Computational Methodology

Composite interval mapping [ZengZeng1993,ZengZeng1994] combines interval mapping with multiple regression. The statistical model is defined as

$${\bf Y}={\bf x^{\ast}}b^{\ast} +{\bf z^{\ast}}d^{\ast} +{\bf XB} +{\bf E}$$ (3.5)

where

• ${\bf Y}$ is a vector of trait values
• $b^{\ast}$ and $d^{\ast}$ are the additive and dominance effects of the putative QTL being tested
• ${\bf x^{\ast}}$ and ${\bf z^{\ast}}$ are indicator variable vectors specifying the probabilities of an individual being in different genotypes for the putative QTL constructed by flanking makers
• ${\bf B}$ is the vector of effects of other selected markers fitted in the model
• ${\bf X}$ is the marker information matrix for those selected markers
• ${\bf E}$ is the error vector.

Estimates of the parameters are obtained by maximum likelihood through an ECM (for Expectation/Conditional Maximization) algorithm [Meng and RubinMeng and Rubin1993]. In each E-step, the probability of an individual being in different genotypes of the putative QTL is updated. In the CM-step, the estimation of parameters $b^{\ast}$ and $d^{\ast}$ is separated from that of ${\bf B}$, and each group is estimated conditional on the others. This procedure is implemented for numerical consideration. As ${\bf x^{\ast}}$ and ${\bf z^{\ast}}$ are separated from ${\bf X}$, ${\bf X}$ is unchanged in each iteration, and its costly recalculation is avoided.

For an $F_2$ population, the hypotheses for testing are $H_{0}: b^{\ast}=0 {\rm and} d^{\ast}=0$ and $H_{3}: b^{\ast}\neq 0 {\rm or} d^{\ast}\neq 0$. This is performed through a likelihood ratio test procedure. In addition, it is possible to test hypotheses on $b^{\ast}$ and $d^{\ast}$ individually. For a backcross data set, dominance cannot be estimated and $d^{\ast}$ is dropped from Equation 3.5.

The trait will have a variance $s^2$. Under the null hypothesis

$$H_0: {\bf Y}= {\bf XB} + {\bf E}$$

the sample variance of the residuals will be $s^2_0$. For a given alternative model, say

$$H_1: {\bf Y}={\bf x^{\ast}}b^{\ast} +{\bf z^{\ast}}d^{\ast} +{\bf XB} +{\bf E}$$

the variance of the residuals would be $s^2_1$. With this in mind we can calculate the proportion of variance explained by a QTL at the test site. The quantity is usually called $r^2$ and estimated by

$$r^2 = \frac{s_0^2 - s_1^2}{s^2}$$

An alternative estimate would use the total variance. Denote it by

$$r^2_t = \frac{s^2 - s_1^2}{s^2}$$

$r^2$ is the proportion of the variance explained by the QTL conditioned on the background markers and any explanatory variables. $r^2_t$ is the proportion of the total variance explained by the QTL and the the background markers and any explanatory variables. Generally, $r^2_t \geq r^2$