Missing Genotypes and Allele-Frequency Bias

Why Missing Genotypes Can Change an Allele Frequency

A missing genotype removes information: there are fewer observed allele copies from which to estimate a frequency. But loss of information and bias are different problems. If genotypes are missing independently of their values, the called individuals remain representative of the sampled population for estimating allele frequency. If the chance of a call differs by genotype, however, the called individuals can have a different genotype distribution from the source population.

For a biallelic variant, let $G$ be the number of alternate allele copies in one diploid individual. Thus $G=0$ denotes two reference alleles, $G=1$ one alternate allele, and $G=2$ two alternate alleles. The allele frequency is the average number of alternate copies per individual divided by two.

Definition: For $n$ diploid individuals with called genotypes $G_1,\ldots,G_n$, the complete-case estimate of the alternate allele frequency is $$ \hat{p}_{\mathrm{cc}}=\frac{\sum_{i=1}^{n}G_i}{2n}. $$ A genotype is complete (or called) if its value is observed; an individual whose genotype is missing at this variant is excluded from this calculation. The denominator is twice the number of called individuals, because each diploid genotype contains two allele copies.

Write $q_g=P(G=g)$ for the source-population probability of genotype $g$, for $g\in\{0,1,2\}$. The source alternate allele frequency is $$ p=\frac{q_1+2q_2}{2}. $$ Let $C=1$ indicate that a genotype is called, and define $r_g=P(C=1\mid G=g)$, the call probability for genotype $g$. These probabilities describe the missingness mechanism at this variant.

Genotype-Independent Missingness: Unbiasedness With Less Information

If $r_0=r_1=r_2=r$, then call status is independent of genotype. Conditional on observing a fixed number $m$ of calls, the called individuals are a random subset with respect to genotype. Each called genotype therefore has the same distribution $q_0,q_1,q_2$ as a source individual, and the expected alternate-copy count per called person is $q_1+2q_2=2p$. Consequently, the complete-case estimate has expectation $p$ under this sampling model.

This statement concerns bias, not precision. Missing calls leave fewer observed individuals and allele copies, so the estimate is less precise than it would have been with more calls. The exact sampling variance depends on the genotype distribution and sampling assumptions; it should not be inferred just by treating all allele copies as independent when that assumption is not appropriate.

Worked Example: Equal Call Rates Preserve the Target Frequency

Suppose a hypothetical sample has 100 diploid individuals, and all three genotypes have the same call probability, $r_0=r_1=r_2=0.80$. There are 80 calls, with 32 individuals having $G=0$, 32 having $G=1$, and 16 having $G=2$. Calculate the complete-case estimate and explain what the equal call-rate assumption implies.

The called genotypes contain $32(0)+32(1)+16(2)=64$ alternate copies among $2(80)=160$ called allele copies. Thus $$ \hat{p}_{\mathrm{cc}}=\frac{64}{160}=0.400. $$

Because every genotype has call probability 0.80, call status does not favor any genotype. The observed value, 0.400, is an estimate based on only 160 called allele copies rather than the 200 copies that would be available if all 100 genotypes were called. Under the stated genotype-independent mechanism, the missingness does not systematically shift the target frequency, although this particular sample estimate can still differ from the source frequency through sampling variation.

Genotype-Dependent Missingness and the Bias Among Calls

When call probabilities vary by genotype, the genotype distribution among called individuals is generally not the source distribution. In a large sample, the called proportion in genotype class $g$ approaches $q_g r_g/\sum_{h=0}^2q_h r_h$. Therefore the alternate allele frequency among calls approaches $$ p_{\mathrm{call}}=\frac{q_1r_1+2q_2r_2}{2\sum_{g=0}^{2}q_g r_g}. $$ This is the target approached by the unweighted complete-case estimate under the specified mechanism, not necessarily the source frequency $p$.

The direction and size of the shift depend on how call probability relates to alternate-copy count. A genotype with a lower call probability is not automatically “underrepresented” in every comparison: representation among calls must be compared with that genotype’s source proportion. The following identity makes the source of bias precise.

