The Causal Pivot: A structural approach to genetic heterogeneity and variant discovery in complex diseases

Summary

We present the Causal Pivot (CP) as a structural causal model (SCM) for analyzing genetic heterogeneity in complex diseases. The CP leverages an established causal factor or factors to detect the contribution of additional suspected causes. Specifically, polygenic risk scores (PRSs) serve as known causes, while rare variants (RVs) or RV ensembles are evaluated as candidate causes. The CP incorporates outcome-induced association by conditioning on disease status. We derive a conditional maximum-likelihood procedure for binary and quantitative traits and develop the Causal Pivot likelihood ratio test (CP-LRT) to detect causal signals. Through simulations, we demonstrate the CP-LRT’s robust power and superior error control compared to alternatives. We apply the CP-LRT to UK Biobank (UKB) data, analyzing three exemplar diseases: hypercholesterolemia (HC, low-density lipoprotein cholesterol ≥4.9 mmol/L; n_c = 24,656), breast cancer (BC, ICD-10 C50; n_c = 12,479), and Parkinson disease (PD, ICD-10 G20; n_c = 2,940). For PRS, we utilize UKB-derived values, and for RVs, we analyze ClinVar pathogenic/likely pathogenic variants and loss-of-function mutations in disease-relevant genes: LDLR for HC, BRCA1 for BC, and GBA1 for PD. Significant CP-LRT signals were detected for all three diseases. Cross-disease and synonymous variant analyses serve as controls. We further develop ancestry adjustment using matching and inverse probability weighting as well as regression and doubly robust methods; we extend this to examine oligogenic burden in the lysosomal storage pathway in PD. The CP reveals an approach to address heterogeneity and is an extensible method for inference and discovery in complex disease genetics.

We present the Causal Pivot as a method to test rare variant contributions conditional on polygenic risk or other known factors. In application to UK Biobank data, we detect causal signals in hypercholesterolemia, breast cancer, and Parkinson disease, offering a generalizable approach to dissect heterogeneity in complex traits.

Introduction

Causal heterogeneity, where root causes of disease vary among affected individuals, is a signature characteristic of complex disease. Affected individuals may reach the disease phenotype through distinct etiologies that may be influenced by alternative genetic factors. Although the causes of disease vary, they ultimately lead to shared pathophysiology or disease state. Moreover, complex disease genetic architectures are known to vary from monogenic to highly polygenic.^1,2 Oligogenic inheritance, environmental phenocopies, and gene × environment interactions are also thought to contribute to complex disease etiologies. This causal heterogeneity impacts both prevention and management because distinct causes may benefit from different approaches for effective intervention or risk mitigation.³

Genome-wide association studies (GWASs) in complex disease are well developed for both rare and common DNA variants,^4,5 but integration across the allele frequency spectrum remains challenging. Moreover, non-familial association approaches are typically case-control designs and may not be applicable to all settings.^6,7 Although GWASs have tools for both rare and common variants, almost all studies are marginal analyses that treat each genetic locus as a separate testing unit; results are summarized by Manhattan plots. The marginal approach avoids confronting heterogeneity with respect to alternative genetic factors differentially present or absent in individuals. Consequently, these methods do not classify individuals into causal groups or subtypes. Although it might be desirable, current GWAS methods do not resolve mechanistic heterogeneity driven by genotype among individual cases or distinguish different subpopulations by mechanistic driver.⁸

Genetic variants—while exhibiting allelic heterogeneity, pleiotropy and potential confounding with ancestry and environment—are explicitly considered to be causal factors that drive downstream disease-relevant processes. Consequently, methods developed for causal inference and structural causal modeling (SCM) in fields such as epidemiology and econometrics^9,10 have been adapted to problems in genetics. Perhaps the most successful example is Mendelian randomization (MR),^11,12,13,14 which uses genetic variants as instrumental variables to evaluate potential causal biomarkers and endophenotypes for common diseases. We reasoned causal analysis methods could have broader applications in genetics^{15,16,17,18,19} and could offer a potential approach to reframe inquiry into heterogeneity in complex disease genetics.

We hypothesized that SCM could serve as a framework to consider multiple genetic factors differentially present or absent in individuals to examine the underlying mechanistic heterogeneity of complex disease. We adopt the strategy to model common variant and rare genetic contributions as separate causal contributions where disease outcome is their common effect. We encapsulate the common variant contribution to disease in a polygenic risk score (PRS). The graphical structure presents a collider pattern between PRS and rare variants (RVs) (Figures 1A and 1E). The general concept that low PRS can be used to prioritize affected individuals for RV sequencing or to classify them as having monogenic disorders has been noted.^20,21 More generally, the use of colliders or v-structures to infer causal graphs is well known.^22,23,24 However, more detailed statistical analysis is still needed to elevate and generalize the procedure, and recognition of the overarching paradigm for PRS-RV causal opposition with respect to complex traits has been limited. Importantly, prior methods investigating the relationship have been non-parametric. These non-parametric methods do not differentiate main effects and interactions, and they lack interval estimation to quantify uncertainty in results. Moreover, the lack of statistical formalism has hampered clarity on the method and extensions to address study design as well as confounding by ancestry and alternative genetic models.

Figure 1.

Core logic of the Causal Pivot

(A) In the base model, polygenic risk score (PRS) is a known causal factor (solid line and arrow) terminating at Y; rare variant (RV) is an independent factor whose effect on Y is unknown (dashed line) and being evaluated.

(B) Causal impact of the PRS on Y. PRS is correlated with Y, manifesting as a higher PRS in cases where Y = 1 than in controls where Y = 0.

(C) Model of unconditional independence between PRS and RV. Unconditional independence results in a lack of association between PRS and RV; there is no correlation in the frequency of the RV⁺ with the PRS tertiles shown in (D).

(E) RV influences phenotype Y. The solid arrow indicates a causal effect of RV on Y. The red box around Y indicates conditioning on the phenotypic state, such as selection of the Y = 1 subpopulation, leading to an induced correlation between PRS and RV. This collider-induced correlation (dashed red line) connects PRS and RV.

(F) The frequency change in RV⁺ when RV is a cause of Y.

(G) The collider-induced correlation between PRS and RV⁺ conditional on Y. This correlation manifests as a Y-conditional shift in PRS between RV⁺ and RV⁻.

(H) Alternatively, the rate of the RV⁺ state covaries with PRS tertiles conditional on Y. This collider-induced correlation between PRS and RV juxtaposes with the unconditional independence of the PRS and RV. The Causal Pivot recruits the collider-induced correlation together with the rate change in the RV as the signal to infer the RV-Y relationship.

