Imputation Quality and Attenuated Association Power

Why Imputation Quality Matters for Power

Imputation lets a study test variants that were not directly measured, but an imputed genotype is not always as informative as an observed genotype. When imputation is uncertain, the association test has less information for distinguishing genetic effects from noise. The key design question is not simply whether a variant can be imputed, but how much association power remains at its achieved imputation quality.

The central approximation is straightforward: if a variant retains a fraction $q$ of the genotype information available from direct measurement, its association-test noncentrality is multiplied by $\sqrt{q}$, and its squared noncentrality is multiplied by $q$. This gives a practical way to adjust power calculations and compare variants. It is an approximation whose meaning depends on the quality metric and the assumptions described below.

Genotype Information and Imputation Quality

For a biallelic variant, let $G$ be the true allele count, coded as 0, 1, or 2 copies of the effect allele. Under Hardy–Weinberg equilibrium, with effect-allele frequency $f$, its variance is $2f(1-f)$. When genotypes are imputed, the association analysis commonly uses a dosage $D$: the expected allele count conditional on the imputation data, between 0 and 2. A dosage can be fractional because it represents uncertainty about the unobserved genotype.

Definition: In the dosage-variance formulation, imputation quality $q$ is the fraction of true-genotype variance retained by the imputed dosage: $$ q=\frac{\operatorname{Var}(D)}{\operatorname{Var}(G)}. $$ Under the ideal conditional-expectation formulation $D=E(G\mid\text{imputation data})$, this is also $\operatorname{Corr}(G,D)^2$. Values near 1 indicate that dosages retain nearly all the genotype information; values near 0 indicate little retained information.

Software may report an INFO score, dosage $R^2$, or another quality statistic. These metrics are not defined identically in every setting, so a power calculation should use a metric that reasonably estimates the retained genotype information in the population being analyzed. In particular, quality can vary with allele frequency and ancestry. A quality value from one reference panel or population should not automatically be assumed to apply to another.

Consider a quantitative-trait association with an additive genetic effect. Let $\beta$ be the change in the trait mean per additional effect allele, and let $\sigma^2$ be the residual variance under the null-scale approximation used for planning. With direct genotypes, the squared noncentrality parameter is approximately

\delta_0^2=\frac{N\beta^2\operatorname{Var}(G)}{\sigma^2}.

Here $N$ is the sample size and $\delta_0$ is the noncentrality parameter under perfect genotype information. With imputed dosages, the approximation becomes

\delta_{\mathrm{imp}}^2=q\delta_0^2, \qquad \delta_{\mathrm{imp}}=\sqrt{q}\,\delta_0.

Thus, it is the squared noncentrality that scales directly with quality. For instance, quality $q=0.64$ retains 64% of the ideal squared noncentrality but retains $\sqrt{0.64}=0.80$, or 80%, of the ideal noncentrality itself.

Why the Square-Root Scaling Appears

The relationship follows from the variance of the score or, equivalently in a large-sample regression approximation, from the information for the genetic effect. Under the ideal dosage definition, $D=E(G\mid\text{imputation data})$. The conditional-expectation identity gives $\operatorname{Cov}(G,D)=\operatorname{Var}(D)$. If $Y=\beta G+\varepsilon$, where the residual $\varepsilon$ has mean zero and is independent of genotype and imputation uncertainty, then

\operatorname{Cov}(Y,D) =\beta\operatorname{Cov}(G,D) =\beta\operatorname{Var}(D).

Consequently, the regression slope of $Y$ on $D$ is $\operatorname{Cov}(Y,D)/\operatorname{Var}(D)=\beta$: under these assumptions, dosage uncertainty does not itself attenuate the expected dosage-regression slope. Instead, the lower dosage variance yields less information about that slope. The approximate information is proportional to $N\operatorname{Var}(D)/\sigma^2$, compared with $N\operatorname{Var}(G)/\sigma^2$ for true genotypes. Their ratio is $q$, so the squared noncentrality ratio is $q$, which establishes the stated scaling.

