Why Compare Dosages With True Genotypes?
Imputation represents uncertainty about a person’s genotype rather than simply assigning a genotype label. For a biallelic variant, let \(G_i\) be person \(i\)’s true genotype, coded as the number of copies of the effect allele: \(0\), \(1\), or \(2\). Let \(D_i\) be the imputed dosage, the expected number of copies given the available imputation information. A dosage can therefore be non-integer. For example, a dosage of \(0.7\) can represent uncertainty between genotypes rather than a hard call of one copy.
To assess how well dosages agree with genotypes, a validation sample needs both the imputed dosages and reliable genotypes at the same variant. Dosage \(r^2\) summarizes their linear association. It is useful because it evaluates the dosage values directly, retaining more information than a comparison that first converts every dosage into a hard call.
The formula is the usual squared correlation: the factors of \(n-1\) in the sample covariance and variances cancel. The value is between 0 and 1. A value near 1 indicates a strong linear relationship in the validation sample; a value near 0 indicates little linear association. Squaring removes the sign, so allele coding must be aligned before calculating it: a reversed allele coding can produce a negative correlation even when the two representations otherwise correspond.
Calculation in a Validation Sample
Worked Example: Calculating Dosage \(r^2\) From Six Validated Genotypes
Suppose a hypothetical validation sample has the following true genotypes and imputed dosages. The genotypes are exact allele counts, and the dosages use the same allele coding.
| Person | True genotype \(G_i\) | Imputed dosage \(D_i\) |
|---|---|---|
| 1 | 0 | 0.1 |
| 2 | 1 | 0.8 |
| 3 | 1 | 1.2 |
| 4 | 2 | 1.7 |
| 5 | 0 | 0.2 |
| 6 | 2 | 1.8 |
The means are \(\bar{G}=6/6=1.000\) and \(\bar{D}=5.8/6=0.9667\). The centered genotype values are \((-1,0,0,1,-1,1)\); the centered dosage values are approximately \((-0.8667,-0.1667,0.2333,0.7333,-0.7667,0.8333)\). Thus:
Substituting into the definition gives:
In this hypothetical validation sample, dosage and true genotype have a strong linear association. The result does not mean that 96.48% of the genotypes are exactly correct; it describes squared correlation, not the fraction of correct calls.
Why Squared Correlation Has a Variance Interpretation
A useful population result explains why dosage \(r^2\) is informative when dosages are calibrated conditional expectations. Let \(I\) denote the imputation information available for an individual, and suppose the dosage is \(D=E[G\mid I]\). In words, the dosage is the expected true genotype given that information. The following derivation concerns population quantities; a validation sample’s squared correlation estimates the corresponding population association, subject to sampling variation.
Because \(D\) is determined by \(I\), \(E[DG]=E[D\,E(G\mid I)]=E[D^2]\). Also, \(E[D]=E[G]\).
Writing \(\mu=E[G]=E[D]\), the covariance is \(\operatorname{Cov}(D,G)=E[DG]-\mu^2=E[D^2]-\mu^2=\operatorname{Var}(D)\).
\(\operatorname{Var}(G)=\operatorname{Var}(E[G\mid I])+E[\operatorname{Var}(G\mid I)]=\operatorname{Var}(D)+E[\operatorname{Var}(G\mid I)]\).
If \(\operatorname{Var}(D)>0\), then \(\operatorname{Corr}(D,G)^2=\operatorname{Cov}(D,G)^2/[\operatorname{Var}(D)\operatorname{Var}(G)]=\operatorname{Var}(D)/\operatorname{Var}(G)\).
Under these conditions, dosage \(r^2\) is the fraction of the population genotype variance represented by dosage variance. Equivalently, the variance not represented by the dosage is the expected conditional genotype variance, the remaining uncertainty after observing the imputation information. The result depends on calibration: it is not a general identity for any numbers labeled “dosage.”
Worked Example: Interpreting the Variance Identity
Consider a hypothetical population in which the allele frequency is \(p=0.5\), genotypes follow Hardy–Weinberg equilibrium, and calibrated dosages have population variance \(0.400\). Under Hardy–Weinberg equilibrium, genotype variance is \(2p(1-p)\), so:
The population squared correlation implied by the calibrated-dosage identity is:
The expected conditional genotype variance is \(0.500-0.400=0.100\), so \(0.400+0.100=0.500\), as required by the total-variance decomposition. This is a population illustration, not a direct validation calculation: in practice, the squared sample correlation is calculated from paired dosage and true-genotype observations.
Assumptions and Practical Checks
The arithmetic definition itself requires paired observations, consistent allele coding, and nonzero variation in both measurements. Its interpretation as imputation performance additionally depends on the validation data being informative about the intended use. In particular, check the following before treating the value as representative:
- Reliable validation genotypes: genotype errors can weaken or distort the observed dosage–genotype relationship.
