The Trans-Ancestry Fine-Mapping Setup

Introduction

Trans-ancestry fine-mapping begins with a set of inputs indexed by both cohort and variant. Each cohort contributes association estimates and their uncertainty, allele frequencies, a sample size, and information about linkage disequilibrium (LD). The variants must also be represented in a shared set so that quantities from different cohorts refer to the same genetic markers.

These inputs describe what was measured in each cohort; they do not by themselves identify a causal variant. Their role is to make the evidence available in a form that a fine-mapping model can use. This setup therefore requires careful attention to units, indexing, and which participants contributed to each estimate. The later task of harmonizing alleles across summary statistics is separate: here, we define the inputs and the consistency they must have.

Association Estimates and Their Uncertainty

The estimate \(\hat{\beta}_{kj}\) summarizes the association between the dosage of variant \(j\) and the phenotype in cohort \(k\), under the association model used for that cohort. Dosage is usually the expected number of copies of a specified allele, on a scale from 0 to 2. The coefficient’s meaning depends on the phenotype model and units. For a quantitative phenotype recorded in its original units, a linear-model coefficient is in phenotype units per additional allele copy. For a binary phenotype analyzed by logistic regression, a coefficient is commonly on the log-odds scale per additional copy.

The standard error \(SE_{kj}\) describes the estimated sampling uncertainty of \(\hat{\beta}_{kj}\), on the same coefficient scale. A coefficient and its standard error must be paired: a value of \(0.08\) is not interpretable as an estimate with a given precision until its units, cohort, variant, and standard error are known. Covariate adjustment, phenotype definition, and the participants included in the analysis are also part of the context of the estimate.

For example, if a hypothetical linear-model estimate is \(0.08\) phenotype units per allele copy with \(SE=0.025\) phenotype units per allele copy, the estimate is positive and its reported standard error is \(0.025\) on that same scale. The standard error is not an effect size, and it does not describe variation in individual phenotypes.

Frequencies, Sample Sizes, and LD

The allele frequency \(f_{kj}\) is the frequency, in cohort \(k\), of the allele used to define the dosage at variant \(j\). It helps describe how common that allele is in the analyzed cohort. Frequencies can differ across cohorts. Frequency is not itself an association estimate, and a frequency difference alone does not show that a variant has a different effect on the phenotype.

Sample size should be recorded with enough detail to explain which observations support each estimate. The cohort total \(N_k\) is the number of participants in the association analysis, but the number with usable genotype and phenotype data can differ by variant. When that happens, record a variant-specific count \(N_{kj}\), or otherwise document the applicable analysis count. For binary outcomes, case and control counts may also be needed to understand the study design. A single cohort-wide number should not silently stand in for a smaller variant-specific sample.

The matrix \(R_k\) describes correlations among variant dosages in cohort \(k\). In the usual dosage-correlation representation, \(R_{k,j\ell}\) is the correlation between dosages at variants \(j\) and \(\ell\). The diagonal entries are 1 for variants with nonzero dosage variance; the matrix is symmetric and must be a valid correlation matrix. Its dimensions are \(M\) by \(M\), and its rows and columns must follow the same variant ordering used for the association estimates. An LD matrix is not a list of association effects: it describes relationships among genotypes.

LD should represent the relevant cohort or a suitable ancestry-matched reference for that cohort. An LD matrix from one population is not automatically interchangeable with another cohort’s matrix. At this setup stage, the important bookkeeping point is that each \(R_k\) is labeled by cohort and variant order. How ancestry-specific LD changes association patterns is a distinct question.

Why Dosage Variance Needs an Explicit Assumption

Allele frequency does not, by itself, determine dosage variance. If \(G\) is the dosage of an allele, then its variance is \(\operatorname{Var}(G)\), which depends on the genotype distribution. Under Hardy–Weinberg equilibrium, the two allele copies are independent Bernoulli variables with success probability \(p\), so \(G\) has a binomial distribution with two trials and probability \(p\). Only under that additional assumption does the familiar expression \(2p(1-p)\) follow.

A useful approximation illustrates why the distinction matters. In a simple linear regression with an intercept and centered genotype dosage, if the residual variance is \(\sigma^2\), then conditional on the observed genotypes the coefficient variance is \(\sigma^2/\sum_i(G_i-\bar{G})^2\). If the sample dosage variance is close to its population value, this gives the approximation \(SE(\hat{\beta})\approx \sigma/\sqrt{N\operatorname{Var}(G)}\). This is an approximation under the stated simple-model conditions, not a universal formula for every association analysis.

A Shared Variant Set and Consistent Indexing

A shared variant set \(\mathcal{J}\) is the set of markers represented across the cohorts for the analysis. It gives each variant a stable identity and index, such as \(j=1,\ldots,M\), so the association estimates, frequencies, and LD matrices can be matched. The set might be determined by the region being analyzed and the availability of variant data. Its exact definition should be documented rather than inferred from a table’s row order.

For the setup to be coherent, each cohort’s quantities must use the same variant identities and order. If one cohort lacks a variant, that absence is missing information; it is not an association estimate of zero and should not be represented as a zero LD row or column. Similarly, two values in the same row position cannot be treated as the same variant unless the marker identities actually agree. The allele labels and effect directions also need to refer to a common convention before estimates are compared; the specific allele-harmonization operations are a separate topic.

Assembling and Checking the Inputs

A practical setup can be organized as a short sequence of checks. These checks establish that the data components refer to one another; they do not perform fine-mapping or resolve biological interpretation.

Common Mistakes and Why the Setup Matters

• Using allele frequency as though it completely specifies dosage variance. The claim \( \operatorname{Var}(G)=2p(1-p) \) requires Hardy–Weinberg equilibrium for a diploid biallelic genotype. Without that assumption, use the actual dosage variance when it is needed.
• Confusing a standard error with a standard deviation of individual outcomes. \(SE_{kj}\) describes uncertainty in an estimated coefficient, not the spread of phenotypes among participants.
• Using one sample size for every variant without checking missingness. Variant-specific counts can differ, so record them when appropriate rather than assuming all variants have identical analysis samples.
• Treating LD as an association effect. An LD entry is a dosage correlation, whereas \(\hat{\beta}_{kj}\) is an association estimate for a phenotype.
• Assuming matching row positions guarantee matching variants. Variant identity and order must be checked explicitly; absent information is not a zero-valued result.
• Calling an associated variant causal from these inputs alone. Association evidence and LD structure inform fine-mapping, but an association estimate is not proof of causality.

Careful setup matters because errors in units, sample counts, or indexing can make otherwise valid calculations answer the wrong question. In particular, a matrix paired with the wrong variant order no longer describes the intended set of associations. Recording the inputs transparently makes later modeling assumptions inspectable and helps distinguish genuine cohort differences from differences in how the data were represented.