Statistical Methods for QTL Analysis

See also QTL Analysis and Quantitative Genetics and  QTL Analysis in R.

In a previous post a gave a general overview of QTL mapping and analysis, and gave the motivation for the use of maximum likelihood to identify the approximate location of a QTL based on RFLP (or marker) variations. Below are more details related to the statistical methods that can be used in this process.

Analysis of Variance and Marker Regression

At each marker (RFLP) loci, compare backcross phenotype distributions for groups that differ according to their marker genotypes. As depicted in Broman (2001)

 

For markers a and c, we see that phenotype distributions differ for genotypes aa and aa’ but not for cc and cc’. Again this indicates that a QTL may be linked to the RFLP genotypes at locus “a” but not "c". 

The differences between genotypes and the associated phenotype distributions for two marker genotypes can be assessed using the t-statistic. For >2 genotypes, analysis of variance may be used. The ANOVA approach allows a flexible experimental design, allowing for the incorporation of covariates, treatment, and environmental effects. 

The model for marker regression, following the notation in Hu and Zu (2009) can be specified as follows:

y_i= X_iβ + Z_iγ +ϵ_i

such that  y_i  is the phenotype of the ith individual, β is the vector of control effects, X_i is design vector,γ is a vector for QTL effects, and Z is a genotype indicator vector.

Z = H₁ for A₁A₁ , H₂ for A₁A₂ , H₃ for A₂A₂  or more generally 

 

Tests on hypotheses related to QTL effects take the form H₀: γ = 0 . Hence we test the null hypothesis of no QTL associated with genotype Z. 

Maximum Likelihood

For maximum likelihood estimation, the following probability density of y_i can be stated as:

f(y_i) = Pr(y_i­|Z_i=H_k)  ‘ the probability of phenotype ‘y’ given genotype H_k­’

= (1/ √ 2π σ)  exp[ 1/2 σ ² (y_i-X_iβ +H_kγ )²

The log likelihood function can then be specified as L(θ) =Σ  ln(f(y_i))

The hypothesis H₀:  γ =0 can be tested using the likelihood ratio test:

λ = -2(L₀-L₁)

where L₀ represents the likelihood under a restricted model. This is equivalent to the general notion presented in Broman (1997) and my previous post:

Likelihood (effect occurs by QTL linkage) / Likelihood(effect occurs by chance)

References:

 Jones, N., H. Ougham, and H. Thomas. Markers and mapping:We are all geneticists now. New Phytol. 137:165–177.1997.

Broman KW. Lab Anim (NY). Review of statistical methods for QTL mapping in experimental crosses.

2001 Jul-Aug;30(7):44-52.

Zhiqiu Hu and Shizhong Xu (2009). PROC QTL - A SAS Procedure for Mapping Quantitative Trait Loci. International Journal of Plant Genomics 2009: 3 doi:10.1155/2009/141234.