Problem Formulation

Given a table of $n$ variants $v_i$ with associated minor allele frequencies (MAF) $P(v_i) = p_i$, is it possible to model the distribution of the total number of variants any certain individual is expected to carry?

Solution

The "ground-truth" distribution can be simulated by generating individuals, iterating through the entire list of variants, sampling a uniform distribution for each variant to determine whether the variant is present in the individual (individual Bernoulli tests) and creating a histogram of the total sum of variants with successful tests.

Considering each variant $X_i$ to be binomial distribution of exactly one test (i.e. does the patient exhibit this variant or no?) with probability $p_i$ of success, the mean and variance of $X_i$ is $\mu_i = p_i$ and $\sigma_i^2 = p_i (1 - p_i)$. Further considering each of these random variables to be independent, the sum of all variants $X = \sum_i^n X_i$ will have a mean and variance of

$$\mu = \sum_i^n \mu_i = \sum_i^n p_i$$ $$\sigma^2 = \sum_i^n \sigma_i^2 = \sum_i^n p_i (1 - p_i)$$

As the Lyapunov condition is met, the Central Limit Theorem (CTL) is applicable. The total number of variants any individual carries is normally distributed with mean $\mu$ and variance $\sigma^2$.

The program runs two simulations: (1) the "ground-truth" with repeated Bernoulli tests for each individual variant and a KS-test for normality against the normal distribution with mean and variance from the "ground-truth" simulated distribution, and (2) a normal distribution with mean and variance from the proposed solution above and a KS-test for similarity against the "ground-truth" distribution. The first KS-test checks whether the Central Limit Theorem is applicable. The second KS-test checks whether the proposed solution is similar to the "ground-truth". With a large enough sample size $n > 1000$, both KS-tests prove similarity with significance $\alpha < 0.05$.

Usage

To retrive ten thousand minor allele frequencies from Ensembl BioMart, use

python get_mafs.py 10000 --outfile mafs.tsv

To perform simulations with ten thousand patients, use

python simulate.py 10000 --mafs mafs.tsv