Worked Dosage Association Analysis For A Copy-Number Variant

From a Deletion Call to an Association Estimate

A copy-number variant (CNV) can affect a phenotype differently according to how many copies are deleted. A dosage association analysis represents that number with a numeric predictor and estimates how the phenotype changes with each additional deleted copy. This is distinct from asking only whether a person carries any deletion: dosage analysis can use the distinction between one and two deleted copies, while a carrier-only analysis combines them.

The regression result is meaningful only if the variant representation is appropriate and the calls are credible. A deletion that is miscalled because of low coverage, inconsistent breakpoints, or a batch artifact can produce an apparently precise but misleading association. The analysis therefore pairs a clearly defined dosage with variant-level quality checks and a model whose assumptions are stated explicitly.

Definition: For a diploid locus, let \(G\) be the number of deletion-bearing homologues: \(G=0\) means no deleted copy, \(G=1\) means one deleted copy, and \(G=2\) means two deleted copies. If genotype probabilities are available, the expected deletion dosage is \(D=P(G=1)+2P(G=2)\). A dosage regression models the phenotype using \(D\) as a numeric predictor. A carrier indicator is \(C=I(G>0)\), equal to 1 for either one or two deleted copies and 0 otherwise.

Dosage Coding and the Regression Model

For a quantitative phenotype \(Y_i\) measured on individual \(i\), a covariate-adjusted linear dosage model is

Y_i=\beta_0+\beta_D D_i+\boldsymbol{\gamma}^{\mathsf T}\mathbf{Z}_i+\varepsilon_i.

Here, \(D_i\) is the deletion dosage, \(\mathbf{Z}_i\) is a vector of covariates such as age, sex, and ancestry principal components, and \(\beta_D\) is the dosage coefficient. The model assumes a linear conditional mean: holding the included covariates fixed, the expected phenotype changes by \(\beta_D\) for each one-copy increase in deletion dosage. This additive assumption is not the same as estimating separate effects for one and two copies.

With expected dosages, the coefficient describes the fitted change per one-unit increase in expected deletion dosage. Using this representation relies on the genotype probabilities being adequately calibrated and on their uncertainty not being systematically related to the phenotype after accounting for the model. If calls are treated as certain when they are not, dosage error can affect the estimate and its uncertainty.

The usual ordinary least-squares standard error and t-based inference assume independent observations, a correctly specified linear conditional mean, and errors with constant variance. Normally distributed errors give exact finite-sample t inference under these conditions; approximate normality may support approximate inference. Relatedness or other dependence among participants can violate independence, and covariate adjustment does not automatically remove every source of confounding.

Worked Example: Calculating Expected Deletion Dosage

Suppose three hypothetical participants have posterior genotype probabilities \((P(G=0),P(G=1),P(G=2))\) of \((0.10,0.80,0.10)\), \((0.70,0.25,0.05)\), and \((0.02,0.18,0.80)\), respectively. Their expected dosages are calculated from \(D=P(G=1)+2P(G=2)\).

For participant A, \(D=0.80+2(0.10)=1.00\) deleted copies. For participant B, \(D=0.25+2(0.05)=0.35\) deleted copies. For participant C, \(D=0.18+2(0.80)=1.78\) deleted copies.

These are expected copy counts, not necessarily hard genotype calls. For example, participant B’s expected dosage of 0.35 does not mean that the participant has 0.35 of a physical copy deleted; it summarizes uncertainty about the underlying genotype. The corresponding carrier probability is \(P(G=1)+P(G=2)\), which is 0.30 for participant B and is not the same quantity as the dosage.

Worked Covariate-Adjusted Analysis

The following numbers are hypothetical. Imagine a study with a quantitative blood-pressure phenotype and a candidate deletion. The analysis plan specifies dosage coding, the covariates, the phenotype scale, and variant-quality checks before examining the association result. The model includes an intercept, deletion dosage, age in years, a sex indicator, and three ancestry principal components.

