A comment on the simple regression method for interval mapping.

H ALEY and KNOTT (1992) developed a simple regression method for mapping quantitative trait loci (QTLs) using F2 populations. MARTINEZ and CURNOW (1992) considered the same approach using backcross populations. Studies that compare the regression method with that of LANDER and BOSTEIN’S (1989) maximum likelihood find very little difference between the two methods. The simple regression method offers a great advantage over the maximum likelihood method in terms of computing speed. MARTINEZ and CURNOW (1992) state that the difference between the two methods arises solely because the residual term in the linear model does not have a normal distribution, being a random variable from one of several normal distributions. Theoretically, the regression method should suffer from the failure of the assumption of normality within marker genotype due to the segregation of the QTL, but the great majority of information is contained in mean differences between marker genotypes with little coming from the within marker genotype distribution (HALEY and KNOTT 1992). Therefore, the efficiency of the simple regression method is unlikely to be substantially less than the maximum likelihood approach (MARTINEZ and CURNOW 1992). The parameters of primary interest in QTL mapping are the positions and effects of QTLs. However, estimation of the residual variance is also important for the following reasons: (1) Effects of QTLs are only meaningful when reported relative to the size of the residual standard deviation; (2) The test statistic (t-test) is determined by the ratio of the estimated QTL effect to its standard deviation while the latter is proportional to the residual standard deviation; ( 3 ) Constructing confidence intervals about the estimated effects of the QTLs requires knowledge of the residual standard deviation (through the t-test statistic). Although the simple regression method provides good estimates of positions and effects of QTLs, estimation of the residual variance tends to be biased. Unfortunately, there has been no attempt to investigate the bias in the literature. This short note reports the result of theoretical derivation