Categorical And Nonlinear Copy-Number Effects

Why Compare Copy States With a Dosage Trend?

A dosage model summarizes copy number with one number and estimates a per-copy trend. This is useful when the phenotype changes at a roughly constant rate as copies are gained or lost. But that summary can conceal important patterns: a deletion may have a different association from a duplication, or only one unusual copy state may be associated with the phenotype. Comparing a categorical copy-state model with a dosage model makes the assumption of a per-copy effect testable.

Let \(C\) denote the observed copy-number state at a structural-variant region, with \(C=2\) representing the reference diploid state in these examples. A dosage variable is \(D=C-2\), so a one-unit increase in \(D\) means one additional copy. The analysis concerns association, not proof that changing copy number causes a phenotype.

Categorical and Per-Copy Models

For a quantitative phenotype \(Y\), a categorical model estimates a separate mean for each observed copy state. With states \(0,1,2,3\), a model using state 2 as the reference is

Y_i=\alpha+\beta_0 I(C_i=0)+\beta_1 I(C_i=1)+\beta_3 I(C_i=3)+\boldsymbol{\gamma}^{T}\mathbf{Z}_i+\varepsilon_i.

Here \(I(C_i=k)\) is 1 when participant \(i\) has copy state \(k\), and 0 otherwise. The vector \(\mathbf{Z}_i\) contains any adjustment covariates. Each \(\beta_k\) is the adjusted mean difference between state \(k\) and state 2, in the units of \(Y\). This model does not require the effects of deletions and duplications to be symmetric or equally spaced.

The corresponding linear dosage model replaces the state indicators with \(D_i=C_i-2\):

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

Under this model, \(\beta_D\) is the mean change in \(Y\) associated with one additional copy, holding the modeled covariates fixed. It imposes equal increments: the mean difference from state 1 to state 2 is the same as from state 2 to state 3. For states 0 through 3, it also requires the difference from state 0 to state 1 to follow that same increment.

Definition: A categorical copy-state effect is an association represented by separate coefficients for copy-number states. A linear per-copy effect is an association represented by one coefficient multiplying copy number (or dosage). A departure from linearity occurs when the state-specific associations cannot be adequately described by the imposed equal per-copy increments.

For a binary phenotype, the same comparison can be made with logistic regression. The categorical model uses copy-state indicators in the log odds; the dosage model uses one dosage coefficient. In that setting, a per-copy coefficient represents a constant change in log odds for each additional copy, not a constant change in probability.

What Is Being Tested?

Three questions are distinct. First, is there any association across copy states? Second, is there a linear per-copy trend? Third, does the categorical pattern improve on the linear trend enough to indicate departure from linearity? A nonsignificant trend test does not rule out a categorical association: effects at lower and higher copy states can cancel in the trend.

For a quantitative phenotype with independent observations and normally distributed errors of common variance, the categorical and dosage models can be compared using a partial \(F\)-test. Both models must include the same covariates and participants. If there are \(K\) observed copy states, the categorical model has \(K\) state means, while the dosage model has an intercept and one dosage coefficient. If both are estimable, the categorical model adds \(K-2\) degrees of freedom.

F=\frac{(RSS_{\mathrm{dosage}}-RSS_{\mathrm{categorical}})/(K-2)} {RSS_{\mathrm{categorical}}/df_{\mathrm{residual,categorical}}}.

Here \(RSS\) is the residual sum of squares. The numerator measures the additional variation explained by allowing separate state means rather than a straight per-copy trend. Under the null hypothesis that the linear dosage model is adequate, this statistic has an \(F\) distribution with \(K-2\) numerator degrees of freedom and the categorical model's residual degrees of freedom. A large statistic is evidence that the categorical model fits better than the linear trend.

This comparison is not the same as testing the categorical model against no copy-state association. The latter asks whether all state means are equal, while the comparison above asks whether the state means follow the particular linear pattern. For binary outcomes, a likelihood-ratio test can compare the nested logistic models: twice the difference in maximized log likelihoods is compared with a chi-square distribution with the difference in model degrees of freedom, under standard large-sample conditions.