We develop a formal statistical approach to genetic heterogeneity, which we call the Causal Pivot (CP). The method can be applied in a cases-only, controls-only, or a case-control design; here we focus on the cases-only application. The CP is based on a structural model of disease (Figure 1) as well as the well-known collider bias concept^25,26,27 in which conditioning on an outcome induces a synthetic correlation (Figure 1E) between otherwise independent causes (Figures 1A–1D and 1F). Under these conditions, if we observe individuals’ disease status and condition on their outcome—either case or control —then, within those outcome groups, the independent causal factors will have induced correlation in well-defined patterns (Figures 1G and 1H). Importantly, model factors will show no induced correlation if the factors are unconditionally independent (Figure 1D) and not mutually causal. We view outcome-induced association between causal factors as a feature to resolve heterogeneity and to drive discovery instead of as a source of bias in the research process.^28,29,30

The CP is a highly general statistical framework built using the tools of SCM and Bayes’ rule. This approach leads naturally to a likelihood framework, maximum-likelihood estimation, confidence intervals, and a CP likelihood ratio test (CP-LRT). The same mathematical structure applies in the context of both binary and quantitative traits. We perform power analyses and compare the CP-LRT to alternatives.^20,21 Importantly, the CP-LRT derives power from both the rate change in RV given the disease as well as the conditionally induced dependency between the causes; alternative methods may fail to recruit both sources of information. This insight leads to demonstration of the robust properties of the CP-LRT in terms of sensitivity and specificity.

Using human data from UK Biobank (UKB), we demonstrate the performance of CP-LRT in demonstration examples of breast cancer (BC), hypercholesterolemia (HC), and Parkinson disease (PD); we focus on the respective genes known to bear disease-causing variants BRCA1, LDLR, and GBA1. These examples are chosen to establish the CP procedure by use of well-known gene-disease pairs to validate and build confidence in the procedure. This ensures that under realistic conditions CP can detect independent causes. We then extend CP to address confounded structures, focusing on ancestry. We successfully demonstrate several alternative approaches to address ancestry confounding using matching, propensity adjusted regression, inverse probability weighting, and doubly robust regression. Finally, we develop a pathway burden test to demonstrate the utility of cases-only CP to investigate oligogenic models for pathway contribution to complex disease using lysosomal storage in PD. Our results demonstrate the generality and potential of CP as an approach to causal discovery and its ability to address heterogeneity in complex disease genetics.

Methods

Causal models and graphs

Structural modeling is also called causal analysis or SCM. Our methods are developed through SCM. In the structural framework directed acyclic graphs are used to express model assumptions where directed arrows depict stochastic dependencies and the lack of a directed path between nodes indicates unconditional independence.

We consider simple models with three or four nodes and two or four edges, respectively (Figures 1A, 1E, and 4A–4C); we then extend the approach to include additional covariates such as age (Figures S9A, S9B, S10, and S12). The disease outcome variable is denoted Y. This outcome represents the phenotypic status, for example as recorded in the UKB metadata by ICD-10 codes or biomarker measurement information. This outcome variable can be binary or continuous. Initially we focus on two genetic exposures: a PRS—a continuous variable—which we will denote as X, representing the combined causal contribution of common variation (minor allele frequency >0.01), and a binary RV state variable denoted as G that represents the presence of a strongly acting RV. This binary RV represents the state that an individual is positive for any of an ensemble of strongly acting RV that in themselves may be vanishingly rare or singleton but in aggregate can have frequency approaching 0.001 or larger in a population or study sample. These X and G are depicted as nodes with edges incident to the Y outcome (Figures 1A and 1D). Later we extend this three-node model to a four-node graph (Figures 4A–4C) where the PRS and RV variables are coordinately influenced by ancestry denoted as A; finally, we show that additional covariates such as age or sex can be accounted for by simple extension.

Figure 4.

Controlling for ancestry

Ancestry is a potential common cause—a source of confounding bias—not only between genetic variants and phenotype but also between alternative genetic causes.

(A) Model scenario where ancestry represented by node A drives both PRS and RV state.

(B) Model in which rare variant (RV) is not a cause of Y. Conditioning on both A and Y, there is no induced or residual correlation between polygenic risk score (PRS) and RV.

(C) RV drives Y, and conditioning on both A and Y yields an induced correlation between PRS and RV.

(D) Each row represents an example phenotype: hypercholesterolemia (HC), breast cancer (BC), and Parkinson disease (PD). The scatterplots show rare variant negative (RV⁻) as black points and RV⁺ (red points) cases in ancestry principal component (PC) space. The scatterplot representation is determined by the first five PCs reduced to two dimensions using t-distributed stochastic neighbor embedding (t-SNE). A zoom-in view of each row is indicated by the excerpted rectangular region in blue. Each RV⁺ case is matched to the nearest (k = 2) RV⁻ cases in the 5-dimensional ancestry space. The mean difference in the PRS of each RV⁺ and its ancestry-matched neighbors and the average of these PRS differences was calculated (dashed vertical red line). Permutation distributions (plotted in the histograms) were generated by randomly permuting the RV states of each sample in the cohort while fixing the ancestry data; the procedure of identifying neighbors and calculating the mean difference in PRS was repeated. The lower 5% threshold value is shown (dashed black lines). The observed value from the true RV⁺ ancestry-matched data is significantly below the 0.05 quantile in all diseases, consistent with the pattern that RV⁺ cases have lower PRS than their RV⁻ ancestry-matched neighbors. Additional analyses using propensity methods are presented in Figure S9 and Table S3.

Causal Pivot

When two causes are incident to the same effect, that effect is called a collider for the two causes. Conditioning on the collider outcome induces an observational—but not causal—correlation between the distinct causes. This situation is a well-known source of bias in epidemiologic studies because researchers may unknowingly condition or select on the outcome and report the induced association between variables X and G as causal between these variables.

The CP exploits collider-induced correlation as a source of signal rather than noise. As we show, when one causal variable incident to the outcome in a collider structure has a known effect on the outcome—such as the effect of the PRS on Y (Figure 1B), then the outcome-conditionally-induced RV-PRS correlation (Figures 1G and 1H) can be used to improve estimation and to test the Y relation to the candidate cause, in this case Y-RV relationship. The null hypothesis is an absence of the Y-RV relationship, and in this situation no induced correlation will appear between X and G conditional on Y. As we show in supplemental methods, the CP is generally applicable and not limited to genetics.

Conditional analysis

Here, we present an overview of the mathematical derivation of the CP approach; more details are in the supplemental methods. In what follows, we refer to the variables PRS as X, RV as G, and disease outcome as Y. The CP derives from manipulation of the probabilistic factorization determined by the graphical model (Figures 1A and 1E). The joint density under the model is

$$f_{\mathrm{\theta }}\left(y,x,g\right)=f_{{\mathrm{\theta }}_R}\left(y\mid x,g\right)f_{{\mathrm{\theta }}_X}\left(x\right)f_{{\mathrm{\theta }}_G}\left(g\right)\text{.}$$ (Equation 1)

In the expression, $f_{\theta }(·)$ represents the respective probability density functions for their arguments, and the bar symbol expresses conditioning on variables to the right-hand side of the bar. The parameter vector θ represents the unknown parameters governing the probabilistic structure of the system; for clarity, we emphasize that the parameter θ comprises the parameters θ_R that determine the forward conditional relationship between Y and X, G—also known as the outcome model; the separate parameters θ_XG determine the distributions of X and G, which in the simple model are exogenous and independent and can be represented separately as θ_X and θ_G.

