1 ## Copyright (C) 1995-2012 Kurt Hornik
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20 ## @deftypefn {Function File} {[@var{theta}, @var{beta}, @var{dev}, @var{dl}, @var{d2l}, @var{p}] =} logistic_regression (@var{y}, @var{x}, @var{print}, @var{theta}, @var{beta})
21 ## Perform ordinal logistic regression.
23 ## Suppose @var{y} takes values in @var{k} ordered categories, and let
24 ## @code{gamma_i (@var{x})} be the cumulative probability that @var{y}
25 ## falls in one of the first @var{i} categories given the covariate
29 ## [theta, beta] = logistic_regression (y, x)
36 ## logit (gamma_i (x)) = theta_i - beta' * x, i = 1 @dots{} k-1
39 ## The number of ordinal categories, @var{k}, is taken to be the number
40 ## of distinct values of @code{round (@var{y})}. If @var{k} equals 2,
41 ## @var{y} is binary and the model is ordinary logistic regression. The
42 ## matrix @var{x} is assumed to have full column rank.
44 ## Given @var{y} only, @code{theta = logistic_regression (y)}
45 ## fits the model with baseline logit odds only.
51 ## [theta, beta, dev, dl, d2l, gamma]
52 ## = logistic_regression (y, x, print, theta, beta)
57 ## in which all output arguments and all input arguments except @var{y}
60 ## Setting @var{print} to 1 requests summary information about the fitted
61 ## model to be displayed. Setting @var{print} to 2 requests information
62 ## about convergence at each iteration. Other values request no
63 ## information to be displayed. The input arguments @var{theta} and
64 ## @var{beta} give initial estimates for @var{theta} and @var{beta}.
66 ## The returned value @var{dev} holds minus twice the log-likelihood.
68 ## The returned values @var{dl} and @var{d2l} are the vector of first
69 ## and the matrix of second derivatives of the log-likelihood with
70 ## respect to @var{theta} and @var{beta}.
72 ## @var{p} holds estimates for the conditional distribution of @var{y}
76 ## Original for MATLAB written by Gordon K Smyth <gks@maths.uq.oz.au>,
77 ## U of Queensland, Australia, on Nov 19, 1990. Last revision Aug 3,
80 ## Author: Gordon K Smyth <gks@maths.uq.oz.au>,
81 ## Adapted-By: KH <Kurt.Hornik@wu-wien.ac.at>
82 ## Description: Ordinal logistic regression
84 ## Uses the auxiliary functions logistic_regression_derivatives and
85 ## logistic_regression_likelihood.
87 function [theta, beta, dev, dl, d2l, p] = logistic_regression (y, x, print, theta, beta)
97 error ("logistic_regression: X and Y must have the same number of observations");
100 ## initial calculations
102 tol = 1e-6; incr = 10; decr = 2;
103 ymin = min (y); ymax = max (y); yrange = ymax - ymin;
104 z = (y * ones (1, yrange)) == ((y * 0 + 1) * (ymin : (ymax - 1)));
105 z1 = (y * ones (1, yrange)) == ((y * 0 + 1) * ((ymin + 1) : ymax));
107 z1 = z1 (:, any(z1));
115 beta = zeros (nx, 1);
118 g = cumsum (sum (z))' ./ my;
119 theta = log (g ./ (1 - g));
123 ## likelihood and derivatives at starting values
124 [g, g1, p, dev] = logistic_regression_likelihood (y, x, tb, z, z1);
125 [dl, d2l] = logistic_regression_derivatives (x, z, z1, g, g1, p);
126 epsilon = std (vec (d2l)) / 1000;
128 ## maximize likelihood using Levenberg modified Newton's method
130 while (abs (dl' * (d2l \ dl) / length (dl)) > tol)
134 tb = tbold - d2l \ dl;
135 [g, g1, p, dev] = logistic_regression_likelihood (y, x, tb, z, z1);
136 if ((dev - devold) / (dl' * (tb - tbold)) < 0)
137 epsilon = epsilon / decr;
139 while ((dev - devold) / (dl' * (tb - tbold)) > 0)
140 epsilon = epsilon * incr;
142 error ("logistic_regression: epsilon too large");
144 tb = tbold - (d2l - epsilon * eye (size (d2l))) \ dl;
145 [g, g1, p, dev] = logistic_regression_likelihood (y, x, tb, z, z1);
146 disp ("epsilon"); disp (epsilon);
149 [dl, d2l] = logistic_regression_derivatives (x, z, z1, g, g1, p);
151 disp ("Iteration"); disp (iter);
152 disp ("Deviance"); disp (dev);
153 disp ("First derivative"); disp (dl');
154 disp ("Eigenvalues of second derivative"); disp (eig (d2l)');
160 theta = tb (1 : nz, 1);
161 beta = tb ((nz + 1) : (nz + nx), 1);
165 printf ("Logistic Regression Results:\n");
167 printf ("Number of Iterations: %d\n", iter);
168 printf ("Deviance: %f\n", dev);
169 printf ("Parameter Estimates:\n");
170 printf (" Theta S.E.\n");
171 se = sqrt (diag (inv (-d2l)));
173 printf (" %8.4f %8.4f\n", tb (i), se (i));
176 printf (" Beta S.E.\n");
177 for i = (nz + 1) : (nz + nx)
178 printf (" %8.4f %8.4f\n", tb (i), se (i));
185 e = ((x * beta) * ones (1, nz)) + ((y * 0 + 1) * theta');
187 e = (y * 0 + 1) * theta';
189 gamma = diff ([(y * 0), (exp (e) ./ (1 + exp (e))), (y * 0 + 1)]')';