sum of squares.

Data Types: single | double

### Fitting Information

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

FieldDescription
StartFormula representing the starting model
LowerFormula representing the lower bound model. The terms in Lower must remain in the model.
UpperFormula representing the upper bound model. The model cannot contain more terms than Upper.
CriterionCriterion used for the stepwise algorithm, such as 'sse'
PEnterThreshold for Criterion to add a term
PRemoveThreshold for Criterion to remove a term
HistoryTable 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.

ColumnDescription
Action

Action taken during the step:

• 'Start' — First step

• '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.

TermsModel specification in a Terms Matrix
DFRegression degrees of freedom after the step
delDFChange in regression degrees of freedom from the previous step (negative for steps that remove a term)
DevianceDeviance (residual sum of squares) at the step (only for a generalized linear regression model)
FStatF-statistic that leads to the step
PValuep-value of the F-statistic

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

Data Types: struct

### Input Data

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

FieldDescription
NameName of the distribution: 'normal', 'binomial', 'poisson', 'gamma', or 'inverse gaussian'
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, the software multiplies the variance function by Dispersion in the computation of the coefficient standard errors.

Data Types: struct

Model information, specified as a LinearFormula object.

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

mdl.Formula

Penalty for the likelihood estimate, specified as "none" or "jeffreys-prior".

• "none" — The likelihood estimate is not penalized during model fitting.

• "jeffreys-prior" — The likelihood estimate is penalized using the Jeffreys prior.

For logistic models, setting LikelihoodPenalty to "jeffreys-prior" is called Firth's regression. To reduce the coefficient estimate bias when you have a small number of samples, or when you are performing binomial (logistic) regression on a separable data set, set LikelihoodPenalty to "jeffreys-prior" during training.

Example: LikelihoodPenalty="jeffreys-prior"

Data Types: char | string

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

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

Data Types: double

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

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.

ColumnDescription
WeightsObservation weights, specified as a numeric value. The default value is 1.
ExcludedIndicator 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.
MissingIndicator of missing observations, specified as a logical value. The value is true if the observation is missing.
SubsetIndicator 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

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 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

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

Data Types: cell

Response variable name, specified as a character vector.

Data Types: char

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

ColumnDescription
ClassVariable 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

InModelIndicator of which variables are in the fitted model, specified as a logical vector. The value is true if the model includes the variable.
IsCategoricalIndicator 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

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

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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 compact Compact generalized linear regression 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
 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
 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
 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 Gather properties of Statistics and Machine Learning Toolbox object from GPU

## Examples

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

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 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.

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## References

[1] McFadden, Daniel. "Conditional logit analysis of qualitative choice behavior." in Frontiers in Econometrics, edited by P. Zarembka,105–42. New York: Academic Press, 1974.

[2] Nagelkerke, N. J. D. "A Note on a General Definition of the Coefficient of Determination." Biometrika 78, no. 3 (1991): 691–92.

[3] Maddala, Gangadharrao S. Limited-Dependent and Qualitative Variables in Econometrics. Econometric Society Monographs. New York, NY: Cambridge University Press, 1983.

[4] Cox, D. R., and E. J. Snell. Analysis of Binary Data. 2nd ed. Monographs on Statistics and Applied Probability 32. London; New York: Chapman and Hall, 1989.

[5] Magee, Lonnie. "R 2 Measures Based on Wald and Likelihood Ratio Joint Significance Tests." The American Statistician 44, no. 3 (August 1990): 250–53.

## Version History

Introduced in R2012a