Multi-Causal Configuration Models

Why Represent Causal Hypotheses as Configurations?

Fine-mapping asks which variant or variants in a genomic region have direct effects on a phenotype. A single-variant hypothesis allows only one causal variant; a multi-causal hypothesis allows more than one. These are different model spaces, not merely different descriptions of the same analysis. If multiple variants truly have direct effects, a model restricted to one may force their combined pattern of association onto a single candidate. Conversely, allowing every subset of variants can create a very large set of candidate models.

A configuration records which variants are included as causal in a candidate model. It does not, by itself, say how strongly the data support that model. That distinction matters: this tutorial defines the candidate configurations and examines how linkage disequilibrium (LD) shapes them. A configuration's probability or relative evidence requires additional calculations.

A useful formal model clarifies what “direct effect” means. Let \(X_j\) be the centered, standardized genotype vector for variant \(j\), with \(X_j^\mathsf{T}X_j/n=1\), and let \(R\) be the \(p\times p\) genotype correlation matrix, so \(R_{j\ell}=X_j^\mathsf{T}X_\ell/n\). Consider the linear model

Here \(y\) is a quantitative phenotype, \(\beta_j\) is the effect conditional on the other variants in the model, and the errors are independent of the genotypes, have mean zero, and have common variance \(\sigma^2\). This formulation assumes an additive effect for each included variant and no interaction terms. In this model, “causal” refers to membership in the specified direct-effect model; an association alone does not establish biological causation.

For example, the marginal regression slope at variant \(\ell\) is \(\hat b_\ell=X_\ell^\mathsf{T}y/n\). Under the model above, its expectation is

Thus, the set of variants with nonzero marginal associations need not equal \(C\). A variant outside \(C\) can have a nonzero marginal association because it is correlated with an included variant. Conversely, effects from multiple included variants can reinforce or partly cancel in the marginal associations.

How Configuration Size Defines the Candidate Space

The number of possible configurations depends on which sizes are allowed. With \(p\) candidate variants, there is one configuration of size zero and \(\binom{p}{k}\) configurations of size \(k\). If the model allows every size from zero through \(K\), where \(K\leq p\), the number of candidate configurations is

The formula counts subsets, not effect-size choices. For example, once a configuration is chosen, the values of its nonzero effects still have to be represented or estimated. If all sizes are allowed, the number of configurations is \(2^p\), since each variant is either included or excluded. Even a moderate limit on size may therefore leave many models to compare.

Common model-space choices include a single-causal space, a fixed-size space, and a bounded-size space. The single-causal space contains only subsets of size one, sometimes with the null added as a separate possibility. A fixed-size-\(k\) space contains only subsets of size \(k\). A bounded-size space includes several sizes, often the null through a chosen maximum \(K\). These choices encode different assumptions about how many direct effects the model permits. They should not be confused with a conclusion about the number of causal variants.

Why LD Changes Distinguishability

The configuration \(C\) identifies the direct-effect terms, but the data typically provide noisy marginal association summaries. Under the linear model, define \(\hat b=(\hat b_1,\ldots,\hat b_p)^\mathsf{T}\). Conditional on the standardized genotypes, the mean and covariance of these marginal slopes are

where \(\beta_j=0\) for variants outside \(C\). The covariance expression follows from \(\hat b=X^\mathsf{T}y/n\), the error covariance \(\sigma^2 I_n\), and \(X^\mathsf{T}X/n=R\). These expressions show two roles of LD: it spreads a direct effect into correlated marginal associations, and it affects the covariance of the association estimates.

If two genotype columns are identical, their correlations with every other column are identical. A single effect at one of those variants can then produce the same expected marginal association pattern as an appropriately matched effect at the other. The data may not distinguish those configurations. When columns are highly, but not perfectly, correlated, the expected patterns can be close rather than identical; distinguishing them then depends on the effect sizes and the amount of information in the data.

Multiple Effects and Marginal Association Patterns

A multi-causal configuration contains more than one direct-effect term. Its marginal pattern depends jointly on the included variants, their effect sizes, and the LD matrix. In particular, the existence of two direct effects does not imply that two large marginal associations will be observed. Effects with opposite signs can partly cancel at correlated variants, while an excluded variant can still show a strong marginal association through LD.

Assumptions, Diagnostics, and Model-Space Choices

A configuration model is interpretable only relative to its assumptions. The examples used a quantitative phenotype, additive direct effects, standardized genotypes, a linear mean model, and independent normal errors with common variance. In practical analyses, additional features such as covariates, phenotype transformations, and uncertainty in LD may matter. The simple equations here should not be carried over unchanged when those features alter the sampling model.

Before interpreting configurations, check that the candidate variants and their allele coding are defined consistently, that the LD matrix corresponds to the variants being analyzed, and that \(R\) is symmetric with unit diagonal and is positive semidefinite. Small numerical deviations from positive semidefiniteness can arise in estimates, but a substantially invalid correlation matrix is a warning that the inputs or their alignment need investigation. Also check whether the allowed maximum configuration size is scientifically and computationally defensible. A limit that is too restrictive excludes plausible models; an unrestricted space can be computationally large and may include many weakly distinguishable candidates.

The main alternatives differ in what they permit. A single-causal model is compact and can be useful when one signal is a defensible approximation, but it cannot represent multiple direct effects. A fixed-size-\(k\) model makes the number of included variants constant, so it excludes both smaller and larger configurations. A bounded-size model allows uncertainty across several sizes but expands the candidate space. These are choices about model structure; none automatically resolves ambiguity caused by LD.