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

## 1. Introduction

This implements the {$C(\alpha)$} test (Neale et al 2011^{1}) for disease traits, to test for the hypothesis of rare variants disease association under the particular assumption that rare variants observed in cases and controls is a mixture of phenotypically deleterious, protective and neutral variants. Instead of using a cumulative dosage (or "burden") based summary statistic over a gene region, it directly contrasts the observed and expected distribution of minor alleles in cases and controls at each locus as an evidence of "unusual distribution", and combine evidences from multiple loci (whether it be an evidence of protective or deleterious) to formulate the {$C(\alpha)$} statistic: {$$T=\sum_{i=1}^m[(y_i-n_ip_0)^2-n_ip_0(1-p_0)]$$}

The original paper evaluates p-value of the test under large sample normal assumption, which usually would not hold for the real world data. Implementation in this program also allows permutation based {$C(\alpha)$} test, if parameter `-p/--permutations`

is set greater than 0.

## 2. Details

### 2.1 Command interface

vtools show test Calpha

Name: Calpha Description: c-alpha test for unusual distribution of variants between cases and controls, Neale et al 2011 usage: vtools associate --method Calpha [-h] [--name NAME] [-q1 MAFUPPER] [-q2 MAFLOWER] [-p N] [--adaptive C] [--moi {additive,dominant,recessive}] c-alpha test for unusual distribution of variants between cases and controls, Neale et al 2011. It tests for deviation of variance of minor allele counts in cases/ctrls from its exception based on binomial distribution. The statistic is asymptotically normally distributed. p-value can be evaluated using either permutation or asymptotic distribution as described in Neale et al 2011, although it is recommended to use permutation to estimate a reliable p-value. Calpha test is a two-tailed test 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 parameters. -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 -p N, --permutations N Number of permutations --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 "Calpha --name Calpha -p 5000" --group_by name2 --to_db cal\ pha -j8 > calpha.txt INFO: 3180 samples are found INFO: 2632 groups are found Loading genotypes: 100% [=====================] 3,180 27.6/s in 00:01:55 Testing for association: 100% [=====================] 2,632/591 11.6/s in 00:03:46 INFO: Association tests on 2632 groups have completed. 591 failed. INFO: Using annotation DB calpha in project test. INFO: Annotation database used to record results of association tests. Created on Wed, 30 Jan 2013 15:54:03 vtools show fields | grep calpha calpha.refGene_name2 refGene_name2 calpha.sample_size_Calpha sample size calpha.num_variants_Calpha number of variants in each group (adjusted for specified MAF calpha.total_mac_Calpha total minor allele counts in a group (adjusted for MOI) calpha.statistic_Calpha test statistic. calpha.pvalue_Calpha p-value head calpha.txt name2 sample_size_Calpha num_variants_Calpha total_mac_Calpha statistic_Calpha pvalue_Calpha std_error_Calpha num_permutations_Calpha AADACL4 3180 5 138 0.0229344 0.407592 1.08434 1000 AAMP 3180 3 35 -0.444631 0.601399 0.896954 1000 ABCD3 3180 3 42 -0.911816 0.93007 1.0528 1000 ABCB6 3180 7 151 -0.751779 0.757243 1.05563 1000 ABCG8 3180 12 152 -0.0149743 0.36963 0.981793 1000 ABHD1 3180 5 29 -0.744439 0.845155 1.0768 1000 ABCB10 3180 6 122 1.14261 0.12094 1.02364 2000 ABL2 3180 4 41 -0.76715 0.966034 0.866904 1000 ACADL 3180 5 65 -0.50523 0.642358 0.943209 1000 |

^{1} Benjamin M. Neale, Manuel A. Rivas, Benjamin F. Voight, David Altshuler, Bernie Devlin, Marju Orho-Melander, Sekar Kathiresan, Shaun M. Purcell, Kathryn Roeder and Mark J. Daly (2011) **Testing for an Unusual Distribution of Rare Variants**. *PLoS Genetics* doi:`10.1371/journal.pgen.1001322`

. http://dx.plos.org/10.1371/journal.pgen.1001322 ⇑