Abstract The general theory of the effective size (Ne) for populations under directional selection is extended to cover linkage. Ne is a function of the association between neutral and selected genes generated by finite sampling. This association is reduced by three factors: the recombination rate, the reduction of genetic variance due to drift, and the reduction of genetic variance of the selected genes due to selection. If the genetic size of the genome (L in Morgans) is not extremely small the equation for Ne is Ne=Nexp(−C2(1−Z)L), where N is the number of reproductive individuals, C2 is the genetic variance for fitness scaled by the squared mean fitness, (1 − Z) = Vm/C2 is the rate of reduction of genetic variation per generation and Vm is the mutational input of genetic variation for fitness. The above predictive equation of Ne is valid for the infinitesimal model and for a model of detrimental mutations. The principles of the theory are also applicable to favorable mutation models if there is a continuous flux of advantageous mutations. The predictions are tested by simulation, and the connection with previous results is found and discussed. The reduction of effective size associated with a neutral mutation is progressive over generations until the asymptotic value (the above expression) is reached after a number of generations. The magnitude of the drift process is, therefore, smaller for recent neutral mutations than for old ones. This produces equilibrium values of average heterozygosity and proportion of segregating sites that cannot be formally predicted from the asymptotic Ne, but both parameters can still be predicted by following the drift along the lineage of genes. The spectrum of gene frequencies in a given generation can also be predicted by considering the overlapping of distributions corresponding to mutations that arose in different generations and with different associated effective sizes.
The ideal reference interval for a variable of clinical interest would be specific for all deterministic factors affecting that variable, including the time of sampling in relation to predictable variations. In particular, growth hormone (GH) is characterized in children by circadian and ultradian variability, with high peaks of secretion occurring mainly during sleep. For clinical applications, the use of tolerance intervals has been recommended as a substitution for prediction limits. In the case of hybrid data (time series collected from a group of subjects), a tolerance interval could be very difficult to determine following a parametric approach similar to the procedure used for the computation of prediction intervals, especially when consideration of both within-subject and between-subject variances is wanted. Accordingly, we have developed a nonparametric method for the computation of tolerance intervals. The method, based on bootstrap techniques, does not require the assumption of normality or symmetry in the data, and is also more appropriate when dealing with small samples. The method was used to establish time-qualified reference limits for several series of GH sampled around the clock in groups of prepubertal children differentiated according to stature. The provision of such tolerance limits introduces time-specification and time-structure evaluation into the prevention, diagnosis and treatment of growth disorders.