The CP procedure follows from application of Bayes’ rule to the probabilistic factorization implied by the graphical model. In notation, application of Bayes’ rule and independence of X and G Equation 1 leads to

$$f_{\mathrm{\theta }}\left(g\mid x,y\right)=\frac{f_{{\mathrm{\theta }}_R}\left(y\mid x,g\right)}{E_G\left(f_{{\mathrm{\theta }}_R}\left(y\mid x,G\right)\right)}{f}_{{\mathrm{\theta }}_G}\left(g\right)\text{.}$$ (Equation 2)

For G representing RV status, this expression states that the conditional probability for a sampled individual to bear an RV—denoted by G = 1—or to not have an RV—denoted G = 0—when X, Y are conditioned upon is given by the ratio of the forward conditional probability of their phenotype Y = y given PRS X = x and G = g divided by the average probability of Y integrating out G multiplied against the prior probability of G = g. This factorization is general and does not depend on the model specification of f(y |x,g) or f(x) or f(g); moreover, the factorizations in Equations 1 and 2 do not depend on a univariate restriction on X, G, or Y, nor does it depend on the binary character of G or the distributional character of X. The form holds under the model in Figure 1.

Model specification

It is useful to provide a parametric model to perform analyses, make calculations, enable parametric testing, and perform power comparisons. Non-parametric approaches also apply, but parametric modeling provides both mathematical insight and yields practical results such as confidence intervals. We adopt a generalized linear model framework for the forward regression model of Y given X and G. The general linear model framework encompasses both logistic regression in the case of binary outcomes as well as linear regression for quantitative traits. The general expression for the forward conditional expectation of Y given X and G is

$$h\left(E\left(Y\mid X,G\right)\right)=\alpha +\beta ·X+\gamma ·G+\eta ·X·G$$ (Equation 3)

In Equation 3, we do not specify values for the random variables X, G, and Y; instead we focus on the structure of the conditional expectation of Y as a function of X and G. Examples for the function h( ) are the logit or log-odds for binary outcomes, such as for a binary trait like BC; the identity function can be used for continuous Y, such as in the case of low-density lipoprotein (LDL) measurements. We assume the parameters α and β are known or estimable from population data; we also assume knowledge of a baseline frequency of RV, the value ω. In many cases the estimate of the X-Y relationship is reported. If not, the easiest way to estimate it is to apply a forward model to unconditional Y including X and exclude G. Although the α parameter may depend on G, we argue that it is negligible because G is rare. We adopt this approach to determine α and β in our demonstration work. The frequency of rare variation (ω) is estimable from population databases such as gnomAD³¹ or from UKB summary information.

Likelihood analysis conditional on Y

To examine the properties of the CP, we construct procedures for the possible effect of RV using the factorization in Equation 2 to determine a likelihood function for observed data conditional on Y = a. We carried out this analysis for both binary Y and for Y > δ for continuous Y. In the case of discrete Y, we have

$$L\left(\gamma ,\eta |\mathit{X},\mathit{G},\mathit{Y}=\mathit{a}\right)=\prod _{i=1}^n{f}_{\theta }\left(g_i|x_i,y_i=a\right)\text{.}$$ (Equation 4)

Taking logarithms and emphasizing that the unknown elements of θ that concern the impact of the RV on outcome are γ, η:

$$l\left(\gamma ,\eta |\mathit{X}\mathbf{,}\mathit{G},\mathit{Y}=\mathit{a}\right)=\sum _{i=1}^{n}l\left(\gamma ,\eta |x_i,g_i,y_i=a\right)\text{.}$$ (Equation 5)

As shown in supplemental methods, this likelihood function can be maximized in the variables γ and η to produce maximum-likelihood estimates (MLEs). This analysis is performed by differentiating the log likelihood with respect to γ and η and solving for the root where the derivative is zero; a check can be performed to ensure the second derivative is negative at the root so that the point is a maximizer. Confidence intervals may be obtained using the observed estimate of the Fisher information using the empirical means of the squared derivatives of the log likelihood (see supplemental methods). We performed this analysis for both the logistic and the liability model for a continuous trait. For the continuous trait we consider the circumstance where the observable sample comprises individuals with a sufficiently extreme quantitative trait Y > δ. Individuals in this class are deemed “cases,” and individuals not in the class are deemed “controls”; we suppose we have access to the trait values.

Likelihood ratio test

As mentioned, the likelihood approach determines MLEs by solving for the roots of the derivatives of log likelihood equations treating these as functions of the unknown parameters. A likelihood ratio test can be constructed by plugging the MLE into the log likelihood function to compute a test statistic. The null hypothesis is that the parameters γ and η are 0. If the MLEs for γ, and η are substituted into the likelihood function, then under the null hypothesis the asymptotic distribution of: −2 ^∗ l(0,0 |X,Y,G) − l(γ, η|X,Y,G) converges to a chi-square on 2° of freedom. The procedure considers the influence of the RV, both individually and in its potential interaction with the PRS. Additional details are provided in supplemental methods.

Logistic model

The logistic model considers the disease outcome to be binary where the log odds of disease is linear in the explanatory variable. We additionally treat RV as a binary variable. In the initial treatment there is no additional stochasticity in the conditional distribution of the outcome Y given X and G except for the coin-flipping variability of a Bernoulli trial; such extensions using random effects and the frailty model are possible but beyond the scope of this paper. The log-likelihood analysis conditional on Y = a produces four different sums: the sum of contributions for Y = 0 where RV = 1, Y = 0 where RV = 0, Y = 1 where RV = 1, and Y = 1 where RV = 0. Breaking the likelihood calculation into these four parts is both computationally practical and conceptually useful. In cases-only analysis, there is no contribution where Y = 0. The details of the log-likelihood analysis and derivations appear in the supplemental methods.

Liability model

The CP liability model considers conditioning on the outcome Y > δ for a quantitative trait that has a linear outcome model as in Equation 3 where h() is the identity function and where outcomes have additional mean zero Gaussian noise. The approach generalizes to non-Gaussian noise, but the Gaussian situation is both canonical and illustrative. Here the model considers the outcome variable Y | Y > δ, x, g. Introducing an indicator variable for the conditional state, the forward model has a conditional density when Y > δ:

$$f\left(y|I_{y>\delta },x,g\right)=\{\begin{array}{c}\frac{f\left(y|x,g\right)}{1-F\left(\delta |x,g\right)}\\ 0\text{otherwise}\end{array}y>\delta \text{.}$$ (Equation 6)

For situations where the outcome value is less than δ, we have

$$f\left(y|I_{y>\delta }=0,x,g\right)=\{\begin{array}{c}\frac{f\left(y|x,g\right)}{F\left(\delta |x,g\right)}\\ 0\text{otherwise}\end{array}y<\delta \text{.}$$ (Equation 7)

To perform analysis with this liability model, we use the general expression in Equation 2 that becomes Equation 5 after applying the log likelihood transformation. As with the logistic model, there are four configurations for binary RV: where the outcome Y < δ and G = 1, where Y < δ and G = 0, where the Y > δ and G = 1, and where Y > δ and G = 0.

Power analysis

To perform analyses, we built a forward simulation procedure to generate outcome data under both the logistic and liability frameworks. For each scenario, we simulated RV as a binary variable with frequency 0.001. For PRS, we generated unit normal outcomes. We fixed β according to values consistent with real data. We let γ and η range across many possible values. For logistic regression, we set α = −2.2 and β = 0.5, and we set γ and η to a collection of alternative values. We generated Y outcomes according to the inverse transformed log-odds. We then applied the likelihood expression in Equation 5 and solved for the root of the derivatives of the log likelihood by numerical procedure to obtain the MLE. We then computed the likelihood ratio test statistic. We performed 1,000 simulations at each model parameterization and recorded the outcomes. To determine power, we calculated the proportion of simulation runs where the observed p value was less than 0.05. For the liability model we generated outcome variables in a parallel fashion, first generating X ∼ N(0,1) and G ∼ ber(0.001) and fixing the parameters α, β and a grid of alternative RV effect sizes. We set δ to be the 92%-ile of the trait distribution.

