The regression on phase (RoP) approach for testing cis and trans effects

For two biallelic loci with alleles {A, a} and {B, b}, there are ten distinct diplotypes, corresponding to four cis and four trans relations. Similar to the genotype of a biallelic locus, we code the phase relationships additively, referred to as phasetypes, which take values in {0, 1, 2} representing the number of a particular phase relationship carried by an individual’s diplotype, as shown in Table 1.

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Table 1. The additive phasetypes for 10 possible diplotypes of two biallelic loci.

G_A and G_B are additive genotypes. D_A and D_B are dominance terms.

https://doi.org/10.1371/journal.pgen.1011887.t001

We assume that either the cis relationship or the trans relationship of two loci contributes to the outcome. The RoP approach tests phase effects based on a generalized linear model:

The phase effects that are in-cis or in-trans are tested separately with two 1 degree of freedom (df) tests: and , respectively. G_A and G_B are the additive genotypes, and D_A and D_B are the dominance terms, which equal 1 for heterozygous genotypes and 0 otherwise (Table 1). These terms are included to adjust for the independent effects from each locus. P_cis and P_trans are phasetypes for cis and trans relationships. Throughout this study, we let P_cis = Cis_AB and P_trans = Trans_AB, i.e., the phasetypes coded based on the alternative alleles for the two loci.

The method is invariant to the reference allele selection for P_cis and P_trans; that is, the test statistics have the same value when P_cis equals Cis_AB, Cis_Ab, Cis_aB or Cis_ab, and when P_trans equals Trans_AB, Trans_Ab, Trans_aB or Trans_ab. This invariant property is due to the linear relationships between the additive genotypes and phasetypes:

and follows from the theorem proven by Chen et al for generalized linear models [22]:

Theorem 1. For two generalized linear model M₁: ⋅  +  ⋅ and M₂: ⋅  +  ⋅ that have the same link function g. is a matrix for p secondary covariates or confounders. and are n × q matrices for q primary covariates of interest. Then testing the association of primary covariates using Score, LRT or Wald tests is identical under M₁ () and M₂ () if 1) The primary covariates of the two models follows a linear relationship: , where and are and transformation matrices; 2) is invertible.

The additive phasetypes satisfy the conditions of the theorem. For example, Cis_ab and Cis_AB have the relation:

Thus, testing for the cis effect with Cis_AB and Cis_ab are equivalent. The same property can be derived between other cis relationship pairs and between trans relationships (S1 Text).

When the reference alleles are correctly specified for the phasetypes, the regression coefficients provide unbiased estimates of phase effects, which is the expected change in phenotype per additional count of the phase relationship carried by the homologous chromosomes. However, if the reference alleles are misspecified, the coefficients retain the correct magnitude but may have an opposite direction relative to the true phase effects. For example, when the true phase effect is driven by Cis_AB or Trans_AB, estimates for P_Cis = Cis_ab and P_trans = Trans_ab remain unbiased, whereas those for Cis_aB, Trans_aB, Cis_Ab, and Trans_Ab have the same magnitude but opposite in sign (S2 text). Since the true risk alleles are typically unknown, the RoP tests focus on detecting the presence of phase effects rather than determining the direction of the effects. Importantly, the predicted phenotype remains unchanged regardless of the choice of reference alleles (S2 Text).

The proposed method can be extended to accommodate multi-allelic variants. For multi-allelic variant A with m alleles {,,...,} and variant B with n alleles {,,...,}, the RoP model is formulated as:

where , , and are the additive and dominance genotypes for allele A_i and B_j, respectively, and and are the corresponding additive phasetypes. The cis and trans effects between the two variants can be tested by

each takes (m–1) (n–1) df. The genotypes and phasetypes for the reference alleles {, } can be omitted due to the linear dependencies between the additive genotypes and phasetypes. This formulation allows RoP to handle LD blocks by treating each distinct haplotype within a block as an allele in the multi-allelic model, which facilitates its application in genome-wide analysis.

Methods for epistasis analysis and haplotype regression

We compared the performance of our Regression on Phase (RoP) method against traditional epistasis tests, which focus on detecting interaction effects between two specific genetic variants while controlling for marginal effects. We focused on a set of benchmark approaches that are commonly used and align with the objectives of our study. Methods designed for exhaustive searches of epistasis signals across the entire genome or large genomic regions are not considered here [23,24].

