MultinomialRegression
Description
MultinomialRegression
is a fitted multinomial regression model
object. A multinomial regression model describes the relationship between predictors and a
response that has a finite set of values.
Use the properties of a MultinomialRegression
object to investigate a
fitted multinomial regression model. The object properties include information about
coefficient estimates, summary statistics, and the data used to fit the model. Use the object
functions to predict responses, and to evaluate and visualize the multinomial regression
model.
Creation
Create a MultinomialRegression
model object with specified parameter
values by using fitmnr
.
Properties
Coefficient Estimates
ClassNames
— Names of response variable categories
categorical array  character array  logical vector  numeric vector  cell array of character vectors
This property is readonly.
Names of the response variable categories used to fit the multinomial regression
model, specified as a kby1 categorical array, character array,
logical vector, numeric vector, or cell array of character vectors.
k is the number of response categories.
ClassNames
has the same data type as the response category
labels. Note that the software treats string arrays as cell arrays of character
vectors. The ClassNames
property is set by the fitmnr
input argument Y
or Tbl
when you create the
model object.
Data Types: single
 double
 logical
 char
 cell
 categorical
CoefficientCovariance
— Covariance matrix for model coefficients
(p+1)by(p+1) matrix of numeric
values
This property is readonly.
Covariance matrix for model coefficients, specified as a (p+1)by(p+1) matrix of numeric values. p is the number of predictor variables.
For details, see Coefficient Standard Errors and Confidence Intervals.
Data Types: single
 double
CoefficientNames
— Coefficient names
cell array of character vectors
This property is readonly.
Coefficient names, specified as a cell array of character vectors, each containing
the name of the corresponding coefficient. Each coefficient name is the name of a
response category appended to the name of a predictor or intercept. This property is
set by the fitmnr
input argument Tbl
or namevalue argument
PredictorNames
when you create the model object.
Data Types: cell
Coefficients
— Coefficient values
table
This property is readonly.
Coefficient values, specified as a table that contains one row for each coefficient and these columns:
Value
— Estimated coefficient valueSE
— Standard error of the estimatetStat
— tstatistic for a twosided test with the null hypothesis that the coefficient is zeropValue
— pvalue for the tstatistic
Use coefTest
or
testDeviance
to perform other tests on the coefficients. Use coefCI
to
find the confidence intervals of the coefficient estimates.
Data Types: table
IncludeClassInteractions
— Indicator for interaction between response categories and coefficients
true
or 1
 false
or 0
This property is readonly.
Indicator for an interaction between response categories and coefficients,
specified as a numeric or logical 1
(true
) or
0
(false
). This property is set by the
fitmnr
namevalue argument IncludeClassInteractions
when you create the
model object.
Data Types: logical
Link
— Link function
'logit'
 'probit'
 'comploglog'
 'loglog'
This property is readonly.
Link function to use for ordinal and hierarchical models, specified as
'logit'
, 'probit'
,
'comploglog'
, or 'loglog'
. For nominal models,
Link
is always 'logit'
. This property is set
by the fitmnr
namevalue argument Link
when you create the model object.
Data Types: char
ModelType
— Type of model
'nominal'
 'ordinal'
 'hierarchical'
This property is readonly.
Type of model, specified as 'nominal'
,
'ordinal'
, or 'hierarchical'
. This property is
set by the fitmnr
namevalue argument ModelType
when you create the model
object.
Data Types: char
NumCoefficients
— Number of model coefficients
positive integer
This property is readonly.
Number of model coefficients, specified as a positive integer.
Data Types: double
Summary Statistics
Deviance
— Deviance of fit
numeric value
This property is readonly.
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 chisquare distribution with degrees of freedom equal to the difference in the number of estimated parameters between the two models. For more information, see Deviance.
Data Types: single
 double
DFE
— Degrees of freedom for error
positive integer
This property is readonly.
Degrees of freedom for the error (residuals), specified as a positive integer. For
nominal and ordinal models, DFE
is given by
$$DFE=n*(k1)N,$$
where n is the number of observations,
k is the number of response categories, and N
is the number of model coefficients. For hierarchical models, DFE
is given by
$$DFE=nN,$$
when IncludeClassInteractions is false
. When
IncludeClassInteractions
is true
,
DFE
for a hierarchical model is given by
$$DFE=\left({\displaystyle \sum _{i=1}^{k1}{n}_{i}}\right)N,$$
where n_{i} is the number of observations corresponding to the ith response category and above.