Worked Examples

Worked Example: A Strong Association That Is Not a Straight Per-Copy Trend

Suppose a hypothetical study measures a quantitative biomarker in 400 unrelated participants, with 100 participants at each copy state \(C=0,1,2,3\). The state-specific sample means are 10, 10, 10, and 14 units, respectively, and each state's sample standard deviation is 2 units. Assume for this calculation that observations are independent and that the within-state errors have a common normal variance. There are no adjustment covariates.

The categorical model estimates the four means directly. Relative to state 2, the estimated differences are 0 units for state 0, 0 units for state 1, and 4 units for state 3. This pattern is not an equal per-copy increase.

For the dosage model, the dosage values are \(-2,-1,0,1\). With equally sized groups, the estimated slope is the regression slope of the four means on dosage. The mean dosage is \(-0.5\), and the mean biomarker value is 11. Therefore,

\widehat{\beta}_D= \frac{(-1.5)(-1)+(-0.5)(-1)+(0.5)(-1)+(1.5)(3)} {(-1.5)^2+(-0.5)^2+(0.5)^2+(1.5)^2} =\frac{6}{5}=1.2\text{ units per copy}.

The fitted dosage-model means are 9.2, 10.4, 11.6, and 12.8 units. The observed group means differ from these fitted values by 0.8, \(-0.4\), \(-1.6\), and 1.2 units. The extra residual sum of squares from imposing the dosage trend is therefore

100\bigl(0.8^2+(-0.4)^2+(-1.6)^2+1.2^2\bigr)=480.

The categorical residual sum of squares is \(4(99)(2^2)=1584\), because each of the four groups has 99 residual degrees of freedom and variance 4. Thus the lack-of-linearity test is

F=\frac{(2064-1584)/2}{1584/396} =\frac{240}{4}=60.

There are \(K-2=2\) numerator degrees of freedom and 396 denominator degrees of freedom. The result, \(F(2,396)=60\), has \(p<0.001\). The data support a pattern that departs from a linear per-copy trend. In particular, state 3 has a higher mean biomarker level than the other states; reporting only the estimated slope of 1.2 units per copy would obscure that pattern.

Worked Example: Deletion and Duplication Associations Differ

Consider a hypothetical cohort study of a binary phenotype. Among participants with one copy, 40 of 200 have the phenotype; among those with two copies, 80 of 400 have it; among those with three copies, 60 of 200 have it. Assume independent participants and use an unadjusted logistic model with copy state as a categorical predictor.

The phenotype proportions are \(40/200=0.20\), \(80/400=0.20\), and \(60/200=0.30\). The deletion-versus-diploid odds ratio is

OR_{1:2}=\frac{40/160}{80/320}=1.00.

The duplication-versus-diploid odds ratio is

OR_{3:2}=\frac{60/140}{80/320}=1.714.

For this latter contrast, the estimated log odds ratio is \(\log(1.714)=0.539\). Its usual large-sample standard error is

SE=\sqrt{\frac{1}{60}+\frac{1}{140}+\frac{1}{80}+\frac{1}{320}} =\sqrt{0.03943}=0.1986.

The Wald statistic is \(0.539/0.1986=2.71\), giving a two-sided \(p\)-value of about 0.0067. A 95% confidence interval for the odds ratio is

\exp\bigl(0.539\pm1.96(0.1986)\bigr) =\exp(0.150,0.928) =(1.16,2.53).

In this hypothetical cohort, the estimated odds of the phenotype are higher for the three-copy state than for the two-copy state, while the estimated odds for the one-copy state are the same as for the two-copy state. The interval describes uncertainty in the population odds ratio under the model; it does not give a probability that the fixed odds ratio lies within this particular interval. A single linear dosage coefficient would impose a common log-odds increment for both transitions and would not represent these state-specific estimates.

