Multinomial logistic regression
returns
a matrix, B
= mnrfit(X
,Y
,Name,Value
)B
, of coefficient estimates for a multinomial
model fit with additional options specified by one or more Name,Value
pair
arguments.
For example, you can fit a nominal, an ordinal, or a hierarchical model, or change the link function.
Fit a multinomial regression for nominal outcomes and interpret the results.
Load the sample data.
load fisheriris
The column vector, species
, consists of iris flowers of three different species, setosa, versicolor, virginica. The double matrix meas
consists of four types of measurements on the flowers, the length and width of sepals and petals in centimeters, respectively.
Define the nominal response variable using a categorical array.
sp = categorical(species);
Fit a multinomial regression model to predict the species using the measurements.
[B,dev,stats] = mnrfit(meas,sp); B
B = 5×2
10^{3} ×
1.8846 0.0426
0.6294 0.0025
0.5310 0.0067
0.4819 0.0094
2.5795 0.0183
This is a nominal model for the response category relative risks, with separate slopes on all four predictors, that is, each category of meas
. The first row of B
contains the intercept terms for the relative risk of the first two response categories, setosa and versicolor versus the reference category, virginica. The last four rows contain the slopes for the models for the first two categories. mnrfit
accepts the third category as the reference category.
The relative risk of an iris flower being species 2 (versicolor) versus species 3 (virginica) is the ratio of the two probabilities (the probability of being species 2 and the probability of being species 3). The model for the relative risk is
$$\mathrm{ln}\left(\frac{{\pi}_{versicolor}}{{\pi}_{virginica}}\right)=42.6+2.5{X}_{1}+6.7{X}_{2}9.4{X}_{3}18.3{X}_{4}.$$
The coefficients express both the effects of the predictor variables on the relative risk and the log odds of being in one category versus the reference category. For example, the estimated coefficient 2.5 indicates that the relative risk of being species 2 (versicolor) versus species 3 (virginica) increases exp(2.5) times for each unit increase in $${X}_{1}$$, the first measurement, given all else is equal. The relative log odds of being versicolor versus virginica increases 2.5 times with a oneunit increase in $${X}_{1}$$, given all else is equal.
If the coefficients are converging toward infinity or negative infinity, the estimated coefficients can vary slightly depending on your operating system.
Check the statistical significance of the model coefficients.
stats.p
ans = 5×2
0 0.0000
0 0.0281
0 0.0000
0 0.0000
0 0.0000
The small $$p$$values indicate that all measures are significant on the relative risk of being a setosa versus a virginica (species 1 compared to species 3) and being a versicolor versus a virginica (species 2 compared to species 3).
Request the standard errors of coefficient estimates.
stats.se
ans = 5×2
12.4038 5.2719
3.5783 1.1228
3.1760 1.4789
3.5403 1.2934
7.1203 2.0967
Calculate the 95% confidence limits for the coefficients.
LL = stats.beta  1.96.*stats.se; UL = stats.beta + 1.96.*stats.se;
Display the confidence intervals for the coefficients of the model for the relative risk of being a setosa versus a virginica (the first column of coefficients in B
).
[LL(:,1) UL(:,1)]
ans = 5×2
10^{3} ×
1.8603 1.9089
0.6224 0.6364
0.5373 0.5248
0.4889 0.4750
2.5935 2.5656
Find the confidence intervals for the coefficients of the model for the relative risk of being a versicolor versus a virginica (the second column of coefficients in B
).
[LL(:,2) UL(:,2)]
ans = 5×2
32.3049 52.9707
0.2645 4.6660
3.7823 9.5795
11.9644 6.8944
22.3957 14.1766
Fit a multinomial regression model for categorical responses with natural ordering among categories.
Load the sample data and define the predictor variables.
load carbig
X = [Acceleration Displacement Horsepower Weight];
The predictor variables are the acceleration, engine displacement, horsepower, and weight of the cars. The response variable is miles per gallon (mpg).
