MultinomialConfidenceIntervals

MultinomialConfidenceIntervals - Simultaneous multinomial confidence intervals.

 Confidence intervals computed simultaneously on all cells using the method of
 Fitzpatrick and Scott (1987). This provides better estimates of the confidence
 intervals than the common (but erroneous) method where each cell is treated
 as an independent binomial variable.

  USAGE

    [p,boudaries] = MultinomialConfidenceIntervals(samples,alpha)

    samples        list of numbers of samples in each cell
    alpha          optional significance level (default = 0.05)

    p              list of estimated cell probabilities
    boundaries     confidence interval boudaries

0001 function [p,boundaries] = MultinomialConfidenceIntervals(samples,alpha)
0002 
0003 %MultinomialConfidenceIntervals - Simultaneous multinomial confidence intervals.
0004 %
0005 % Confidence intervals computed simultaneously on all cells using the method of
0006 % Fitzpatrick and Scott (1987). This provides better estimates of the confidence
0007 % intervals than the common (but erroneous) method where each cell is treated
0008 % as an independent binomial variable.
0009 %
0010 %  USAGE
0011 %
0012 %    [p,boudaries] = MultinomialConfidenceIntervals(samples,alpha)
0013 %
0014 %    samples        list of numbers of samples in each cell
0015 %    alpha          optional significance level (default = 0.05)
0016 %
0017 %    p              list of estimated cell probabilities
0018 %    boundaries     confidence interval boudaries
0019 
0020 % Copyright (C) 2004-2011 by Michaël Zugaro
0021 %
0022 % This program is free software; you can redistribute it and/or modify
0023 % it under the terms of the GNU General Public License as published by
0024 % the Free Software Foundation; either version 3 of the License, or
0025 % (at your option) any later version.
0026 
0027 % Parameter checking and default values
0028 
0029 if nargin < 1,
0030   error('Incorrect number of parameters (type ''help <a href="matlab:help MultinomialConfidenceIntervals">MultinomialConfidenceIntervals</a>'' for details).');
0031 end
0032 
0033 if nargin < 2
0034     alpha = 0.05;
0035 end
0036 
0037 % Process
0038 
0039 n = sum(samples);
0040 p = samples/n;
0041 
0042 % There is a problem in the paper, so we cannot compute the intervals in the general case
0043 % Thus, the following code is commented out
0044 
0045 %  if alpha < 0.016,
0046 %      alpha = alpha/2;
0047 %  elseif alpha < 0.15,
0048 %  log(sqrt(2*pi)*(1-alpha/6))
0049 %      x = sqrt(-4/9*log(sqrt(2*pi)*(1-alpha/6)));
0050 %      alpha = 2*normcdf(x)
0051 %  %      alpha = 6*normcdf(3*norminv(alpha/2)/sqrt(8))-5;
0052 %  else
0053 %      error('Cannot compute confidence intervals for this alpha level');
0054 %  end
0055 %  b = abs(norminv(alpha/2)/(2*sqrt(n)));
0056 
0057 if alpha == 0.1,
0058     k = 1;
0059 elseif alpha == 0.05,
0060     k = 1.13;
0061 elseif alpha == 0.01,
0062     k = 1.40;
0063 else
0064     error('Cannot compute confidence intervals for this alpha level');
0065 end
0066 
0067 b = k/sqrt(n);
0068 boundaries = [max(p-b,0);min(p+b,1)];