Cases-only analysis vs. case-control conditional analysis

Our CP procedure can be applied to cases-only data, to controls-only data, or to case-control data taken together. In all these situations, data analysis proceeds according to a conditional analysis using the fundamental factorization in Equation 2 and to application of the maximum-likelihood procedure in Equation 5 followed by a likelihood ratio test. We focused on comparison of cases-only CP analysis against the traditional forward regression approach and alternative cases-only analyses.

Comparator tests and robustness analyses

The CP-LRT benefits from the full information in the stochastic system as described by the structural causal graph and as encapsulated in the conditional likelihood function. This conditional likelihood gives the expected rate change in G given both X and Y. We consider two comparator non-parametric models to our CP-LRT. First, we consider a Z-test procedure where we assume the frequency of RV status in the source population—deemed ω—is exactly known. Under the null hypothesis where RV status has no effect on outcome, the expected number of RV in the outcome group Y = a would be binomial with expectation N_Y=aω and variance N_Y=aω(1 − ω). We take Z to be the difference between the observed RV⁺ disease individuals (N_Y=a,RV+) and the expected under H₀ standardizing by square root of variance. This comparator focuses on the rate change of a functional variant state in the disease population and ignores the PRS. Alternatively, prior work²⁰ proposed a Wilcoxon rank-sum test (also known as Mann-Whitney U test) on the PRS values among RV⁺ and RV⁻ individuals conditional on Y = a. The Wilcoxon is a non-parametric test for a shift in means between two populations, in this case RV⁺ and RV⁻ conditional on Y = a or Y > δ. In contrast to the alternatives, the Wilcoxon test does not utilize prior knowledge of RV frequency information.

Robustness analyses

Using simulation, we compared the type 1 error of the CP-LRT and Z test when G has no effect and scrutinized the impact of underspecification of the rate ω. The type 1 error is the rate at which the test falsely identifies an effect of G when there is none. These analyses were performed across a range of underspecification of the RV rate, which we deem ω^∗; the true simulation parameter values in these analyses are α = −2.2, β = 0.6, ω = 0.001, γ = 0, and η = 0.

We then consider the power (1 − type 2 error) impact of overspecifying the RV rate. We examine this situation in the context where α = −2.2, β = 0.6, ω = 0.001, γ = 2.2, and η = 0. In this case there is an RV effect, but oversetting the value of ω^∗ makes it more difficult for the statistical procedures to detect. We examine a range of values ω^∗ up to and beyond the expected conditional frequency of RV.

Impact of PRS effect size

We examined the impact of the known causal factor’s effect strength on CP performance. We focused on the binary trait outcome context, and we evaluated the impact of PRS effect by letting the β parameter vary. We considered both the impact on the conditional mean of PRS given RV and trait outcome Y as well as the conditional mean of the RV state, which equates to the probability of RV⁺. The conditional mean is derived by taking the expectation with respect to the conditional density as determined by the factorization in Equation 2. We consider the odds ratio (OR) for RV⁺ conditional on Y and set the PRS value X = −1 and X = 1. For the conditional mean of PRS given RV and Y, we note there are four conditional mean functions to consider conditioning on states of trait Y and RV denoted G: Y = 1, G = 1; Y = 1, G = 0; Y = 0, G = 1; Y = 0, G = 0. We also evaluated the impact of PRS effect size on statistical power. To emphasize the role of induced correlation we misspecified the RV rate.

UKB data

We analyzed data drawn from the UKB under an approved application (Project 98786) titled “Genetic Heterogeneity in Diseases with Complex Inheritance.” We used HC (LDL direct | instance 0 ≥ 4.9 mmol/L; data field 30780), BC (ICD-10 C50; data field 40006), and PD (ICD-10 G20; data field 131022) as disease models with expected positive findings. Cases were filtered based on inferred European ancestry, availability of exome and genotyping data, female-only for breast cancer, and availability of a phenotype-specific polygenic risk score. The final samples included HC (cases 24,656, controls 347,889); BC (cases 12,479, controls 198,311); and PD (cases 2,949, controls 388,198). As this study exclusively utilized de-identified data from the UKB, it qualifies for exemption from ethical review under the guidelines for research involving non-identifiable human data. Cases where an individual had more than one of these disease phenotypes were excluded.

PRS

We used PRS made available through the UKB (field ID 26220 standard PRS for BC; 26250 standard PRS for LDL cholesterol; and 26260 standard PRS for PD).³² Samples without PRS values were removed, and the values were shifted and scaled to have mean 0 and standard deviation 1 (Figure S4).

Exome variant extraction and classification

We used PLINK2 to read genotypes from the UKB exome BGEN files (data field 23159) and extract variants within our target gene lists (see Data S1) of the disease-defined cohort samples. The extracted genotype files were converted to Matrix Market format using a custom Python script. Variants were then annotated with OpenCRAVAT; annotators included those pulled from the OpenCRAVAT respository (gnomAD v3.0, REVEL, and CADD exome) as well as custom annotators (ClinVar, Alpha Missense, and ESM1b) produced from publicly available data. We excluded variants that were absent (homozygous-reference) across all samples.

We focused on three example genes for the demonstration diseases: BRCA1 for BC, LDLR for HC, and GBA1 for PD. We used the OpenCRAVAT annotations to classify rare deleterious and/or pathogenic variants within these three genes. Pathogenic variants are ClinVar “pathogenic” or “likely pathogenic” without conflicting or qualifying annotations. Additionally, we include rare stop-gain or frameshift variants that are not found in ClinVar and have a gnomAD v3 non-Finnish European allele frequency of less than 0.1%. We evaluate as RV⁺ as those individuals with at least one alternate allele for any rare loss-of-function (LoF) or ClinVar pathogenic variant within each gene (Figure S5). To ensure independence between our two causal factors (RV status and PRS), we exclude all variants for each gene that associate with the respective PRS for that gene’s demonstration disease as determined by logistic regression with a p value of less than 0.05. We extracted relevant cohort metadata (age, sex, disease indicators, LDL, and ancestry components) from the UKB cohort browser.

Controlling for covariates

We consider potential confounding between PRS and RV arising from shared ancestry, denoted A (Figures 4A–4C), and additional factors (Figures S11A–S11C). We examine several approaches to address the influence of covariates including matching, inverse probability weighting, and propensity informed regression adjustment (Table S3). We focus on the conditional outcome where Y = 1, cases-only analysis. We expect that if RV increases the risk for Y = 1 then a negative association will appear between RV and PRS such that affected individuals (Y = 1) with RV⁺ will have an average lower PRS than affected individuals with RV⁻ after controlling for additional factors. To perform matching, samples in each of the three cohort groups were differentiated based on positive disease (case) status. We determine a distance matrix between all samples using the “distances” package in R. In the absence of eigenvalues for the available ancestry principal components (PCs) in UKB, we employ an exponentially decreasing weighting function (scaled to unity) (decay = e^r∗5, weights = decay/sum(decay), r = 0.637) to weight-squared differences between observations for each ancestry PC in descending order, summing these to determine a distance value. RV⁺ cases were matched to the closest (k = 2) RV⁻ cases using these calculated distances. For each RV⁺ case, the difference between the observed PRS value and the mean PRS value for the matched RV⁻ cases is calculated. An overall test statistic is then calculated by taking the average of these values over all RV⁺ cases. Permutation testing is performed using the RV⁺ status as the permuted variable to determine a p value. A one-tailed permutation test of means is then performed using the observed test statistic against the permutation distribution.