Create an ordinal response variable categorizing MPG
into four levels from 9 to 48 mpg by labeling the response values in the range 919 as 1, 2029 as 2, 3039 as 3, and 4048 as 4.
miles = ordinal(MPG,{'1','2','3','4'},[],[9,19,29,39,48]);
Fit an ordinal response model for the response variable miles
.
[B,dev,stats] = mnrfit(X,miles,'model','ordinal'); B
B = 7×1
16.6895
11.7208
8.0606
0.1048
0.0103
0.0645
0.0017
The first three elements of B
are the intercept terms for the models, and the last four elements of B
are the coefficients of the covariates, assumed common across all categories. This model corresponds to parallel regression, which is also called the proportional odds model, where there is a different intercept but common slopes among categories. You can specify this using the 'interactions','off'
namevalue pair argument, which is the default for ordinal models.
[B(1:3)'; repmat(B(4:end),1,3)]
ans = 5×3
16.6895 11.7208 8.0606
0.1048 0.1048 0.1048
0.0103 0.0103 0.0103
0.0645 0.0645 0.0645
0.0017 0.0017 0.0017
The link function in the model is logit ('link','logit'
), which is the default for an ordinal model. The coefficients express the relative risk or log odds of the mpg of a car being less than or equal to one value versus greater than that value.
The proportional odds model in this example is
$$\begin{array}{l}\mathrm{ln}\left(\frac{P\left(mpg\le 19\right)}{P\left(mpg>19\right)}\right)=16.6895+0.1048{X}_{A}+0.0103{X}_{D}+0.0645{X}_{H}+0.0017{X}_{W}\\ \\ \mathrm{ln}\left(\frac{P\left(mpg\le 29\right)}{P\left(mpg>29\right)}\right)=11.7208+0.1048{X}_{A}+0.0103{X}_{D}+0.0645{X}_{H}+0.0017{X}_{W}\\ \\ \mathrm{ln}\left(\frac{P\left(mpg\le 39\right)}{P\left(mpg>39\right)}\right)=8.0606+0.1048{X}_{A}+0.0103{X}_{D}+0.0645{X}_{H}+0.0017{X}_{W}\end{array}$$
For example, the coefficient estimate of 0.1048 indicates that a unit change in acceleration would impact the odds of the mpg of a car being less than or equal to 19 versus more than 19, or being less than or equal to 29 versus greater than 29, or being less than or equal to 39 versus greater than 39, by a factor of exp(0.01048) given all else is equal.
Assess the significance of the coefficients.
stats.p
ans = 7×1
0.0000
0.0000
0.0000
0.1899
0.0350
0.0000
0.0118
The $$p$$values of 0.035, 0.0000, and 0.0118 for engine displacement, horsepower, and weight of a car, respectively, indicate that these factors are significant on the odds of mpg of a car being less than or equal to a certain value versus being greater than that value.
Fit a hierarchical multinomial regression model.
Load the sample data.
load('smoking.mat');
The data set smoking
contains five variables: sex, age, weight, and systolic and diastolic blood pressure. Sex is a binary variable where 1 indicates female patients, and 0 indicates male patients.
Define the response variable.
Y = categorical(smoking.Smoker);
The data in Smoker
has four categories:
0: Nonsmoker, 0 cigarettes a day
1: Smoker, 1–5 cigarettes a day
2: Smoker, 6–10 cigarettes a day
3: Smoker, 11 or more cigarettes a day
Define the predictor variables.
X = [smoking.Sex smoking.Age smoking.Weight...
smoking.SystolicBP smoking.DiastolicBP];
Fit a hierarchical multinomial model.
[B,dev,stats] = mnrfit(X,Y,'model','hierarchical'); B
B = 6×3
43.8148 5.9571 44.0712
1.8709 0.0230 0.0662
0.0188 0.0625 0.1335
0.0046 0.0072 0.0130
0.2170 0.0416 0.0324
0.2273 0.1449 0.4824
The first column of B
includes the intercept and the coefficient estimates for the model of the relative risk of being a nonsmoker versus a smoker. The second column includes the parameter estimates for modeling the log odds of smoking 1–5 cigarettes a day versus more than five cigarettes a day given that a person is a smoker. Finally, the third column includes the parameter estimates for modeling the log odds of a person smoking 6–10 cigarettes a day versus more than 10 cigarettes a day given he/she smokes more than 5 cigarettes a day.
