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# cl_multinom

-*- texinfo -*-

## SYNOPSIS

function CL = cl_multinom( x, N, b, calculation_type )

## DESCRIPTION

``` -*- texinfo -*-

@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.

@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

Confidence level.

CL = cl_multinom( [27;43;19;11], 10000, 0.05 )
returns 0.69 confidence level.

'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```

## CROSS-REFERENCE INFORMATION

This function calls:
This function is called by:

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