Worked Example: A Zero Linear Trend Can Hide a Copy-State Association

Suppose a hypothetical quantitative-phenotype study has 50 participants in each of states \(C=1,2,3\). The state means are 12, 10, and 12 units, and each state's sample standard deviation is 3 units. Assume independent observations and a common normal error variance. Use state 2 as the reference.

The estimated categorical contrasts are 2 units for state 1 versus state 2 and 2 units for state 3 versus state 2. For either contrast, the standard error is

SE=\sqrt{\frac{3^2}{50}+\frac{3^2}{50}}=0.600\text{ units}.

Each contrast has a \(t\)-statistic of \(2/0.600=3.33\), with approximately 147 residual degrees of freedom. The two-sided \(p\)-value is about 0.0011 for each unadjusted contrast. This is evidence that both the deletion state and the duplication state differ from the diploid state in this hypothetical sample.

The dosage values are \(-1,0,1\). Because the outer-state means are equal, their fitted linear dosage slope is 0 units per copy. For the test of any categorical association against a model with only an intercept, the between-state sum of squares is

50\left[\left(12-\frac{34}{3}\right)^2+ \left(10-\frac{34}{3}\right)^2+ \left(12-\frac{34}{3}\right)^2\right] =133.33.

The categorical residual sum of squares is \(3(49)(3^2)=1323\), so the overall categorical test is \(F=(133.33/2)/(1323/147)=7.41\), with \(p<0.001\). Thus, there is evidence of a categorical association even though the estimated linear trend is zero. Here the pattern is symmetric: both non-diploid states have higher means. A zero per-copy slope alone would miss that pattern.

Model Conditions and Interpretation

The quantitative-outcome \(F\)-comparison assumes independent observations, a correctly specified mean model, and errors with a common variance; normal errors support the exact small-sample reference distribution. In practice, covariates used to control confounding should be included in both models, with the same participants and the same covariate terms, so the comparison isolates the copy-state modeling choice. If the outcome is binary, logistic regression instead assumes an appropriate logit mean model and independent observations; likelihood-ratio approximations require adequate information for the fitted parameters.

The copy states themselves must be defined and measured consistently. A state indicator represents the copy state assigned to a participant, not certainty that every structural-variant call is correct. Misclassification, unmodeled population structure, or technical differences related to copy state can distort an association. Those issues are not fixed merely by choosing categorical rather than dosage coding.

A categorical model uses more parameters than a dosage model. It can reveal departures from linearity, but may be less precise when some states have few observations. If a state is absent, its separate effect cannot be estimated; if it is very uncommon, conventional estimates and tests may be unstable. The model comparison should therefore be interpreted alongside state counts and uncertainty, rather than as an automatic reason to prefer the more complex model.

Common Mistake

Common mistake: “The dosage trend is not significant, so copy number is not associated with the phenotype.” This conclusion does not follow: opposite or curved state-specific associations can cancel in a single slope. Fit or inspect state-specific contrasts and test whether the categorical pattern is supported; then report the pattern and its uncertainty rather than treating a linear trend as the only possible effect.

The reverse mistake is to conclude that a significant categorical-versus-dosage comparison proves a particular biological mechanism. The test shows evidence against the specified equal-increment trend, not why the pattern occurs. State-specific estimates, confidence intervals, phenotype scale, and study design are needed to describe what the data support.

Key takeaway: A per-copy coefficient is a substantive shape assumption: each additional copy has the same modeled increment. Categorical contrasts relax that assumption and allow deletions and duplications to differ. Compare the models directly when the shape matters, and distinguish a linear-trend test from tests of overall copy-state association.

Check Your Understanding

Answer each question by identifying the model or interpretation that fits.

With copy states 1, 2, and 3, what equal-increment assumption does a linear per-copy model impose on the three state means?
In a categorical model using state 2 as the reference, what does the coefficient for state 1 represent for a quantitative phenotype?
Why can a dosage-trend test be nonsignificant even when copy state is associated with the phenotype?
When comparing categorical and dosage models, why should the same covariates and participants be used in both?
For a binary phenotype, what does a constant per-copy logistic coefficient imply about log odds, and what does it not imply about probability?