A Simple Linear Regression Method for Quantitative Trait Loci Linkage Analysis with Censored Observations
Anderson, Carl A., McRae, Allan F., Visscher, Peter M., Genetics
Standard quantitative trait loci (QTL) mapping techniques commonly assume that the trait is both fully observed and normally distributed. When considering survival or age-at-onset traits these assumptions are often incorrect. Methods have been developed to map QTL for survival traits; however, they are both computationally intensive and not available in standard genome analysis software packages. We propose a grouped linear regression method for the analysis of continuous survival data. Using simulation we compare this method to both the Cox and Weibull proportional hazards models and a standard linear regression method that ignores censoring. The grouped linear regression method is of equivalent power to both the Cox and Weibull proportional hazards methods and is significantly better than the standard linear regression method when censored observations are present. The method is also robust to the proportion of censored individuals and the underlying distribution of the trait. On the basis of linear regression methodology, the grouped linear regression model is computationally simple and fast and can be implemented readily in freely available statistical software.
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DOMESTIC animals and experimental species provide a unique resource for the understanding of quantitative genetic variation. Quantitative trait analysis of experimental crosses has provided many important insights into the genetics of complex traits (MORGANTE and SALAMINI 2003; reviewed in ANDERSSON and GEORGES 2004). Several genes underlying quantitative genetic variation have been identified in the fields of animal and crop science, many of which have signifi- cant commercial potential (e.g., JEON et al. 1999; NEZER et al. 1999; FRARY et al. 2000; FRIDMAN et al. 2000; GRISART et al. 2002).
Most current quantitative trait loci (QTL) mapping techniques utilize an interval-mapping approach first put forward by LANDER and BOTSTEIN (1989). The approach places a hypothetical trait locus at fixed incremental positions (for example, every 1-2 cM) along a map of known marker positions and tests for its effect on the trait using information from flanking markers. For a given location the basic linear model is
y^sub ij^ = m^sub j^ + e^sub ij^,
where y^sub ij^ is the trait value for individual i with genotype j, m^sub j^ is the mean effect of genotype j, and e^sub ij^ is random error (e^sub ij^ ~ N(0, σ^sub e^^sup 2^ )). The genotype of an individual at the position being tested is rarely known so the probability of an individual being each of the possible genotypes is calculated from the available marker information. LANDER and BOTSTEIN (1989) implement their method using a maximum-likelihood approach. The maximum-likelihood method takes into account heterogeneous variances within marker classes to estimate genotype probabilities. The model parameters are estimated under both the null (no QTL) and alternative (with QTL) hypotheses. An advantage of maximum likelihood is that it uses all of the available observations on marker genotypes and trait values. The disadvantage of maximum likelihood is that it is computationally intensive and usually requires specialized software.
An alternative method, least-squares regression, uses expected genotype probabilities calculated from flanking markers rather than the more complex approximation viamaximum likelihood (HALEY and KNOTT 1992). For this approach least-squares linear regression is used to estimate the effect of genotype on the trait of interest at each test position along the genome. The asymptotic equivalence of least-squares regression with maximumlikelihood interval mapping has been shown through simulation (HALEY and KNOTT 1992) and by theoretical calculations of power (REBAI et al. 1995). The leastsquares approach has been shown to be robust to deviations from normality in all but themost extreme situations (VISSCHER et al. …