fitmnr

Fit multinomial regression model

Since R2023a

Syntax

MnrMdl = fitmnr(X,Y)

MnrMdl = fitmnr(Tbl,Y)

MnrMdl = fitmnr(Tbl,ResponseVarName)

MnrMdl = fitmnr(Tbl,Formula)

MnrMdl = fitmnr(___,Name=Value)

Description

MnrMdl = fitmnr(X,Y) fits a multinomial regression model and returns a MultinomialRegression object MnrMdl for the predictor data in the matrix X and the response data in Y. The input Y can be a matrix of counts or a vector of response variable classes.

MnrMdl = fitmnr(Tbl,Y) uses the variables in Tbl as the predictor data. Each table variable corresponds to a different predictor variable.

MnrMdl = fitmnr(Tbl,ResponseVarName) uses the variables in Tbl as the predictor and response data. The ResponseVarName argument specifies which variable contains the response data.

MnrMdl = fitmnr(Tbl,Formula) specifies the multinomial regression model in Wilkinson Notation. The terms in Formula use only the variable names in Tbl.

MnrMdl = fitmnr(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in the previous syntaxes. For example, you can specify the link function and the model type.

Examples

Fit Nominal Multinomial Regression Model

Load the fisheriris data set.

load fisheriris

The column vector species contains iris flowers of three different species: setosa, versicolor, virginica. The double matrix meas contains of four types of measurements for the flowers: the length and width of sepals and petals in centimeters. The data in species is nominal, meaning that no natural order exists among the iris species.

Use the unique function to display the iris species in order of their first appearance in the species vector.

unique(species)

ans = 3x1 cell
    {'setosa'    }
    {'versicolor'}
    {'virginica' }

The output shows that virginica is the last species in the species vector.

Fit a nominal multinomial regression model for the predictor data in meas and the response data in species. By default, the fitmnr function uses virginica as the reference category because it appears last in species.

MnrModel = fitmnr(meas,species)

MnrModel = 
Multinomial regression with nominal responses

                               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


150 observations, 290 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 317.6851, p-value = 7.0555e-64

MnrModel is a multinomial regression model object that contains the results of fitting a nominal multinomial regression model to the data. The output shows 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.

Investigate the general form of the model equations by using the Formula property of MnrModel.

MnrModel.Formula

ans = 
logit(y) ~ 1 + x1 + x2 + x3 + x4

The output shows that the link function for MnrModel is the default multinomial logit link function. You can use the fitted model coefficients and the general form of the model equations to write these fitted model equations:

$$\ln \left(\frac{{\pi }_{setosa}}{{\pi }_{virginica}}\right)=2018.4+673.9x_1-568.2x_2-516.4x_3-2760.9x_4$$

$$\ln \left(\frac{{\pi }_{versicolor}}{{\pi }_{virginica}}\right)=42.6+2.5x_1+6.7x_2-9.4x_3-18.3x_4$$

The equations describe the effects of the predictor variables on the quotient of the probability of a flower being in a category by the probability of a flower being in the reference category. For example, the estimated coefficient 2.5 in the second equation indicates that the probability of a flower being versicolor divided by the probability of a flower being virginica increases exp(2.5) times for each unit increase in $$x_1$$. For more information about nominal multinomial regression model equations, see Multinomial Models for Nominal Responses.

Fit Multinomial Regression Model with Interactions

Load the fisheriris data set.

load fisheriris

The column vector species contains iris flowers of three different species: setosa, versicolor, virginica. The double matrix meas contains of four types of measurements for the flowers: the length and width of sepals and petals in centimeters.

Create a table containing the flower measurements and the species data by using the table function.

tbl = table(meas(:,1),meas(:,2),meas(:,3),meas(:,4),species,...
    VariableNames=["Sepal_Length","Sepal_Width","Petal_Length","Petal_Width","Species"])

tbl=150×5 table
    Sepal_Length    Sepal_Width    Petal_Length    Petal_Width     Species  
    ____________    ___________    ____________    ___________    __________

