Combining probabilities from independent tests of significance into a single aggregate figure
use CombinePvals;
my $obj = CombinePvals->new ($reference_to_list_of_pvals);
my $pval = $obj->method_name;
my $pval = $obj->method_name (@arguments);
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 \*(L"\s-1REFERENCES\s0\*(R" 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.
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.
Michael C. Wendl
Copyright (C) 2009 Washington University
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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.
These methods return an object in the CombinePvals class.
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.
When all the individual p-vals are derived from tests based on discrete distributions, the \*(L"standard\*(R" 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.
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 cummulative 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 \*(L"all-by-all\*(R" multiplication, so it is the least efficient, although entirely exact.
This routine is designed for combining a set of p-values that all come from a single probability distribution.
NOT IMPLEMENTED YET
The mathematical literature furnishes several straightforward options for combining p-vals if all of the distributions underlying all of the individual tests are continuous.
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.
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 \*(L"spread\*(R" 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 \s-1CDF\s0 (the p-val of each test), but the \s-1CDF\s0 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.
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.
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.
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.
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.