Why Low-Pass Sequencing Changes the Imputation Input
Imputation uses observed genetic data and reference haplotypes to infer genotypes at unobserved variants. With an array, the input commonly includes genotype calls at selected markers. With low-pass sequencing, the input can instead be a small number of reads at many positions. A read is evidence about a genotype, not necessarily a definitive genotype call: sequencing errors, low coverage, and sampling variation can leave several genotypes plausible.
The key statistical distinction is therefore between observations and calls. An imputation procedure can use the number of reads supporting each allele, rather than first converting that count to a single hard genotype. It combines that evidence with the haplotype patterns represented in a reference panel. The result is a posterior distribution over possible genotypes, from which a dosage and a measure of uncertainty can be obtained.
From Read Counts to Genotype Evidence
A simple model makes the source of read evidence explicit. Suppose \(d\) reads cover a biallelic site, and \(K\) of them report the alternate allele. Assume reads are independent conditional on the true genotype, the probability of a base-call error is a known \(e\), and errors are symmetric: a true reference base is read as alternate with probability \(e\), and a true alternate base is read as reference with probability \(e\). Also assume no allele-specific mapping bias, contamination, or incorrect alignment at the site.
Under this model, the probability that one read reports the alternate allele is \(q_g=e+(1-2e)g/2\) when \(G=g\). Thus \(K\mid G=g\) has a binomial distribution. This likelihood describes how compatible the reads are with each genotype; it is not itself the probability of that genotype.
The haplotype context contributes prior probabilities \(\pi_g=P(G=g\mid H)\). These probabilities summarize how often each genotype is compatible with the observed flanking markers and reference haplotypes, under the imputation model. Combining prior and read likelihood gives
The expression follows from Bayes’ rule: the joint probability of genotype \(g\) and the observed read count, conditional on the haplotype context, is the likelihood times the prior; dividing by the sum over all possible genotypes normalizes the probabilities to one. This calculation assumes the read data are conditionally described by the stated likelihood and that the haplotype-derived prior is appropriate for the sample. In practical imputation, the haplotype model may jointly use many markers and reads; the simple calculation isolates the central logic.
Worked Example: One Alternate Read and a Haplotype Prior
Suppose a site has \(d=2\) reads, of which \(k=1\) reports the alternate allele. Let the error rate be \(e=0.01\), and suppose the haplotype context gives prior probabilities \((\pi_0,\pi_1,\pi_2)=(0.80,0.18,0.02)\). Use the binomial model to calculate the posterior genotype probabilities, dosage, and posterior variance.
For \(G=0,1,2\), the alternate-read probabilities are \(q_0=0.01\), \(q_1=0.50\), and \(q_2=0.99\). The likelihoods for exactly one alternate read are \(L_0=2(0.01)(0.99)=0.0198\), \(L_1=2(0.50)(0.50)=0.5\), and \(L_2=2(0.99)(0.01)=0.0198\).
Multiplying each likelihood by its prior gives unnormalized weights \(w_0=0.0198(0.80)=0.01584\), \(w_1=0.5(0.18)=0.09\), and \(w_2=0.0198(0.02)=0.000396\). Their sum is \(0.106236\), so the posterior probabilities are approximately \(0.149102\), \(0.847170\), and \(0.003728\), respectively.
The dosage is \(E(G\mid K,H)=0(0.149102)+1(0.847170)+2(0.003728)\approx0.854626\) alternate alleles. For the variance, first calculate \(E(G^2\mid K,H)=0^2(0.149102)+1^2(0.847170)+2^2(0.003728)\approx0.862081\). Therefore, \(\operatorname{Var}(G\mid K,H)=0.862081-(0.854626)^2\approx0.131706\) squared alternate-allele units. The reads favor one alternate allele, but uncertainty remains: the posterior probability of \(G=0\) is about 0.149.
What the Posterior Output Means
The three genotype probabilities retain information that a single hard call would discard. The dosage is useful in downstream analyses because it expresses the expected allele count, but it does not assert that a fractional genotype exists. A dosage of 0.855 means the posterior mean allele count is 0.855, not that the individual carries 0.855 of an allele. The posterior variance describes remaining uncertainty about the genotype under the model.
A useful justification for the dosage is that it minimizes expected squared prediction error. If a numeric prediction \(a\) is used for \(G\), then \(E[(G-a)^2\mid R,H]=\operatorname{Var}(G\mid R,H)+(a-E(G\mid R,H))^2\). The second term is nonnegative and is zero when \(a=E(G\mid R,H)\). Thus the posterior mean is the best prediction under squared-error loss. A hard call instead selects the genotype with the largest posterior probability; that can be appropriate when a discrete decision is required, but it loses uncertainty and can differ from the dosage.
Worked Example: More Reads Can Sharpen Evidence
At another hypothetical site, suppose \(d=4\), \(k=0\), \(e=0.01\), and the haplotype prior is \((0.20,0.60,0.20)\). Calculate the posterior probabilities and dosage under the same independent-read model.
The likelihoods for observing no alternate reads are \(L_0=0.99^4=0.96059601\), \(L_1=0.50^4=0.0625\), and \(L_2=0.01^4=0.00000001\). The prior-weighted values are \(w_0=0.192119202\), \(w_1=0.0375\), and \(w_2=0.000000002\), with total \(0.229619204\).
After normalization, \(P(G=0\mid K,H)\approx0.836686\), \(P(G=1\mid K,H)\approx0.163314\), and \(P(G=2\mid K,H)\approx0.000000009\). The dosage is approximately \(0(0.836686)+1(0.163314)+2(0.000000009)=0.163314\) alternate alleles. Although the prior placed probability 0.60 on \(G=1\), four reference reads shift the posterior toward \(G=0\). They do not make \(G=0\) certain: the nonzero probability of \(G=1\) reflects the possibility of missing alternate reads.
