1## Copyright (C) 1995-2017 Kurt Hornik 2## 3## This program is free software: you can redistribute it and/or 4## modify it under the terms of the GNU General Public License as 5## published by the Free Software Foundation, either version 3 of the 6## License, or (at your option) any later version. 7## 8## This program is distributed in the hope that it will be useful, but 9## WITHOUT ANY WARRANTY; without even the implied warranty of 10## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 11## General Public License for more details. 12## 13## You should have received a copy of the GNU General Public License 14## along with this program; see the file COPYING. If not, see 15## <http://www.gnu.org/licenses/>. 16 17## -*- texinfo -*- 18## @deftypefn {} {[@var{pval}, @var{z}] =} z_test_2 (@var{x}, @var{y}, @var{v_x}, @var{v_y}, @var{alt}) 19## For two samples @var{x} and @var{y} from normal distributions with unknown 20## means and known variances @var{v_x} and @var{v_y}, perform a Z-test of the 21## hypothesis of equal means. 22## 23## Under the null, the test statistic @var{z} follows a standard normal 24## distribution. 25## 26## With the optional argument string @var{alt}, the alternative of interest 27## can be selected. If @var{alt} is @qcode{"!="} or @qcode{"<>"}, the null 28## is tested against the two-sided alternative 29## @code{mean (@var{x}) != mean (@var{y})}. If alt is @qcode{">"}, the 30## one-sided alternative @code{mean (@var{x}) > mean (@var{y})} is used. 31## Similarly for @qcode{"<"}, the one-sided alternative 32## @code{mean (@var{x}) < mean (@var{y})} is used. The default is the 33## two-sided case. 34## 35## The p-value of the test is returned in @var{pval}. 36## 37## If no output argument is given, the p-value of the test is displayed along 38## with some information. 39## @end deftypefn 40 41## Author: KH <Kurt.Hornik@wu-wien.ac.at> 42## Description: Compare means of two normal samples with known variances 43 44function [pval, z] = z_test_2 (x, y, v_x, v_y, alt) 45 46 if (nargin < 4 || nargin > 5) 47 print_usage (); 48 endif 49 50 if (! (isvector (x) && isvector (y))) 51 error ("z_test_2: both X and Y must be vectors"); 52 elseif (! (isscalar (v_x) && (v_x > 0) 53 && isscalar (v_y) && (v_y > 0))) 54 error ("z_test_2: both V_X and V_Y must be positive scalars"); 55 endif 56 57 n_x = length (x); 58 n_y = length (y); 59 mu_x = sum (x) / n_x; 60 mu_y = sum (y) / n_y; 61 z = (mu_x - mu_y) / sqrt (v_x / n_x + v_y / n_y); 62 cdf = stdnormal_cdf (z); 63 64 if (nargin == 4) 65 alt = "!="; 66 endif 67 68 if (! ischar (alt)) 69 error ("z_test_2: ALT must be a string"); 70 elseif (strcmp (alt, "!=") || strcmp (alt, "<>")) 71 pval = 2 * min (cdf, 1 - cdf); 72 elseif (strcmp (alt, ">")) 73 pval = 1 - cdf; 74 elseif (strcmp (alt, "<")) 75 pval = cdf; 76 else 77 error ("z_test_2: option %s not recognized", alt); 78 endif 79 80 if (nargout == 0) 81 s = ["Two-sample Z-test of mean(x) == mean(y) against ", ... 82 "mean(x) %s mean(y),\n", ... 83 "with known var(x) == %g and var(y) == %g:\n", ... 84 " pval = %g\n"]; 85 printf (s, alt, v_x, v_y, pval); 86 endif 87 88endfunction 89 90%!test 91%! ## Two-sided (also the default option) 92%! x = randn (100, 1); v_x = 2; x = v_x * x; 93%! [pval, zval] = z_test_2 (x, x, v_x, v_x); 94%! zval_exp = 0; pval_exp = 1.0; 95%! assert (zval, zval_exp, eps); 96%! assert (pval, pval_exp, eps); 97 98%!test 99%! ## Two-sided (also the default option) 100%! x = randn (10000, 1); v_x = 2; x = v_x * x; n_x = length (x); 101%! y = randn (20000, 1); v_y = 3; y = v_y * y; n_y = length (y); 102%! [pval, z] = z_test_2 (x, y, v_x, v_y); 103%! if (mean (x) >= mean (y)) 104%! zval = abs (norminv (0.5*pval)); 105%! else 106%! zval = -abs (norminv (0.5*pval)); 107%! endif 108%! unew = zval * sqrt (v_x/n_x + v_y/n_y); 109%! delmu = mean (x) - mean (y); 110%! assert (delmu, unew, 100*eps); 111 112%!test 113%! x = randn (100, 1); v_x = 2; x = v_x * x; 114%! [pval, zval] = z_test_2 (x, x, v_x, v_x, ">"); 115%! zval_exp = 0; pval_exp = 0.5; 116%! assert (zval, zval_exp, eps); 117%! assert (pval, pval_exp, eps); 118 119%!test 120%! x = randn (10000, 1); v_x = 2; x = v_x * x; n_x = length (x); 121%! y = randn (20000, 1); v_y = 3; y = v_y * y; n_y = length (y); 122%! [pval, z] = z_test_2 (x, y, v_x, v_y, ">"); 123%! zval = norminv (1-pval); 124%! unew = zval * sqrt (v_x/n_x + v_y/n_y); 125%! delmu = mean (x) - mean (y); 126%! assert (delmu, unew, 100*eps); 127 128%!test 129%! x = randn (100, 1); v_x = 2; x = v_x * x; 130%! [pval, zval] = z_test_2 (x, x, v_x, v_x, "<"); 131%! zval_exp = 0; pval_exp = 0.5; 132%! assert (zval, zval_exp, eps); 133%! assert (pval, pval_exp, eps); 134 135%!test 136%! x = randn (10000, 1); v_x = 2; x = v_x * x; n_x = length (x); 137%! y = randn (20000, 1); v_y = 3; y = v_y * y; n_y = length (y); 138%! [pval, z] = z_test_2 (x, y, v_x, v_y, "<"); 139%! zval = norminv (pval); 140%! unew = zval * sqrt (v_x/n_x + v_y/n_y); 141%! delmu = mean (x) - mean (y); 142%! assert (delmu, unew, 100*eps); 143