The coefficients differ across categories. You can specify this using the 'interactions','on'
namevalue pair argument, which is the default for hierarchical models. So, the model in this example is
$$\mathrm{ln}\left(\frac{P\left(y=0\right)}{P\left(y>0\right)}\right)=43.8148+1.8709{X}_{S}+0.0188{X}_{A}+0.0046{X}_{W}0.2170{X}_{SBP}0.2273{X}_{DBP}$$
$$\mathrm{ln}\left(\frac{P\left(1\le y\le 5\right)}{P\left(y>5\right)}\right)=5.95710.0230{X}_{S}+0.0625{X}_{A}0.0072{X}_{W}+0.0416{X}_{SBP}0.1449{X}_{DBP}$$
$$\mathrm{ln}\left(\frac{P\left(6\le y\le 10\right)}{P\left(y>10\right)}\right)=44.0712+0.0662{X}_{S}+0.1335{X}_{A}0.0130{X}_{W}0.0324{X}_{SBP}0.4824{X}_{DBP}$$
For example, the coefficient estimate of 1.8709 indicates that the likelihood of being a smoker versus a nonsmoker increases by exp(1.8709) = 6.49 times as the gender changes from female to male given everything else held constant.
Assess the statistical significance of the terms.
stats.p
ans = 6×3
0.0000 0.5363 0.2149
0.3549 0.9912 0.9835
0.6850 0.2676 0.2313
0.9032 0.8523 0.8514
0.0009 0.5187 0.8165
0.0004 0.0483 0.0545
Sex, age, or weight don’t appear significant on any level. The $$p$$values of 0.0009 and 0.0004 indicate that both types of blood pressure are significant on the relative risk of a person being a smoker versus a nonsmoker. The $$p$$value of 0.0483 shows that only diastolic blood pressure is significant on the odds of a person smoking 0–5 cigarettes a day versus more than 5 cigarettes a day. Similarly, the $$p$$value of 0.0545 indicates that diastolic blood pressure is significant on the odds of a person smoking 6–10 cigarettes a day versus more than 10 cigarettes a day.
Check if any nonsignificant factors are correlated to each other. Draw a scatterplot of age versus weight grouped by sex.
figure() gscatter(smoking.Age,smoking.Weight,smoking.Sex) legend('Male','Female') xlabel('Age') ylabel('Weight')
The range of weight of an individual seems to differ according to gender. Age does not seem to have any obvious correlation with sex or weight. Age is insignificant and weight seems to be correlated with sex, so you can eliminate both and reconstruct the model.
Eliminate age and weight from the model and fit a hierarchical model with sex, systolic blood pressure, and diastolic blood pressure as the predictor variables.
X = double([smoking.Sex smoking.SystolicBP... smoking.DiastolicBP]); [B,dev,stats] = mnrfit(X,Y,'model','hierarchical'); B
B = 4×3
44.8456 5.3230 25.0248
1.6045 0.2330 0.4982
0.2161 0.0497 0.0179
0.2222 0.1358 0.3092
Here, a coefficient estimate of 1.6045 indicates that the likelihood of being a nonsmoker versus a smoker increases by exp(1.6045) = 4.97 times as sex changes from male to female. A unit increase in the systolic blood pressure indicates an exp(–.2161) = 0.8056 decrease in the likelihood of being a nonsmoker versus a smoker. Similarly, a unit increase in the diastolic blood pressure indicates an exp(–.2222) = 0.8007 decrease in the relative rate of being a nonsmoker versus being a smoker.
