The ALLELE Procedure

Frequency Estimates

A marker locus M can have a series of alleles $\text{[math]}$ , $\text{[math]}$ . A sample of $\text{[math]}$ individuals can therefore have several different genotypes at the locus, with $\text{[math]}$ copies of type $\text{[math]}$ . The number $\text{[math]}$ of copies of allele $\text{[math]}$ can be found directly by summation: $\text{[math]}$ . The sample frequencies are written as $\text{[math]}$ and $\text{[math]}$ . The $\text{[math]}$ ’s are unbiased maximum likelihood estimates (MLEs) of the population proportions $\text{[math]}$ .

The variance of the sample allele frequency $\text{[math]}$ is calculated as

$\text{[math]}$

and can be estimated by replacing $\text{[math]}$ and $\text{[math]}$ with their sample values $\text{[math]}$ and $\text{[math]}$ . The variance of the sample genotype frequency $\text{[math]}$ is not generally calculated; instead, an MLE of the HWD coefficient $\text{[math]}$ for alleles $\text{[math]}$ and $\text{[math]}$ is calculated as

$\text{[math]}$

and the MLE’s variance is estimated using one of the following formulas, depending on whether the two alleles are the same or different:

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

The standard error, the square root of the variance, is reported for the sample allele frequencies and the disequilibrium coefficient estimates. When the BOOTSTRAP= option of the PROC ALLELE statement is specified, bootstrap confidence intervals are formed by resampling individuals from the data set and are reported for these estimates, with the $\text{[math]}$ % confidence level given by the ALPHA= $\text{[math]}$ option (or $\text{[math]}$ by default).