# Gibbs in genetics

Tags: Gibbs

## Introduction

1. essential feature of genetic studies: involve related individuals.
2. standard analysis methods of epidemiology is inappropriate which assume independence.

## Standard methods in genetics

### Genetic terminology

1. genes, alleles, diallelic, genotype,
2. homozygous, heterozygous
3. dominant trait, recessive, codominant
4. fully penetrant, locus, partially penetrant
5. polygene
6. Hardy-Weinberg equilibrium

### Genetic models

1. $y_i$: phenotype for subject $i=1,\ldots,I$
2. $x_i$: measured risk factors
3. $z_i$: polygene
4. $P$: a probability

a genetic model is specified in terms of two submodels.

1. penetrance model $P(y\mid G, x,\Omega)$
2. genotype model $P(G,z\mid\Theta)$

penetrance model:

1. $y_i$ are conditionally independent given their genotype
2. consider late-onset disease traits, characterized by a dichotomous disease status indicator $d$.
3. an age variable $t$
4. hazard function: $\lambda(t)$
5. the penetrance for an unaffected individual is the probability of surviving to age $t$ free of the disease, $$S(t) = \exp[-\Lambda(t)]\,,$$ where $\Lambda(t)=\int_0^t\lambda(u)du$
6. the penetrance for an affected individuals is the density function $\lambda(t)S(t)$
7. proportional hazards model: $$\lambda(t, G, x, z)=\lambda_0(t)\exp\{\beta x+\gamma\cdot \mathrm{dom}(G)+\eta z+\ldots\}$$
8. genotype model: $$P(G\mid\Theta)=P(G_1\mid\Theta)\prod\limits_{i=2}^IP(G_i\mid G_1,\ldots,G_{i-1};\Theta)$$

segregation analysis.

## New words

1. nuisance
2. frailty
3. meiosis
4. sperm
5. pedigree
6. daunting
7. spouse

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