Why the Representation of an Imputed Genotype Matters
An imputation result is not always a single genotype. For a diploid, biallelic variant, the genotype \(G\) counts copies of the effect allele and can be 0, 1, or 2. Imputation provides posterior probabilities for these possibilities, conditional on the observed data and the imputation model. An analysis must decide whether to retain those probabilities through a dosage or replace them with a hard call.
These choices are not interchangeable. A dosage summarizes the posterior distribution by its mean, whereas a hard call selects one genotype. Applying a confidence threshold can also turn an uncertain result into a missing value. The choice can therefore affect genotype counts, allele-frequency summaries, and which samples contribute to downstream analyses.
The earlier material on dosage as the expected genotype established the meaning of \(d\). Here the focus is what is lost when the posterior distribution is replaced by a call. Calculations assume consistent effect-allele coding, valid posterior probabilities summing to one, and the same samples and variant in each comparison. A threshold is a chosen analysis rule, not a guarantee that every retained call is correct. Its meaning depends on the quality of the posterior probabilities.
What Dosages and Hard Calls Preserve
The dosage preserves the posterior mean number of effect-allele copies. It can be fractional because it is an expectation, not a claim that an individual carries a fraction of an allele. For a fixed set of \(n\) diploid individuals, the sum of dosages is the posterior expected total effect-allele count, and dividing it by \(2n\) gives the corresponding posterior expected allele frequency.
This expectation follows from linearity of conditional expectation; it does not require the individuals’ genotypes to be independent. In contrast, the sum of hard calls counts the alleles in the chosen call set. It is generally not the posterior expected allele count. If thresholding removes individuals, the denominator changes too: an allele frequency computed among retained calls describes that subset, not automatically the original sample.
Neither representation contains everything. A dosage records the posterior mean but not the full distribution or its spread. A hard call records a selected category but discards both the fractional expectation and the uncertainty. In particular, knowing a dosage alone does not determine the probabilities \(q_0,q_1,q_2\), the most probable genotype, or the probability of being heterozygous.
Worked Example: Four Posterior Distributions, Three Summaries
Suppose four hypothetical individuals have the following posterior probabilities, listed in the order \(q_0,q_1,q_2\). Use a threshold \(\tau=0.90\), retaining a call when its largest posterior probability is at least 0.90.
| Individual | \((q_0,q_1,q_2)\) | Dosage \(q_1+2q_2\) | Maximum-posterior call |
|---|---|---|---|
| A | (0.90, 0.09, 0.01) | 0.09 + 2(0.01) = 0.11 | 0 |
| B | (0.05, 0.90, 0.05) | 0.90 + 2(0.05) = 1.00 | 1 |
| C | (0.10, 0.45, 0.45) | 0.45 + 2(0.45) = 1.35 | 1, under a tie rule favouring 1 |
| D | (0.02, 0.18, 0.80) | 0.18 + 2(0.80) = 1.78 | 2 |
The four dosages sum to \(0.11+1.00+1.35+1.78=4.24\) expected effect-allele copies. Thus, for these four individuals, the dosage-based allele-frequency summary is \(4.24/(2\times4)=0.530\). The maximum-posterior hard calls sum to \(0+1+1+2=4\) copies, giving \(4/8=0.500\) if all four calls are used.
With the stated threshold, A and B are retained: their largest probabilities are 0.90. C and D are missing because their largest probabilities are 0.45 and 0.80. The retained hard calls sum to one copy among two individuals, so the frequency among those retained calls is \(1/(2\times2)=0.250\). This is not an estimate for the original four-person set: thresholding changed both the data included and the denominator. The example also shows why a threshold rule must be reported alongside any summary based on retained calls.
Thresholding and Posterior Uncertainty
A confidence threshold is a decision rule applied to the largest posterior probability. It is distinct from dosage calculation: the dosage uses all three probabilities, while the thresholded call uses their maximum to decide whether to report a category. Thresholding can be useful when a workflow requires discrete genotypes, but it turns a probability distribution into either one selected category or missing data.
The posterior variance makes one aspect of the lost information explicit. Since \(G\) takes values 0, 1, and 2, \(E[G^2\mid D]=q_1+4q_2\). Therefore,
This variance cannot in general be recovered from the dosage alone. Two different posterior distributions can have the same mean but different spread, and even different most probable genotypes. A hard call cannot resolve that ambiguity either; it only chooses a category according to the specified rule.
Worked Example: Equal Dosages, Different Uncertainty
Compare two hypothetical posterior distributions for the same effect-allele coding. Individual A has \((q_0,q_1,q_2)=(0.60,0,0.40)\); individual B has \((0.20,0.80,0)\).
