Sex linked recessive traits: Optimal designs for parameter estimation

In the literature there have been a lot of studies concerning probabilistic modelling in genetics.
These have mainly studied model building and the statistical estimation of gene frequencies.
However, according to our opinion experimental design problems have not been studied
enough. Against this background this study is performed. We study the estimation of the gene
frequencies of sex linked recessive traits. E.g. the genes for the colourblindness
and for blood
group Xg are located on the X chromosome. Our basic assumption is that the trait is monogenic
and recessive. Such a trait has quite different phenotype frequencies in the male and the female
part of the population. This is caused by the fact that if the trait has the gene frequency p in the
total population, then the frequency of affected individuals is p among the males and p2 among
the females. Consequently, direct comparisons of phenotype frequencies between males and
females are of no value.
We discuss and compare the maximum likelihood estimators of the gene frequency for mixed,
male and female samples (cf. Haldane, Annals of Human Genetics, 1963:27:107111).
For pvalues
below 1/3, male samples and for pvalues
over 1/3, the female samples yield most
efficient estimates. Estimates obtained by mixed samples are never efficient. Among the
geneticists there is a common agreement that the colourblindness
is not a monogenic trait and
consequently, hypothesis testing plays a central role. Therefore, the possibility for model
testing is crucial, but model testing demands mixed samples causing a reduction in the
efficiency of the parameter estimates. The obtained results are applied to empirical data found
in the literature.