Data Types: double
Dispersion
— Variance
numeric scalar
This property is readonly.
Variance, specified as a numeric scalar. If you set the fitmnr
EstimateDispersion
namevalue argument to true
when you create the model object, the function estimates the standard error as the
Dispersion
value. Otherwise, fitmnr
assigns the default theoretical value of 1 to Dispersion
.
Data Types: single
 double
EstimateDispersion
— Indicator for whether dispersion is estimated
false
 true
This property is readonly.
Indicator for whether dispersion is estimated, specified as a logical
false
or true
. This property is set by the
fitmnr
EstimateDispersion
namevalue argument when you create the model
object.
Data Types: single
 double
 logical
Fitted
— Fitted response values based on input data
categorical array  character array  logical vector  numeric vector  cell array of character vectors
This property is readonly.
Fitted (predicted) response values based on the input data, specified as an
nby1 categorical array, character array, logical vector,
numeric vector, or cell array of character vectors. n is the number
of observations in the input data. Fitted
has the same data type
as the response category labels. Note that the software treats string arrays as cell
arrays of character vectors. Use predict
to
compute the predictions for other predictor values, or to compute the confidence
bounds on Fitted
.
Data Types: single
 double
 logical
 char
 cell
 categorical
LogLikelihood
— Loglikelihood of fitted model
numeric value
This property is readonly.
Loglikelihood of the fitted model, specified as a numeric value, based on the
assumption that each response value follows a multinomial distribution. When you
create the model object, fitmnr
calculates the loglikelihood of the model by taking the sum of the log probabilities
for the response data.
Data Types: single
 double
ModelCriterion
— Criterion for model comparison
structure
This property is readonly.
Criterion for model comparison, specified as a structure with these fields:
AIC
— Akaike information criterion.AIC = –2*lnL + 2*m
, wherelnL
is the loglikelihood andm
is the number of estimated parameters.AICc
— Akaike information criterion corrected for the sample size.AICc = AIC + (2*m*(m + 1))/(n – m – 1)
, wheren
is the number of observations.BIC
— Bayesian information criterion.BIC = –2*lnL + m*ln(n)
.CAIC
— Consistent Akaike information criterion.CAIC = –2*lnL + m*(ln(n) + 1)
.
Information criteria are model selection tools you can use to compare multiple models that are fit to the same data. These criteria are likelihoodbased 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 bestfitting model. The bestfitting model can vary depending on the criterion used for model comparison.
Data Types: struct
Residuals
— Residuals for fitted model
table
This property is readonly.
Residuals for the fitted model, specified as a table in which each variable contains one row for each observation and one column for each response class.
Column  Description 

Raw  Raw residuals. Observed minus fitted values, $${r}_{ij}={y}_{ij}{\widehat{\pi}}_{ij}*{m}_{i},\text{\hspace{1em}}\{\begin{array}{c}i=1,\cdots ,n\\ j=1,\cdots ,N\end{array}.$$

Pearson  Raw residuals divided by the root mean squared error (RMSE) 
Deviance  Deviance residuals given by the formula $$r{d}_{i}=2*{\displaystyle {\sum}_{j}^{k}{y}_{ij}*\mathrm{log}\left(\frac{{y}_{ij}}{{\widehat{\pi}}_{ij}*{m}_{i}}\right)},\text{\hspace{1em}}i=1,\cdots ,n.$$ 
Rows not used in the fit because of missing values contain NaN
values. To inspect missing values, see ObservationInfo.
Use plotResiduals
to create a plot of the residuals. For
details, see Residuals.
Data Types: table
Rsquared
— Pseudo Rsquared values for the fitted model
structure
This property is readonly.
Pseudo Rsquared values for the fitted model, specified as a structure. Each field
of Rsquared
contains a pseudo Rsquared value calculated with a
different formula [1].
Field  Description 

'Ordinary'  The ordinary pseudo Rsquared value is $${R}^{2}=1\frac{\mathrm{ln}({L}_{Full})}{\mathrm{ln}({L}_{Null})},$$ where $${L}_{Full}$$ is the loglikelihood of the fitted model and $${L}_{Null}$$ is the loglikelihood of a model with no predictors. 
