Estimation of genotype distributions and posterior genotype probabilities for beta-mannosidosis in Salers cattle

beta-Mannosidosis is a lethal lysosomal storage disease inherited as an autosomal recessive in man, cattle and goats. Laboratory assay data of plasma beta-mannosidase activity represent a mixture of homozygous normal and carrier genotype distributions in a proportion determined by genotype frequency. A maximum likelihood approach employing data transformations for each genotype distribution and assuming a diallelic model of inheritance is described. Estimates of the transformation and genotype distribution parameters, gene frequency, genotype fitness and carrier probability were obtained simultaneously from a sample of 2,812 observations on U.S. purebred Salers cattle with enzyme activity, age, gender, month of pregnancy, month of testing, and parents identified. Transformations to normality were not required, estimated gene and carrier genotype frequencies of 0.074 and 0.148 were high, and the estimated relative fitness of heterozygotes was 1.36. The apparent overdominance in fitness may be due to a nonrandom sampling of progeny genotypes within families. The mean of plasma enzyme activity was higher for males than females, higher in winter months, lower in summer months and decreased with increased age. Estimates of carrier probabilities indicate that the test is most effective when animals are sampled as calves, although effectiveness of the plasma assay was less for males than females. Test effectiveness was enhanced through averaging repeated assays of enzyme activity on each animal. Our approach contributes to medical diagnostics in several ways. Rather than assume underlying normality for the distributions comprising the mixture, we estimate transformations to normality for each genotype distribution simultaneously with all other model parameters. This process also excludes potential biases due to data preadjustment for systematic effects. We also provide a method for partitioning phenotypic variation within each genotypic distribution which allows an assessment of the value of repeat measurements of the predictive variable for genotype assignment.