Assess the statistical significance of the terms.
stats.p
ans = 4×3
0.0000 0.4715 0.2325
0.0210 0.7488 0.6362
0.0010 0.4107 0.8899
0.0003 0.0483 0.0718
The $$p$$values of 0.0210, 0.0010, and 0.0003 indicate that the terms sex and both types of blood pressure are significant on the relative risk of a person being a nonsmoker versus a smoker, given the other terms in the model. Based on the $$p$$value of 0.0483, diastolic blood pressure appears significant on the relative risk of a person smoking 1–5 cigarettes versus more than 5 cigarettes a day, given that this person is a smoker. Because none of the $$p$$values on the third column are less than 0.05, you can say that none of the variables are statistically significant on the relative risk of a person smoking from 6–10 cigarettes versus more than 10 cigarettes, given that this person smokes more than 5 cigarettes a day.
X
— Observations on predictor variablesObservations on predictor variables, specified as an nbyp matrix. X
contains n observations
for p predictors.
mnrfit
automatically includes a constant
term (intercept) in all models. Do not include a column of 1s in X
.
Data Types: single
 double
Y
— Response valuesResponse values, specified as a column vector or a matrix. Y
can
be one of the following:
An nbyk matrix, where 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 number of observations are made at each predictor combination.
An nby1 column vector of scalar integers from 1 to k indicating the value of the response for each observation. In this case, all sample sizes are 1.
An nby1 categorical array indicating the nominal or ordinal value of the response for each observation. In this case, all sample sizes are 1.
Specify optional
commaseparated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
'Model','ordinal','Link','probit'
specifies
an ordinal model with a probit link function.'Model'
— Type of model to fit'nominal'
(default)  'ordinal'
 'hierarchical'
Type of model to fit, specified as the commaseparated pair
consisting of 'Model'
and one of the following.
'nominal'  Default. There is no ordering among the response categories. 
'ordinal'  There is a natural ordering among the response categories. 
'hierarchical'  The choice of response category is sequential/nested. 
Example: 'Model','ordinal'
'Interactions'
— Indicator for interaction between multinomial categories and coefficients'on'
 'off'
Indicator for an interaction between the multinomial categories
and coefficients, specified as the commaseparated pair consisting
of 'Interactions'
and one of the following.
In all cases, the model has different intercepts across categories.
The choice of 'Interactions'
determines the dimensions
of the output array B
.
Example: 'Interactions','off'
'Link'
— Link function'logit'
(default)  'probit'
 'comploglog'
 'loglog'
Link function to use for ordinal and hierarchical models, specified
as the commaseparated pair consisting of 'Link'
and
one of the following.
'logit'  Default. f(γ) = ln(γ/(1 –γ)) 
'probit'  f(γ) = Φ^{1}(γ) — error term is normally distributed with variance 1 
'comploglog' 
Complementary loglog 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, Xβ. The link functions might be functions of cumulative or conditional probabilities based on whether the model is for an ordinal or a sequential/nested response. For example, for an ordinal model, γ represents the cumulative probability of being in categories 1 to j and the model with a logit link function as follows:
$$\mathrm{ln}\left(\frac{\gamma}{1\gamma}\right)=\mathrm{ln}\left(\frac{{\pi}_{1}+{\pi}_{2}+\cdots +{\pi}_{j}}{{\pi}_{j+1}+\cdots +{\pi}_{k}}\right)={\beta}_{0j}+{\beta}_{1}{X}_{1}+{\beta}_{2}{X}_{2}+\cdots +{\beta}_{p}{X}_{p},$$
where k represents the last category.
You cannot specify the 'Link'
parameter for
nominal models; these always use a multinomial logit link,
$$\mathrm{ln}\left(\frac{{\pi}_{j}}{{\pi}_{r}}\right)={\beta}_{j0}+{\beta}_{j1}{X}_{j1}+{\beta}_{j2}{X}_{j2}+\cdots +{\beta}_{jp}{X}_{jp},\text{\hspace{1em}}j=1,\text{\hspace{0.17em}}\dots ,k1,$$
where π stands for a categorical
probability, and r corresponds to the reference
category. mnrfit
uses the last category as the
reference category for nominal models.
Example: 'Link','loglog'
'EstDisp'
— Indicator for estimating dispersion parameter'off'
(default)  'on'
Indicator for estimating a dispersion parameter, specified as
the commaseparated pair consisting of 'EstDisp'
and
one of the following.
'off'  Default. Use the theoretical dispersion value of 1. 
