Provided by: libgenome-model-tools-music-perl_0.04-5_all bug

NAME

       CombinePvals - combining probabilities from independent tests of significance into a single aggregate
       figure

SYNOPSIS

               use CombinePvals;

               my $obj = CombinePvals->new ($reference_to_list_of_pvals);

               my $pval = $obj->method_name;

               my $pval = $obj->method_name (@arguments);

DESCRIPTION

       There are a variety of circumstances under which one might have a number of different kinds of tests
       and/or separate instances of the same kind of test for one particular null hypothesis, where each of
       these tests returns a p-value.  The problem is how to properly condense this list of probabilities into a
       single value so as to be able to make a statistical inference, e.g. whether to reject the null
       hypothesis.  This problem was examined heavily starting about the 1930s, during which time numerous
       mathematical contintencies were treated, e.g. dependence vs. independence of tests, optimality, inter-
       test weighting, computational efficiency, continuous vs. discrete tests and combinations thereof, etc.
       There is quite a large mathematical literature on this topic (see "REFERENCES" below) and any one
       particular situation might incur some of the above subtleties.  This package concentrates on some of the
       more straightforward scenarios, furnishing various methods for combining p-vals.  The main consideration
       will usually be the trade-off between the exactness of the p-value (according to strict frequentist
       modeling) and the computational efficiency, or even its actual feasibility.  Tests should be chosen with
       this factor in mind.

       Note also that this scenario of combining p-values (many tests of a single hypothesis) is fundamentally
       different from that where a given hypothesis is tested multiple times.  The latter instance usually calls
       for some method of multiple testing correction.

REFERENCES

       Here is an abbreviated list of the substantive works on the topic of combining probabilities.

       •   Birnbaum, A. (1954) Combining Independent Tests of Significance, Journal of the American Statistical
           Association 49(267), 559-574.

       •   David, F. N. and Johnson, N. L. (1950) The Probability Integral Transformation When the Variable is
           Discontinuous, Biometrika 37(1/2), 42-49.

       •   Fisher, R. A. (1958) Statistical Methods for Research Workers, 13-th Ed. Revised, Hafner Publishing
           Co., New York.

       •   Lancaster, H. O. (1949) The Combination of Probabilities Arising from Data in Discrete Distributions,
           Biometrika 36(3/4), 370-382.

       •   Littell, R. C. and Folks, J. L. (1971) Asymptotic Optimality of Fisher's Method of Combining
           Independent Tests, Journal of the American Statistical Association 66(336), 802-806.

       •   Pearson, E. S. (1938) The Probability Integral Transformation for Testing Goodness of Fit and
           Combining Independent Tests of Significance, Biometrika 30(12), 134-148.

       •   Pearson, E. S. (1950) On Questions Raised by the Combination of Tests Based on Discontonuous
           Distributions, Biometrika 37(3/4), 383-398.

       •   Pearson, K. (1933) On a Method of Determining Whether a Sample Of Size N Supposed to Have Been Drawn
           From a Parent Population Having a Known Probability Integral Has Probably Been Drawn at Random
           Biometrika 25(3/4), 379-410.

       •   Van Valen, L. (1964) Combining the Probabilities from Significance Tests, Nature 201(4919), 642.

       •   Wallis, W. A. (1942) Compounding Probabilities from Independent Significance Tests, Econometrica
           10(3/4), 229-248.

       •   Zelen, M. and Joel, L. S. (1959) The Weighted Compounding of Two Independent Significance Tests,
           Annals of Mathematical Statistics 30(4), 885-895.

AUTHOR

       Michael C. Wendl

       mwendl@wustl.edu

       Copyright (C) 2009 Washington University

       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, write
       to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

GENERAL REMARKS ON METHODS

       The available methods are listed below.  Each of computational techniques assumes that tests, as well as
       their associated p-values, are independent of one another and none considers any form of differential
       weighting.

CONSTRUCTOR METHODS

       These methods return an object in the CombinePvals class.

   new
       This is the usual object constructor, which takes a mandatory, but otherwise un-ordered (reference to a)
       list of the p-values obtained by a set of independent tests.

               my $obj = CombinePvals->new ([0.103, 0.078, 0.03, 0.2,...]);

       The method checks to make sure that all elements are actual p-values, i.e. they are real numbers and they
       have values bounded by 0 and 1.