The most widely employed strategy for testing epistasis effects between two variants is to include genotype interactions in a regression framework [13,14]. Using this approach, each variant’s genotype is modelled additively, and an interaction term tests for deviation from additivity:

where the presence of an epistasis effect is assessed through a 1 df test of .

Some extensions incorporate dominance terms:

where epistasis is evaluated using a 4 df test .

When phase information is available, Schaid [18] proposed a saturated model that includes an additional phase term:

the phase term when both loci are heterozygous and the alternative alleles are in-cis (i.e. ) and otherwise. Both epistasis and phase effects can be evaluated simultaneously using a 5 df test .

In addition to regression-based tests, other methods have been proposed that incorporate haplotype frequencies to improve power in case-control studies. Among these, we consider the haplotype odds ratio (OR) test [16] as a benchmark. This approach leverages phase information, albeit indirectly through haplotype frequencies, making it a relevant comparator for our RoP method, which directly models phase effects.

Specifically, the haplotype frequencies for cases and for controls are used to define the log-odds ratio:

The OR test tests for the same null hypothesis as the 1 df interaction test, , using the test statistic:

where

The resulting test statistic follows a distribution with 1 df under the null hypothesis. Notably, under the rare disease assumption, log(OR) approximates the genotype interaction effect [25], and as we show later, it also approximates cis effects .

When phase information is available, a natural alternative to RoP is haplotype regression, in which each haplotype of two biallelic variants is treated as an allele of a single multi-allelic marker and modelled additively in a generalized linear model. This approach is equivalent to jointly testing the effect of the four cis relationships without adjusting for the marginal effects:

and the effects of haplotype are evaluated by the 3 df test: . Cis_ab is used as the reference.

This approach has two key limitations for detecting coordinated phase effects. First, it is sensitive to the marginal effects, and a significant result cannot reveal whether the signal arises from independent main effects or from their coordinated phase (S1A Fig). Second, because only the four cis relationships are modelled, the method cannot identify which phase configuration drives the association, and it is underpowered in detecting trans effects (S1B and S1C Fig). Due to these limitations, haplotype regression was not included as a benchmark in our simulations.

Phase analysis at cystic fibrosis modifier loci

We obtained phased whole genome sequence from 564 individuals with CF from the Canadian CF Gene Modifier Study (CGMS) using 10XG linked-read technology, with the intent of investigating the complex variation at the trypsinogen and SLC6A14 CF modifier loci. Details for the sample processing and variant calling are described in [2]. For the association analysis of MI at the trypsinogen locus with phase effects between the 20 kb deletion polymorphism and the missense variant, we focused on the same n = 307 individuals as in [2]. At the SLC6A14 locus, we analyzed the phase effects of the MI-associated promoter variant and the lung disease-associated enhancer variant on gene expression in CF human nasal epithelial (HNE) cells (n = 79), on the age at the first Pseudomonas aeruginosa (PsA) infection (n = 41), and on lung function (n = 413) from individuals enrolled in the CGMS. Gene expression of SLC6A14 from naive HNE cells was obtained by RNA sequencing [26,27], and expression levels were measured using transcript per million counts (TPM). The RNA integrity number (RIN) was included in the regression model to account for variation in RNA quality. The age at the first PsA infection was defined as in [28] and [29]. The residuals after regressing age at infection on the current calendar age were used as the outcome for the association analysis [29]. Lung disease severity was measured by SakNorm as in the CF lung GWAS study [20], which was calculated by the quantile of averaged forced expiratory volume in 1 second (FEV1) measurements over a three-year period, adjusted for age, height, sex, and CF-specific survival rate [30]. The quantiles were generated using lung function reference equations based on the CGMS cohort [31].

Limitations of conditional analysis to distinguish allelic heterogeneity from phase effects

We evaluated the ability of conditional analysis to distinguish between allelic heterogeneity and phase effects through a simulation study. Haplotypes for two biallelic loci were generated under Hardy-Weinberg equilibrium, considering two LD scenarios (D’ = 0 and D’ = 0.8). The frequency of the primary variant () was fixed at 0.2, while the frequency of the secondary variant () was varied from 0.05 to 0.95. A continuous outcome was simulated where the loci had independent effects and where their effects were coordinated through their cis or trans relationships (Table 2).

