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

• Want to figure out what is going on
  • preliminary search: find important story
  • need strategies to uncover patterns

• Want to tell story in publication

• How to accomplish QTL mapping goal
  • organic search for patterns
  • organize methods as you go
  • document steps (so you can redo)

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

Genetic map for Osborn's Brassica napus study

• Also known as ANOVA
• Split sample into groups
  • by genotype at marker
  • red = missing genotype
• Do a t-test or ANOVA
• Repeat for each marker

Soller et al. (1976)

\[ y = \mu_m + e \]

• \(y\) = phenotypic trait
• \(m\) = marker genotype (0,1)
• \(\mu_m\) = mean for genotype \(m\)
• \(e\) = error = unexplained variation

Marker regression:

• fit model for each marker across genome
• pick most significant marker

• Advantages
  • simple; no need for genetic map
  • easy to add covariates
  • easily extended to more complex models
  • ignores marker position on genome
• Disadvantages
  • excludse individuals with missing genotype data
  • imperfect information about QTL location
  • suffers in low density scans
  • only considers one QTL at a time

• missing data problem: Markers \(\longleftrightarrow\) QTL
• model selection problem: QTL, covariates \(\longrightarrow\) phenotype

• Assume a single QTL model.
• posit each genome position \(\lambda\), one at a time, as putative QTL
  • \(q\) = genotypes at locus \(\lambda\)

\[ \texttt{pr}(y | q): y = \mu_q + e \]

• mixing proportions over flanking markers \[ \texttt{pr}(q | m): \texttt{table of proportions}\]

• model is mixture over possible QTL genotypes \(q\)

• mixture of normals

Lander & Botstein (1989) Genetics

Calculate \(pr(q|m)\) assuming

• no crossover interference
• no genotyping errors

Or use the hidden Markov model (HMM)} technology

• to allow for genotyping errors
• to incorporate dominant markers

Calculate \(pr(q|m)\) assuming

• no crossover interference
• no genotyping errors

Or use the hidden Markov model (HMM)} technology

• to allow for genotyping errors
• to incorporate dominant markers

Calculate \(pr(q|m)\) assuming

• no crossover interference
• no genotyping errors

Or use the hidden Markov model (HMM)} technology

• to allow for genotyping errors
• to incorporate dominant markers

Calculate \(pr(q|m)\) assuming

• no crossover interference
• no genotyping errors

Or use the hidden Markov model (HMM)} technology

• to allow for genotyping errors
• to incorporate dominant markers

Calculate \(pr(q|m)\) assuming

• no crossover interference
• no genotyping errors

Or use the hidden Markov model (HMM)} technology

• to allow for genotyping errors
• to incorporate dominant markers

Calculate \(pr(q|m)\) assuming

• no crossover interference
• no genotyping errors

Or use the hidden Markov model (HMM)} technology

• to allow for genotyping errors
• to incorporate dominant markers

\(\texttt{pr}(y|m) = \sum \texttt{pr}(y|q)\texttt{pr}(q|m)\)

• 2 markers separated by 20 cM
  • QTL closer to left marker
• phenotype distribution
  • given marker genotypes
• mixture components
  • dashed curves

think marker regression with fuzzy groups

QTL genotype given markers: \(\texttt{pr}(q|m)\)

phenotype given QTL: \(\texttt{pr}(y|q)= \text{N}(y|\mu_q,\sigma^2)\) (normal density)

\[\texttt{pr}(y|m) = \sum_q \texttt{pr}(y|q)\texttt{pr}(q|m)\]

log likelihood over individuals:

\[l(\mu_0,\mu_1,\sigma) = \sum_i \log \texttt{pr}(y_i | m_i)\]

find \(\hat{\mu}_0\), \(\hat{\mu}_1\), \(\hat{\sigma}\) to maximize \(l(\mu_0,\mu_1,\sigma)\) (MLEs)

E step: (pseudo)weights for individual \(i\), QTL genotype \(q\) \[w_{iq} = \texttt{pr}(q|m_i,y_i,\hat{\mu},\hat{\sigma}) = c_i * \texttt{pr}(q|m_i) \text{N} (y_i|\hat{\mu}_q,\hat{\sigma})\] \(c_i\) set so that \(\sum_q w_{iq} = 1\)

M step: (pseudo)values for QTL group means and variance \[\hat{\mu}_q = \sum_i y_i w_{iq} / \sum_i w_{iq}\]

\[\hat{\sigma}^2 = \sum_i \sum_q w_{iq} (y_i-\hat{\mu}_q)^2/n\]

EM algorithm: set \(w_{iq} = \texttt{pr}(q|m_i)\); iterate E&M to converge

Idea: just run one iteration of EM algorithm

• becomes marker regression on genotype probabilities
• ignores mixture of normals issue
• now widely used for dense marker maps (high throughput)

Haley, Knott (1992
Martinez, Curnow (1992

LOD score measures strength of evidence for QTL at locus \(\lambda\)
\(\log_{10}\) likelihood ratio of models:

• model with QTL at \(\lambda\) (mean depends on QTL genotype \(q\) at \(\lambda\))
• model with no QTL (common mean for all individuals)

\[\texttt{lod}(\lambda) = [l(\hat{\mu}_{0\lambda}, \hat{\mu}_{1\lambda}, \hat{\sigma}_\lambda) - l(\hat{\mu}, \hat{\sigma})]/\log(10)\]

QTL model: means are MLEs \(\hat{\mu}_{0\lambda}, \hat{\mu}_{1\lambda}\) with QTL at \(\lambda\)

No QTL model: mean is unconditional MLE \(\hat{\mu}=\bar{y}\)

SD computed given model means: \(\hat{\sigma}_\lambda, \hat{\sigma}\)

LOD and means by genotype scans on chr N2

• Advantages
  • takes proper account of missing data
  • allows examination of positions between markers
  • gives improved estimates of QTL effects
  • provides pretty graphs (important!)
• Disadvantages
  • increased computation time
  • requires specialized software
  • difficult to generalize and extend
  • only one QTL at a time

Large LOD scores = evidence for presence of a QTL
LOD threshold = 95 %ile of histogram of max LOD genome-wide (if there are no QTLs anywhere)

Derivation:

• Analytical calculations (Lander & Botstein 1989)
• Simulations (Lander & Botstein 1989)
• Permutation tests (Churchill & Doerge 1994)

• Null distribution from simulation
  • backcross with typical size genome
• Dashed curve:
  • LOD score histogram for any one point
• Solid curve:
  • max LOD histogram, genome-wide

shuffle phenotypes independent of genotype data
repeat 10,000 times

significant area is quite broad …

but 1.5 LOD support interval is narrower

• 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 …

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

• 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.