This distinction matters: describing the result as “the effect estimate is always shrunk by $q$” is not generally correct for dosage regression. Under other error-prone genotype representations or analysis procedures, bias can arise, but the standard power adjustment concerns the information loss under the dosage model and assumptions above.

From Noncentrality to Power

For a two-sided association test with a large-sample normal approximation, the test statistic under the alternative is approximately $Z\sim N(\delta_{\mathrm{imp}},1)$, with $\delta_{\mathrm{imp}}\geq 0$ for a positive effect. If the test rejects when $|Z|>c$, its power is

P(|Z|>c) =\Phi(-c-\delta_{\mathrm{imp}}) +1-\Phi(c-\delta_{\mathrm{imp}}),

where $\Phi$ is the standard normal cumulative distribution function. For a genome-wide threshold near $5\times10^{-8}$, a commonly used two-sided critical value is approximately $c=5.45$. The second tail is usually tiny for a well-powered positive association, but the complete expression includes both tails. Use the threshold and approximation appropriate to the study’s testing plan.

Worked Example: Power at Moderate Imputation Quality

Suppose a hypothetical quantitative-trait study has ideal squared noncentrality $\delta_0^2=60$ for a variant if it were directly genotyped. Assume dosage quality $q=0.80$, a two-sided threshold with $c=5.45$, and the normal approximation above. Calculate the imputed noncentrality and power.

First, the imputed squared noncentrality and noncentrality are

\delta_{\mathrm{imp}}^2=0.80(60)=48, \qquad \delta_{\mathrm{imp}}=\sqrt{48}=6.9282.

Substitute this value into the two-sided power formula:

\begin{aligned} \text{Power} &=\Phi(-5.45-6.9282)+1-\Phi(5.45-6.9282)\\ &=\Phi(-12.3782)+1-\Phi(-1.4782)\\ &\approx 0+0.9303\\ &\approx 0.9303. \end{aligned}

Under these assumptions, the study has approximately 93.03% power to detect this association at the stated threshold. The imputation quality has reduced the ideal squared noncentrality from 60 to 48; it has not reduced power by 20 percentage points.

Comparing Discovery Prospects Across Variants

A variant’s quality is only one part of its power. Even with the same sample size, variants can have different ideal noncentralities because their allele frequencies, genetic effects, and—for some designs—case-control information differ. For a quantitative trait under the model above, $\delta_0^2$ includes $\operatorname{Var}(G)=2f(1-f)$; an effect size and quality value alone are not enough to determine power unless the other relevant quantities are held fixed.

The following comparison isolates the contribution of imputation quality by holding all other design features constant. The ideal squared noncentrality is 60 for each variant, and the test uses $c=5.45$. The numbers are hypothetical.

Worked Example: Comparing Two Imputed Variants

Variant A has $q=0.64$, while Variant B has $q=0.70$. Calculate each variant’s approximate power.

For Variant A:

\delta_A^2=0.64(60)=38.4, \qquad \delta_A=\sqrt{38.4}=6.1968.

Its power is

\begin{aligned} \text{Power}_A &=\Phi(-5.45-6.1968)+1-\Phi(5.45-6.1968)\\ &=\Phi(-11.6468)+1-\Phi(-0.7468)\\ &\approx 0+0.7724\\ &\approx 0.7724. \end{aligned}

For Variant B:

\delta_B^2=0.70(60)=42, \qquad \delta_B=\sqrt{42}=6.4807.

Therefore,

\begin{aligned} \text{Power}_B &=\Phi(-5.45-6.4807)+1-\Phi(5.45-6.4807)\\ &=\Phi(-11.9307)+1-\Phi(-1.0307)\\ &\approx 0+0.8487\\ &\approx 0.8487. \end{aligned}

Variant B has higher expected power under the specified assumptions: approximately 84.87% versus 77.24%. The difference arises from quality alone in this comparison. In an actual analysis, different allele frequencies or effect sizes could change the ordering, so one should compare the resulting noncentralities or powers rather than rank variants by quality alone.