Both dosages are 0.80: for A, \(0+2(0.40)=0.80\); for B, \(0.80+2(0)=0.80\). But A’s maximum-posterior call is 0, while B’s is 1. With \(\tau=0.80\), A is missing because its maximum probability is 0.60, whereas B is called 1 because its maximum is 0.80.
Their posterior variances differ as well. For A, \(E[G^2\mid D]=4(0.40)=1.60\), so \(\operatorname{Var}(G\mid D)=1.60-(0.80)^2=0.96\). For B, \(E[G^2\mid D]=0.80\), so \(\operatorname{Var}(G\mid D)=0.80-(0.80)^2=0.16\). The equal dosage captures the same expected allele count, but not the same uncertainty or the same most probable genotype.
Worked Comparison of Aggregate Genotype Summaries
The difference between dosage and hard calls can be modest or substantial in an aggregate summary; it depends on the posterior probabilities. Even when the summaries happen to be close, they answer different questions: one sums expected allele counts, while the other sums selected genotype categories.
Worked Example: Expected Allele Count Versus Called Count
For three hypothetical individuals, suppose the posterior probabilities are \((0.70,0.25,0.05)\), \((0.15,0.70,0.15)\), and \((0.05,0.35,0.60)\), respectively. The dosages are:
Individual 1: \(0.25+2(0.05)=0.35\). Individual 2: \(0.70+2(0.15)=1.00\). Individual 3: \(0.35+2(0.60)=1.55\). Their total is \(0.35+1.00+1.55=2.90\) expected copies, and the dosage-based allele-frequency summary is \(2.90/(2\times3)=0.4833\).
The maximum-posterior calls are 0, 1, and 2. They sum to 3 copies, giving a hard-call frequency of \(3/6=0.500\). Both calculations are arithmetically correct for their respective representations. The dosage summary is the posterior expectation of the total allele count for these three individuals; the hard-call summary counts the selected categories. A small difference here does not make the methods equivalent.
Choosing and Checking a Representation
For a summary based on expected allele counts, dosages retain information from all posterior genotype probabilities and avoid a threshold-induced loss of samples. For an analysis that requires discrete genotype categories, hard calls may be operationally necessary, but the rule for assigning calls and handling uncertain results should be explicit. A maximum-posterior call without a threshold still discards uncertainty; a thresholded call additionally discards individuals whose maximum probability falls below the cutoff.
Useful checks include confirming that posterior probabilities sum to one, verifying effect-allele coding, and comparing dosage summaries with hard-call summaries and call rates. Examine how many samples become missing under the chosen threshold and whether those exclusions are concentrated in particular groups or subsets. A change in the analysed sample set can alter a result even if the threshold does not change any retained genotype. Where a downstream method needs more than the posterior mean, retain or recover the posterior probabilities rather than assuming the dosage contains the entire distribution.
A common mistake is to say, “The dosage is 0.8, so the person has genotype 0, with 80% certainty.” This is incorrect: a dosage is an expected allele count, not a genotype label or a confidence probability. As the equal-dosage example shows, a dosage of 0.8 can accompany a most probable genotype of 0 or 1 and different posterior uncertainty. The correction is to report the dosage as an expectation, and use the posterior probabilities to describe uncertainty or to apply a clearly stated hard-call rule.
Check Your Understanding
Answer each question, then check the solutions below.
- For posterior probabilities \((0.20,0.50,0.30)\), calculate the dosage and maximum-posterior hard call.
- For the same probabilities, would a threshold of 0.60 retain a hard call? What is the reason?
- Two individuals have the same dosage. Does that imply they have the same most probable genotype or posterior variance?
- Three diploid individuals have dosages 0.40, 1.10, and 1.50. Calculate their dosage-based allele-frequency summary.
- State one consequence of applying a confidence threshold before calculating a hard-call allele frequency.
Solutions. 1. The dosage is \(0.50+2(0.30)=1.10\); the largest probability is 0.50, so the maximum-posterior call is 1. 2. No. The largest probability, 0.50, is less than 0.60, so the thresholded result is missing. 3. No. Equal means do not determine the posterior distributions; the most probable genotype and the variance can differ. 4. The total dosage is \(0.40+1.10+1.50=3.00\), so the summary is \(3.00/(2\times3)=0.500\). 5. Thresholding can remove individuals, changing the sample set and the denominator; the resulting frequency describes retained calls unless additional assumptions justify a broader interpretation.