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Table 2. Models for simulations.

Continuous outcomes are used to assess power in conditional analysis, and binary outcomes are used to compare T1E and the power of RoP with epistasis tests. For logistic models, corresponds to a baseline prevalence of P(Y = 1) = 0.12. I are indicator variables corresponding to dominant and recessive inheritance patterns. For T1E, we set . For power, for additive and dominant models and for the recessive cis effect.

https://doi.org/10.1371/journal.pgen.1011887.t002

Conditional analysis alone is insufficient to reliably distinguish between allelic heterogeneity and phase effects. The power to detect the secondary variant (SNP B) when conditioning on the primary variant (SNP A) is primarily influenced by the LD between the two variants, regardless of whether their effects were independent or coordinated through their phase (Fig 2). When the two variants were not in LD, conditioning on the primary SNP increased the power of detecting the secondary SNP. In contrast, when the two variants were in LD, the conditional analysis reduced the power of detecting the secondary variant, especially in the presence of phase effects. This indicates that conditional analysis may fail to identify variants that exert phase effects within a high LD region. The RoP method explicitly models cis and trans configurations while adjusting for marginal effects. As a result, it is sensitive to phase effects (Fig 2B, 2C, 2E, and 2F) but not to allelic heterogeneity (Fig 2A and 2D), enabling the method to distinguish these two mechanisms. Other non-conditional methods, such as epistasis tests, also adjust for marginal effects and detect the coordinated interactions. However, as demonstrated by simulation results in the following section, these methods cannot determine which phase configuration drives the association signal.

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Fig 2. The power of detecting the secondary SNP B with (conditional) and without (non-conditional) adjusting for the genotype of primary SNP A, compared with the power of RoP in detecting cis and trans effects.

From left to right: (A) (D) allelic heterogeneity, (B) (E) cis and (C) (F) trans effects; (A)-(C): the variants are not in LD (D’ = 0) and (D)-(F) in LD (D’ = 0.8). A reference line at T1E = 0.05 was added to illustrate that RoP maintains nominal type I error under allelic heterogeneity.

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RoP outperforms epistasis tests in detecting phase effects

To evaluate and compare methods suitable for detecting phase effects between two selected variants, we conducted simulations to assess T1E and power for the RoP method and traditional epistasis tests. Haplotypes for two biallelic loci were simulated as in the previous section. While the RoP method can be applied to quantitative outcomes, we simulated binary traits to compare its performance with the haplotype OR test. Case-control outcomes were generated by logistic models with 1,000 unrelated individuals per group, following the scenarios detailed in Table 2. To assess T1E, we examined six configurations in which independent effects originated from one or both loci under additive, dominant, and recessive inheritance patterns. For power analysis, we modelled additive and dominant phase effects for both cis and trans relationships, and recessive phase effects for cis relationships only. Both T1E and power were calculated over 1,000 iterations.

The 1 df genotype interaction test and the haplotype OR test exhibited inflated T1E in the presence of independent effects (Figs 3 and S2 and S1 Table). Specifically, the 1 df interaction test showed inflated T1E when the two loci were in LD and one of them carried either dominant or recessive marginal effects. The OR test maintained T1E control when only one locus had a marginal effect (S2 Fig), but inflation occurred if both loci had marginal effects, irrespective of their LD relationship (Fig 3). This is consistent with previous analytical results, which demonstrated that the OR test is only invariant to marginal effects contributed by a single locus [25]. In contrast, the RoP test, the saturated test, and the 4 df genotype interaction test maintained correct T1E across all scenarios, demonstrating their robustness in distinguishing true phase effects from independent contributions of individual variants.

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Fig 3. The type I error when independent effects occurred at both loci.

Additive (A)(D), dominant (B)(E) and recessive (C)(F) inheritance models were examined; (A)-(C): the variants are not in LD (D’ = 0) and (D)-(F): in LD (D’ = 0.8). Dashed lines correspond to a type I error of 0.05.