'on'  Estimate a dispersion parameter for the multinomial distribution in computing standard errors. 
Example: 'EstDisp','on'
B
— Coefficient estimatesCoefficient estimates for a multinomial logistic regression
of the responses in Y
, returned as a vector or
a matrix.
If 'Interaction'
is 'off'
,
then B
is a k – 1 + p vector.
The first k – 1 rows of B
correspond
to the intercept terms, one for each k –
1 multinomial categories, and the remaining p rows
correspond to the predictor coefficients, which are common for all
of the first k – 1 categories.
If 'Interaction'
is 'on'
,
then B
is a (p + 1)by(k –
1) matrix. Each column of B
corresponds to the
estimated intercept term and predictor coefficients, one for each
of the first k – 1 multinomial categories.
The estimates for the kth category are taken
to be zero as mnrfit
takes the last category as
the reference category.
dev
— Deviance of the fitDeviance of the fit, returned as a scalar value. It is twice the difference between the maximum achievable log likelihood and that attained under the fitted model. This corresponds to the sum of deviance residuals,
$$dev=2*{\displaystyle \sum _{i}^{n}{\displaystyle \sum _{j}^{k}{y}_{ij}*\mathrm{log}\left(\frac{{y}_{ij}}{{\widehat{\pi}}_{ij}*{m}_{i}}\right)}}={\displaystyle \sum _{i}^{n}r{d}_{i}},$$
where rd_{i} are
the deviance residuals. For deviance residuals see stats
.
stats
— Model statisticsModel statistics, returned as a structure that contains the following fields.
beta  The coefficient estimates. These are the same as B . 
dfe 
Degrees of freedom for error

sfit  Estimated dispersion parameter. 
s 
Theoretical or estimated dispersion parameter.

estdisp  Indicator for a theoretical or estimated dispersion parameter. 
se  Standard errors of coefficient estimates, B . 
coeffcorr  Estimated correlation matrix for B . 
covb  Estimated covariance matrix for B . 
t  t statistics for B . 
p  pvalues for B . 
resid 
Raw residuals. Observed minus fitted values, $${r}_{ij}={y}_{ij}{\widehat{\pi}}_{ij}*{m}_{i},\text{\hspace{1em}}\{\begin{array}{c}i=1,\cdots ,n\\ j=1,\cdots ,k\end{array},$$ where π_{ij} is the categorical, cumulative or conditional probability, and m_{i} is the corresponding sample size. 
residp 
Pearson residuals, which are the raw residuals scaled by the estimated standard deviation: $$r{p}_{ij}=\frac{{r}_{ij}}{{\widehat{\sigma}}_{ij}}=\frac{{y}_{ij}{\widehat{\pi}}_{ij}*{m}_{i}}{\sqrt{{\widehat{\pi}}_{ij}*\left(1{\widehat{\pi}}_{ij}\right)*{m}_{i}}},\text{\hspace{1em}}\{\begin{array}{c}i=1,\cdots ,n\\ j=1,\cdots ,k\end{array},$$ where π_{ij} is the categorical, cumulative, or conditional probability, and m_{i} is the corresponding sample size. 
residd 
Deviance residuals: $$r{d}_{i}=2*{\displaystyle {\sum}_{j}^{k}{y}_{ij}*\mathrm{log}\left(\frac{{y}_{ij}}{{\widehat{\pi}}_{ij}*{m}_{i}}\right)},\text{\hspace{1em}}i=1,\cdots ,n.$$ where π_{ij} is the categorical, cumulative, or conditional probability, and m_{i} is the corresponding sample size. 
mnrfit
treats NaN
s in
either X
or Y
as missing values,
and ignores them.
[1] McCullagh, P., and J. A. Nelder. Generalized Linear Models. New York: Chapman & Hall, 1990.
[2] Long, J. S. Regression Models for Categorical and Limited Dependent Variables. Sage Publications, 1997.
[3] Dobson, A. J., and A. G. Barnett. An Introduction to Generalized Linear Models. Chapman and Hall/CRC. Taylor & Francis Group, 2008.
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