In fboehm/reducedscan: QTL Scans with Simpler Statistical Models in Multiparental Populations

knitr::opts_chunk$set(echo = TRUE)

Overview

The goal is to infer, for a known QTL position, the allelic series.

Model

$$Y = XCa + \epsilon$$

where Y is a n by 1 vector of trait values, X is a n by 8 matrix of founder allele probabilities for a single marker, C is a 8 by l allelic series configuration matrix, and a is a l-long vector of founder allele effects.

The number of alleles, then, is $l$. $\epsilon$ is a random error term, n-long vector, with distribution:

$$\epsilon \sim N(0, \sigma^2I)$$

The observed quantities are $Y$ and marker founder allele probabilities, $X$.

Unobserved quantities are $C|l$, $l$, $a$, and $\sigma^2$.

Conditional distribution

$$p(C, l, a, \sigma^2 | y, X) \propto p(y | C, l, a, \sigma^2)\pi(C, l)\pi(a)\pi(\sigma^2) \ \propto p(y | C, l, a, \sigma^2)\pi(C| l)\pi(l) \pi(a)\pi(\sigma^2)$$

The term $p(y | C, l, a, \sigma^2)$ is the likelihood. $\pi$ specifies a prior for the argument, ie, $\pi(a)$ is the prior distribution for $a$.

We'll use a Metropolis-Hastings algorithm to draw samples from the posterior distribution.

At each step in a single iteration, we condition on current values of the unknowns.

1. update allelic effects $a$
2. update residual variance $\sigma^2$
3. update $C | l$
4. update $l$

fboehm/reducedscan documentation built on April 30, 2021, 8:31 p.m.