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
• GWA (genome-wide association mapping)
  • adjust for population structure
  • capture "missing heritability"
  • genome-wide selection

Satagopan JM, Yandell BS, Newton MA, Osborn TC (1996) Genetics

• examples: treatment, location, age, weight, height
• account for important design structure (location)
• adjust for important predictors
• reduce residual variation to increase power
  • covariate with strong effect

\(\text{H}_0: y = \mu + \beta_x x + e\)
\(\text{H}_a: y = \mu_q + \beta_x x + e\)

• more complicated (ignored here)
  • QTL * covariate interactions
  • covariate as ratio \(y/x\)

• use care when the covariate is another phenotype
• permutations: keep phenotype & covariates together

• Goals
  • identify QTL (and possible interactions among QTL)
  • estimate interval for QTL location
  • estimate QTL effects
• Challenges
  • how many QTL? which ones?
  • more complicated to fit each multiple QTL model
  • need rules to search across many QTL models

• benefits
  • reduce residual variation
  • increased power
  • separate linked QTL
  • identify interactions among QTL (epistasis)
• shortcomings
  • only includes significant loci
  • gets complicated very quickly
  • selection bias: overestimate effects of included loci
  • many loci of small effect ignored …

basic model looks the same

\[ y = \mu_q + e \] but now QTL has parts: \(q = (q_1, q_2, \cdots)\) \[ \mu_q = \mu(q_1, q_2, \cdots) = \mu+q_1\beta_1+q_2\beta_2+\cdots\]

• allows for multiple loci
• can add epistasis (here for BC)

\[ \mu_q = \mu+\beta_1q_1+\beta_2q_2+\gamma q_1 q_2 \] - more terms for F2 …

For all pairs of positions, fit the following models:

\(\texttt{H}_f: y = \mu + \beta_1 q_1 + \beta_2 q_2 + \gamma q_1 q_2 + e\)
\(\texttt{H}_a: y = \mu + \beta_1 q_1 + \beta_2 q_2 + e\)
\(\texttt{H}_1: y = \mu + \beta_1 q_1 + e\)
\(\texttt{H}_0: y = \mu + e\)

log\(_{10}\) likelihoods for QTL positions \(\lambda_1\) (for \(q_1\)) and \(\lambda_2\) (for \(q_2\))

\(l_f(\lambda_1,\lambda_2)\)
\(l_a(\lambda_1,\lambda_2)\)
\(l_1(\lambda_1)\)
\(l_0\)

full (interactive) vs no QTL (\(f-0\)): \[\texttt{lod}_f(\lambda_1,\lambda_2) = l_f(\lambda_1,\lambda_2) - l_0\]

additive vs no QTL (\(a-0\)): \[\texttt{lod}_a(\lambda_1,\lambda_2) = l_a(\lambda_1,\lambda_2) - l_0\]

interaction, or full vs additive (\(f-a\)): \[\texttt{lod}_i(\lambda_1,\lambda_2) = l_f(\lambda_1,\lambda_2) - l_a(\lambda_1,\lambda_2)\]

usual 1 QTL vs no QTL (\(1-0\)): \[\texttt{lod}_1(\lambda_1) = l_1(\lambda_1) - l_0\]

R/qtl tools: sim.geno, makeqtl, addqtl
fancier form of using first QTL as covariate

• no evidence for 2 QTL at 4 week
• modest evidencde for 2 QTL at 8 week
• no evidence for interaction

Schranz et al. Osborn (2002)

• QTL mapping began as hypothesis testing: is this a QTL?
  • much focus on adjusting for multiple testing
• better to view problem as model selection
  • what set of QTL are well supported?
  • is there evidence for QTL-QTL interactions?

Model = an identified set of QTL and QTL-QTL interactions
(and possibly covariates and QTL-covariate interactions)

• Class of models: begin with additive models
  • add pairwise or higher interactions?
  • other approaches?
• Model fit (MLE, Haley-Knott, …)
• Model comparison
  • estimated prediction error
  • model effects criterion: AIC, BIC, penalized likelihood
  • Bayes method (prior across model space)
• Model search
  • forward, backward, stepwise selection
  • randomized algorithm

Goal: identify major players

• selected model has two types of errors:
  • miss important terms (QTLs or interactions)
  • include extraneous terms
• both errors likely at the same time
  • identify as many correct terms as possible
  • while controlling rate of inclusion of extra terms
• hypothesis testing only has one error at a time
  • pick no QTL model, but there is really a QTL at \(\lambda_1\)
  • pick 1 QTL model, but there is really no QTL

What is special here?

• continuum of ordinal-valued predictors (the genetic loci)
• association among these QTL predictors
• loci on different chromosomes are independent
• along chromosome:
  • simple (and known) correlation structure

See Broman MultiQTL talk for more details

• Benefits
  • reduce residual variation
  • increased power
  • separate linked QTL
  • identify interactions among QTL (epistasis)
• Shortcomings
  • only includes significant loci
  • gets complicated very quickly
  • selection bias: overestimate effects of included loci
  • many loci of small effect ignored …

• estimated QTL effect QTL varies from true effect
• detect QTL when estimated effect is large
• experiments with detected QTL often have larger estimated than true effect
• selection bias largest in QTLs with small or moderate effects
• true QTL effects smaller than those observed

• estimated % variance explained by identified QTLs: too high
• repeating an experiment: different QTL (Beavis effect)
• congenics (or near isogenic lines): off base
• marker-assisted selection: missed effect

See Broman (2003) and Haley, Knott (1992).
Beavis WD (1994). The power and deceit of QTL experiments: Lessons from comparative QTL studies. In DB Wilkinson, (ed) 49th Ann Corn Sorghum Res Conf, pp 252–268. Amer Seed Trade Asso, Washington, DC.