Overview

Standard genetic association studies assume additive effects, where each copy of an allele contributes equally to a phenotype. However, many genetic effects are non-additive -- for example, recessive variants only manifest when two copies are present.

arcade provides a suite of tools to detect and test for these non-additive effects.

The model

We model the phenotype as a combination of additive and non-additive effects 1 2 3 4 5. Non-additive effects (also called dominance deviation) capture any deviation from a purely additive genetic model:

\[ y = X^A \beta^A + X^D \beta^D + \varepsilon \]

where \(y\) is the phenotype vector for \(n\) individuals, \(X^A\) and \(X^D\) are the additive and non-additive genotype encoding matrices (\(n \times m\) for \(m\) loci), \(\beta^A\) and \(\beta^D\) are the corresponding effect size vectors, and \(\varepsilon\) is the residual noise with variance \(1 - h^2_A - h^2_D\).

The key requirement is that \(X^A\) and \(X^D\) are orthogonal (\(X^A \cdot X^D = 0\)), so that additive and non-additive effects can be estimated independently in a joint test.

Genetic encodings

Under Hardy-Weinberg Equilibrium

Assuming HWE, the non-additive encoding maps genotypes as:

\[ [0, 1, 2] \rightarrow \left[-\frac{p}{1-p},\; 1,\; -\frac{1-p}{p}\right] \]

where \(p\) is the minor allele frequency 1 2 3 4 5. This encoding is uncorrelated with the additive encoding by construction.

General case (without assuming HWE)

To extend this approach beyond common variants 1, arcade uses observed genotype proportions directly. Let \(r\), \(h\), and \(a\) be the proportions of homozygous reference, heterozygous, and homozygous alternate genotypes respectively, with \(r + h + a = 1\). At the gene level, these correspond to wildtype, monoallelic, and biallelic carrier proportions.

We start with the raw additive and non-additive vectors:

\[ {X^A}' = \begin{bmatrix}0\\1\\2\end{bmatrix}, \quad {X^D}' = \begin{bmatrix}0\\1\\0\end{bmatrix} \]

and define an inner product weighted by genotype proportions:

\[ \langle X, Y \rangle = r\, X_0 Y_0 + h\, X_1 Y_1 + a\, X_2 Y_2 \]

Applying the Gram-Schmidt process to orthogonalize these vectors yields the non-additive encoding:

\[ X^D = \frac{1}{\sqrt{h \cdot a \cdot r \cdot \bigl(a + r - (a - r)^2\bigr)}} \begin{bmatrix} -ha\\ 2ar\\ -hr \end{bmatrix} \]

Under HWE (substituting \(r = p^2\), \(h = 2pq\), \(a = q^2\)), this recovers the standard HWE encoding above. In both cases, orthogonality holds: \(X^A \cdot X^D = 0\). The resulting values are linearly rescaled to the \([0, 2]\) dosage range for compatibility with standard VCF tools, which preserves orthogonality.

Encoding summary

Encoding Hom Ref (0/0) Het (0/1) Hom Alt (1/1) Tests for
Additive 0 1 2 Linear dose-response
Non-additive 0 d 0 Heterozygote deviation (orthogonalized)
Recessive 0 0 2 Recessive effects

The non-additive encoding captures the deviation of heterozygotes from the additive expectation. The value \(d\) depends on the allele frequency (or genotype proportions) and is computed by arcade automatically.

Two pipelines

Gene-level pipeline (interpret_phase + make_pseudo_vcf)

For identifying compound heterozygotes and homozygous carriers within genes, then generating gene-level pseudo-variant VCFs for set-based or burden-style GWAS.

A compound heterozygote is an individual who carries two different rare variants on separate haplotypes within the same gene. These can mimic homozygous loss-of-function by disrupting both copies of a gene, even when neither variant alone is common enough to produce homozygotes. With phasing information, compound heterozygotes (variants on different haplotypes) can be distinguished from in cis pairs (variants on the same haplotype).

VCF → bcftools query → interpret_phase → make_pseudo_vcf → GWAS
  1. Extract genotypes from a VCF using bcftools query
  2. Run interpret_phase to identify compound hets and homozygous carriers per gene
  3. Convert results to pseudo-variant VCFs using make_pseudo_vcf (additive + non-additive)
  4. Run GWAS jointly: Y ~ Additive + Non-additive

Variant-level pipeline (recode)

For recoding individual variant genotypes for non-additive testing, without gene-level collapsing.

VCF → recode → GWAS
  1. Take an existing VCF with standard genotypes
  2. Run recode to produce non-additive or recessive encodings
  3. Run GWAS jointly: Y ~ Additive + Non-additive

Tools

Tool Description
interpret_phase Identify compound heterozygous and homozygous variants from phased (or unphased) genotypes within gene regions
make_pseudo_vcf Convert interpret_phase output into pseudo-variant biallelic VCFs with additive, non-additive, or recessive dosage encodings
recode Orthogonalize or recode existing VCFs for non-additive or recessive genotype encodings

Downstream integration

The output VCFs from arcade integrate directly with standard GWAS software:

References

  1. Palmer DS, Zhou W, Abbott L, et al. Analysis of genetic dominance in the UK Biobank. Science 379(6639):1341-1348 (2023). doi:10.1126/science.abn8455 ↩↩↩

  2. Vitezica ZG, Varona L, Legarra A. On the Additive and Dominant Variance and Covariance of Individuals Within the Genomic Selection Scope. Genetics 195(4):1223-1230 (2013). doi:10.1534/genetics.113.155176 ↩↩

  3. Zhu Z, Bakshi A, Vinkhuyzen AAE, et al. Dominance Genetic Variation Contributes Little to the Missing Heritability for Human Complex Traits. Am J Hum Genet 96(3):377-385 (2015). doi:10.1016/j.ajhg.2015.01.001 ↩↩

  4. Pazokitoroudi A, Chiu AM, Burch KS, Pasaniuc B, Sankararaman S. Quantifying the contribution of dominance deviation effects to complex trait variation in biobank-scale data. Am J Hum Genet 108(5):799-808 (2021). doi:10.1016/j.ajhg.2021.03.018 ↩↩

  5. Hivert V, Sidorenko J, Rohart F, et al. Estimation of non-additive genetic variance in human complex traits from a large sample of unrelated individuals. Am J Hum Genet 108(5):786-798 (2021). doi:10.1016/j.ajhg.2021.02.014 ↩↩