# ASCI 896 Statistical Genomics

Gota Morota
2/23/16

### Ritland (1996)

\begin{align} \hat{r} &= 2 \times \frac{\sum^m_{k=1} \sum^2_{ki} \frac{f_{M_{ki}} - p^{2}_{ki}}{p_{ki}}}{ m } \end{align} where $$f_{M_{ki}}$$ is the molecular coancestry contributed by allele $$i$$ at the $$k\text{th}$$ locus, $$p_i$$ is the frequency of allele $$i$$, and $$m$$ is the total number of SNPs.

### Nejati-Javaremi et al. (1997)

\begin{align*} \hat{r} &= 2 \times \frac{\sum^m_{k=1} \frac{\sum^2_{i=1} \sum^2_{j=1} I_{k,ij}}{4}}{m} \end{align*} where $$\frac{\sum^2_{i=1} \sum^2_{j=1} I_{k,ij}}{4}$$ is the molecular similarity between two individuals at the $$k\text{th}$$ locus.

\begin{align*} \mathbf{G} = \frac{\mathbf{X_cX_c}^T }{2 \sum p_k(1-p_k)} \end{align*} where $$\mathbf{X_c}$$ is the centered genotype matrix and $$p_k$$ is the frequency of reference allele at the $$k\text{th}$$ locus.
\begin{align*} \mathbf{G} = \frac{\mathbf{X_{cs}X^{'}_{cs}} }{m} \\ \end{align*} where $$\mathbf{X_cs}$$ is the centered and scaled genotype matrix, and $$m$$ is the number of SNPs.
Alternatively, \begin{align*} \mathbf{G = X_cDX_c'} \end{align*} where $$\mathbf{D}$$ is diagonal with $$d_{ii} = 1/(m[2p_k (1 - p_k) ])$$.
Or \begin{align*} G_{ij} &= \frac{1}{m} \sum^m_{k=1} \frac{(x_{ik} - 2p_k)(x_{jk} - 2p_k) }{ 2p_k (1 - p_k)} \end{align*}