Gibbs in genetics

Posted on Aug 24, 2018

The note is for Gilks, W. R., Richardson, S., & Spiegelhalter, D. (Eds.). (1995). Markov chain Monte Carlo in practice. CRC press..

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
7. Linkage analysis

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

比较表型和性状：

性状是指生物体所有特征的总和，由基因决定，必须是可以遗传的。而表型则是这些基因决定的性状在环境作用下的具体表现，与性状的概念有着本质区别，表型是不可遗传的。所以说表型又称性状的观点是不符合遗传学概念的。

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