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Fig 4 presents the power to detect additive phase effects. The RoP tests outperformed other methods and effectively distinguished between cis and trans effects: when a cis relationship between two variants contributed to the outcome, the power to detect the trans effect was controlled at a level near 0.05, and vice versa. Similar results were also observed for dominant phase effects (S3A, S3C, S3D, and S3F Fig). However, the RoP tests showed reduced power in detecting recessive cis effects, whereas the 4 df genotype interaction test exhibited superior performance (S3B and S3E Fig). The haplotype OR test, which incorporates phase information indirectly, demonstrated identical power to the RoP test in detecting additive and dominant cis effects but was ineffective for detecting trans effects. Analytical findings (S3 Text) confirm that the log haplotype frequency OR approximates additive cis effects but equals zero under trans effects, explaining its lack of power in trans scenarios. The saturated test, which incorporates an additional phase term, outperforms the 4 df genotype interaction test in detecting additive and dominant phase effects but exhibits lower power compared to RoP. In fact, the additive cis and trans effects can be written as linear combinations of the four interaction terms plus the phase term (S4 Text), indicating that the saturated model does capture these effects, but the additional degree of freedom reduces its power relative to the RoP framework. The saturated test cannot distinguish the cis and trans mechanisms, as the direction of the phase term coefficient relies on both the selection of reference alleles for phasetypes and the contribution of cis or trans relationships (S4 Text).

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Fig 4. The power of detecting additive phase effects that are in cis (A)(C) and in trans (B)(D).

(A) and (B): the variants are not in LD (D’ = 0); (C) and (D): the variants are in LD (D’ = 0.8). Note, the OR test (blue) and 1 df genotype interaction test (grey dashed) have inflated T1E when there is a main effect, as in Fig 3.

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While the RoP method performed strongly for additive and dominant phase effects, detecting recessive cis effects remains challenging. This mirrors the findings from single-locus association studies, where the additive genotype coding often lacks power for recessive variants [32,33]. Zhou and Dizier [34,35] showed that the power can be improved by jointly testing for additive and dominance terms. To enhance the RoP approach under recessive inheritance, we introduced the dominance phase terms and , which equal 1 when P_cis and P_trans are 1, respectively. We then tested recessive cis effects using . Contrary to the expectations based on single-locus results, this adjustment reduces the power of recessive cis effect detection (Fig 5 Model 1). This is because the phenotypic variation attributed to the recessive cis effect can be effectively explained by the additive and dominance trans terms in the model (S4 Text). Removing trans terms can improve the power for detecting recessive cis effects (Fig 5 Model 2) but risk T1E inflation if the true effect is in-trans. The 4 df genotype interaction test demonstrated the best power to detect the recessive cis effect. This is because the recessive cis model can be written as the linear combination of the interactions between additive and dominance genotype terms, even though these terms did not incorporate phase information (S4 Text). The saturated test exhibited slightly lower power than the 4 df interaction test as a result of the extra 1 df for the phase term, which does not contribute to detecting recessive cis effects. Based on these observed tradeoffs, when the RoP method is not significant for both cis and trans effects, we suggest applying the 4 df genotype interaction test to assess the possibility of a recessive cis relationship with the phenotype.

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Fig 5. Power to detect recessive cis effects.

(A) D’ = 0; (B) D’ = 0.8. Model 1: ; Model 2: . Recessive cis effects were tested by .

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LD between variants can influence the power of RoP tests. Overall, LD reduces power for both cis and trans effect detection, with a more pronounced effect on trans effects when the two variants share similar allele frequencies. To further elucidate how LD influences the ability of the RoP method to detect phase effects at different allele frequencies, we performed detailed numerical analyses under a linear regression framework (Fig 6). We modelled a quantitative outcome with a sample size of n = 1,000, residual standard error σ = 2, and coefficient = 1 for additive phase effects. Power was evaluated at varying allele frequencies and compared between scenarios with no LD (D’= 0) and strong LD (D’ = 0.8). For cis effects (Fig 6A–6C), LD can either enhance or reduce power depending on allele frequencies. When both loci are rare, LD increases the power to detect cis effects. In contrast, when one allele is rare and the other common, LD introduces a stronger correlation between the cis term and the rare variant’s genotype, thus diminishing power. Under complete LD (D’ = 1), the cis term for a rare-common pair essentially became indistinguishable from the genotype of the rare variant, making it impossible to separate cis effects from marginal effects. The trans effect detection showed greater power reduction when allele frequencies were similar, regardless of whether the variants were rare or common, and it was less affected when allele frequencies were different (Figs 4B, 4D, and 6D–6F). This is because the three diplotypes (Ab/Ab, aB/aB, Ab/aB) became infrequent when two variants had similar frequencies and are in LD, inducing stronger multicollinearity between the trans term and the main effect terms in the remaining diplotypes. Given these complexities, especially in smaller samples or when the underlying mechanism of phase effects is unknown, focusing on variant pairs that are not in strong LD may be more efficient.

