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### 24.5 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 (`x`)` be the cumulative probability that 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 (`y`)`. If 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.