Common Errors In Allele And Genotype Frequency Calculations

Why Frequency Calculations Go Wrong

Allele- and genotype-frequency calculations are simple once the quantity and its denominator are clear. Errors often arise because the same sample can be described in two different ways: by the genotypes of individuals or by the alleles carried in their chromosome copies. A count of individuals with at least one copy of an allele, for example, is not generally a count of that allele.

The distinction matters when summarizing data and when calculating Hardy–Weinberg expectations. Those expectations are calculated from allele frequencies, not by reusing a genotype frequency as though it were an allele frequency. Before calculating, identify what is being counted, what the denominator is, and whether the model matches the data.

Count Allele Copies, Not Individuals

Each individual in a diploid sample contributes two allele copies at the locus. An \(AA\) individual contributes two copies of \(A\); an \(Aa\) individual contributes one copy of \(A\) and one of \(a\); and an \(aa\) individual contributes two copies of \(a\). Thus the total number of allele copies is \(2N\), and the sample allele-frequency estimates are:

Here \(\hat{p}\) estimates the population frequency \(p\) of allele \(A\), and \(\hat{q}\) estimates the population frequency \(q\) of allele \(a\). The hat indicates an estimate from the sample; it does not mean the frequency is known exactly for the population. The genotype frequencies are instead \(n_{AA}/N\), \(n_{Aa}/N\), and \(n_{aa}/N\). These have individuals, not allele copies, in their denominator.

A Short Check on the Counting Formula

The formula follows by counting the copies of \(A\) in each genotype class. Summing over the classes gives \(2n_{AA}+n_{Aa}\) copies of \(A\). The sample contains \(2N\) copies altogether, so dividing gives \(\hat{p}\). The corresponding count of \(a\) copies is \(2n_{aa}+n_{Aa}\). Adding the two allele-frequency estimates checks that no allele copies have been lost or counted twice:

This check depends on the genotype counts covering all \(N\) individuals and on each individual contributing exactly two copies at the locus. If some genotypes are missing, the denominator should reflect the individuals with valid genotype data. If ploidy differs, the total number of allele copies must be counted directly rather than assumed to equal \(2N\).

Genotype Frequencies Are Not Allele Frequencies

A genotype frequency describes a proportion of individuals. An allele frequency describes a proportion of chromosome copies. A heterozygote is one individual but contains one copy of each allele, which is why its contribution to an allele frequency is half its genotype frequency. Neither an \(AA\) frequency nor the proportion of individuals carrying \(A\) can be substituted for \(\hat{p}\) without checking the genotype composition.

The genotype frequencies and allele frequencies are connected, but they answer different questions. For example, the frequency of \(A\) is \(f_{AA}+\frac{1}{2}f_{Aa}\), while the proportion of individuals carrying at least one \(A\) is \(f_{AA}+f_{Aa}\). These expressions differ whenever heterozygotes are present. Confusing them can produce a number between zero and one that looks plausible but represents the wrong quantity.

Using Allele Frequencies for Hardy–Weinberg Expectations

For a biallelic autosomal locus, the Hardy–Weinberg model gives expected genotype proportions \(p^2\), \(2pq\), and \(q^2\), where \(p\) and \(q\) are the allele frequencies and \(p+q=1\). A basic interpretation is that two allele copies are drawn from the population with probabilities \(p\) and \(q\), and paired to form a diploid genotype. This calculation describes model expectations; it does not turn the observed genotype proportions into those expectations.

The arithmetic also provides a useful check on the inputs. If \(p+q=1\), then \(p^2+2pq+q^2=(p+q)^2=1\). The three expected genotype proportions therefore add to one. If an allele frequency was calculated from valid biallelic diploid genotype data, substituting \(\hat{p}\) and \(\hat{q}\) gives sample-based expected proportions. These are not the observed genotype frequencies, even when they happen to be close.

Check Whether the Inputs Fit the Calculation

A correct formula can still yield a misleading result if the input data do not meet its basic requirements. Before calculating, check the locus, the individuals, and the denominator. The ordinary formula with denominator \(2N\) presumes that all \(N\) individuals are diploid at the locus and have usable genotype calls. It is not automatically appropriate for a sex-linked locus with different ploidy across individuals, a mixture of populations with different allele frequencies, or data in which the denominator changes across genotype categories.

Likewise, applying Hardy–Weinberg proportions requires more than having three genotype labels. The standard proportions concern diploid genotypes at a biallelic autosomal locus under a population model. A calculation based on pooled data may be inappropriate if the pooled individuals do not share the same relevant population allele frequencies or do not have the same ploidy. In such cases, first define the appropriate group and count the allele copies for that group; do not force the data into the standard formula.

Common Mistakes and a Practical Audit

• Counting carriers instead of allele copies. The proportion of individuals with at least one \(A\) is not generally the \(A\) allele frequency. Count two copies for \(AA\), one for \(Aa\), and zero for \(aa\).
• Using a genotype frequency as \(p\) or \(q\). A genotype frequency has individuals in its denominator. Calculate allele frequency from the genotype composition, including half the heterozygote frequency.
• Mixing denominators. Genotype frequencies divide by the number of individuals with valid calls; allele frequencies divide by the number of relevant allele copies. Check that missing data or ploidy differences do not change the denominator.
• Calling model expectations observed data. Hardy–Weinberg proportions are calculated from allele frequencies under a model. They are not a relabeling of the observed genotype frequencies.
• Applying the standard formula to unsuitable inputs. Check that the data concern the intended population and diploid autosomal locus before using \(2N\) or the standard three-genotype expectations.

A useful audit is to write down the target quantity before doing arithmetic: genotype frequency among individuals, or allele frequency among copies. Then state the count and denominator, verify that genotype frequencies sum to one and that allele frequencies sum to one, and only then calculate any model expectations. These checks catch many errors without requiring a more complicated analysis.