Low-Pass Sequencing and Array-Based Imputation
Both approaches rely on haplotype relationships between observed markers and the target variant, but their inputs differ. An array typically supplies calls at a designed set of markers, often with high confidence at well-performing probes. Low-pass sequencing can supply read counts across many more positions, but at any one site the count may be sparse and uncertain. Array genotypes can also be uncertain, and sequencing data can be deep enough to support near-certain calls; these are tendencies, not guarantees.
| Feature | Array-based input | Low-pass sequencing input |
|---|---|---|
| Observed data | Genotype calls or genotype probabilities at selected markers | Read-level allele evidence, commonly summarized by counts and likelihoods |
| Uncertainty at observed sites | Often concentrated in uncertain or poorly performing calls | Can be substantial at low-depth sites and should be represented by read likelihoods |
| Target-variant output | Posterior genotype probabilities and dosages from marker haplotypes | Posterior genotype probabilities and dosages using read evidence as well as haplotype context |
| Common information loss | Hard-calling uncertain marker genotypes before imputation | Reducing read counts to hard calls before imputation |
The comparison is not “certain array data” versus “uncertain sequencing data.” Both have input and output uncertainty. The important practice is to pass uncertainty forward where the software permits it: arrays can provide genotype probabilities, while low-pass sequencing can provide genotype likelihoods. At imputed sites, posterior probabilities and dosages are more informative than a hard call alone.
Worked Example: A Target Genotype Imputed From Array Markers
Suppose array genotypes at several flanking markers are confidently observed, and their haplotype pattern gives a target-site prior \((0.05,0.90,0.05)\). Assume the target itself was not directly measured and no target-site reads are available. What are the imputed dosage and posterior variance?
With no direct target-site evidence, the posterior probabilities equal this haplotype-based distribution. The dosage is \(0(0.05)+1(0.90)+2(0.05)=1.00\) alternate allele. Also, \(E(G^2)=0(0.05)+1(0.90)+4(0.05)=1.10\), so the posterior variance is \(1.10-(1.00)^2=0.10\) squared alternate-allele units.
This example illustrates the array-based input: confidently called flanking markers inform the target through reference haplotypes. A dosage of 1.00 does not imply certainty; here the posterior still assigns probability 0.10 to each homozygous genotype. Low-pass reads at the target could add direct evidence and change these probabilities, while uncertain array calls should not automatically be treated as error-free.
Assumptions, Diagnostics, and Failure Modes
The numerical read model above is intentionally simple. Its interpretation depends on assumptions that should be checked or addressed by a suitable analysis pipeline. In particular, errors may differ by strand or allele, reads may not be independent, and alignment or mapping bias may make alternate and reference reads unequally likely. A reference haplotype panel can also be a poor match to the analyzed population or fail to represent the relevant haplotype. If any of these features are substantial, the computed posterior probabilities may be overconfident or systematically shifted.
Several diagnostics help reveal trouble. Examine read depth and allele balance at observed heterozygous sites; an extreme imbalance can indicate bias or technical problems. Compare read-based likelihoods with haplotype predictions: widespread disagreement may signal strand, allele, or build mismatches, or a panel that is not suitable. Check whether posterior probabilities are unexpectedly concentrated despite very low depth, and inspect imputation quality across variants rather than relying only on a few high-confidence examples. The earlier material on imputation quality metrics and validation by masking provides complementary ways to assess imputed outputs.
A further failure mode is treating the simple binomial calculation as a complete sequencing model. It ignores, among other things, mapping quality, base quality variation, local alignment ambiguity, and dependence among reads. Operational methods may incorporate richer likelihoods and joint haplotype models. The binomial expression is valuable for understanding the logic, but its assumptions should not be mistaken for universal properties of sequencing data.
- Hard-calling first: choosing the most likely genotype at a low-depth site discards uncertainty before haplotype evidence can be combined with it. Retaining likelihoods or probabilities avoids this unnecessary loss when supported by the method.
- Calling the likelihood a posterior: a read count can be compatible with more than one genotype. The posterior also depends on the haplotype prior and must be normalized across possible genotypes.
- Over-interpreting dosage: a fractional dosage is an expected allele count, not a fractional biological genotype or proof of a particular genotype.
- Assuming more markers guarantee better inference: additional data help only when they are correctly aligned, adequately modeled, and informative about the target haplotypes.
Check Your Understanding
Try each question, then compare your reasoning with the solutions below.
- Under the symmetric-error model with \(e=0.02\), what is the probability that a read reports the alternate allele when \(G=1\)?
- In a haplotype-based imputation, why is a read likelihood not the same as a posterior genotype probability?
- For genotype probabilities \((0.2,0.5,0.3)\) for \(G=0,1,2\), calculate dosage and posterior variance.
- Name one condition under which the independent binomial read model may give misleadingly confident results.
Solutions. (1) \(q_1=0.02+(1-0.04)(1/2)=0.50\). (2) The likelihood gives \(P(K\mid G=g)\); the posterior also uses the haplotype prior and normalizes across genotypes, as in Bayes’ rule. (3) The dosage is \(0(0.2)+1(0.5)+2(0.3)=1.1\). Also \(E(G^2)=0.5+4(0.3)=1.7\), so the variance is \(1.7-(1.1)^2=0.49\) squared alternate-allele units. (4) For example, correlated reads or allele-specific mapping bias violates the model’s assumptions and can make the likelihood evidence too strong or systematically distorted.