original ideas focused on rare & costly markers
models & methods refined as technology advanced

• single marker regression
• QTL (quantitative trait loci)
  • single locus models: interval mapping for QTL
  • QTL model search: QTLs & epistasis
• polygenes (association mapping)
  • adjust for population structure
  • capture "missing heritability"
• genome-wide selection

• only detect some genetic effects
  • significant QTL
• effects of modest or small effect ignored
  • non-significant QTL
  • effects too small to observe or test
• these other effects have two sources
  • many small effects on phenotype
  • population admixture reflected in genome structure

"actual" genetic model \(y = \mu_q + e\)
with \(J\) genetic effects (recode \(q_j\) to have variance 1)

\[\mu_q = \mu+\sum_{j=1}^J \beta_j q_j\]

actual heritability: \[h^2 = \frac{\sum_{j=1}^J \beta_j^2}{\sigma^2 + \sum_{j=1}^J \beta_j^2}\] (consider backcross and ignore epistasis from here forward)

but "best" QTL model has \(2\) terms

\[ \mu_q = \mu+ \beta_1 q_1 + \beta_2 q_2\] with heritability: \[h_\texttt{QTL}^2 = \frac{\beta_1^2 + \beta_2^2}{\sigma^2 + \sum_{j=1}^J \beta_j^2}\]

missing genetic variability: \[h^2 - h_\texttt{QTL}^2 > 0\]

Eskin (2105) http://dx.doi.org/10.1145/2817827

• spurious association of phenotype
• population structure affects
  • some phenotypes
  • some genetic loci
• but genotype may not affect phenotype
  • if we adjust for population structure

Eskin (2106 draft) Review doi:10.1101/092106

Eskin (2106 draft) Review doi:10.1101/092106

\[y = \mu_q + g + e\]

\(\mu_q\) = QTL effects (fixed)
\(g\) = polygenic effects (random)
\[g\sim N(0,\sigma_g^2 K)\]
e = unexplained variation (random)
\[e\sim N(0,\sigma^2 I)\]

\(K\) = kinship matrix
\(I\) = identity matrix (1s on diagonal, 0s off diagonal)

• estimate kinship \(K\) via pedigree: all we had in past
  • average / predicted relationship
  • works globally, might be inaccurate locally
  • think siblings vs parent/offspring
• estimate kinship \(K\) via SNP or GBS
  • estimate \(K\) from marker data \(M\)
  • in past, selected "neutral" markers
  • now use markers away from QTL \(q\)

\[K = c*M^{\text{T}}M\] \(c\) set so diagonal of \(K\) is 1

distribution of phenotype \[y = \mu_q + g + e\] \[y \sim N(\mu_q,V), V =\sigma_g^2K + \sigma^2I\]

iterate to solve (similar to EM idea)

• get MLE of \(\mu_q\) given \(V\): \(\hat{\mu}_q=(V^{\text{T}}V)^{-1}V^{\text{T}}y\)
• estimate \(\sigma_g\) and \(\sigma^2\) given \(\mu_q\)

• 283 mice
• Diversity Outbred cross
• generations 4 & 5
• 320 (of 7851) SNP markers
• phenotype = OF_immobile_pct
• Data: https://github.com/rqtl/qtl2data/
  
• Recla, Robledo, Gatti, Bult, Churchill, Chesler (2014)

• allele-based genome scan: LOD maps
  • continuous curve across loci
  • interval mapping for missing data
  • model effect of founder alleles
• DO founder alleles: A,B,C,D,E,F,G,H
• response ~ sum of effects of alleles
• predict allele effects
  • naive: allele means based on geno probs
  • BLUP: predicted allele effects using kinship

• SNP-based genome scan: GWAS Manhattan plots
  • discrete tests of SNPs or other features
  • typically 2 SNP alleles
  • model effect of number of non-ref SNP copies
• SNP recorded as pair of DNA base pairs (A,C,G,T)
  • SNPs typically have two values (G/T)
  • individual has genotype GG, GT or TT
• account for other associations with kinship
  • effects of other SNPs
  • population structure

• consider SNPs within region of LOD peak
• use SNPs to refine search
  • relate SNPs to genomic features
  • compare SNP pattern across founders
  • DO reference is B = B6
  • \(s\) = 0,1,2 copies of non-reference nucleotide
• mixed model \(y= a+bs+g+e\)
  • \(\mu_q\) replaced by \(a+bs\)
• test slope \(b=0\) using LOD score