noble (3) Genome::Model::Tools::Music::PathScan::CombinePvals.3pm.gz
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)