Interval Mapping and the LOD Score
Lander and Botstein (1989) introduced interval mapping, which evaluates QTL presence at each position $\lambda$ along a chromosome using a likelihood ratio:
\[\text{LOD}(\lambda) = \log_{10}\!\left(\frac{L(\text{QTL at }\lambda)}{L(\text{no QTL})}\right)\]Under the QTL model, phenotype $y_i$ given flanking marker genotypes has a mixture distribution. A LOD $> 3$ (corresponding roughly to $p < 0.001$ genome-wide) is the traditional threshold.
Marker Regression at a Single Marker
The simplest form regresses phenotype on marker genotype class. For an $F_2$ intercross with QTL genotypes $QQ$, $Qq$, $qq$:
| Genotype | Frequency | Mean |
|---|---|---|
| $QQ$ | $\tfrac{1}{4}$ | $\mu + a$ |
| $Qq$ | $\tfrac{1}{2}$ | $\mu + d$ |
| $qq$ | $\tfrac{1}{4}$ | $\mu - a$ |
where $a$ is the additive effect and $d$ the dominance deviation. The $F$-test on the regression recovers the marker–trait association.
Composite Interval Mapping
Composite interval mapping (CIM) conditions on flanking markers to reduce residual variance and control for linked QTL:
\[y_i = \mu + \beta^* x^*_i + \sum_{k \in \mathcal{C}} b_k m_{ik} + \varepsilon_i\]where $x^*_i$ is the QTL genotype probability at position $\lambda$ (computed from flanking markers via recombination fractions), and the cofactor markers $\mathcal{C}$ absorb variance from elsewhere in the genome.
Permutation Thresholds
Because the LOD profile is correlated across positions, analytical $p$-value thresholds are conservative or anti-conservative. Churchill and Doerge (1994) proposed genome-wide thresholds from permutation:
- Permute phenotype–genotype labels (breaks QTL signal, preserves marker correlation).
- Record maximum LOD across the genome for each permutation.
- Use the 95th percentile of the empirical maximum-LOD distribution as the significance threshold.
Typically $B = 1000$ permutations suffice; this is the gold standard for single-QTL tests.