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### 25.3 Models

— Functio 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.

Stting 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.