• Testing associations are as simple as t-test and ANOVA

• A more generalized framework: Likelihood test

\[LRT = L_0 / L_1 = -2(logL_1 - logL_0)\]

For the model:

\[y=Xb+Zu+e\\ y\sim N(Xb,V)\]

REML function is given by:

\[L(\sigma^2_u,\sigma^2_e)=-0.5( ln|V|+ln|X'V^{-1}X|+y'Py)\]

Where \(V=ZKZ'\sigma^2_u+I\sigma^2_e\) and \(y'Py=y'e\)

Types of allele coding

1. Add. (1 df): {-1,0,1} or {0,1,2} - Very popular (Lines, GCA)
2. Dom. (1 df): {0,1,0} - Popular (Trees, clonals and Hybrids)
3. Jointly A+D (2 df): Popular on QTL mapping in F2s
4. Complete dominance (1 df): {0,0,1} or {0,1,1} - Very unusual
5. Interactions (X df): Marker x Factor (epistasis and GxE)

Power

• Key: Number of individuals & allele frequency
• More DF = lower power
• Multiple testing: Bonferroni and FDR
• Tradeoff: Power vs false positives

Resolution

• Genotyping density
• LD blocks
• Recombination

GWAS tests \(m\) hypothesis:

• No correction: \(\alpha = 0.05/m\)
• Bonferroni: \(\alpha = 0.05/m\)
• FDR (25%): \(\alpha = 0.05/(m\times0.75)\)

• Generally on experimental pops (F2, DH, RIL, BC)
• Based on single-marker analysis or interval mapping
• Recombination rates would increase power

• Use of historical LD between marker and QTL
• AM allowed studies on random panels
• Dense SNP panels would increase resolution

1. Confounding associations with sub-populations
2. Major limitation of association mapping
3. Structure: PCs, clusters, subpopulation (eg. race)

Less missing values = more obs. = more detection power

• Markov models: Based on flanking markers
• Random forest: Multiple decision trees capture LD
• kNN & Projections: Fill with similar haplotypes

• Full model (\(L_1\)): \[y = Xb + m_ja + e\]
• Null model (\(L_0\)): \[y = Xb + e\]

1. Advantage: Fast, not restricted to Gaussian traits
2. Popular methodology on human genetic studies
3. \(Xb\) includes (1) environment, (2) structure and (3) covariates

• Full model (\(L_1\)): \[y = Xb + Zu + m_ja + e\]
• Null model (\(L_0\)): \[y = Xb + Zu + e\]

1. The famous "Q+K model"
2. Advantage: Better control of false positives, no need for PCs
3. Polygenic effect (\(u\)) assumes \(u\sim N(0,K\sigma^2_u)\)
4. Faster if we don't reestimate \(\lambda = \sigma^2_e/\sigma^2_u\) for each SNP

1. Uses the same base model as MLM
2. Advantage: Faster than MLM
3. Based on clustered individuals:

• \(Z\) is indicates the subgroup
• \(K\) is the relationship among subgroup
• Often needs PCs to complement \(K\)

1. Tests all markers at once
2. Advantage: No double-fitting, no PCs, no Bonferroni

• Model (BayesB, BayesC, SSVS): \[y = Xb + Ma + e\]
• Marker effects are from a mixture of distributions

\(a_j \sim Binomial\) with \(p(\pi) = 0\) and \(p(1-\pi) = a_j\)

1. Screen a set (\(s\)) of low MAF markers on NGS data
2. Advantage: Detect signals from low power SNPs
3. Applied to uncommon diseases (not seen in plant breeding)
4. Two possible model

• Full model 1 (\(L_1\)): \(y = Xb + M_sa + e\)
• Full model 2 (\(L_2\)): \(y = Xb + PC_1(M_s) + e\)
• Null model (\(L_0\)): \(y = Xb + e\)

Test either \(LR(L_1,L_0)\) or \(LR(L_2,L_0)\)

• QTLs detected with 3 methods, across 3 mapping pops
• Validations made on 3 unrelated populations