Worked Example: Estimating the Dosage Effect

Among 800 hypothetical participants, 792 have usable phenotype, covariate, and deletion-dosage data. Of those 792, 752 have \(G=0\), 38 have \(G=1\), and 2 have \(G=2\). Thus, there are 40 carriers, and the total hard-call dosage is \(38+2(2)=42\) deleted copies. Suppose the covariate-adjusted regression output for blood pressure, measured in mmHg, gives \(\hat{\beta}_D=-0.420\) mmHg per deleted copy and \(SE(\hat{\beta}_D)=0.170\) mmHg per deleted copy.

The model has seven fitted coefficients: the intercept, dosage, age, sex, and three principal components. Thus, the residual degrees of freedom are \(df=792-7=785\). The test of no dosage association uses \(H_0:\beta_D=0\) against \(H_a:\beta_D\ne 0\). The observed t statistic is

t=\frac{\hat{\beta}_D-0}{SE(\hat{\beta}_D)} =\frac{-0.420}{0.170} =-2.471.

With \(df=785\), the two-sided p-value is approximately 0.0137. At a pre-specified significance level of \(\alpha=0.05\), this result is evidence against a zero linear dosage coefficient under the model assumptions. The p-value is the probability, assuming the null model and its assumptions, of obtaining a t statistic at least as extreme in either direction; it is not the probability that the null hypothesis is true.

Using a 95% t-based confidence interval with critical value \(t_{0.975,785}\approx1.963\), the interval is

-0.420\ \mathbin{\pm}\ 1.963(0.170) =-0.420\ \mathbin{\pm}\ 0.33371 =(-0.75371,\,-0.08629)\ \text{mmHg per deleted copy}.

The fitted association is a 0.420 mmHg lower adjusted mean blood pressure per additional deleted copy, under the assumed additive model. The interval is a range of values compatible with the data and model at the 95% confidence level under repeated sampling; it does not assign a 95% probability to this particular interval containing the fixed coefficient. The result is an association, not evidence by itself that the deletion causes a change in blood pressure.

Why the Regression Coefficient Has This Interpretation

The coefficient is a comparison conditional on the included covariates, not simply the difference between the raw averages of carriers and noncarriers. One way to see this is to remove the linear contribution of the covariates from both the outcome and dosage, then regress the resulting residualized outcome on residualized dosage. The slope of that regression is the adjusted dosage coefficient.

For a short justification, write \(\widetilde{D}_i\) for dosage after subtracting its fitted linear prediction from the covariates, and \(\widetilde{Y}_i\) for the outcome after the same covariate adjustment. The least-squares slope minimizes the sum of squared residuals in the residualized regression. Differentiating that sum with respect to the slope and setting the derivative to zero gives

\frac{\partial}{\partial b}\sum_i(\widetilde{Y}_i-b\widetilde{D}_i)^2 =-2\sum_i\widetilde{D}_i(\widetilde{Y}_i-b\widetilde{D}_i)=0, \qquad \hat{\beta}_D= \frac{\sum_i\widetilde{D}_i\widetilde{Y}_i} {\sum_i\widetilde{D}_i^2}.

Thus, the estimate reflects the relationship between the parts of dosage and phenotype not linearly explained by the included covariates. If dosage has little variation after adjustment, the denominator is small and the estimate can be imprecise. In a rare-variant analysis, very few carriers—and especially very few individuals with two deleted copies—can make the additive trend difficult to assess even when the model returns a coefficient.

Variant-Quality Checks Before Interpreting Association

Quality checks address whether the variant and its genotypes are credible, not whether the association p-value is small. Review the call set using pre-specified criteria suited to the assay and region. Useful checks include call rate, read depth and other supporting evidence, consistency of breakpoint definitions, and whether calls cluster by sequencing batch or other technical factors. Also inspect sample-level diagnostics and confirm that the phenotype and covariates are available for the analyzed participants.

Thresholds should not be chosen after seeing which version gives the preferred association. A variant can pass a call-rate rule yet still have a systematic artifact, and a satisfactory technical review cannot establish that the association model is correct. For a rare deletion, apparent enrichment in a phenotype group should prompt scrutiny of technical and ancestry patterns rather than an automatic biological interpretation.