Result: Let $E[r_G]=\sum_gq_gr_g$, and let $\operatorname{Cov}(G,r_G)=E[G r_G]-E[G]E[r_G]$, where expectations are over source genotypes. Then $$ p_{\mathrm{call}}-p=\frac{\operatorname{Cov}(G,r_G)}{2E[r_G]}. $$ Thus genotype-independent call probabilities imply zero bias. Genotype-dependent call probabilities can produce bias when their covariance with alternate-copy count is nonzero; dependence alone does not guarantee a nonzero shift.

To verify the identity, use $E[G]=q_1+2q_2=2p$ and $E[G r_G]=q_1r_1+2q_2r_2$. The numerator of $p_{\mathrm{call}}$ is $E[G r_G]$, while its denominator is $2E[r_G]$. Subtracting $p=E[G]/2$ gives $$ p_{\mathrm{call}}-p =\frac{E[G r_G]}{2E[r_G]}-\frac{E[G]}{2} =\frac{E[G r_G]-E[G]E[r_G]}{2E[r_G]} =\frac{\operatorname{Cov}(G,r_G)}{2E[r_G]}. $$ The denominator is positive whenever at least some genotypes can be called. If higher alternate-copy counts tend to have lower call probabilities, the covariance is negative and the called-sample frequency tends to be below the source frequency. Other patterns can shift the estimate in either direction.

Worked Example: Genotype-Specific Call Rates Shift the Estimate

Consider a hypothetical source population of 1,000 individuals with 400 having $G=0$, 400 having $G=1$, and 200 having $G=2$. Suppose the genotype-specific call probabilities are $r_0=0.90$, $r_1=0.80$, and $r_2=0.50$. Use expected called counts to compare the source and called-sample frequencies.

The source alternate allele frequency is $$ p=\frac{400(0)+400(1)+200(2)}{2(1000)} =\frac{800}{2000}=0.400. $$

The expected called counts are $400(0.90)=360$ for $G=0$, $400(0.80)=320$ for $G=1$, and $200(0.50)=100$ for $G=2$, for a total of 780 calls. The expected alternate-copy count among calls is $320(1)+100(2)=520$. Hence $$ p_{\mathrm{call}}=\frac{520}{2(780)} =\frac{520}{1560}\approx0.3333. $$ The unweighted complete-case estimate approaches about 0.3333 in this example, below the source frequency of 0.400.

The called genotype proportions are $360/780\approx0.4615$ for $G=0$, $320/780\approx0.4103$ for $G=1$, and $100/780\approx0.1282$ for $G=2$. Compared with source proportions 0.400, 0.400, and 0.200, respectively, $G=0$ and $G=1$ are overrepresented among calls, while $G=2$ is underrepresented. The call probability for $G=1$ (0.80) is lower than that for $G=0$ (0.90), even though both genotypes are overrepresented among calls relative to their source proportions. This illustrates why call probabilities and representation among calls should not be conflated.

Correcting With Known Genotype-Specific Call Probabilities

If the genotype-specific call probabilities are known or can be estimated credibly, called genotypes can be reweighted to compensate for their unequal chances of being observed. A called individual with genotype $g$ receives weight $1/r_g$. In the population, the expected weighted count for class $g$ is proportional to $q_g r_g(1/r_g)=q_g$, recovering its source representation.

Definition: Given called genotypes $G_i$ and known call probabilities $r_{G_i}>0$, the inverse-probability weighted allele-frequency estimate is $$ \hat{p}_{\mathrm{IPW}}= \frac{\sum_{i:C_i=1}G_i/r_{G_i}} {2\sum_{i:C_i=1}1/r_{G_i}}. $$ The positivity condition $r_g>0$ is required for every genotype class in the target population. This is a ratio estimate: under the stated model it recovers the source frequency in the population limit, but it is not generally exactly unbiased in every finite sample.