        5.1             3.5            1.4             0.2        {'setosa'}
        4.9               3            1.4             0.2        {'setosa'}
        4.7             3.2            1.3             0.2        {'setosa'}
        4.6             3.1            1.5             0.2        {'setosa'}
          5             3.6            1.4             0.2        {'setosa'}
        5.4             3.9            1.7             0.4        {'setosa'}
        4.6             3.4            1.4             0.3        {'setosa'}
          5             3.4            1.5             0.2        {'setosa'}
        4.4             2.9            1.4             0.2        {'setosa'}
        4.9             3.1            1.5             0.1        {'setosa'}
        5.4             3.7            1.5             0.2        {'setosa'}
        4.8             3.4            1.6             0.2        {'setosa'}
        4.8               3            1.4             0.1        {'setosa'}
        4.3               3            1.1             0.1        {'setosa'}
        5.8               4            1.2             0.2        {'setosa'}
        5.7             4.4            1.5             0.4        {'setosa'}
      ⋮

Fit a multinomial regression model to the flower data using the measurements as the predictor data and the species as the response data. By default, fitmnr uses virginica as the reference category because it appears last the Species column of tbl. Specify the formula for the regression model to investigate whether the interaction between sepal width and sepal length is statistically significant.

MnrModel = fitmnr(tbl,"Species ~ Petal_Length*Sepal_Length + Petal_Width + Sepal_Width")

MnrModel = 
Multinomial regression with nominal responses

                                             Value       SE       tStat        pValue   
                                            _______    ______    ________    ___________

    (Intercept_setosa)                       232.64    50.281      4.6268     3.7137e-06
    Sepal_Length_setosa                      11.918     10.31       1.156        0.24768
    Sepal_Width_setosa                       27.965    4.5863      6.0975     1.0775e-09
    Petal_Length_setosa                     -5.2318     13.66    -0.38302        0.70171
    Petal_Width_setosa                      -260.77    9.3403     -27.919    1.5677e-171
    Sepal_Length:Petal_Length_setosa        -10.453    2.2141     -4.7211     2.3452e-06
    (Intercept_versicolor)                  -231.52    43.626     -5.3068     1.1158e-07
    Sepal_Length_versicolor                  48.459     8.178      5.9255     3.1133e-09
    Sepal_Width_versicolor                   7.5712    1.7854      4.2406     2.2288e-05
    Petal_Length_versicolor                  47.602    9.5989      4.9591     7.0832e-07
    Petal_Width_versicolor                  -20.603    2.7859     -7.3955     1.4089e-13
    Sepal_Length:Petal_Length_versicolor    -9.5086    1.7102     -5.5598     2.7005e-08


150 observations, 288 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 318.3928, p-value = 1.9971e-62

MnrModel is a multinomial regression model object that contains the results of fitting a (default) nominal multinomial regression model to the data. The output shows coefficient statistics for each predictor in meas. The small p-value for the interaction term in each log odds equation indicates that the interaction between sepal length and petal length has a statistically significant effect on the log quotient of the probability of a flower being in a category by the probability of a flower being in the reference category. For more information about nominal multinomial regression model equations, see Multinomial Models for Nominal Responses.

Fit Ordinal Multinomial Regression Model

Load the carbig data set.

load carbig

The variables Acceleration, Displacement, Horsepower, and Weight contain data for car acceleration, engine displacement, horsepower, and weight, respectively. The variable MPG contains car mileage data.

Sort the data in MPG into four categories using the discretize function. Create a table of predictor data using the table function.

mileage = discretize(MPG,[9,19,29,39,48],"categorical");
X = table(Acceleration,Displacement,Horsepower,Weight,mileage,VariableNames=["Acceleration","Displacement","Horsepower","Weight","Mileage"])

X=406×5 table
    Acceleration    Displacement    Horsepower    Weight      Mileage  
    ____________    ____________    __________    ______    ___________

          12            307            130         3504     [9, 19)    
        11.5            350            165         3693     [9, 19)    
          11            318            150         3436     [9, 19)    
          12            304            150         3433     [9, 19)    
        10.5            302            140         3449     [9, 19)    
          10            429            198         4341     [9, 19)    
           9            454            220         4354     [9, 19)    
         8.5            440            215         4312     [9, 19)    
          10            455            225         4425     [9, 19)    
         8.5            390            190         3850     [9, 19)    
        17.5            133            115         3090     <undefined>
        11.5            350            165         4142     <undefined>
          11            351            153         4034     <undefined>
        10.5            383            175         4166     <undefined>
          11            360            175         3850     <undefined>
          10            383            170         3563     [9, 19)    
      ⋮

The Mileage column of the table X contains entries that are undefined.

