Variance components linkage analysis is a powerful method to detect quantitative trait loci (QTLs) for complex diseases. It has the advantages of easy applicability to large extended pedigrees and provides a good flexible framework to accommodate more complicated models like gene-gene, gene-environmental interactions. This dissertation consists of two major parts. In the first part, I propose two approaches for deriving relative-to-relative covariances that are indispensable for expanding the applications of standard variance components linkage approach to more complicated genetic models such as those involving genomic imprinting. In the first approach, I extend 'Li and Sacks' ITO method to model ordered genotypes and derive some generalized linear functions of the extended transition matrices. I demonstrate the wide applicability of this extension by applying it to calculate the covariance in unilineal and bilineal relatives under genomic imprinting. In the second approach, I derive a general formula for calculating the genetic covariance using ordered genotypes for any type of relative pair, which does not have the limitation of extended ITO method to biallelic loci and to unilineal and bilineal relatives. I also propose a recursive algorithm to calculate necessary coefficients in the formula, which opens up the possibility of calculating even inbred relative-to-relative covariance.In the second part of my dissertation, I discuss linkage evidence for susceptibility loci for adiposity-related phenotypes in the Samoan population, an extensive summary of our multicenter study "Genome-scan for Obesity Susceptibility Loci in Samoans". Obesity, BMI greater than or equal to 30 kg/m^2, in the U.S. has become a major and serious public health problem, affecting 33% of adults in 2002. Obesity increases risks for serious diet-related diseases, such as cardiovascular disease, type-2 diabetes, and certain forms of cancers. Obesity is a typical multi-factorial disease with overwhelming evidence of genetic effects, yet their roles in obesity are largely unknown. Our current research findings will help further understand the whole picture of the genetics of obesity, which may have great influence on early prevention and later interventions of human obesity, making it a fundamentally important contribution to public health
