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

## 1.  Introduction

This implements the {$C(\alpha)$} test (Neale et al 20111) 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]

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
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