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  1. 1. Introduction
  2. 2. Details
    1. 2.1 Command interface
    2. 2.2 Application

1.  Introduction

The method proposed by Madsen and Browning 20091 first introduced the idea of assigning "weights" to rare variants within a genetic region before they are collapsed. In this case the variants having higher weights will have more substantial contribution to the collapsed variant score. In the Madsen & Browning paper the "weights" are defined as {$\sqrt{n_iq_i(1-q_i)}$} with the assumption that the "rarer" the variant, the larger the risk effect it is to a phenotype. The {$q_i$} in the original paper was based on observed control sample, which might result in inflated type I error2. Implementation of the WSS statistic in the WSSRankTest method uses the same definition for {$q_i$} but the Mann-Whitney U test (definition and C++ implementation for this program) now relies on a full permutation procedure rather than normal approximation, such that the bias is correctly accounted for.

As with the Varible Thresholds strategy, the idea of weighting can be applied to many other rare variant methods. The WeightedBurdenBt and WeightedBurdenQt methods implements the Madsen & Browning weighting based on controls (or samples with low quantitative phenotypic values) or the entire population, and tests for association for both case control and quantitative traits with/without presence of phenotype co-variates.

2.  Details

2.1  Command interface

vtools show test WSSRankTest
Name:          WSSRankTest
Description:   Weighted sum method using rank test statistic, Madsen & Browning 2009
usage: vtools associate --method WSSRankTest [-h] [--name NAME] [-q1 MAFUPPER]
                                             [-q2 MAFLOWER]
                                             [--alternative TAILED] [-p N]
                                             [--adaptive C]
                                             [--moi {additive,dominant,recessive}]

Weighted sum method using rank test statistic, Madsen & Browning 2009. p-value
is based on the significance level of the Wilcoxon rank-sum test. Two methods
are available for evaluating p-value: a semi-asymptotic p-value based on
normal distribution, or permutation based p-value. Variants will be weighted
by 1/sqrt(nP*(1-P)) and the weighted codings will be summed up for rank test.
Two-sided test is available for the asymptotic version, which will calculate
two p-values based on weights from controls and cases respectively, and use
the smaller of them with multiple testing adjustment. For two-sided
permutation based p-value please refer to "vtools show test WeightedBurdenBt"

optional arguments:
  -h, --help            show this help message and exit
  --name NAME           Name of the test that will be appended to names of
                        output fields, usually used to differentiate output of
                        different tests, or the same test with different
  -q1 MAFUPPER, --mafupper MAFUPPER
                        Minor allele frequency upper limit. All variants
                        having sample MAF<=m1 will be included in analysis.
                        Default set to 0.01
  -q2 MAFLOWER, --maflower MAFLOWER
                        Minor allele frequency lower limit. All variants
                        having sample MAF>m2 will be included in analysis.
                        Default set to 0.0
  --alternative TAILED  Alternative hypothesis is one-sided ("1") or two-sided
                        ("2"). Note that two-sided test is only available for
                        asymptotic version of the test. Default set to 1
  -p N, --permutations N
                        Number of permutations. Set it to zero to use the
                        asymptotic version. Default is zero
  --adaptive C          Adaptive permutation using Edwin Wilson 95 percent
                        confidence interval for binomial distribution. The
                        program will compute a p-value every 1000 permutations
                        and compare the lower bound of the 95 percent CI of
                        p-value against "C", and quit permutations with the
                        p-value if it is larger than "C". It is recommended to
                        specify a "C" that is slightly larger than the
                        significance level for the study. To disable the
                        adaptive procedure, set C=1. Default is C=0.1
  --moi {additive,dominant,recessive}
                        Mode of inheritance. Will code genotypes as 0/1/2/NA
                        for additive mode, 0/1/NA for dominant or recessive
                        model. Default set to additive

2.2  Application

Example using snapshot vt_ExomeAssociation

% vtools associate rare status -m "WSSRankTest --name wss -p 5000" --group_by name2 --to_db w\
ss -j8 > wss.txt
INFO: 3180 samples are found
INFO: 2632 groups are found
INFO: Starting 8 processes to load genotypes
Loading genotypes: 100% [=========================================] 3,180 33.7/s in 00:01:34
Testing for association: 100% [================================================] 2,632/591 10.7/s in 00:04:06
INFO: Association tests on 2632 groups have completed. 591 failed.
INFO: Using annotation DB wss in project test.
INFO: Annotation database used to record results of association tests. Created on Wed, 30 Jan 2013 16:18:43
% vtools show fields | grep wss
wss.name2                    name2
wss.sample_size_wss          sample size
wss.num_variants_wss         number of variants in each group (adjusted for specified MAF
wss.total_mac_wss            total minor allele counts in a group (adjusted for MOI)
wss.statistic_wss            test statistic.
wss.pvalue_wss               p-value
wss.std_error_wss            Empirical estimate of the standard deviation of statistic
wss.num_permutations_wss     number of permutations at which p-value is evaluated
% head wss.txt
name2	sample_size_wss	num_variants_wss	total_mac_wss	statistic_wss	pvalue_wss	std_error_wss	num_permutations_wss
AADACL4	3180	5	138	34206	0.911089	11215.6	1000
ABCD3	3180	3	42	12967	0.63037	6602.73	1000
ABCG5	3180	6	87	37794	0.248751	8912.03	1000
AAMP	3180	3	35	16160	0.290709	5777.64	1000
ABCB10	3180	6	122	56091	0.145854	10409.2	1000
ABHD1	3180	5	29	9825	0.605395	5363.56	1000
ABCB6	3180	7	151	49949	0.608392	11831.6	1000
ABL2	3180	4	41	16097	0.438561	6499.52	1000
ACADM	3180	4	103	19070	0.967033	9782.51	1000


1 Bo Eskerod Madsen and Sharon R. Browning (2009) A Groupwise Association Test for Rare Mutations Using a Weighted Sum Statistic. PLoS Genetics doi:10.1371/journal.pgen.1000384. http://dx.plos.org/10.1371/journal.pgen.1000384

2 Mathieu Lemire (2011) Defining rare variants by their frequencies in controls may increase type I error. Nature Genetics doi:10.1038/ng.818. http://www.nature.com/doifinder/10.1038/ng.818