'Adjusted'  The adjusted pseudo Rsquared value is $${R}^{2}=1\frac{\mathrm{ln}({L}_{Full})K}{\mathrm{ln}({L}_{Null})},$$ where K is the number of model coefficients in $${L}_{Full}$$. 
Data Types: struct
Input Data
Formula
— Regression model
LinearFormula
object
This property is readonly.
Regression model, specified as a LinearFormula
object. This
property is set by the fitmnr
input argument Formula
when you create the model object.
NumObservations
— Number of observations
positive integer
This property is readonly.
Number of observations used by the fitting algorithm to fit the model, specified
as a positive integer. NumObservations
is the number of
observations supplied in the original table or matrix, minus any rows with missing
values.
Data Types: double
NumPredictors
— Number of predictor variables
positive integer
This property is readonly.
Number of predictor variables used by the fitting algorithm to fit the model, specified as a positive integer.
Data Types: double
NumVariables
— Number of variables
positive integer
This property is readonly.
Number of variables in the input data, specified as a positive integer.
NumVariables
includes any variables that are not used as
predictors or as the response to fit the model.
Data Types: double
ObservationInfo
— Observation information
nby3 table
This property is readonly.
Observation information, specified as an nby3 table containing the following columns, where n is the number of observations.
Column  Description 

Weights  Observation weights, specified as a numeric value. The default value is
1 . 
Missing  Indicator of missing observations, specified as a logical value. The
value is true if the observation is missing. 
Subset  Indicator of whether fitmnr uses the observation, specified as a logical value. The
value is true if the observation is not missing, meaning
fitmnr uses the observation. 
Data Types: table
ObservationNames
— Observation names
cell array of character vectors
This property is readonly.
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, the
ObservationNames
property contains those names.Otherwise,
ObservationNames
is an empty cell array.
This property is set by the fitmnr
input argument Tbl
when you create the model object and assign
row names to Tbl
.
Data Types: cell
PredictorNames
— Names of predictors used to fit model
cell array of character vectors
This property is readonly.
Names of the predictors used to fit the model, specified as a cell array of
character vectors. This property is set by one of the following fitmnr
arguments when you create the model object:
Tbl
input argumentX
input argument together with thePredictorNames
namevalue argument
Data Types: cell
ResponseName
— Response variable name
character vector
This property is readonly.
Response variable name, specified as a character vector. This property is set by
one of the following fitmnr
arguments when you create the model object:
ResponseName
namevalue argumentTbl
input argument together with theResponseVarName
input argumentTbl
input argument together with theFormula
input argument
Data Types: char
VariableInfo
— Information about variables
table
This property is readonly.
Information about the variables contained in the Variables property, specified as a table with one row for each variable and the following columns.
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

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 as
predictors or as the response to fit the model.
Data Types: table
VariableNames
— Names of variables
cell array of character vectors
This property is readonly.
Names of the variables, specified as a cell array of character vectors. Elements
of this property are set by one of the following fitmnr
arguments when you create the model object:
The
Tbl
input argument specifies the names of the predictor variables, response, and unused variables.The
PredictorNames
namevalue argument specifies the names of the predictor variables.The
ResponseVarName
namevalue argument specifies the name of the response variable.
VariableNames
also includes any variables that are not used as
predictors or as the response to fit the model.
Data Types: cell
Variables
— Input data
table
This property is readonly.
Input data, specified as a table. Variables
contains both
predictor and response values. Elements of this property are set by one of the
following fitmnr
arguments when you create the model object:
If you specify
X
, thenVariables
contains all variables in the columns ofX
.If you specify
Tbl
, thenVariables
contains all variables inTbl
, including variables not used as predictor or response data to fit the model.If you specify
Y
, thenVariables
also contains the response data inY
.
