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GeneralizedLinearModel class

Superclasses: CompactGeneralizedLinearModel

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.

Input Arguments

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Input data, specified as a table or dataset array. When modelspec is a formula, the formula specifies the predictor and response variables. Otherwise, if you do not specify the predictor and response variables, the last variable in tbl is the response variable and the others are the predictor variables by default.

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'.

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.

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 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 | logical | categorical

Properties

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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

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

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

• tStatt-statistic for a test that the coefficient is zero

• pValuep-value for the t-statistic

Use anova (only for a linear regression model) or coefTest to perform other 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

This property is read-only.

Deviance of the fit, specified as a numeric value. 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 on deviance, see Deviance.

Data Types: single | double

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

This property is read-only.

Diagnostic information for the model, specified as a table. Diagnostics can help identify outliers and influential observations. Diagnostics 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. For example, in the equation that defines the hat matrix:

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

Data Types: table

This property is read-only.

Scale factor of the variance of the response, specified as a numeric value. 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 set is “overdispersed” relative to the binomial distribution.

Data Types: double

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 'binomial' or 'poisson' distributions.

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

Data Types: logical

This property is read-only.

Generalized distribution information, specified as a structure with the following fields relating to the generalized distribution.

FieldDescription
NameName of the distribution, one of '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, Dispersion multiplies the variance function in the computation of the coefficient standard errors.

Data Types: struct

This property is read-only.

Fitted (predicted) values based on the input data, specified as 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.

Data Types: table

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

This property is read-only.

Log likelihood 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

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

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

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

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

This property is read-only.

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

Data Types: double

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

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. The ObservationInfo contains the columns described in this table.

ColumnDescription
WeightsObservation weight, specified as a numeric value. The default value is 1.
ExcludedIndicator of excluded observation, 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 observation, specified as a logical value. The value is true if the observation is missing.
SubsetIndicator of whether or not a fitting function uses the observation, specified as a logical value. The value is true if the observation is not excluded or missing, meaning that 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

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

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. 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.

Data Types: double

This property is read-only.

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

Data Types: cell

This property is read-only.

Residuals for the fitted model, specified as a table with one row for each observation and the following columns.

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.

Data Types: table

This property is read-only.

Response variable name, specified as a character vector.

Data Types: char

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 — Log-likelihood ratio

• Deviance — Deviance

The R-squared value is the 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.

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

Data Types: struct

This property is read-only.

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

The Pythagorean theorem implies

SST = SSE + SSR,

where SST is the total sum of squares, SSE is the sum of squared errors, and SSR is the regression sum of squares.

Data Types: single | double

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.

The Pythagorean theorem implies

SST = SSE + SSR,

where SST is the total sum of squares, SSE is the sum of squared errors, and SSR is the regression sum of squares.

Data Types: single | double

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).

The Pythagorean theorem implies

SST = SSE + SSR,

where SST is the total sum of squares, SSE is the sum of squared errors, and SSR is the regression sum of squares.

Data Types: single | double

This property is read-only.

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

• '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.

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

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.

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

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

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 response the vector y.

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

Data Types: table

Methods

 addTerms Add terms to generalized linear model compact Compact generalized linear regression model fit (Not Recommended) Create generalized linear regression model plotDiagnostics Plot diagnostics of generalized linear regression model plotResiduals Plot residuals of generalized linear regression model removeTerms Remove terms from generalized linear model step Improve generalized linear regression model by adding or removing terms stepwise (Not Recommended) Create generalized linear regression model by stepwise regression

Inherited Methods

 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 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

Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects (MATLAB).

Examples

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

Load the hospital dataset array.

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 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

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