We also perform analyses using propensity methods. In brief, propensity methods are a class of approaches to adjust for unequal representation of the causal exposure that correlates with observable additional factors. In this situation the outcome variable is taken to be the PRS among the affected individuals (Y = 1), and the exposure is the binary RV state; one common cause is ancestry, and we also incorporate additional factors such as age and sex. We use the random forest method to perform a machine-learning predictive analysis for RV⁺ using the ancestry PCs as well as age and sex in each disease cohort. The outcome of random forest is a probability of RV⁺ for each observation. We then compute the inverse probability weighted sum of PRS among RV⁺ and subtract the sum of propensity-weighted values among the RV⁻ cases, using 1 minus the probability of RV⁺ as the weights of the RV⁻ cases. We use permutation of assignment of the RV status to determine a null distribution and a one-sided p value. We also perform analyses using propensity predictions as an additional regressor in linear models for PRS, as a weight for PRS regression, and by doubly robust regression where we include the propensity as a weight and also as a regressor (Table S3).

Test of RV load in individuals

We tabulate a load of variants in a biological pathway as an alternative count-valued candidate cause to test using CP. For this oligogenic load analysis, we focus on the lysosomal storage pathway in PD. We enumerate qualifying variants (see method above) in 54 lysosomal storage genes that were also present in UKB exomes. We count the number of variants in each individual. To examine the potential induced association of this count variable with PRS conditional on disease, we use Poisson regression with a log link to consider the average RV count as a rate that depends on PRS. In brief, in Poisson regression the logarithm of the rate of RV is modeled as a linear function of the PRS. We use the likelihood ratio test for Poisson regression as implemented in the R glm() method to determine p values.

Results

Comparative power analyses

We hypothesized that our CP-LRT could have promising statistical properties to discover RV causal candidates by exploiting a PRS as a known cause. The model assumes the known factor PRS is previously established and has a defined effect size. In our power analyses, the candidate cause is a binary variable representing the presence of at least one qualifying RV in an individual. This variable is assumed to have a known sparse rate of occurrence in the general population, and we used the value 0.001. Simulation-based power analyses are presented in Figure 2A in the context of a binary outcome trait considering the possibility of a main effect of RV as well as a PRS-RV interaction. In all simulations the forward logistic regression using the full data of both cases and controls had the highest power; this test is using approximately an order of magnitude more subjects than cases-only analyses because n = 500,000 for case-control vs. n_c of approximately 50,000. Importantly, using only cases—approximately 10% of the population—CP-LRT achieves 90% of the power of the case-control analysis for the CP-LRT when the main effect of the RV is at least as large as the effect of the PRS. In the situation of cases-only analyses with no interaction effect, the Z test looking for over-representation of RV⁺ individuals had the highest power. When the RV main effect is sufficient and there is an interaction between RV and PRS, the CP-LRT has superior power compared to the Z test. The CP-LRT has superior power over the Wilcoxon test (purple curve) when there is no interaction effect, as shown by the CP-LRT (red curve above solid purple line), but the Wilcoxon test has higher power for weaker acting RV when there is an interaction.

Figure 2.

Simulation analyses

Simulation analyses comparing the Causal Pivot likelihood ratio test (CP-LRT) to alternatives focusing on the binary trait context. Monte Carlo methods are used to generate the results in (A)–(D); these simulations generate polygenic risk scores (PRS) and rare variant (RV) status, which are then used to simulate trait outcomes under the corresponding logistic model. Alternative analyses are then performed to detect the effect of the RV and to compare alternative approaches on the same simulated data.

(A) Traits are generated using a logistic model with causal effects from PRS and RV status. The simulated population size was n = 500,000, comparable to the UK Biobank (UKB) scale. The intercept was set to α = −2.2, corresponding to a disease prevalence of 10%, and the PRS effect was set to β = 0.5, consistent with three UKB demonstration diseases. The RV frequency was ω = 0.001. The x axis represents the main effect of the RV (γ), and the y axis shows statistical power determined as the proportion of simulations that reject the null hypothesis at a nominal type I error rate of 0.05 (i.e., p value <0.05). Power comparisons include the CP-LRT (orange), Wilcoxon rank-sum test (purple), Z test (blue), and forward logistic regression using both cases and controls (green). Dashed curves indicate a PRS-RV interaction effect (η = −0.4). In the absence of interaction effects, the Wilcoxon test had markedly lower power, and the Z test outperformed CP-LRT but did not differentiate between main effects and interaction effects nor produce parameter estimates.

(B) Type I error robustness. The y axis shows the fold increase in type I error relative to the nominal level of 0.05. Simulations held γ = 0, η = 0, and α, β, and ω as in (A), while underspecifying the RV frequency (ω^∗). The CP-LRT (orange) retained type I error robustness with values near 0.05 despite a 10% reduction in ω^∗, while the Z test (blue) experienced a sharp increase in type I error as ω^∗ decreased.

(C) Sensitivity to RV frequency overspecification. We set α = −2.2, β = 0.6, and γ = 2.2; the x axis represents RV frequency (ω^∗) used in analysis, exceeding the true frequency (ω = 0.001). Power comparisons between CP-LRT (orange) and Z test (blue) show that overspecifying ω^∗ diminishes Z-test power. At ω^∗ = 0.045, the expected RV frequency in the Y = 1 population, the Z test power approaches 0.05, while CP-LRT maintains ∼75% power by leveraging structurally induced correlation between PRS and RV.

(D) Impact of the strength of the PRS. The x axis ranges the RV effect size γ, and the other parameters are fixed (α = −2.2, η = −0.4, ω = 0.001); ω^∗ is misspecified to its empirical frequency in the disease population. The Z test (blue) has no power in this context. Both the Wilcoxon (purple) and CP-LRT (orange) show power increase as the strength of the PRS increases. The Wilcoxon has superior power but does not distinguish the main effect from interaction. Results for the liability model are included in Figure S3.

We also performed robustness analysis focusing on the binary trait context in cases-only analysis. In this situation both the Z test and the CP-LRT rely on prior information concerning RV frequency. Therefore, we explore the impact of errors in the specification of this frequency. Figure 2B considers the situation where there is no RV effect, with both γ = 0 and η = 0; the unconditional RV⁺ rate is incorrectly specified lower than its true value across a decreasing range. Results show that the CP-LRT had lower type 1 error rate compared to the Z test, as indicated by the cyan Z-test curve dominating the red CP-LRT curve. The CP-LRT retained a type 1 error of 0.05 when the RV rate was misspecified by 10%.

