Why Imputation Quality Needs More Than One Metric
Imputation supplies estimated genotypes at variants that were not directly measured. A single imputed variant can have a dosage for nearly every person while still being uncertain for many of them. Conversely, a file can contain a high fraction of nonmissing genotype entries without those entries being accurate. Quality metrics summarize different aspects of this problem, so their names and values should not be treated as interchangeable.
The earlier material on genotype posteriors and dosage established two useful quantities. For person \(i\), let \(G_i\in\{0,1,2\}\) be the unobserved number of copies of the counted allele, and let the imputation posterior probabilities be \(P(G_i=g\mid\text{data})\). The dosage \(D_i\) is the posterior expected allele count. Quality metrics use these probabilities or dosages to summarize uncertainty or variation across people.
Dosage \(r^2\): Variation in Dosages Relative to Expected Genotype Variation
If true genotypes were available in an independent validation sample, dosage accuracy could be summarized by the squared correlation between imputed dosages and true allele counts. That is, \(r^2=\operatorname{Corr}(D,G)^2\). It ranges from 0 to 1, with larger values indicating a stronger linear relationship between dosage and true genotype in that validation setting. Without observed truth, the correlation cannot be calculated directly. A common imputation estimate instead compares the across-person dosage variance with the genotype variance expected under Hardy–Weinberg equilibrium.
The denominator \(2\hat p(1-\hat p)\) is the genotype variance under Hardy–Weinberg equilibrium at allele frequency \(\hat p\). The ratio asks whether dosages vary across people almost as much as genotypes would be expected to vary under that model. It is an estimate of imputation quality, not a direct observation of the correlation with true genotypes. Its interpretation depends on the reference variance being appropriate and on the dosage distribution being estimated reliably.
Worked Example: Calculating a Variance-Based Dosage \(r^2\)
Suppose four hypothetical people have dosages \(0.1,\ 0.4,\ 1.6,\ 1.9\) for a counted allele. Use the population-variance convention, dividing by the number of people, to calculate the variance-based estimate.
The mean dosage is \(\bar D=(0.1+0.4+1.6+1.9)/4=1.0\). Thus the estimated allele frequency is \(\hat p=1.0/2=0.5\). The expected genotype variance in the denominator is \(2(0.5)(1-0.5)=0.5\).
The dosage variance is \(\operatorname{Var}(D)=[(0.1-1.0)^2+(0.4-1.0)^2+(1.6-1.0)^2+(1.9-1.0)^2]/4\) \(=(0.81+0.36+0.36+0.81)/4=0.585\).
Therefore the variance ratio is \(0.585/0.5=1.17\). This small, hypothetical sample produces an estimate above 1, which is a warning not to interpret the unbounded finite-sample ratio as a literal squared correlation. Implementations may use different variance conventions or constrain reported values. The calculation is not evidence that imputation is more than perfectly accurate; it illustrates why the definition used by the software and the stability of the estimate matter.
This calculation also shows why a reported dosage \(r^2\) is not automatically identical to a validation correlation squared. A validation correlation uses paired observed dosages and true genotypes. The variance ratio uses dosages and a model-based reference variance, so it is an indirect estimate and can be sensitive to sampling variation and departures from its reference assumptions.
INFO: A Summary of Posterior Uncertainty
A common INFO-style metric summarizes how much genotype uncertainty remains across people. One widely used definition starts with each person's posterior genotype variance, averages those variances, and compares the result with the expected genotype variance \(2\hat p(1-\hat p)\). Under this definition, a smaller average posterior variance produces a larger INFO value. The name INFO does not guarantee that every program uses precisely this formula: algorithms and software may define or estimate their reported information statistic differently.
Worked Example: Calculating Posterior-Variance INFO
Consider four hypothetical people with posterior probabilities for genotypes \(G=0,1,2\), respectively: \((0.9,0.1,0)\), \((0.5,0.5,0)\), \((0,0.5,0.5)\), and \((0,0.1,0.9)\). Calculate INFO using the definition above.
The dosages, found by summing genotype value times its posterior probability, are \(0.1,\ 0.5,\ 1.5,\ 1.9\). Their mean is \(1.0\), so \(\hat p=1.0/2=0.5\) and the denominator is \(2(0.5)(0.5)=0.5\).
For each person, \(\operatorname{Var}(G_i\mid\text{data})=E(G_i^2\mid\text{data})-D_i^2\). The four posterior variances are \(0.1-0.1^2=0.09\), \(0.5-0.5^2=0.25\), \(2.5-1.5^2=0.25\), and \(3.7-1.9^2=0.09\). Their mean is \((0.09+0.25+0.25+0.09)/4=0.17\).
Thus \(\mathrm{INFO}=1-(0.17/0.5)=0.66\). Under this definition, the value indicates that posterior uncertainty, averaged over these four people, is 34% of the reference variance in the denominator. It is not a statement that 66% of individual genotype calls are correct.
Why Dosage \(r^2\) and INFO Are Related but Not Interchangeable
The connection between the two metrics can be seen from the law of total variance. For a genotype \(G\) and the imputation data, this identity states that the total variance of \(G\) equals the variance of its conditional expectation plus the expected conditional variance:
Since \(E(G\mid\text{data})=D\), dosage variation represents the first term, and average posterior uncertainty represents the second. If the total genotype variance is represented by the same reference value \(2p(1-p)\) in both metrics, dividing the identity by that value gives a dosage-variance fraction plus a posterior-uncertainty fraction equal to 1. This explains why the variance-based dosage ratio and the posterior-variance INFO form can be similar under compatible definitions and assumptions.