The correction requires that the call probabilities correspond to the relevant sampling process and that genotype missingness is adequately described by those probabilities. If calls also depend on measured covariates, the probabilities may need to condition on those covariates. If an entire genotype class has zero call probability, its source contribution cannot be recovered from called genotypes alone without additional information or assumptions. Very small call probabilities also create large weights and can make the corrected estimate unstable.

Worked Example: Weighting Called Genotypes by Their Call Probabilities

In a hypothetical sample, the called counts are 45 for $G=0$, 36 for $G=1$, and 9 for $G=2$. Suppose the specified call probabilities are $r_0=0.90$, $r_1=0.60$, and $r_2=0.30$. Calculate both the unweighted estimate and the inverse-probability weighted estimate.

The unweighted estimate is based on 90 called people and $36+2(9)=54$ alternate copies: $$ \hat{p}_{\mathrm{cc}}=\frac{54}{2(90)}=\frac{54}{180}=0.300. $$

For the weighted estimate, the adjusted genotype-class counts are $45/0.90=50$, $36/0.60=60$, and $9/0.30=30$. The adjusted total is $50+60+30=140$, and the weighted alternate-copy total is $60(1)+30(2)=120$. Therefore $$ \hat{p}_{\mathrm{IPW}}=\frac{120}{2(140)} =\frac{120}{280}\approx0.4286. $$ The correction raises the estimate because the $G=2$ class, which carries two alternate copies per person, has a lower call probability and is upweighted. The value 0.4286 is a model-based corrected estimate, not a guarantee that the unobserved genotypes in this particular sample had exactly that frequency.

What the Correction Does—and Does Not—Establish

A correction is only as reliable as its missingness model. Known probabilities may come from a specified measurement model or a validated estimate of call rates. If the probabilities are misspecified, weighting can introduce rather than remove bias. Since weights $1/r_g$ grow as $r_g$ becomes small, a few observations can dominate the corrected estimate. It is therefore useful to report the unweighted and corrected estimates together, state the call-rate assumptions, and examine how sensitive the result is to plausible alternatives.

Missingness can also vary with factors other than genotype, such as sample quality or laboratory batch. If those factors are associated with genotype frequencies and are not accounted for, the called set may be unrepresentative even when missingness is not directly caused by genotype. Conversely, a variable call probability need not change allele frequency if the resulting covariance between $G$ and call probability is zero. The relevant question is whether the observation process changes the average alternate-copy count in the called sample.

Common Mistakes

Assuming every missing call causes bias. Equal call probabilities across genotypes reduce the number of observations but do not systematically change the target allele frequency under the stated sampling model.
Calling the genotype with the lowest call rate “underrepresented” without checking proportions. Underrepresentation means its proportion among calls is below its proportion in the source population; compare those proportions directly.
Interpreting a corrected estimate as exact recovery. Inverse-probability weighting relies on specified call probabilities and positive call rates. A finite-sample ratio estimate remains subject to sampling variation and model error.
Using unweighted called counts when call rates differ. The unweighted estimate describes allele frequency among calls, which can differ from the source frequency.

Key takeaway: Genotype-independent missingness primarily costs precision. Genotype-dependent missingness can shift the complete-case frequency by changing the genotype mix among calls. When genotype-specific call probabilities are known, inverse-probability weighting can correct that shift under explicit positivity and model assumptions.

Check Your Understanding

Answer each question by distinguishing the source population from the set of called genotypes.

For a diploid sample with 24 called individuals of $G=1$ and 8 of $G=2$, calculate the complete-case alternate allele frequency.
Why can equal call probabilities across all three genotype classes reduce precision without systematically biasing the complete-case estimate?
A genotype class is 10% of the source population and 8% of the called sample. Is it overrepresented or underrepresented among calls? Explain.
State the covariance condition under which genotype-dependent call probabilities produce no shift in the population allele frequency among calls.
Why does inverse-probability weighting require positive call probabilities for genotype classes present in the target population?