Conversely, the rate parameter for the RV might be falsely specified above its true value. Figure 2C shows this situation when the RV effect is set to α = −2.2, with γ = 2.2 and the PRS parameter β = 0.6—values consistent with the UKB analyses for BC. In this situation, the CP-LRT has higher power to detect an effect while the Z-test power is sharply reduced, as indicated by the red CP-LRT curve dominating the Z-test cyan curve. Interestingly, the CP-LRT retains a power of ∼75% when the RV frequency is set to the expected RV frequency among cases of approximately 0.0045; in this circumstance the Z-test power is the nominal error rate of 0.05.

In Figure 2D, we explore the impact of the strength of the PRS on power for the alternative approaches; this was done by varying the parameter β and looking at power as RV effect changes. As in Figure 2C, we set ω to its value in the affected population to focus on the induced correlation between PRS and RV as PRS strength varies. In this case the Z test has no power, as shown. We ranged the RV effect size and evaluated at three choices of β. In all analyses, power increases as β increases. Results show that in this case of misspecification and variation in β, the Wilcoxon test has highest power. The CP-LRT has power greater than 0.7 when β = 0.5 and γ > 1.5.

In addition to power analysis, we performed investigations concerning the impact of PRS explanatory power on the conditional mean of PRS given RV status and outcome as well as the conditional probability of RV⁺ given PRS and outcome. These results are presented in Figures S1 and S2. The results show that as the PRS effect size increases, there is a strongly increasing impact on the odds of RV⁺ status among individuals with low PRS scores. The conditional mean of the PRS is also influenced by increasing the effect size of the PRS; interestingly, the conditional mean is more differential between RV⁺ and RV⁻ among controls rather than cases. The differential between RV⁺ and RV⁻ in the conditional PRS mean among the controls is increasing in the PRS effect size (Figure S2).

For the liability model (Figure S3), we simulated according to a linear model for continuous quantitative outcomes. We modeled our analyses on the example with parameters α = 0, β = 0.3, and ω = 0.001, and we used a cutoff value of δ = 1.13, corresponding to the 92%-ile of the trait distribution, consistent with clinically relevant LDL-cholesterol cutoffs. Conditional cases-only analysis corresponds to the situation where observed samples are restricted to those with trait values over a cutoff. In this situation, the CP-LRT has greater power than the Wilcoxon test when there is no interaction effect. When the interaction effect is strong, the Wilcoxon test has power of unity. The forward linear regression model using both case and control data has the strongest power, and the Z test also has high power.

Application to UKB data

To assess relevance of CP in real data, we used UKB exomes and PRS. We used well-studied complex traits with both validated PRS and established single-gene contributors. The latter allowed us to recruit ClinVar annotations to select known disease-causing RVs as positive controls. Disease cohorts and variant selection are as described in methods. Most variants identified as RV⁺ were present in only a single individual, as shown in Figure 3B. The variants had a range of coding impact, as shown in Figure S5. A detailed enumeration of the variants identified for the CP analysis is included as Data S1.

Figure 3.

Demonstration of the CP in UKB data

(A) Count of rare variant positive (RV⁺) samples in each polygenic risk score (PRS) tertile. Vertical bars represent the count of RV⁺ samples within each PRS tertile (x axes). Each row represents a different disease cohort paired with its disease specific PRS: hypercholesterolemia (HC), breast cancer (BC), and Parkinson disease (PD), respectively. Each column presents a different aggregation of disease-causing variants and their associated gene: LDLR, BRCA1, and GBA1. The diagonal shows the negative correlation between the number of RV⁺ samples and PRS scores within each PRS-disease-gene grouping. CP-LRT was used to determine p values for all nine analyses, and the p values for all tests are highly significant along the diagonal. Off-diagonal entries are not significant except for the gene LDLR, which shows modest association pattern with both breast cancer (BC) and Parkinson disease (PD) with p < 0.05. More detailed results including confidence intervals for gγ and η appear in Table S2.

(B) Multiplicity and allele type of the RV included in (A). Most RVs occur in only a single sample. A detailed listing of the variants that inform the analysis is provided in Data S1.

Cross-cohort comparisons

We employed a cross-cohort comparative approach to evaluate the CP-LRT. We extracted affected individuals in each disease population and their respective PRS values as well as an RV determination by the presence of at least one qualifying RV (see methods). For visualization of the relationship of RV⁺ against PRS, we split the respective cohorts into tertiles by their PRS. For each PRS tertile in each cohort we counted the rate of RV⁺ in each PRS tertile bin; this analysis is repeated for each of the three example genes BRCA1, LDLR, and GBA1, respectively. We performed statistical testing by application of our CP-LRT (Figure 3A). The three analyses along the diagonal of the graphic are all highly significant (p < 1 × 10⁻⁶), indicating that the CP-LRT found an effect in each disease against disease-causing RV⁺ status in the respective gene. Cross-cohort analyses are mostly not significant, indicating that RV⁺ status for BRCA1 and GBA1 are not associated with either PD state or BC state, respectively. The LDLR showed a weak but significant CP-LRT association with both BC and PD (Table S1). The MLE estimates of the RV and interaction parameters, including the point estimates and confidence intervals, are provided Table S2. The point estimates computed using the CP MLE strongly agreed with the results from case-control analyses for both the main effect of RV and the RV-PRS interaction effect (Figure S6). We performed additional analyses stratifying by age, with consistent results (Figure S12). Rare synonymous variants did not exhibit the CP effect (Figure S7). Similar CPE MLE results were obtained with the liability model for LDL (Figure S8).

Controlling for the effect of ancestry

Ancestry may confound the PRS-RV relationship by acting as a common cause for an individual to have a functional RV as well as a high or low PRS. We call this scenario the diamond graph (Figures 4A–4C and S11). We used a variety of methods to consider the outcome conditional RV-PRS relationship while controlling for confounding effects. For matching analyses, we identified ancestry neighborhoods of affected individuals, and we took the difference of the PRS between matched RV⁺ and RV⁻ affected individuals. We averaged these PRS differences. The results show that RV⁺ affected individuals had significantly lower PRS scores on average than ancestry matched RV⁻ individuals (Figure 4D).

Propensity scoring and inverse probability weighting are alternative approaches to adjust for covariates. Conditioning on outcome, we treated RV⁺ status as a binary exposure, and we constructed random forest models for the propensity of RV⁺ using the ancestry PCs age and sex. The random forest models were successfully predictive of RV⁺ status, and the results again showed that PRS values are significantly lower in RV⁺ compared to RV⁻ individuals. High-order PCs were the most predictive of RV⁺ status. We extended this to other methods for adjustment of covariates including propensity-weighted regression adjustment and doubly robust regression, with similar results (Table S3). We also examined robustness of the CP to alternative polygenic scores (Table S4) which showed consistency in the CP effect.

Generalization to pathway burden scores

We considered the lysosomal storage pathway in the context of PD as a demonstration pathway for oligogenic load analysis using the CP. The lysosomal storage pathway is the system that supports recycling of macromolecules in the brain, and there are many lines of evidence that point to lysosomal storage as a central pathway in neurodegeneration.³³ We established a set of 54 human lysosomal storage pathway genes (Data S1). For each individual, we tabulated the number of qualifying RVs in the lysosomal storage pathway. We then computed the mean PRS of individuals with disease, stratifying on the number of pathway RVs. The results are presented in Figure 5A; Poisson regression based on the CP demonstrated a significant negative trend in PRS with increasing pathway burden of RVs among affected individuals. To emphasize the combinatorial sparsity of the RV data, we analyzed the count of two-variant combinations among cases Figure 5B. No single pair of lysosomal genes accounted for the majority of the observed oligogenic load.