Data Types: table
Object Functions
coefCI  Confidence intervals for coefficient estimates of multinomial regression model 
coefTest  Linear hypothesis test on multinomial regression model coefficients 
feval  Predict responses of multinomial regression model using one input for each predictor 
partialDependence  Compute partial dependence 
plotPartialDependence  Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots 
plotResiduals  Plot residuals of multinomial regression model 
plotSlice  Plot of slices through fitted multinomial regression surface 
predict  Predict responses of multinomial regression model 
random  Generate random responses from fitted multinomial regression model 
testDeviance  Deviance test for multinomial regression model 
Examples
Analyze Nominal Model Coefficients
Load the fisheriris
sample data set.
load fisheriris
The column vector species
contains iris flowers of three different species: setosa, versicolor, virginica. The matrix meas
contains four types of measurements for the flower: the length and width of sepals and petals in centimeters.
Fit a multinomial regression model to predict the iris flower species using the measurements. Display the results of the fit using the Coefficients
property of the fitted model.
MnrModel = fitmnr(meas,species); MnrModel.Coefficients
ans=10×4 table
Value SE tStat pValue
_______ ______ _______ __________
(Intercept_setosa) 2018.4 12.404 162.72 0
x1_setosa 673.85 3.5783 188.32 0
x2_setosa 568.2 3.176 178.9 0
x3_setosa 516.44 3.5403 145.87 0
x4_setosa 2760.9 7.1203 387.75 0
(Intercept_versicolor) 42.638 5.2719 8.0878 6.0776e16
x1_versicolor 2.4652 1.1228 2.1956 0.028124
x2_versicolor 6.6809 1.4789 4.5176 6.2559e06
x3_versicolor 9.4294 1.2934 7.2906 3.086e13
x4_versicolor 18.286 2.0967 8.7214 2.7476e18
MnrModel
is a multinomial regression model object that contains the results of fitting a nominal multinomial regression model to the data. The Coefficients
property contains coefficient statistics for each predictor in meas
. The small pvalues in the column pValue
indicate that all coefficients are statistically significant at the 95% confidence level. fitmnr
sorts the categories in species
in order of their first appearance. The last category is the default reference category.
To display the sorted names of the response variable categories, use the ClassNames
property of MnrModel
.
MnrModel.ClassNames
ans = 3x1 cell
{'setosa' }
{'versicolor'}
{'virginica' }
The output shows that the last category, 'virginica'
, is the reference category by default.
To get 95% confidence intervals for the fitted coefficient estimates, call the object function coefCI
.
coefCI(MnrModel)
ans = 10×2
10^{3} ×
1.9940 2.0428
0.6668 0.6809
0.5745 0.5620
0.5234 0.5095
2.7749 2.7469
0.0323 0.0530
0.0003 0.0047
0.0038 0.0096
0.0120 0.0069
0.0224 0.0142
The output shows 95% confidence intervals for the 10 coefficients in the Value
column of the Coefficients
table. None of the confidence intervals cross zero, confirming that all coefficients affect the log odds at the 95% confidence level.
Predict Response Categories
Load the fisheriris
sample data set.
load fisheriris
The column vector species
contains three iris flowers species: setosa, versicolor, and virginica. The matrix meas
contains four types of measurements for the flower: the length and width of sepals and petals in centimeters.
Divide the species and measurement data into training and test data by using the cvpartition
function. Get the indices of the training data rows by using the training
function.
n = length(species);
partition = cvpartition(n,'Holdout',0.05);
idx_train = training(partition);
Create training data by using the indices of the training data rows to create a matrix of measurements and a vector of species labels.
meastrain = meas(idx_train,:); speciestrain = species(idx_train,:);
Fit a multinomial regression model using the training data.
mdl = fitmnr(meastrain,speciestrain)
mdl = Multinomial regression with nominal responses Value SE tStat pValue _______ ______ ________ __________ (Intercept_setosa) 86.305 12.541 6.8817 5.9158e12 x1_setosa 1.0728 3.5795 0.29971 0.7644 x2_setosa 23.846 3.1238 7.6336 2.2835e14 x3_setosa 27.289 3.5009 7.795 6.4409e15 x4_setosa 59.58 7.0214 8.4855 2.1472e17 (Intercept_versicolor) 42.637 5.2214 8.1659 3.1906e16 x1_versicolor 2.4652 1.1263 2.1887 0.028619 x2_versicolor 6.6808 1.474 4.5325 5.829e06 x3_versicolor 9.4292 1.2946 7.2837 3.248e13 x4_versicolor 18.286 2.0833 8.7775 1.671e18 143 observations, 276 error degrees of freedom Dispersion: 1 Chi^2statistic vs. constant model: 302.0378, pvalue = 1.5168e60
mdl
is a multinomial regression model object that contains the results of fitting a nominal multinomial regression model to the data. The table output shows coefficient statistics for each predictor in meas
. By default, fitmnr
uses virginica
as the reference category.