Fit an ordinal multinomial regression model using Acceleration, Displacement, Horsepower, and Weight as predictor variables and Mileage as the response variable. fitmnr ignores the rows of X containing the undefined entries in the Mileage column.

MnrModel = fitmnr(X,"Mileage",ModelType="ordinal")

MnrModel = 
Multinomial regression with ordinal responses

                              Value          SE         tStat       pValue  
                            _________    __________    _______    __________

    (Intercept_[9, 19))        -16.69        1.9529    -8.5459    1.2757e-17
    (Intercept_[19, 29))      -11.721         1.768    -6.6296    3.3667e-11
    (Intercept_[29, 39))      -8.0606        1.7297    -4.6601    3.1603e-06
    Acceleration              0.10476      0.079916     1.3109       0.18989
    Displacement             0.010336     0.0049035     2.1078      0.035045
    Horsepower                0.06452       0.01476     4.3712    1.2354e-05
    Weight                  0.0016638    0.00066089     2.5175      0.011821


392 observations, 1169 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 503.6344, p-value = 1.0964e-107

MnrModel is a multinomial regression model object that contains the results of fitting an ordinal multinomial regression model to the data. The output shows coefficient statistics for each predictor in X. By default, fitmnr uses the logit link function and calculates common coefficients among the ordinal response categories. This type of model is sometimes called a proportional odds model. At the 95% confidence level, the $$p$$-values of 0.035045, 1.2354e-05, and 0.011821 indicate that each of the variables Displacement, Horsepower, and Weight has a statistically significant effect on a log quotient of probabilities. The numerator is the probability of a car's mileage being less than or equal to a certain value, and the denominator is the probability of a car's mileage being greater than that value.

You can use the fitted model coefficients and the general form of the model equations to write these fitted model equations:

$$\begin{array}{l}\ln \left(\frac{P\left(X_M\le 19\right)}{P\left(X_M>19\right)}\right)=-16.69+0.10476X_A+0.010336X_D+0.06452X_H+0.0016638X_W\end{array}$$

$$\begin{array}{l}\ln \left(\frac{P\left(X_M\le 29\right)}{P\left(X_M>29\right)}\right)=-11.721+0.10476X_A+0.010336X_D+0.06452X_H+0.0016638X_W\end{array}$$

$$\begin{array}{l}\ln \left(\frac{P\left(X_M\le 39\right)}{P\left(X_M>39\right)}\right)=-8.0606+0.10476X_A+0.010336X_D+0.06452X_H+0.0016638X_W\end{array}$$

$$X_M$$, $$X_A$$, $$X_D$$, $$X_H$$, and $$X_W$$ represent the Mileage, Acceleration, Displacement, Horsepower, and Weight variables in X, respectively. For more information about the equations and the significance of terms in an ordinal multinomial regression model, see Multinomial Models for Ordinal Responses.

Input Arguments

X — Predictor data

n-by-p numeric matrix

Predictor data, specified as an n-by-p numeric matrix, where n is the number of observations in the data set and p is the number of predictor variables. Each row of X corresponds to an observation, and each column corresponds to a predictor variable.

Note

fitmnr automatically includes a constant term (intercept) in all models. Do not include a column of 1s in X.

Data Types: single | double

Y — Response category labels

n-by-k matrix | n-by-1 numeric vector | n-by-1 logical vector | n-by-1 string vector | n-by-1 categorical array | n-by-1 cell array of character vectors

Response category labels, specified as an n-by-k matrix, or an n-by-1 numeric vector, logical vector, string vector, categorical array, or cell array of character vectors.

• If Y is an n-by-k matrix, n is the number of observations in the data set, and k is the number of response categories. Y(i,j) is the number of outcomes of the multinomial category j for the predictor combinations given by X(i,:). In this case, the function determines the number of observations at each predictor combination.

• If Y is an n-by-1 numeric vector, logical vector, string vector, categorical array, or cell array of character vectors, the value of the vector or array indicates the response for each observation. In this case, all sample sizes are 1.

Data Types: single | double | logical | char | string | cell | categorical

Tbl — Predictor and response data

table

Predictor and response data, specified as a table. The variables of Tbl can contain numeric or categorical data. When you specify Tbl, you must also specify the response data using the input argument Y, ResponseVarName, or Formula.

• If you specify the response data in Y, the table variables represent only the predictor data for the multinomial regression. A row of predictor values in Tbl corresponds to the observation in Y at the same position. Tbl must have the same number of rows as the length of Y.

