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 read-only.
Names of the response variable categories used to fit the multinomial regression
model, specified as a k-by-1 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 read-only.
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 read-only.
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 name-value argument
PredictorNames
when you create the model object.
Data Types: cell
Coefficients
— Coefficient values
table
This property is read-only.
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
— t-statistic for a two-sided test with the null hypothesis that the coefficient is zeropValue
— p-value for the t-statistic
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 read-only.
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
name-value argument IncludeClassInteractions
when you create the
model object.
Data Types: logical
Link
— Link function
'logit'
| 'probit'
| 'comploglog'
| 'loglog'
This property is read-only.
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
name-value argument Link
when you create the model object.
Data Types: char
ModelType
— Type of model
'nominal'
| 'ordinal'
| 'hierarchical'
This property is read-only.
Type of model, specified as 'nominal'
,
'ordinal'
, or 'hierarchical'
. This property is
set by the fitmnr
name-value argument ModelType
when you create the model
object.
Data Types: char
NumCoefficients
— Number of model coefficients
positive integer
This property is read-only.
Number of model coefficients, specified as a positive integer.
Data Types: double
Summary Statistics
Deviance
— Deviance of fit
numeric value
This property is read-only.
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
DFE
— Degrees of freedom for error
positive integer
This property is read-only.
Degrees of freedom for the error (residuals), specified as a positive integer. For
nominal and ordinal models, DFE
is given by
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
when IncludeClassInteractions is false
. When
IncludeClassInteractions
is true
,
DFE
for a hierarchical model is given by
where ni is the number of observations corresponding to the ith response category and above.
Data Types: double
Dispersion
— Variance
numeric scalar
This property is read-only.
Variance, specified as a numeric scalar. If you set the fitmnr
EstimateDispersion
name-value 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 read-only.
Indicator for whether dispersion is estimated, specified as a logical
false
or true
. This property is set by the
fitmnr
EstimateDispersion
name-value 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 read-only.
Fitted (predicted) response values based on the input data, specified as an
n-by-1 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 read-only.
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 read-only.
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 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.
Data Types: struct
Residuals
— Residuals for fitted model
table
This property is read-only.
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,
|
Pearson | Raw residuals divided by the root mean squared error (RMSE) |
Deviance | Deviance residuals given by the formula |
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 R-squared values for the fitted model
structure
This property is read-only.
Pseudo R-squared values for the fitted model, specified as a structure. Each field
of Rsquared
contains a pseudo R-squared value calculated with a
different formula [1].
Field | Description |
---|---|
'Ordinary' | The ordinary pseudo R-squared value is where is the loglikelihood of the fitted model and is the loglikelihood of a model with no predictors. |
'Adjusted' | The adjusted pseudo R-squared value is where K is the number of model coefficients in . |
Data Types: struct
Input Data
Formula
— Regression model
LinearFormula
object
This property is read-only.
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 read-only.
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 read-only.
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 read-only.
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
n-by-3 table
This property is read-only.
Observation information, specified as an n-by-3 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 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, 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 read-only.
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
name-value argument
Data Types: cell
ResponseName
— Response variable name
character vector
This property is read-only.
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
name-value 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 read-only.
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 read-only.
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
name-value argument specifies the names of the predictor variables.The
ResponseVarName
name-value 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 read-only.
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 6.0776e-16
x1_versicolor 2.4652 1.1228 2.1956 0.028124
x2_versicolor 6.6809 1.4789 4.5176 6.2559e-06
x3_versicolor -9.4294 1.2934 -7.2906 3.086e-13
x4_versicolor -18.286 2.0967 -8.7214 2.7476e-18
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 p-values 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
103 ×
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 5.9158e-12 x1_setosa -1.0728 3.5795 -0.29971 0.7644 x2_setosa 23.846 3.1238 7.6336 2.2835e-14 x3_setosa -27.289 3.5009 -7.795 6.4409e-15 x4_setosa -59.58 7.0214 -8.4855 2.1472e-17 (Intercept_versicolor) 42.637 5.2214 8.1659 3.1906e-16 x1_versicolor 2.4652 1.1263 2.1887 0.028619 x2_versicolor 6.6808 1.474 4.5325 5.829e-06 x3_versicolor -9.4292 1.2946 -7.2837 3.248e-13 x4_versicolor -18.286 2.0833 -8.7775 1.671e-18 143 observations, 276 error degrees of freedom Dispersion: 1 Chi^2-statistic vs. constant model: 302.0378, p-value = 1.5168e-60
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 4.4321e-08 (Intercept_5) 27.528 4.9738 5.5346 3.1195e-08 (Intercept_6) 45.346 7.8292 5.7919 6.9593e-09 Acceleration -0.063533 0.1041 -0.6103 0.54167 Displacement -0.16731 0.027885 -6 1.9726e-09 406 observations, 1618 error degrees of freedom Dispersion: 1 Chi^2-statistic vs. constant model: 786.5846, p-value = 1.5679e-171
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 p-values 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 eight-cylinder 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 p-value 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 4.4321e-08 (Intercept_5) 27.528 4.9738 5.5346 3.1195e-08 (Intercept_6) 45.346 7.8292 5.7919 6.9593e-09 Acceleration -0.063533 0.1041 -0.6103 0.54167 Displacement -0.16731 0.027885 -6 1.9726e-09 406 observations, 1618 error degrees of freedom Dispersion: 1 Chi^2-statistic vs. constant model: 786.5846, p-value = 1.5679e-171
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 p-values 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 eight-cylinder 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 M1 is twice the difference between the loglikelihood of the model M1 and the saturated model Ms. 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 yi, 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 M1 is
where b1 and bs contain the estimated parameters for the model M1 and the saturated model, respectively. 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 the model M1.
Assume you have two different multinomial regression models M1 and M2, and M1 has a subset of the terms in M2. You can evaluate the fit of the models by comparing the deviances D1 and D2 of the two models. The difference of the deviances is
Asymptotically, the difference D has a chi-square distribution with
degrees of freedom v equal to the difference in the number of
parameters estimated in M1 and
M2. You can obtain the
p-value for this test by using
1 – chi2cdf(D,v,"upper")
.
Typically, you examine D using a model M2 with a constant term and no predictors. Therefore, D has a chi-square 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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