Worked Example: Interpreting a Hypothetical Quality Review

Suppose a hypothetical cohort has 800 participants and dosage calls are available for 792. The call rate is

\frac{792}{800}=0.990=99.0\%.

The review also finds similar median supporting read depth for carriers and noncarriers (31-fold and 30-fold, respectively), and the deletion calls have consistent breakpoint definitions. A plot of calls by sequencing batch shows no obvious batch-specific cluster. These observations support proceeding to association analysis under the study’s pre-specified quality criteria; they do not prove that every genotype is correct or that residual technical bias is absent.

If instead the calls were concentrated in one batch, or carrier calls had substantially weaker read support, the association result would need to be treated as potentially artifactual until that pattern was resolved. A small p-value cannot substitute for this review. Conversely, high-quality calls do not guarantee an association: technical validity and statistical evidence answer different questions.

Dosage Versus Carrier-Only Analysis

A carrier-only model replaces \(D_i\) with \(C_i=I(G_i>0)\), while retaining the same covariates. Its coefficient compares the adjusted mean phenotype among carriers with that among noncarriers. It does not estimate a per-copy change. In particular, one-copy and two-copy deletions contribute equally to the carrier indicator, whereas dosage coding assigns them values of one and two.

Worked Example: Comparing With a Carrier-Only Model

Using the same 792 hypothetical participants and covariates, suppose the carrier-only regression estimates a carrier coefficient of \(-0.310\) mmHg, with standard error \(0.180\) mmHg. Its test statistic is

t=\frac{-0.310}{0.180}=-1.722.

With \(df=785\), the two-sided p-value is approximately 0.085. The 95% confidence interval is

-0.310\ \mathbin{\pm}\ 1.963(0.180) =-0.310\ \mathbin{\pm}\ 0.35334 =(-0.66334,\,0.04334)\ \text{mmHg}.

This estimate compares carriers as a group with noncarriers and is not directly comparable in scale to the dosage coefficient of \(-0.420\) mmHg per copy. The carrier interval includes zero, while the dosage interval does not, but that contrast in p-values does not establish that the two effects differ. The models encode different comparisons; a formal comparison would require an analysis designed to test the relevant difference in effects.

Common Mistakes

Calling a dosage coefficient a carrier effect. The dosage coefficient is a fitted change per additional deleted copy under an additive model. A carrier coefficient compares any carrier with a noncarrier.
Treating expected dosage as a literal fractional genotype. An expected dosage such as 0.35 summarizes genotype uncertainty; it does not mean that 0.35 of a chromosome is deleted.
Assuming covariate adjustment eliminates bias. Adjustment controls only the included covariates as represented in the model. It cannot automatically resolve unmeasured confounding or technical artifacts.
Choosing the model with the smaller p-value. Dosage and carrier coding answer different questions. The coding should follow the scientific question and analysis plan, not which result appears more favorable.
Reading non-significance as evidence of no effect. A carrier-only interval that includes zero indicates uncertainty compatible with zero as well as some nonzero effects; it does not prove the effect is absent.

Key takeaway: Define deletion dosage clearly, check that the variant calls are technically credible, and interpret the adjusted coefficient in phenotype units under its additive-model assumptions. A carrier-only analysis is a distinct comparison, not a simpler restatement of the dosage effect.

Check Your Understanding

Try these questions before reviewing the worked examples.

A participant has probabilities \(P(G=0)=0.60\), \(P(G=1)=0.30\), and \(P(G=2)=0.10\). What is the expected deletion dosage, and what is the carrier probability?
In a dosage model, what does a coefficient of \(-0.25\) mmHg per deleted copy mean, holding the included covariates fixed?
Why does a carrier-only coefficient not estimate the same effect as a dosage coefficient?
Name two technical patterns that could make a deletion association appear credible statistically but questionable biologically.
Why does a smaller p-value from dosage coding than from carrier coding not prove that the dosage effect differs from the carrier effect?