Combining Correlated Functional Annotations Without Double Counting

Why Combining Annotations Requires Care

A candidate variant may overlap accessible chromatin, a histone-mark peak, and a region labelled as an enhancer. These observations can look like three separate pieces of support. But the tracks may be recording related aspects of the same regulatory state: for example, accessibility and a histone mark may both be likely where an active enhancer is present. Counting each as independent evidence can therefore make the combined support appear stronger than it is.

The issue is not that correlated annotations are useless. Their joint pattern can still distinguish variants with different functional profiles. The issue is how to summarize that pattern. A multiplication of separate evidence measures is justified only under a particular conditional-independence assumption. When that assumption is not credible, use a joint pattern, a pre-specified grouped summary, or conditional probabilities that account for the other annotations.

Bayes' rule says that posterior odds equal prior odds multiplied by a likelihood ratio:

This update is meaningful only when the conditional probabilities describe a relevant and appropriately calibrated reference setting. An annotation label by itself is not a likelihood ratio, and a large likelihood ratio is not proof that a particular variant is causal. It is evidence to combine with prior information under the stated model.

When Multiplication Is Justified

For a single binary annotation, such as \(A=1\), its likelihood ratio is \(P(A=1\mid C=1)/P(A=1\mid C=0)\). Multiplying the separate likelihood ratios for \(A\), \(H\), and \(E\) to assess the joint pattern is justified if the annotations are conditionally independent both among causal variants and among noncausal variants. In this setting, “conditionally independent” means that within each value of \(C\), knowing one annotation does not change the probabilities of the others.

Under that assumption, and when the relevant probabilities are positive, the likelihood ratio for the joint pattern factors into the product of the three marginal likelihood ratios. Without the assumption, the product generally does not equal the joint likelihood ratio. Correlation within either causal-status group is enough to break that equality. The dependence can arise because tracks measure overlapping biological features, share data or processing steps, or use related criteria to assign labels.

Worked Example: Three Positive Tracks and Double Counting

Dependence-Aware Summaries

There is no single summary that is best for every annotation set. The choice depends on the scientific question, the available reference data, and what information the summary is intended to preserve. Three common approaches are to use the joint pattern, combine annotations into a defined group, or factor the joint probability into conditional contributions.

A grouped summary must be defined before interpreting its likelihood ratio. “Any positive track” is different from “all tracks positive,” and both differ from a score that assigns distinct weights to patterns. Grouping can be useful when the individual tracks are highly redundant or when data are too limited to estimate every joint pattern reliably. Its cost is information loss: two variants with different patterns may be assigned the same group value.

Conditional Factorization and Its Conditions

The chain rule can express a joint probability using conditional probabilities. For the all-positive pattern, the factorization is

This equality uses conditional probabilities only when the conditioning events have positive probability in the relevant causal-status group. In particular, \(P(A=1\mid C=c)\) must be positive, and \(P(A=1,H=1\mid C=c)\) must be positive for the last factor to be defined. If a conditioning event has probability zero, its conditional probability is undefined; the joint probability can still be discussed directly. Do not use a chain-rule factorization with undefined factors.

When all factors are defined and the joint probabilities in both groups are positive, dividing the two chain-rule expressions yields a product of conditional likelihood ratios. This is a valid identity, not an independence assumption. Each later factor is conditional on the earlier annotations in the chosen order; it does not represent an independent, context-free contribution.

A Short Justification for the Product Rule

The distinction between a valid conditional factorization and an unjustified product of marginal ratios follows directly from the chain rule. If the required conditioning events have positive probability in each group, write the joint probability in each group as its chain-rule product and divide. The result is a product of conditional ratios. To replace those conditional probabilities with marginal probabilities, however, one must additionally assume conditional independence.

Under conditional independence given each value of \(C\), the second and third conditional probabilities equal the corresponding marginal probabilities within each group. Only then does this expression reduce to the product of marginal likelihood ratios. If any required conditioning event has probability zero in either group, do not form the corresponding conditional ratio; use the joint probabilities directly and treat zero-probability cases separately.

Choosing and Interpreting a Summary

A joint model is most informative when there are enough well-matched reference observations to estimate the relevant patterns. With three binary tracks there are eight possible patterns, so estimating separate probabilities for all patterns may be difficult when some combinations are rare. Grouping can reduce that burden, but the grouping should have a clear interpretation and should not be selected after looking for the most favorable result without accounting for that selection.

Conditional summaries can be useful when the scientific question is explicitly incremental: for example, whether an enhancer label adds information after accessibility and histone marks are known. The order matters for interpreting the individual conditional factors, even though a correctly calculated joint likelihood ratio does not depend on the order chosen for a chain-rule factorization. Conditional probabilities also need adequate data for the relevant subgroups.

For locus interpretation, these calculations concern evidence for a candidate variant's causal status under a specified model. They do not by themselves identify a target gene, establish a molecular mechanism, or show that an annotation caused the association. Interpret the updated odds in light of how the reference probabilities were obtained, the tissue or cell type represented, and the annotation definitions.

Common Mistakes

• Multiplying marginal ratios because there are several tracks. The correction is to check conditional independence within both causal-status groups, or use joint or conditional probabilities instead.
• Treating related labels as independent confirmation. Accessibility, a histone mark, and an enhancer label may overlap biologically or operationally. Their count is not automatically their amount of independent evidence.
• Using a conditional factor when its conditioning event has zero probability. Conditional probabilities are undefined in that case. Work with joint probabilities, and handle zero-probability cases directly rather than writing an invalid factorization.
• Calling a likelihood ratio a posterior probability. A likelihood ratio multiplies prior odds. It is not itself the probability that a variant is causal.
• Assuming a grouped label preserves the full pattern. “Any annotation” collapses distinct combinations. Use a grouped likelihood ratio only when that is the observation or summary being analyzed.