- Comparable samples: the validation group should resemble the target group in ancestry, allele frequency, and other features relevant to imputation. A result in one group need not transfer to another.
- Consistent variant definition: confirm the genome location, reference and effect alleles, and genotype coding. A strand or allele mismatch can make the comparison meaningless.
- Enough variation: if all validated genotypes or all dosages are identical, the denominator is zero and \(r^2\) is undefined. Very little genotype variation also makes the estimate unstable.
- Appropriate sampling: a small or selectively chosen validation set may not represent the target population. Related participants or repeated measurements can also make simple uncertainty calculations that assume independent observations inappropriate.
A single value should be accompanied by checks of the paired data. A scatterplot of dosage against genotype can reveal reversed coding, outliers, unusual clusters, or a relationship that is not well summarized by a straight line. Check the range and spread of both variables and verify that each person’s dosage was paired with that person’s genotype. When the validation sample is small, report that limitation rather than presenting the estimate as precise.
What Dosage \(r^2\) Does Not Measure
Worked Example: High \(r^2\) Can Hide Miscalibration
In a second hypothetical validation set, let the true genotypes be \(G=(0,1,1,2)\) and the dosages be \(D=(0.2,0.7,0.7,1.2)\). Here \(D=0.2+0.5G\) for every person. The means are \(\bar{G}=1\) and \(\bar{D}=0.7\), so the centered vectors are \((-1,0,0,1)\) and \((-0.5,0,0,0.5)\). Their cross-product sum is \(1\), and their sums of squares are \(2\) and \(0.5\). Therefore:
Despite perfect squared correlation, the dosages are not equal to the true genotypes. The mean squared dosage error is \((0.2^2+(-0.3)^2+(-0.3)^2+(-0.8)^2)/4=0.215\), in squared allele copies, and the root mean squared error is \(\sqrt{0.215}=0.4637\) allele copies. The example shows why \(r^2\) should not be treated as a direct measure of absolute dosage error or calibration.
Squaring the correlation also discards its direction. A negative correlation can produce a high \(r^2\), so inspect the signed correlation and confirm allele orientation. Further, \(r^2\) is not genotype-call accuracy: it does not count how often a selected hard call matches the validated genotype. Hard-call concordance answers a different question and depends on how dosages are converted into calls. A direct error summary, such as mean squared error, can complement \(r^2\) when the magnitude of dosage discrepancies matters.
A common mistake is to read \(r^2=0.80\) as “80% of genotypes are correct” or “there is an 80% probability that a dosage is correct.” Neither conclusion follows. The correction is to describe it as a squared linear association in the validation sample, and to use a separate measure when the question concerns exact calls, absolute errors, or uncertainty for an individual.
Limitations and Choice of Comparison
The main advantage of dosage \(r^2\) over hard-call concordance is that it uses the graded dosage values rather than discarding uncertainty at a threshold. Its main limitation is that correlation is insensitive to positive changes of scale and shifts: as the previous example showed, even systematically compressed and offset dosages can have \(r^2=1\). It is also sensitive to the composition of the validation sample because the amount of genotype variation affects the association. Comparisons across variants or populations should therefore be interpreted with allele frequency and validation design in mind.
No single summary answers every quality question. Use dosage \(r^2\) when the goal is agreement in linear variation between dosage and validated genotype. Use hard-call concordance when the goal is the fraction of selected genotype calls that match. Examine dosage errors when calibration or the size of discrepancies matters. These measures are complementary, not interchangeable. In particular, a good value in one validation sample does not establish equal performance for every ancestry group or intended analysis.
Check Your Understanding
Work each item using the definitions above. Brief solutions follow the questions.
- If every dosage is the same value, can dosage \(r^2\) be calculated? Explain.
- A validation sample has \(\operatorname{Var}(G)=0.60\) and calibrated dosages with \(\operatorname{Var}(D)=0.45\). Under the stated population assumptions, what is the squared correlation?
- What does a dosage \(r^2\) of 0.81 say, and what does it not say?
- Why can reversed allele coding be a problem even though dosage \(r^2\) is squared?
Solutions. (1) No. If every dosage is constant, \(\sum(D_i-\bar{D})^2=0\), so the denominator of the sample correlation is zero. (2) The calibrated-dosage identity gives \(0.45/0.60=0.75\). (3) It indicates a squared linear association of 0.81 in the specified validation sample; it does not mean that 81% of genotypes are correct or that an individual dosage has an 81% probability of being correct. (4) Reversing coding can reverse the signed correlation. Squaring hides that direction, so a high squared value alone may not reveal the mismatch; harmonize alleles and inspect the signed relationship.