Figure 5.

Pathway burden analysis

Rare variants (RVs) aggregated by functional pathway can be evaluated as an independent cause by pivot against the polygenic risk score (PRS), demonstrating that CP-LRT is generalizable to count variables.

(A) The PRS values (y axis) against the count of pathogenic/likely pathogenic RVs in the lysosomal storage pathway among Parkinson disease (PD) cases from UKB. Each boxplot presents cases with the tabulated count of RV⁺ per case. There is a decreasing trend in PRS, and a Poisson regression analysis that models the rate of RV as a function of PRS demonstrates a statistically significant negative association between the rate of RV⁺ and the PRS value (p = 0.03).

(B) Pairwise co-occurrence of RV⁺ in the lysosomal genes. No single pair accounts for the majority of instances showing the sparse occupancy of positivity for two variants.

Discussion

To date, the general principles of causal inference and structural causal models in complex trait genetics have largely focused on MR,^14,16 which exploits DNA variants as instrumental variables. MR methods are typically used to investigate mechanisms that flow through a candidate biomarker and are not used to differentiate subgroups based on multiple independent causes or causal classes. Given the success of MR, it is likely that SCM can be used more broadly to model genetic effects, suggesting several new avenues for investigating common complex diseases and traits.³⁴

To develop the CP, we reason that individuals reach the complex disease state through alternative mechanisms that are differentially present or absent among individuals. In most research and diagnostic scenarios investigators have some prior knowledge—although incomplete—about causal factors and the strength of their relationship to outcomes. CP explores the utility of induced correlation between a known factor and a candidate cause conditional on outcomes as an additional source of signal for candidate causes. This approach could enable discovery in cases-only analysis because the induced correlation would be present even when appropriate control individuals are not available. This approach is most clearly aligned to the Causes of Effects scholarship,^35,36,37 which focuses on evaluation of the necessity and sufficiency of alternative causal contributors conditional on outcomes. In this study we use PRSs^20,21 as a proxy for the aggregate genome-wide causal contribution of weakly acting common variants to complex disease. This allows us to treat the PRS as a pivot to search for new causes by incorporating induced correlation between alternative disease drivers and PRS by conditioning on outcomes. Although CP exploits the aggregate causal effect captured by the PRS, it does not depend on the validity of individually weak effects or even the actual causal variants. We note that any known cause, including non-genetic factors, could be either substituted for or added to PRS as a known factor for pivot analysis. The effects detected in the CP relationship are closely related to the heritability captured by polygenic scores and RVs. For example, in the circumstance of rare variation, we can interpret the RV coefficient estimated with the LRT in terms of heritability by taking the square of the coefficient multiplied by the frequency and dividing by the trait variance, assuming that the frequency is close to zero so that 1 minus the frequency is approximately 1.

We use simulation-based power analyses for controlled comparison of the CP-LRT approach and alternatives in the cases-only context. The Wilcoxon rank sum and Z test are chosen as non-parametric comparators that also apply in cases-only analysis. The Wilcoxon test has been proposed previously^20,21 and focuses exclusively on the induced correlation between PRS and RV state conditional on disease. Alternatively, the Z test focuses exclusively on the rate change in RV in the disease population compared to its observed background frequency in the overall population, and it does not use the PRS information. The CP-LRT approach—derived from the structural causal model factorization and Bayes’ rule—incorporates both the frequency information from the general population and the PRS-RV conditional dependency.

The results show that in the absence of an interaction effect, the CP-LRT has stronger power than the Wilcoxon test but is weaker than the Z test. When there is an interaction effect the power of the CP-LRT exceeds the Z test, but the Wilcoxon test surpasses it among weakly acting RV with smaller effects. Importantly, the CP likelihood approach results in parameter estimates and confidence intervals; these interval estimates are an important advantage of the procedure over the non-parametric methods, which do not quantify uncertainty, produce parameter estimates, or distinguish between main effects and PRS-RV interactions.

Robustness analyses for CP-LRT reveal key properties of the method. In the cases-only context, prior information concerning RV frequency is a precondition to apply either the Z test or the CP-LRT. Importantly, the CP-LRT is informed by the structural model and includes the PRS as part of its procedure, while the Z test does not. The results show that this difference confers a robustness advantage to the CP-LRT. In terms of specificity, the CP-LRT is more stable against false positives when the population rate of RVs is underestimated; the Z test is highly vulnerable to this underspecification. Conversely, the CP-LRT is also more sensitive when the RV rate is overestimated, and it demonstrates good power when the RV frequency was set to the frequency in the affected population without requiring prior information for RV rate. Taken together, the CP-LRT has an advantage in sensitivity and specificity because the procedure is drawing information from the structural model and the conditionally induced correlation.

We also examined the impact of the strength of the PRS on CP performance. We studied the characteristics of the conditional means of both PRS and RVs. We see that increasing the PRS effect leads to an increase in the response of the conditional mean and corresponding increase in power; this property is most clearly revealed in the context of misspecification of the RV rate. The Z test does not incorporate the PRS and is independent of its effect size. The Wilcoxon power also increases with increasing PRS effect. The patterns of the conditional mean functions are consistent with increasing power driving larger magnitude changes in conditional mean. Interestingly, the differential between RV⁺ and RV⁻ conditional mean of the PRS is larger in controls than in cases. There are circumstances for larger PRS effects where the difference in the conditional PRS mean between RV⁺ and RV⁻ in cases is zero; this pattern does not hold in controls.

Our analyses of data from the UKB reinforce the validity of the CP model. We chose demonstration diseases with well-established disease-causing variants that are mechanistically associated with their respective complex diseases^38,39,40; these diseases also have established PRSs with demonstrated risk associations.^41,42,43 In all cases we observed a strong collider-induced association between the PRS and RV status within diseases and significant CP-LRT results, consistent with our motivating structural model. We also performed parallel analysis using synonymous RV in these same subjects; these synonymous variants do not demonstrate significant collider-induced patterns and lack significant CP-LRT findings. Taken together, these results are consistent with our CP model. Cross-disease analyses are also revealing. The induced association with functional RVs does not appear between GBA1 variants and BC nor between BRCA1 variants and PD. Interestingly, LDLR variation does show a modest induced correlation effect with both BC and PD. Previous studies have mechanistically implicated cholesterol in diverse disease processes including cancer⁴⁴ and neurodegeneration.^45,46 These findings may warrant further investigation.

An additional contribution of our work is to address ancestry confounding through causal approaches. We extend our graph to consider ancestry as a common cause node for both PRS and RV status. We note that PRSs themselves are constructed controlling for ancestry by regression methods,⁴⁷ and in our work we avoid RVs that demonstrate marginal association with PRSs. However, these analyses are limited in power when individual RVs are sparse and could miss cryptic dependency. To overcome these limitations, we performed matching of RV⁺ to RV⁻ cases in the space of the global ancestry PCs. We then compared the PRS of RV⁺ cases to the average PRS of their ancestry-matched RV⁻ neighbors. Neighbor matching is a well-established method to non-parametrically correct for complex confounding variables, but this method has not been extensively deployed in genetic epidemiology. We also took a propensity-modeling approach. We used machine learning by random forest to predict the probability for an individual to be RV⁺ based on ancestry PCs. We then used inverse probability weighting using this propensity model to construct a weighted difference of the PRS between RV⁺ and RV⁻ individuals; this analysis also produced significant results showing lower PRS in RV⁺ cases, consistent with the CP model for all three example diseases. We note that ancestry can also be modeled as a common cause of genetic variation (whether RV or PRS) and disease. In that sense, ancestry can be viewed as a potential confounder even if it acts as a proxy for non-genetic mechanisms; this is an area for future inquiry.