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Fig 6. The impact of LD on RoP.

(A) and (D): The ratio of the power when D’ = 0.8 over the power when D’ = 0 to detect cis and trans effects, respectively. Green areas correspond to scenarios where LD reduces power, and red areas correspond to improved power. (B) and (C): the power to detect cis effects when A allele is common ( = 0.25) and rare ( = 0.05); (E) and (F): same as (B) and (C) but for trans effects.

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Investigating phase effects at cystic fibrosis modifier loci: Allelic heterogeneity and tissue specificity

We used the 10XG linked-read data from the CGMS cohort to investigate whether GWAS-identified variants at the trypsinogen and SLC6A14 loci contribute to CF disease severity independently or through coordinated phase effects.

At the trypsinogen locus, the missense variant rs62473563 and the 20 kb common deletion, which reside in different LD blocks, are associated with MI and serve as eQTLs for PRSS2 expression in the pancreas. Conditional analysis suggested that accounting for one variant reduced the p-value for the other’s association with MI, leading to the conclusion that the two variants contribute to MI risk independently [2]. Our simulation study showed that the changes in p-value in the conditional analysis are strongly influenced by LD and cannot disentangle allelic heterogeneity and phase effects. Instead, we evaluated phase effects between the deletion and the missense variant at the trypsinogen locus by the RoP method. The cis configuration involving the deletion and rs62473563 T allele did not occur in our dataset, and the remaining cis terms were linearly dependent on genotypes. Consequently, we focused on trans effects using the RoP method, and complementary analyses using the haplotype odds ratio and the 4 df genotype interaction tests for cis effects. None of these tests revealed significant associations with MI risk ( = 0.95, = 0.83, = 0.95), supporting the original conclusion that these two variants independently influence MI risk rather than acting through a phase-dependent mechanism.

The SLC6A14 locus displayed tissue-specific associations in previous CF GWASs (Fig 1B). A cluster of variants located in an intergenic region approximately 200 kb upstream of the SLC6A14 transcription start site (TSS), annotated closer to AGTR2 than SLC6A14, were GWAS-significantly associated with CF lung function. One of the associated variants in the cluster, rs4446858, disrupts an IRF1/IRF2 binding site [36] annotated within a region with a distal enhancer-like signature (E2765449/enhD) [37–39], indicating possible immune-related regulatory mechanisms. The top MI-associated SNP, rs3788766, which resides ∼2.4 kb upstream of SLC6A14 TSS, showed no individual effect on lung disease (Figure 3 in [19]). To evaluate the potential role of the rs3788766 SLC6A14 promoter variant in CF lung disease, we analyzed the phase effects of rs3788766 and rs4446858 on CF lung disease severity (SakNorm), the age at first PsA infection and the expression level of SLC6A14 in CF HNE (Table 3). Analyses were stratified by sex to address the impact of X chromosome inactivation [40,41]. Replicating GWAS findings with our 10XG data, we confirmed that rs4446858 alone, but not rs3788766, was associated with SLC6A14 expression and CF lung function in females. Neither variant alone was significantly associated with the age at first PsA infection at the 0.05 level. However, when analyzing their phase relationship, we found that their cis configuration significantly affected SLC6A14 expression and the age at first PsA infection in both males and females, whereas the trans configuration showed no significant effect.

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Table 3. The association p-values for the SLC6A14 locus with lung phenotypes.

The trans effects were tested only in females. The age at 1st PsA infection phenotype was measured by the residual of the age at the first infection regressed on the calendar age [29]. Lung disease severity was measured by SakNorm [20,30].

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