MImapqtl

MImapqtl implements QTL mapping analysis for multiple QTL in multiple intervals for a single trait in a single environment. It can begin analysis from an initial model specifying the positions of QTL, or de novo, that is with no initial model. If given an initial model, the program will estimate the parameters, refine the estimates of QTL positions within intervals, test the significance of all parameters, search for more QTL, search for epistatic interactions and finally calculate the $r^2$ and breeding values for the model. If analysis is initiated de novo, then there will be a search for QTL, a search for interactions and calculation of $r^2$ and breeding values. MImapqtl can also do a single pass search for a new QTL and create a likelihood ratio profile for a new putative QTL given any initial model.

For $m$ putative QTL, the model is

$$y_i = \mu + \sum_{r=1}^m \alpha_r x^*_{ir} + \sum_{r=1}^m \delta_r z^*_... ...^{AA}_{rs} x^*_{ir} x^*_{is} + \sum_{r\neq s} \beta^{AD}_{rs} x^*_{ir} z^*_{is}$$

$$\mbox{} + \sum_{r\neq s} \beta^{DA}_{rs} z^*_{ir} x^*_{is} + \sum_{r\neq s} \beta^{DD}_{rs} z^*_{ir} z^*_{is} + e_i$$ (3.6)

where $y_i$ is the trait and $e_i$ the residual for individual $i$. The parameters $\alpha_r, \delta_r$ are the additive and dominance effects of QTL $r$. The $\beta$'s are epistatic interactions. The superscripts on the $\beta$'s are for the type of interaction: We distinguish between additive by additive (AA), additive by dominance (AD), dominance by additive (DA) and dominance by dominance (DD) interactions. There is also a mean denoted by $\mu$ and a variance for the residuals ($\sigma^2$) which are assumed to have a normal distribution and mean zero. The $x^*$ and $z^*$ are coded variables denoting the genotype of the putative QTL. If there are only two marker genotypic classes, the $z^*$'s are all set to zero. Otherwise, $z^*_{ir} = 0.5$ for a heterozygote and $-0.5$ for a homozygote. The $x^*_{ir}$ are defined differently for different crosses and are shown in Table 3.7. The sum over $r\neq s$ means over all unordered QTL pairs and we use the convention $r > s$.

Table 3.7: Coded variables

		$x^*$
	$z^*$	$B_1$	$B_2$	$SF_i$	$RF_i$	$RI_i$
QQ	-1/2	1/2	0	1	1	1
Qq	1/2	-1/2	1/2	0	0	0
qq	-1/2	0	-1/2	-1	-1	-1

The likelihood function of the data given the model is a mixture of normal distributions

$$L( \mathbf{E}, \mu, \sigma^2 \vert \mathbf{Y}, \mathbf{X}) =... ...ij} \phi(y_i \vert \mu + \mathbf{D}_j \mathbf{E}, \sigma^2 ) ]$$ (3.7)

In (3.7), the $p_{ij}$ are the probabilities of the multilocus genotypes conditioned on marker data. The variable $g$ is the number of genotypic classes for the experimental design: For backcrosses and recombinant inbred lines, $g=2$ , while for intercrosses it is $g=3$. In practice, it is often infeasable to do the sum over all $g^m$ multilocus genotypes: We use a subset of the most frequent genotypes. The parameters are in $\mathbf{E}$ while the coded indicator variables are in $\mathbf{D}$. $\phi(y_i \vert \mu, \sigma^2 )$ is the normal density function with mean $\mu$ and variance $\sigma^2$. We use an EM (expectation maximization) algorithm to obtain maximum likelihood parameter estimates [Kao and ZengKao and Zeng1997,Zeng, Kao, and BastenZeng et al.1999].