Finally, we consider a pathway load as a candidate cause and again utilize the PRS as a pivot. We chose the count of qualifying variants in the lysosomal storage pathway variants in PD because such variation is mechanistically associated with neurodegenerative disease processes both by experimental investigation and epidemiologic evidence.³³ We found significant induced association with the PRS among cases. This finding is encouraging for further investigation. Oligogenic disease models can be difficult to interpret, and the CP framework places them in a causal perspective that reinforces a mechanistic interpretation of statistical findings; we note that if many alternative oligogenic mechanisms or pathways are considered simultaneously with the CP-LRT, multiple testing correction should be applied.

The study of independent causes, in classical genetics called locus and allelic heterogeneity, is central to understanding the genetic architecture of monogenic disorders.⁴⁸ Parallel concepts of causal heterogeneity are needed for complex traits where current methods focus on the average causal contribution of variants rather than their operation in potentially heterogeneous subgroups.⁴⁹ There are already well-known examples in certain complex traits in which the total population of affected individuals includes both those with high polygenic risk and individuals affected by monogenic forms. In this work, we have developed and evaluated an approach to address this form of causal heterogeneity using a general SCM framework. By examining the possible conditional relationship between a known cause and a distinct candidate cause conditional on the phenotypic outcome, we demonstrate that one can establish the candidate’s contribution. Moreover, we show that this approach applies in cases-only, controls-only, or combined case-control analyses. The terminology CP is adopted to reinforce that the known cause is used to leverage the conclusion.

The CP is not a foreign concept in medical research. Differential diagnosis is an example of CP thinking. Given that an individual is observed to have some phenotype, when certain causes are ruled out, other causes become more likely. In genomic medicine the pivot has been observed in somatic cancer, where driver mutations are observed to exclude each other.^50,51,52 This exclusion principle is also at the heart of complementation-group analysis in experimental genetics. This approach has the goal of deconvolving patient populations into causal groups or subtypes; it has the additional benefit of providing individual-level insights following the diagnostic process where evaluating and eliminating alternative potential causes is fundamental to personalized medicine. Our focus in this work is the utility of this approach as a statistical method for discovery in genetic data. In Figure S1 we show the relevance of our work in a more diagnostic context, where we consider the odds ratio of RV⁺ for individuals with low and high PRS values. Future work will explore the validity of individual-level causal attribution using the lens of counterfactuals.

We used simulation and real data to consider CP analyses in a cases-only context. The cases-only test has slightly less power than a comparator case-control analysis, indicating that most of the information about the causal effect of the RV is carried in the cases. It is important to note that such a cases-only analysis is not possible using traditional association tests. The new strategy is therefore ideal when selection on the phenotype is unavoidable. There are many such scenarios relevant to complex diseases. Genotype × environment interaction studies investigate whether the genetic variant and environmental exposure jointly influence the risk, which cannot be tested in those who do not have the outcome. In case-crossover designs the study is restricted to individuals who have already experienced the outcome, and the focus is on understanding the exposure timing in relation to the event within the same individual. Cases-only analysis also applies in the context of somatic mutation outcomes in cancer, where analysis is confined to individuals who already have cancer and the focus is on the correlation between mutations and exposures or germline genetic variation. Another cases-only context occurs in drug safety studies that investigate the association between drug-exposure-specific adverse outcomes; the design also applies to pharmacogenetic studies that assess the drug response among patients who have already experienced a particular outcome, such as an adverse drug reaction. In some datasets, testing for certain specific biomarkers, multi-omics profiles, and specialized imaging studies may only be triggered by the pre-occurrence of phenotypes suggestive of disease and these phenotypes may not be available in controls. In all these circumstances, cases-only genetic analyses are required.

The present work has several limitations that will need further clarification. One limitation of the use of real-world data in this study is the dependence on selection by diagnostic code (ICD-9,10). It is well known that diagnostic codes are subject to a variety of errors including misdiagnosis and underdiagnosis. Misclassification of phenotype in the CP would be expected to reduce power but not inflate type I error. In future investigations of CP, we plan to address the role of biomarkers to improve phenotype specificity. Another limitation is the uncertainty of variant classification. Important progress has been made in the last several years on prediction of deleteriousness of missense substitutions in protein coding sequences. In addition, consensus methods for pathogenicity classification are expected to transition from harmonized terms⁵³ to a points-weighted system that may further enhance the consistency of classification in clinical practice. In our work we recruited stop-gain and frameshift variants that were unknown to ClinVar as LoF variants in our analyses (see Figure S5). Another limitation in the present study is that we have not specifically addressed epistatic interactions among rare genetic variants. Epistatic interaction is an interesting possible explanation for components of heritability that are not captured in the per marker marginal effects. Remarkably, the causal graph provides a straightforward approach to modeling causal interaction in the strict sense. This can be accomplished by the introduction of an interaction in the disease model by adding an interaction term. We showed a test of a single pathway burden test using CP. In any exploratory analysis of causative pathways, a multiple testing correction would be necessary. Finally, our model assumes no unmeasured confounding between RV and outcome (Figure S13).

The study of causal heterogeneity in complex disease genetics is a fertile area for research. Significant progress has been made in identifying genes and mechanistic processes that drive complex disease. In this context comparably less effort has been assigned to deconvolving the disease population into subgroups. Our CP and CP-LRT method exploits prior research from GWASs as well as surveys of genetic variation to drive discovery and moves toward assignment of causal types. In principle, it may be possible to partition GWAS results to identify mechanistic subgroups and those more likely to respond to targeted therapies within specific diseases. Additional research in the area of causal heterogeneity using SCM is likely to prove productive.

Data and code availability

The code for the methods developed during this study is available at chadashaw/causal-pivot: https://github.com/chadashaw/causal-pivot/.

Acknowledgments

This research was generously supported by the Ting Tsung and Wei Fong Chao Foundation, the Huffington Foundation, and the Jan and Duncan Neurological Research Institute at Texas Children's Hospital. This work was also supported by a grant from Genetics & Genomics Services, Inc. as well as funding from Texas Genomics Consulting.

Author contributions

C.A.S.: conceptualization, formal analysis, funding acquisition, methodology, project administration, software, resources, supervision, writing – original draft, and writing – review & editing. C.J.W.: data curation, software, validation, visualization, writing – original draft, and writing – review & editing. T.T.T.: formal analysis and methodology. D.I.: formal analysis and methodology. N.D.: formal analysis and methodology. J.M.S.: conceptualization, data curation, funding acquisition, and writing – review & editing. J.W.B.: conceptualization, data curation, funding acquisition, project administration, software, resources, writing – original draft, and writing – review & editing.

Declaration of interests

J.W.B. and C.A.S. are co-owners of Texas Genomics Consulting.

Published: August 18, 2025