Planning for a Target Power

The same relationship can be used in reverse to find an approximate minimum quality for a target power. When the lower tail $\Phi(-c-\delta_{\mathrm{imp}})$ is negligible and the effect direction is specified as positive, target power $1-\beta_{\mathrm{II}}$ requires approximately

\delta_{\mathrm{imp}}\approx c+z_{1-\beta_{\mathrm{II}}},

where $z_{1-\beta_{\mathrm{II}}}$ is the corresponding standard-normal quantile. Since $\delta_{\mathrm{imp}}^2=q\delta_0^2$, this gives

q_{\min}\approx \frac{\left(c+z_{1-\beta_{\mathrm{II}}}\right)^2}{\delta_0^2}.

This is a planning approximation, not a guarantee of achieved power. It assumes the same test threshold and normal approximation and should be checked with the full two-sided formula when the lower tail is not negligible.

Worked Example: Quality Needed for 80% Power

Use the hypothetical variant with ideal squared noncentrality $\delta_0^2=60$, and ask what imputation quality is needed for approximately 80% power at $c=5.45$. The standard normal quantile for 80% cumulative probability is approximately $z_{0.80}=0.8416$.

The required imputed noncentrality is approximately

\delta_{\mathrm{imp}}\approx5.45+0.8416=6.2916.

Thus the approximate minimum quality is

q_{\min}\approx\frac{6.2916^2}{60} =\frac{39.5842}{60} =0.6597.

So quality of about 0.660 is needed under this planning approximation. The preceding comparison checks the implication: $q=0.64$ gives power about 0.7724, below 80%, whereas $q=0.70$ gives power about 0.8487, above 80%.

Effective Sample Size and Limits of the Approximation

Because the squared noncentrality is proportional to sample size, multiplying it by $q$ is algebraically equivalent to replacing $N$ by $qN$ in this particular power calculation. This is sometimes called an effective-sample-size interpretation. For example, if $N=10{,}000$ and $q=0.80$, the association information is approximately the same as that from 8,000 participants with perfectly observed genotypes, all else equal.

This is an equivalence for the test’s expected information under stated assumptions, not a claim that 20% of participants have been removed or that every statistical feature of the data matches a smaller study. In case-control planning, one can similarly apply the quality factor to the relevant ideal association noncentrality, including the design’s effective sample size. The underlying case-control information calculation remains essential; quality does not replace it.

The adjustment is most defensible when dosages are used appropriately, imputation errors are nondifferential with respect to the phenotype conditional on the true genotype and covariates, the quality measure estimates retained genotype information in the analyzed sample, and the association-test approximation is suitable. It can be misleading if the score is poorly calibrated, imputation quality differs across ancestry groups in ways not reflected by a single value, or missingness and genotype error are associated with phenotype or study site. These situations can affect bias or calibration as well as power, so multiplying by $q$ alone may not describe the problem.

Common Mistake

Wrong reasoning: “Quality is 0.64, so power is 64% of the power with perfect genotypes.”
Correction: Quality scales the squared noncentrality, not power directly. First calculate $\delta_{\mathrm{imp}}=\sqrt{q}\delta_0$, then evaluate the power formula at the test threshold. Power is a nonlinear function of noncentrality, so its reduction depends on the starting power and threshold.
Another pitfall: Treating $qN$ as a literal participant count or comparing quality values without considering allele frequency, effect size, and study design. It is an information-equivalence shortcut under assumptions, not a substitute for a complete power model.

Key takeaway: For a standard dosage-based association power approximation, imputation quality $q$ multiplies ideal squared noncentrality: $\delta_{\mathrm{imp}}^2=q\delta_0^2$. Calculate power using the resulting noncentrality and the specified significance threshold, and interpret an effective-sample-size adjustment only within its assumptions.

Check Your Understanding

Answer each question using the information in this tutorial.

A variant has ideal squared noncentrality 25 and imputation quality 0.81. What are its imputed squared noncentrality and noncentrality?
Why does dosage uncertainty reduce the standard error or information without necessarily attenuating the expected dosage-regression slope under the stated model?
Two variants have the same quality but different allele frequencies. Why might their powers differ?
What does the effective-sample-size interpretation mean, and what does it not mean?
For a two-sided test, why is it useful to state the full power expression even when one tail is negligible?