The identity does not make every software-reported INFO value identical to every dosage \(r^2\) estimate. Different frequency estimates, variance conventions, posterior approximations, and adjustments can break exact equality. If two reported metrics disagree, first compare their documented definitions rather than assuming one is wrong.
Call Rate Measures Availability, Not Imputation Confidence
A genotype call rate is a count-based measure: the number of nonmissing genotype entries divided by the number of people assessed. It can be useful for describing completeness of directly measured genotypes, or the fraction of imputed records that receive a hard call after applying a specified rule. But imputed dosages can be available even when no hard call is assigned, and a nonmissing hard call may still have substantial uncertainty. The calling rule must therefore be stated when a call rate is reported.
Worked Example: A High Call Rate Does Not Guarantee High Imputation Quality
At a hypothetical imputed variant, 980 of 1,000 people have a nonmissing hard genotype call after a stated calling rule. The call rate is \(980/1{,}000=0.98\), or 98%. The imputation file also contains dosages for all 1,000 people, but many posterior distributions are diffuse.
The 98% figure describes the fraction with nonmissing hard calls under that rule. It does not describe the fraction of calls that match the unknown true genotypes, nor does it measure average posterior uncertainty. INFO or a dosage \(r^2\) estimate addresses an aspect of imputation quality; validation concordance, when suitable truth data are available, directly compares imputed results with observed genotypes. These are different questions, so the call rate alone cannot settle whether imputation is reliable.
Assumptions, Diagnostics, and Choice of Metric
These metrics require careful interpretation. The variance-based dosage ratio uses a reference variance derived from allele frequency under Hardy–Weinberg equilibrium. The posterior-variance INFO formula also relies on its stated denominator and on posterior probabilities that meaningfully represent uncertainty. Neither metric guarantees accuracy for every person or every population. In particular, the summaries can conceal differences among individuals and can be unstable when the estimated allele frequency or effective number of informative observations is small.
A practical diagnostic sequence is to verify which allele is counted and how its frequency is estimated, identify the exact software definition and variance convention, and inspect the range and distribution of the metric across variants. Unexpected values, such as a variance ratio above 1 or an INFO value outside the range expected for the implementation, warrant checking the definition, data, and numerical handling rather than assigning a quality interpretation immediately. Also check whether missing entries were excluded consistently from the denominator and whether any hard-call threshold affected the reported call rate.
| Quantity | What it summarizes | What it does not establish by itself |
|---|---|---|
| Dosage \(r^2\) | Dosage variation relative to a reference genotype variance, or squared dosage–genotype correlation when truth is observed | That every person's imputed genotype is accurate |
| INFO | Posterior uncertainty or information, under the specific software definition | A universal percentage of correct calls |
| Call rate | Fraction with a nonmissing genotype entry under a stated rule | How uncertain or accurate the nonmissing entries are |
| Validation concordance | Agreement between imputed results and observed genotypes in a validation set | Performance in populations or settings not represented by that validation set |
The main alternative to model-based summary metrics is empirical validation against genotypes withheld from imputation or measured independently. This comparison can directly assess agreement, but it requires appropriate truth data and its result applies to the validation sample and design. Dosage \(r^2\) and INFO can be calculated without such truth data, but their estimates depend on modeling and implementation choices. Call rate is simpler to compute, but answers only a completeness question. Reporting more than one appropriately defined quantity can therefore be informative, provided each is named and interpreted correctly.
Common Pitfalls
- Wrong: “INFO is 0.8, so 80% of the genotypes are correct.” Correction: INFO summarizes information or posterior uncertainty under a particular definition; it is not generally a percentage of correct individual calls.
- Wrong: “The call rate is 99%, so imputation quality is excellent.” Correction: Call rate measures nonmissing entries under a calling rule. It does not measure uncertainty or agreement with truth.
- Wrong: “The dosage \(r^2\) formula is identical in every program.” Correction: Software may differ in its variance convention, allele-frequency estimate, and adjustments. Check the documented definition before comparing values.
- Wrong: “A value above 1 proves imputation is better than perfect.” Correction: A finite-sample variance ratio can exceed 1 under some conventions. Treat it as a diagnostic of the estimate and its assumptions, not as a literal squared correlation above 1.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A variant has 450 nonmissing hard calls among 500 people. What is its call rate, and what does that number not tell you?
- For a variant with \(\hat p=0.2\) and dosage variance \(0.20\), calculate the variance-based dosage \(r^2\).
- Under the posterior-variance INFO definition, the mean posterior variance is \(0.08\) and \(2\hat p(1-\hat p)=0.32\). Calculate INFO and interpret it carefully.
- Why can a high call rate coexist with low imputation quality?
Solutions. (1) The call rate is \(450/500=0.90\), or 90%. It gives the fraction with nonmissing calls under the stated rule; it does not give their accuracy or posterior uncertainty. (2) The denominator is \(2(0.2)(0.8)=0.32\), so the ratio is \(0.20/0.32=0.625\). This is a variance-based estimate, not necessarily a validation correlation squared. (3) \(\mathrm{INFO}=1-(0.08/0.32)=0.75\). Under the specified definition, average posterior uncertainty is one quarter of the reference variance; 0.75 is not a claim that 75% of calls are correct. (4) Call rate counts nonmissing entries, whereas imputation quality concerns uncertainty or agreement with true genotypes. A hard call can be present despite considerable uncertainty, depending on the calling rule.