/usr/share/octave/packages/3.2/statistics-1.0.10/cl_multinom.m is in octave-statistics 1.0.10-1.
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#
# @deftypefn {Function File} {@var{CL}} cl_multinom( @var{x},@var{N},@var{b},@var{calculation_type} ) - Confidence level of multinomial portions
# Returns confidence level of multinomial parameters estimated @math{ p = x / sum(x) } with predefined confidence interval @var{b}.
# Finite population is also considered.
#
# This function calculates the level of confidence at which the samples represent the true distribution
# given that there is a predefined tolerance (confidence interval).
# This is the upside down case of the typical excercises at which we want to get the confidence interval
# given the confidence level (and the estimated parameters of the underlying distribution).
# But once we accept (lets say at elections) that we have a standard predefined
# maximal acceptable error rate (e.g. @var{b}=0.02 ) in the estimation and we just want to know that how sure we
# can be that the measured proportions are the same as in the
# entire population (ie. the expected value and mean of the samples are roghly the same) we need to use this function.
#
# @subheading Arguments
# @itemize @bullet
# @item @var{x} : int vector : sample frequencies bins
# @item @var{N} : int : Population size that was sampled by x. If N<sum(x), infinite number assumed
# @item @var{b} : real, vector : confidence interval
# if vector, it should be the size of x containing confence interval for each cells
# if scalar, each cell will have the same value of b unless it is zero or -1
# if value is 0, b=.02 is assumed which is standard choice at elections
# otherwise it is calculated in a way that one sample in a cell alteration defines the confidence interval
# @item @var{calculation_type} : string : (Optional), described below
# "bromaghin" (default) - do not change it unless you have a good reason to do so
# "cochran"
# "agresti_cull" this is not exactly the solution at reference given below but an adjustment of the solutions above
# @end itemize
#
# @subheading Returns
# Confidence level.
#
# @subheading Example
# CL = cl_multinom( [27;43;19;11], 10000, 0.05 )
# returns 0.69 confidence level.
#
# @subheading References
#
# "bromaghin" calculation type (default) is based on
# is based on the article
# Jeffrey F. Bromaghin, "Sample Size Determination for Interval Estimation of Multinomial Probabilities", The American Statistician vol 47, 1993, pp 203-206.
#
# "cochran" calculation type
# is based on article
# Robert T. Tortora, "A Note on Sample Size Estimation for Multinomial Populations", The American Statistician, , Vol 32. 1978, pp 100-102.
#
# "agresti_cull" calculation type
# is based on article in which Quesenberry Hurst and Goodman result is combined
# A. Agresti and B.A. Coull, "Approximate is better than \"exact\" for interval estimation of binomial portions", The American Statistician, Vol. 52, 1998, pp 119-126
#
# @end deftypefn
# Copyright (C) 2009 Levente Torok / TorokLev@gmail.com
# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 2 of the License, or
# (at your option) any later version.^
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License"
# along with this program; If not, see <http://www.gnu.org/licenses/>.
#
function CL = cl_multinom( x, N, b, calculation_type )
k = rows(x);
nn = sum(x);
p = x / nn;
if (nargin < 3)
b = .05;
endif
if (isscalar( b ))
if (b==0) b=0.02; endif
b = ones( rows(x), 1 ) * b;
if (b<0) b=1 ./ max( x, 1 ); endif
endif
bb = b .* b;
if (N==nn)
CL = 1;
return;
endif
if (N<nn)
fpc = 1;
else
fpc = (N-1) / (N-nn); # finite population correction tag
endif
beta = p.*(1-p);
if ( nargin < 4 )
calculation_type = "bromaghin";
endif
switch calculation_type
case {"cochran"}
t = sqrt( fpc * nn * bb ./ beta )
alpha = ( 1 - normcdf( t )) * 2
case {"bromaghin"}
t = sqrt( fpc * (nn * 2 * bb )./ ( beta - 2 * bb + sqrt( beta .* beta - bb .* ( 4*beta - 1 ))) );
alpha = ( 1 - normcdf( t )) * 2;
case {"agresti_cull"}
ts = fpc * nn * bb ./ beta ;
if ( k<=2 )
alpha = 1 - chi2cdf( ts, k-1 ); % adjusted Wilson interval
else
alpha = 1 - chi2cdf( ts/k, 1 ); % Goodman interval with Bonferroni argument
endif
endswitch
CL = 1 - max( alpha );
endfunction
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