• If you do not specify Y, you must indicate which variable in Tbl contains the response data by using the ResponseVarName or Formula input argument. You can also choose a subset of factors in Tbl to use in the multinomial regression by setting the PredictorNames name-value argument. The fitmnr function associates the values of the predictor variables in Tbl with the response data in the same row.

Example: snake=table(weight,length,species); fitmnr(snake,"species")

Data Types: table

ResponseVarName — Name of variable to use as response

string scalar | character vector

Name of the variable to use as the response, specified as a string scalar or character vector. ResponseVarName indicates which variable in Tbl contains the response data. When you specify ResponseVarName, you must also specify the Tbl input argument.

Example: snake=table(weight,length,species); fitmnr(snake,"species")

Data Types: char | string

Formula — Multinomial regression model to fit

string scalar | character vector

Multinomial regression model to fit, specified as a string scalar or a character vector in Wilkinson notation.

Example: Specify the multinomial regression model as a sum of the variables meas1, meas2, and meas3 and their three-way interaction by using the formula "species ~ meas1 + meas2 + meas3 + meas1:meas2:meas3".

Data Types: char | string

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: fitmnr(X,Y,Link="probit",ModelType="ordinal",CategoricalPredictors=[1,2]) specifies "probit" as the link function for an ordinal multinomial regression model, and specifies the first two columns in X as categorical variables.

CategoricalPredictors — Predictors to treat as categorical

numeric vector | string vector | cell array of character vectors | "all"

Predictors to treat as categorical, specified as a numeric or string vector, a cell array of character vectors, or "all".

Specify CategoricalPredictors as one of the following:

• Numeric vector of indices between 1 and p, where p is the number of predictor variables. The index of a predictor variable is the order in which it appears in the columns of the matrix X or table Tbl.

• Logical vector of length p, where a true entry indicates that the corresponding predictor is categorical.

• String array or cell array of character vectors, where each element in the array is the name of a predictor variable. The predictor names must match the names in PredictorNames or Tbl.

• "all", indicating that all predictor variables are categorical.

By default, fitmnr treats numeric predictors as continuous, and predictor variables of other types as categorical. For more information about categorical data, see Natural Representation of Categorical Data.

Example: CategoricalPredictors=["Location" "Smoker"]

Example: CategoricalPredictors=[1 3 4]

Data Types: single | double | logical | string | cell

EstimateDispersion — Indicator for estimating dispersion parameter

false or 0 (default) | true or 1

Indicator for estimating a dispersion parameter, specified as a numeric or logical 0 (false) or 1 (true).

Value	Description
0 (false)	Use the theoretical dispersion value of 1 (default).
1 (true)	Estimate a dispersion parameter for the multinomial distribution for use in computing standard errors.

Example: EstimateDispersion=1

Data Types: single | double | logical

IncludeClassInteractions — Indicator for an interaction between response categories and coefficients

true or 1 | false or 0

Indicator for an interaction between response categories and coefficients, specified as a numeric or logical 1 (true) or 0 (false).

Value	Description
1 (true)	Default for nominal and hierarchical models. Fit a model with different coefficients across classes.
0 (false)	Default for ordinal models. Fit a model with a common set of coefficients for the predictor variables, across all multinomial classes. This type of model is often called a parallel regression or proportional odds model. For nominal models, fitmnr returns an intercept-only model if you set IncludeClassInteractions to false.

In all cases, the model has different intercepts across classes. The IncludeClassInteractions value determines the dimensions of the output array MultinomialRegression.Coefficients.

Example: IncludeClassInteractions=1

Data Types: single | double | logical

IterationLimit — Maximum number of iterations

100 (default) | positive integer

Maximum number of iteratively reweighted least squares (IRLS) iterations used to fit the model, specified as a positive integer. The default value for IterationLimit is 100.

For more information about the IRLS algorithm, see Iteratively Reweighted Least Squares.

Example: IterationLimit=400

Data Types: single | double

Link — Link function

"logit" (default) | "probit" | "comploglog" | "loglog"

Link function to use for ordinal and hierarchical models, specified as one of the values in the following table.