Get the indices of the test data rows by using the test
function. Create test data by using the indices of the test data rows to create a matrix of measurements and a vector of species labels.
idx_test = test(partition); meastest = meas(idx_test,:); speciestest = species(idx_test,:);
Predict the iris species for the measurements in meastest
.
speciespredict = predict(mdl,meastest)
speciespredict = 7x1 cell
{'setosa' }
{'setosa' }
{'setosa' }
{'setosa' }
{'setosa' }
{'versicolor'}
{'versicolor'}
Compare the predictions in speciespredict
with the category names in speciestest
.
speciestest
speciestest = 7x1 cell
{'setosa' }
{'setosa' }
{'setosa' }
{'setosa' }
{'setosa' }
{'versicolor'}
{'versicolor'}
The output shows that the model accurately predicts the iris species for the measurements in meastest
.
Plot Engine Cylinder Probabilities
Load the carbig
sample data set.
load carbig;
The vectors Acceleration
and Displacement
contain data for car acceleration and displacement, respectively. The vector Cylinders
contains data for the number of cylinders in each car engine.
Fit an ordinal multinomial regression model using Acceleration
and Displacement
as predictor variables and Cylinders
as the response variable.
MnrModel = fitmnr([Acceleration,Displacement],Cylinders,Model="ordinal",... PredictorNames=["Acceleration" "Displacement"])
MnrModel = Multinomial regression with ordinal responses Value SE tStat pValue _________ ________ _______ __________ (Intercept_3) 11.949 3.1817 3.7555 0.00017299 (Intercept_4) 27.08 4.9481 5.4727 4.4321e08 (Intercept_5) 27.528 4.9738 5.5346 3.1195e08 (Intercept_6) 45.346 7.8292 5.7919 6.9593e09 Acceleration 0.063533 0.1041 0.6103 0.54167 Displacement 0.16731 0.027885 6 1.9726e09 406 observations, 1618 error degrees of freedom Dispersion: 1 Chi^2statistic vs. constant model: 786.5846, pvalue = 1.5679e171
MnrModel
is a multinomial regression model object that contains the results of fitting an ordinal multinomial regression model to the data. The table output shows coefficient statistics for each predictor variable. The pvalues in the column pValue
indicate that there is not enough evidence to conclude that the coefficient for the Acceleration
term is statistically significant. However, enough evidence exists to conclude that Displacement
has a statistically significant effect at the 99% confidence level.
Display the possible quantities for car engine cylinders using the ClassNames
property.
MnrModel.ClassNames
ans = 5×1
3
4
5
6
8
The last category in the output is the default reference category. The output shows that the reference category corresponds to cars with eightcylinder engines.
Use plotSlice
to plot stacked histograms of the probabilities of a car having each number of cylinders as the value of the predictor variable Displacement
changes. By default, plotSlice
fixes the value of Acceleration
at its training data mean.
plotSlice(MnrModel,"stackedhist",PredictorToVary="Displacement") hold on lgd = legend; title(lgd, "Number of cylinders");
The plot shows that the probability of a car having more cylinders increases as the car displacement increases, which is consistent with the small pvalue for the Displacement
model term.
Analyze Effect of Car Displacement on Reference Category Probability
Load the carbig
sample data set.
load carbig;
The vectors Acceleration
and Displacement
contain data for car acceleration and displacement, respectively. The vector Cylinders
contains data for the number of cylinders in each car engine.
Fit an ordinal multinomial regression model using Acceleration
and Displacement
as predictor variables and Cylinders
as the response variable.
