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## 25.5 Models

__Function File:__[`theta`,`beta`,`dev`,`dl`,`d2l`,`p`] =**logistic_regression***(*`y`,`x`,`print`,`theta`,`beta`)Perform ordinal logistic regression.

Suppose

`y`takes values in`k`ordered categories, and let`gamma_i (`

be the cumulative probability that`x`)`y`falls in one of the first`i`categories given the covariate`x`. Then[theta, beta] = logistic_regression (y, x)

fits the model

logit (gamma_i (x)) = theta_i - beta' * x, i = 1 … k-1

The number of ordinal categories,

`k`, is taken to be the number of distinct values of`round (`

. If`y`)`k`equals 2,`y`is binary and the model is ordinary logistic regression. The matrix`x`is assumed to have full column rank.Given

`y`only,`theta = logistic_regression (y)`

fits the model with baseline logit odds only.The full form is

[theta, beta, dev, dl, d2l, gamma] = logistic_regression (y, x, print, theta, beta)

in which all output arguments and all input arguments except

`y`are optional.Setting

`print`to 1 requests summary information about the fitted model to be displayed. Setting`print`to 2 requests information about convergence at each iteration. Other values request no information to be displayed. The input arguments`theta`and`beta`give initial estimates for`theta`and`beta`.The returned value

`dev`holds minus twice the log-likelihood.The returned values

`dl`and`d2l`are the vector of first and the matrix of second derivatives of the log-likelihood with respect to`theta`and`beta`.`p`holds estimates for the conditional distribution of`y`given`x`.