Value	Function
"logit" (default)	f(γ) = ln(γ/(1 –γ))
"probit"	f(γ) = Φ-1(γ) — error term is normally distributed with variance 1
"comploglog"	f(γ) = ln(–ln(1 – γ))
"loglog"	f(γ) = ln(–ln(γ))

The link function defines the relationship between response probabilities and the linear combination of predictors. γ is a cumulative probability when the model has an ordinal response, and a conditional probability when the model is has a hierarchical response.

For an ordinal model, π_j is the probability of an observation being in category j, and γ represents the cumulative probability of an observation being in categories 1 to j. The model with a logit link function has the form

$$\ln \left(\frac{\gamma }{1-\gamma }\right)=\ln \left(\frac{{\pi }_1+{\pi }_2+\cdots +{\pi }_j}{{\pi }_{j+1}+\cdots +{\pi }_k}\right)={\beta }_{0j}+{\beta }_1{X}_1+{\beta }_2{X}_2+\cdots +{\beta }_p{X}_p,$$

where k represents the last category.

For a hierarchical model, π_j and γ represent the probability that an observation is in category j given that it is not in the previous categories. The model with a logit link function has the form

$$\ln \left(\frac{\gamma }{1-\gamma }\right)=\ln \left(\frac{{\pi }_j}{{\pi }_{j+1}+\cdots +{\pi }_k}\right)={\beta }_{0j}+{\beta }_1{X}_1+{\beta }_2{X}_2+\cdots +{\beta }_p{X}_p,$$

where k represents the last category.

You cannot specify Link for nominal models because they always use a multinomial logit link

$$\ln \left(\frac{{\pi }_j}{{\pi }_r}\right)={\beta }_{j0}+{\beta }_{j1}{X}_{j1}+{\beta }_{j2}{X}_{j2}+\cdots +{\beta }_{jp}{X}_{jp},j=1,\dots ,k-1,$$

where π_j is the probability of an observation being in category j, and r corresponds to the reference category. fitmnr uses the last category as the reference category for nominal models.

Example: Link="loglog"

Data Types: char | string

ModelType — Type of model to fit

"nominal" (default) | "ordinal" | "hierarchical"

Type of model to fit, specified as one of the values in the following table.

Value	Description
"nominal"(default)	The response classes are not ordered. Multinomial regression is sometimes called softmax regression when ModelType is "nominal".
"ordinal"	The response classes have a natural order.
"hierarchical"	The response classes are nested.

Example: ModelType="hierarchical"

Data Types: char | string

PredictorNames — Names of predictor variables

string vector (default) | cell array of character vectors

Predictor names, specified as a string vector or a cell array of character vectors.

• If you specify Tbl in the call to fitmnr, then PredictorNames must be a subset of the table variables in Tbl. fitmnr uses only the predictors specified in PredictorNames. In this case, the default value of PredictorNames is the collection of names of the predictor variables in Tbl.

• If you specify the matrix X in the call to fitmnr, you can specify any names for PredictorNames. In this case, the default value of PredictorNames is the 1-by-p cell array ['x1','x2',…,'xp'], where p is the number of predictor variables.

When you specify Formula, fitmnr ignores PredictorNames.

Example: PredictorNames=["time","latitude"]

Data Types: string | cell

ResponseName — Name to assign to response variable

string scalar | character vector

Name to assign to the response variable, specified as a string scalar or character vector. If you specify ResponseVarName or Formula, fitmnr ignores ResponseName.

Example: ResponseName="City"

Data Types: char | string

Tolerance — Termination tolerance

1e-6 (default) | numeric scalar

Termination tolerance for the Iteratively Reweighted Least Squares (IRLS) fitting algorithm, specified as a numeric scalar. If the relative difference of each coefficient is less than Tolerance across two iterations of the IRLS algorithm, then fitmnr considers the model to be converged. The default value for Tolerance is 1e-6.

For more information about the IRLS algorithm, see Iteratively Reweighted Least Squares.

Example: Tolerance=1e-4

Data Types: single | double

Weights — Observation weights

n-by-1 numeric vector

Observation weights, specified as an n-by-1 numeric vector, where n is the number of observations. The default value for Weights is an n-by-1 vector of ones.

Example: Weights

Data Types: single | double

Output Arguments

MnrMdl — Multinomial regression model

MultinomialRegression model object

Multinomial regression model, returned as a MultinomialRegression model object.

Algorithms

If X or Tbl contains NaN values, <undefined> values, empty characters, or empty strings, the fitmnr function ignores the corresponding observations in Y.

Version History

Introduced in R2023a