MnrModel = fitmnr([Acceleration,Displacement],Cylinders,Model="ordinal",... PredictorNames=["Acceleration" "Displacement"])
MnrModel = Multinomial regression with ordinal responses Value SE tStat pValue _________ ________ _______ __________ (Intercept_3) 11.949 3.1817 3.7555 0.00017299 (Intercept_4) 27.08 4.9481 5.4727 4.4321e08 (Intercept_5) 27.528 4.9738 5.5346 3.1195e08 (Intercept_6) 45.346 7.8292 5.7919 6.9593e09 Acceleration 0.063533 0.1041 0.6103 0.54167 Displacement 0.16731 0.027885 6 1.9726e09 406 observations, 1618 error degrees of freedom Dispersion: 1 Chi^2statistic vs. constant model: 786.5846, pvalue = 1.5679e171
MnrModel
is a multinomial regression model object that contains the results of fitting an ordinal multinomial regression model to the data. The table output shows coefficient statistics for each of the predictor variable. The pvalues in the column pValue
indicate that there is not enough evidence to conclude that the coefficient for the Acceleration
term is statistically significant. However, enough evidence exists to conclude that Displacement
has a statistically significant effect at the 99% confidence level.
Display the possible quantities for car engine cylinders using the ClassNames
property.
MnrModel.ClassNames
ans = 5×1
3
4
5
6
8
The reference category corresponds to cars with eightcylinder engines.
Plot the partial dependence of the reference category probability on the Displacement
predictor by using the plotPartialDependence
object function.
plotPartialDependence(MnrModel,2,8)
The plot shows that the probability of a car being in the reference category increases sharply when the value of Displacement
reaches approximately 250.
More About
Deviance
Deviance is a generalization of the residual sum of squares. It measures the goodness of fit compared to a saturated model.
The deviance of a model M_{1} is twice the difference between the loglikelihood of the model M_{1} and the saturated model M_{s}. A saturated model is a model with the maximum number of parameters that you can estimate.
For example, if you have n observations with potentially different response values y_{i}, i = 1, 2, ..., n, then you can define a saturated model (with n parameters) that perfectly predicts the responses. Let L(b,y) denote the maximum value of the likelihood function for a model with the parameters b. Then the deviance of the model M_{1} is
$$2\left(\mathrm{log}L\left({b}_{1},y\right)\mathrm{log}L\left({b}_{S},y\right)\right),$$
where b_{1} and b_{s} contain the estimated parameters for the model M_{1} and the saturated model, respectively. The deviance has a chisquare distribution with n – p degrees of freedom, where n is the number of parameters in the saturated model and p is the number of parameters in the model M_{1}.
Assume you have two different multinomial regression models M_{1} and M_{2}, and M_{1} has a subset of the terms in M_{2}. You can evaluate the fit of the models by comparing the deviances D_{1} and D_{2} of the two models. The difference of the deviances is
$$\begin{array}{l}D={D}_{2}{D}_{1}=2\left(\mathrm{log}L\left({b}_{2},y\right)\mathrm{log}L\left({b}_{S},y\right)\right)+2\left(\mathrm{log}L\left({b}_{1},y\right)\mathrm{log}L\left({b}_{S},y\right)\right)\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=2\left(\mathrm{log}L\left({b}_{2},y\right)\mathrm{log}L\left({b}_{1},y\right)\right).\end{array}$$
Asymptotically, the difference D has a chisquare distribution with
degrees of freedom v equal to the difference in the number of
parameters estimated in M_{1} and
M_{2}. You can obtain the
pvalue for this test by using
1 – chi2cdf(D,v,"upper")
.
Typically, you examine D using a model M_{2} with a constant term and no predictors. Therefore, D has a chisquare distribution with p – 1 degrees of freedom. If the dispersion is estimated, the difference divided by the estimated dispersion has an F distribution with p – 1 numerator degrees of freedom and n – p denominator degrees of freedom.
References
[1] Allison, P. D. "Measures of Fit for Logistic Regression." Statistical Horizons LLC and the University of Pennsylvania, 2014.
[2] McCullagh, P., and J. A. Nelder. Generalized Linear Models. New York: Chapman & Hall, 1990.
[3] Long, J. S. Regression Models for Categorical and Limited Dependent Variables. Sage Publications, 1997.
[4] Dobson, A. J., and A. G. Barnett. An Introduction to Generalized Linear Models. Chapman and Hall/CRC. Taylor & Francis Group, 2008.
Version History
Introduced in R2023a
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