mnrfit - Multinomial logistic regression

Syntax

Description

B = mnrfit(X,Y) returns a matrix B of coefficient estimates for a multinomial logistic regression of the responses in Y on the predictors in X. X is an n-by-p matrix of p predictors at each of n observations. Y is an n-by-k matrix, where Y(i,j) is the number of outcomes of the multinomial category j for the predictor combinations given by X(i,:). The sample sizes for each observation are given by the row sums sum(Y,2).

Alternatively, Y can be an n-by-1 column vector of scalar integers from 1 to k indicating the value of the response for each observation, and all sample sizes are taken to be 1.

The result B is a (p+1)-by-(k–1) matrix of estimates, where each column corresponds to the estimated intercept term and predictor coefficients, one for each of the first k–1 multinomial categories. The estimates for the k^th category are taken to be zero.

Note   mnrfit automatically includes a constant term in all models. Do not enter a column of 1s directly into X.

mnrfit treats NaNs in either X or Y as missing values, and ignores them.

B = mnrfit(X,Y,param1,val1,param2,val2,...) allows you to specify optional parameter name/value pairs to control the model fit. Parameters are:

• 'model' — The type of model to fit; one of the text strings 'nominal' (the default), 'ordinal', or 'hierarchical'

• 'interactions' — Determines whether the model includes an interaction between the multinomial categories and the coefficients. Specify as 'off' to fit a model with a common set of coefficients for the predictor variables, across all multinomial categories. This is often described as parallel regression. Specify as 'on' to fit a model with different coefficients across categories. In all cases, the model has different intercepts across categories. Thus, B is a vector containing k–1+p coefficient estimates when 'interaction' is 'off', and a (p+1)-by-(k–1) matrix when it is 'on'. The default is 'off' for ordinal models, and 'on' for nominal and hierarchical models.

• 'link' — The link function to use for ordinal and hierarchical models. The link function defines the relationship g(μ_ij) = x_ib_j between the mean response for the i^th observation in the j^th category, μ_ij , and the linear combination of predictors x_ib_j. Specify the link parameter value as one of the text strings 'logit'(the default), 'probit', 'comploglog', or 'loglog'. You may not specify the 'link' parameter for nominal models; these always use a multivariate logistic link.

• 'estdisp' — Specify as 'on' to estimate a dispersion parameter for the multinomial distribution in computing standard errors, or 'off' (the default) to use the theoretical dispersion value of 1.

[B,dev] = mnrfit(...) returns the deviance of the fit dev.

[B,dev,stats] = mnrfit(...) returns a structure stats that contains the following fields:

• dfe — Degrees of freedom for error

• s — Theoretical or estimated dispersion parameter

• sfit — 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 — p-values for B

• resid — Residuals

• residp — Pearson residuals

• residd — Deviance residuals

Examples

Fit multinomial logistic regression models to data with one predictor variable and three categories in the response variable:

x = [-3 -2 -1 0 1 2 3]';
Y = [1 11 13; 2 9 14; 6 14 5; 5 10 10; 5 14 6; 7 13 5;...
    8 11 6];
bar(x,Y,'stacked'); ylim([0 25]);

% Now fit a nominal model for the individual response 
% category probabilities, with separate slopes on the
% single predictor variable, x, for each
% category:

% The first row of betaHatNom contains the intercept terms 
% for the first two response categories.  The second row 
% contains the slopes.
betaHatNom = mnrfit(x,Y,'model','nominal',...
'interactions','on')

% Compute the predicted probabilities for the three 
% response categories:
xx = linspace(-4,4)';
pHatNom = mnrval(betaHatNom,xx,'model','nominal',...
'interactions','on');
line(xx,cumsum(25*pHatNom,2),'LineWidth',2);

Fit a "parallel" ordinal model for the cumulative response category probabilities, with a common slope on the single predictor variable, x, across all categories:

% The first two elements of betaHatOrd are the
% intercept terms for the first two response categories.
% The last element of betaHatOrd is the common slope.
betaHatOrd = mnrfit(x,Y,'model','ordinal',...
	'interactions','off')

% Compute the predicted cumulative probabilities for the
% first two response categories.  The cumulative
% probability for the third category is always 1.
pHatOrd = mnrval(betaHatOrd,xx,'type','cumulative',...
'model','ordinal','interactions','off');
bar(x,cumsum(Y,2),'grouped'); ylim([0 25]);
line(xx,25*pHatOrd,'LineWidth',2);

References

[1] McCullagh, P., and J. A. Nelder. Generalized Linear Models. New York: Chapman & Hall, 1990.

See Also

glmfit | glmval | mnrval | regress | regstats

How To

• Multinomial Distribution

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