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

tbl — Input data

table | dataset array

Input data including predictor and response variables, specified as a table or dataset array. The predictor variables and response variable can be numeric, logical, categorical, character, or string. The response variable can have a data type other than numeric only if 'Distribution' is 'binomial'.

• By default, GeneralizedLinearModel takes the last variable as the response variable and the others as the predictor variables.

• To set a different column as the response variable, use the ResponseVar name-value pair argument.

• To use a subset of the columns as predictors, use the PredictorVars name-value pair argument.

• To define a model specification, set the modelspec argument using a formula or terms matrix. The formula or terms matrix specifies which columns to use as the predictor or response variables.

The variable names in a table do not have to be valid MATLAB^® identifiers. However, if the names are not valid, you cannot use a formula when you fit or adjust a model; for example:

• You cannot specify modelspec using a formula.

• You cannot use a formula to specify the terms to add or remove when you use the addTerms function or the removeTerms function, respectively.

• You cannot use a formula to specify the lower and upper bounds of the model when you use the step or stepwiseglm function with the name-value pair arguments 'Lower' and 'Upper', respectively.

You can verify the variable names in tbl by using the isvarname function. The following code returns logical 1 (true) for each variable that has a valid variable name.

cellfun(@isvarname,tbl.Properties.VariableNames)

If the variable names in tbl are not valid, then convert them by using the matlab.lang.makeValidName function.

tbl.Properties.VariableNames = matlab.lang.makeValidName(tbl.Properties.VariableNames);

X — Predictor variables

matrix

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

y — Response variable

vector | matrix

Response variable, specified as a vector or matrix.

• If 'Distribution' is not 'binomial', then y must be an n-by-1 vector, where n is the number of observations. Each entry in y is the response for the corresponding row of X. The data type must be single or double.

• If 'Distribution' is 'binomial', then y can be an n-by-1 vector or n-by-2 matrix with counts in column 1 and BinomialSize in column 2.

Data Types: single | double | logical | categorical

Properties

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 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

Deviance — Deviance of the fit

numeric value

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

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 — Diagnostic information

table

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.

Field	Meaning	Utility
Leverage	Diagonal elements of HatMatrix	Leverage 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.
CooksDistance	Cook's measure of scaled change in fitted values	CooksDistance 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.
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.

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

Dispersion — Scale factor of the variance of the response

numeric value

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

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 'binomial' or 'poisson' distributions.

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

Data Types: logical

Distribution — Generalized distribution information

structure

This property is read-only.

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

Field	Description
Name	Name of the distribution, one of '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

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 with one row for each observation and the following columns.

Field	Description
Response	Predicted values on the scale of the response.
LinearPredictor	Predicted values on the scale of the linear predictor. These are the same as the link function applied to the Response fitted values.
Probability	Fitted 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

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 following fields.

Field	Description
Name	Name of the link function, or '' if you specified the link as a function handle rather than a character vector.
LinkFunction	The function that defines f, a function handle.
DevianceFunction	Derivative of f, a function handle.
VarianceFunction	Inverse of f, a function handle.

The link 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

LogLikelihood — Log likelihood

numeric value

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

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

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

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. The ObservationInfo contains the columns described in this table.

Column	Description
Weights	Observation weight, specified as a numeric value. The default value is 1.
Excluded	Indicator 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.
Missing	Indicator of missing observation, specified as a logical value. The value is true if the observation is missing.
Subset	Indicator 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

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. 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

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

Residuals — Residuals for fitted model

table

This property is read-only.

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

Field	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 chosen to remove skewness.
Deviance	Residuals 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

ResponseName — Response variable name

character vector

This property is read-only.

Response variable name, specified as a character vector.

Data Types: char

Rsquared — R-squared value for the 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 — Log-likelihood ratio

• Deviance — Deviance

• AdjGeneralized — Adjusted generalized R-squared

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

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.

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

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.

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

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).

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

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

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 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

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

Canonical Link Function

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

Canonical Link Functions for Generalized Linear Models

Distribution	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

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:

$$w_i=\frac{g^{\prime }\left({\mu }_i\right)}{\sqrt{V\left({\mu }_i\right)}},$$

where

• g is the link function mapping y_i to x_ib.

• $$g^{\prime }$$ 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

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

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 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.

Cook’s Distance

The Cook’s distance D_i of observation i is

$$D_i=w_i\frac{e_i^2}{p\widehat{\phi }}\frac{h_{ii}}{{\left(1-h_{ii}\right)}^2},$$

where

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

• e_i is the linear predictor residual, $$g\left(y_i\right)-x_i\widehat{\beta }$$, where

  • g is the link function.

  • y_i is the observed response.

  • x_i is the observation.

  • $$\widehat{\beta }$$ 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.

Deviance

Deviance of a model M₁ is twice the difference between the loglikelihood of that model and the saturated model, M_S. The saturated model is the model with the maximum number of parameters that can be estimated. For example, if there are 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. Then the deviance of model M₁ is

$$-2\left(\log L\left(b_1,y\right)-\log L\left(b_S,y\right)\right),$$

where b₁ are the estimated parameters for model M₁ and b_S are the estimated parameters for the saturated model. 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 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

$$\begin{array}{l}D=D_2-D_1=-2\left(\log L\left(b_2,y\right)-\log L\left(b_S,y\right)\right)+2\left(\log L\left(b_1,y\right)-\log L\left(b_S,y\right)\right)\\ =-2\left(\log L\left(b_2,y\right)-\log L\left(b_1,y\right)\right).\end{array}$$

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. 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 = D₂ – D₁.

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 A, B, and C and the response variable Y in the order A, B, C, and Y. Each row of T represents one term:

• [0 0 0 0] — Constant term or intercept

• [0 1 0 0] — B; equivalently, A^0 * B^1 * C^0

• [1 0 1 0] — A*C

• [2 0 0 0] — A^2

• [0 1 2 0] — B*(C^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.