EXACT ENUMERATIVE PROCEDURES FOR STRICTLY DISCRETE DISTRIBUTIONS

       When all the individual p-vals are derived from tests based on discrete distributions, the "standard"
       continuum methods cannot be used in the strictest sense.  Both Wallis (1942) and Lancaster (1949) discuss
       the option of full enumeration, which will only be feasible when there are a limited number of p-values
       and their range is not too large.  Feasibility experiments are suggested, depending upon the type of
       hardware and size of calculation.

   exact_enum_arbitrary
       This routine is designed for combining p-values from completely arbitrary discrete probability
       distributions.  It takes a list-of-lists data structure, each list being the probability tails ordered
       from most extreme to least extreme (i.e. as a probability cumulative density function) associated with
       each individual test.  However, the ordering of the lists themselves is not important.  For instance,
       Wallis (1942) gives the example of two binomials, a one-tailed test having tail values of 0.0625, 0.3125,
       0.6875, 0.9375, and 1, and a two-tailed test having tail values 0.125, 0.625, and 1.  We would then call
       this method using

               my $pval = $obj->exact_enum_arbitrary (
                  [0.0625, 0.3125, 0.6875, 0.9375, 1],
                  [0.125, 0.625, 1]
               );

       The internal computational method is relatively straightforard and described in detail by Wallis (1942).
       Note that this method does "all-by-all" multiplication, so it is the least efficient, although entirely
       exact.

   exact_enum_identical
       This routine is designed for combining a set of p-values that all come from a single probability
       distribution.

               NOT IMPLEMENTED YET

TRANSFORMS FOR CONTINUOUS DISTRIBUTIONS

       The mathematical literature furnishes several straightforward options for combining p-vals if all of the
       distributions underlying all of the individual tests are continuous.

   fisher_chisq_transform
       This routine implements R.A. Fisher's (1958, originally 1932) chi-square transform method for combining
       p-vals from continuous distributions, which is essentially a CPU-efficient approximation of K. Pearson's
       log-based result (see e.g. Wallis (1942) pp 232).  Note that the underlying distributions are not
       actually relevant, so no arguments are passed.

               my $pval = $obj->fisher_chisq_transform;

       This is certainly the fastest and easiest method for combining p-vals, but its accuracy for discrete
       distributions will not usually be very good.  For such cases, an exact or a corrected method are better
       choices.

CORRECTION PROCEDURES FOR DISCRETE DISTRIBUTIONS: LANCASTER'S MODELS

       Enumerative procedures quickly become infeasible if the number of tests and/or the support of each test
       grow large.  A number of procedures have been described for correcting the methodologies designed for
       continuum testing, mostly in the context of applying so-called continuity corrections.  Essentially,
       these seek to "spread" dicrete data out into a pseudo-continuous configuration as appropriate as
       possible, and then apply standard transforms.  Accuracy varies and should be suitably established in each
       case.

       The methods in this section are due to H.O. Lancaster (1949), who discussed two corrections based upon
       the idea of describing how a chi-square transformed statistic varies between the points of a discrete
       distribution.  Unfortunately, these methods require one to pass some extra information to the routines,
       i.e. not only the CDF (the p-val of each test), but the CDF value associated with the next-most-extreme
       statistic.  These two pieces of information are the basis of interpolating.  For example, if an
       underlying distribution has the possible tail values of 0.0625, 0.3125, 0.6875, 0.9375, 1 and the test
       itself has a value of 0.6875, then you would pass both 0.3125 and 0.6875 to the routine.  In all cases,
       the lower value, i.e. the more extreme one, precedes higher value in the argument list.  While there
       generally will be some extra inconvenience in obtaining this information, the accuracy is much improved
       over Fisher's method.

   lancaster_mean_corrected_transform
       This method is based on the mean value of the chi-squared transformed statistic.

               my $pval = $obj->lancaster_mean_corrected_transform (@cdf_pairs);

       Its accuracy is good, but the method is not strictly defined if one of the tests has either the most
       extreme or second-to-most-extreme statistic.

   lancaster_median_corrected_transform
       This method is based on the median value of the chi-squared transformed statistic.

               my $pval = $obj->lancaster_median_corrected_transform (@cdf_pairs);

       Its accuracy may sometimes be not quite as good as when using the average, but the method is strictly
       defined for all values of the statistic.

   lancaster_mixed_corrected_transform
       This method is a mixture of both the mean and median methods.  Specifically, mean correction is used
       wherever it is well-defined, otherwise median correction is used.

               my $pval = $obj->lancaster_mixed_corrected_transform (@cdf_pairs);

       This will be a good way to handle certain cases.

   additional methods
       The basic functionality of this package is encompassed in the methods described above.  However, some
       lower-level functions can also sometimes be useful.

       exact_enum_arbitrary_2

       Hard-wired precursor of exact_enum_arbitrary for 2 distributions.  Does no pre-checking, but may be
       useful for comparing to the output of the general program.

       exact_enum_arbitrary_3

       Hard-wired precursor of exact_enum_arbitrary for 3 distributions.  Does no pre-checking, but may be
       useful for comparing to the output of the general program.

       binom_coeffs

       Calculates the binomial coefficients needed in the binomial (convolution) approximate solution.

               $pmobj->binom_coeffs;

       The internal data structure is essentially the symmetric half of the appropriately-sized Pascal triangle.
       Considerable memory is saved by not storing the full triangle.

perl v5.30.3                                       2020-Genome::Model::Tools::Music::PathScan::CombinePvals(3pm)