Reading PP.H0 Through PP.H4

What the Five Posterior Probabilities Summarize

A colocalization analysis asks whether genetic association with two traits in a genomic region is consistent with a shared causal variant or with distinct causal variants. In this setting, the five reported posterior probabilities, PP.H0 through PP.H4, assign posterior support to five competing hypotheses. They are a compact summary, but their interpretation depends on both which hypotheses are being compared and how much evidence there is that either trait is associated at all.

The usual five-hypothesis formulation considers a specified region and a set of variants represented in the association data. Its basic model treats each trait as having at most one causal variant in that region. The variants and summary statistics must refer to the same genomic alleles and region; mismatched or poorly aligned data can make the posterior misleading. As in the earlier material on LD and colocalization, correlated variants can make it difficult to distinguish the variant underlying an association from other variants that track it.

These hypotheses are mutually exclusive within the model, so their posterior probabilities sum to one, apart from rounding. “Only trait 1” under \(H_1\), for example, means that the model assigns the association to trait 1 and not trait 2 in this region. It does not establish that trait 2 has no association anywhere, nor does it prove a biological absence of effect. Likewise, \(H_4\) means that the model favors the same variant accounting for association in both traits; it does not establish that one trait causes the other.

Two Questions to Ask When Reading the Results

A useful interpretation separates two questions. First, is there evidence that either trait is associated in the region? Under the five hypotheses, the posterior probability of some association is \(1-PP.H0\). The posterior probability that trait 1 is associated is \(PP.H1+PP.H3+PP.H4\); for trait 2 it is \(PP.H2+PP.H3+PP.H4\). These sums follow by adding the hypotheses in which the relevant trait is associated.

Second, conditional on the model placing the analysis in a both-traits-associated hypothesis, how much posterior support favors a shared variant rather than distinct variants? That comparison is between \(PP.H4\) and \(PP.H3\). When their sum is positive, the posterior probability of \(H_4\) conditional on \(H_3\) or \(H_4\) is:

This conditional quantity is not a replacement for the full posterior table. It deliberately excludes \(H_0\), \(H_1\), and \(H_2\), and therefore answers only the shared-versus-distinct question among the two hypotheses in which both traits are associated. A large value can coexist with appreciable posterior probability on \(H_0\), for example. Always inspect the full distribution before making a concise claim.

Why the Shared-versus-Distinct Comparison Works

The posterior probability of a hypothesis is proportional to its prior probability multiplied by the likelihood of the observed data under that hypothesis. In a configuration-based analysis, the posterior weight for each hypothesis is obtained by summing the weights of the variant configurations that belong to it. Write the resulting aggregate weight for \(H_k\) as \(W_k\). Normalizing the weights gives \(PP.H_k=W_k/\sum_{j=0}^{4}W_j\).

The common normalization cancels. Thus, the conditional comparison uses the relative posterior weight of the shared-variant and distinct-variant hypotheses, without letting the other three hypotheses change that ratio. But this cancellation also explains its limitation: the ratio gives no information about how much total weight is assigned to \(H_3\) and \(H_4\). Those hypotheses could together have most of the posterior mass, or very little of it.

For example, if \(PP.H4=0.36\) and \(PP.H3=0.04\), the conditional shared probability is \(0.36/(0.36+0.04)=0.90\). This is strong preference for \(H_4\) over \(H_3\) among those two hypotheses, but \(PP.H4+PP.H3=0.40\) means that 60% of posterior mass lies elsewhere. It would be incomplete to report only “90% support for colocalization” without specifying the conditioning and the remaining posterior mass.

Worked Interpretations

Assumptions, Diagnostics, and Failure Modes

The posterior interpretation is only as appropriate as the model and data supplied to it. The standard five-hypothesis formulation assumes at most one causal variant per trait in the analyzed region. It also relies on comparable variant coverage and correctly harmonized alleles and effect directions, and on priors that represent the analysis being performed. The posterior is conditional on these choices; it is not a property of the locus independent of the data and model.

A practical reading should check the following before interpreting a high \(PP.H4\):

• Posterior accounting: confirm that the five values sum to approximately one, allowing for rounding, and identify which hypotheses contain most of the probability mass.
• Association evidence: inspect \(1-PP.H0\), and the trait-specific sums, not just the ratio \(PP.H4/(PP.H3+PP.H4)\).
• Variant and allele alignment: verify that the traits are analyzed over a comparable region and that variant identities and alleles have been harmonized. A strand or allele mismatch can undermine the comparison.
• LD and resolution: correlated variants can produce similar association patterns. The result may support a shared signal without precisely identifying the causal variant. Conversely, differences in LD patterns across the analyzed data can affect the relative evidence.
• Model fit: if a region plausibly contains multiple causal signals for either trait, a single-signal result may not represent the data well. The five probabilities should then not be treated as a complete description of all possible causal configurations.

A useful diagnostic question is whether the conclusion would remain sensible if the posterior mass outside \(H_3\) and \(H_4\) were stated explicitly. If an interpretation depends only on a high conditional ratio, but the analysis has substantial probability on \(H_0\), \(H_1\), or \(H_2\), report that uncertainty rather than compressing the result to a single “colocalized” label. If the posterior is spread across \(H_3\) and \(H_4\), the data have not resolved shared versus distinct signals, even when both traits show association.

How This Interpretation Differs From Nearby Summaries

This distinction is why overlapping association peaks or small p-values are not substitutes for colocalization. Nearby variants in LD can each tag a signal, and two traits can both be associated in the same region while having distinct causal variants. Conversely, a colocalization posterior is not a direct test of mediation: a shared variant can affect two traits through separate biological pathways. The posterior probabilities describe relative support among specified hypotheses, not a causal mechanism.

Common Mistake and Key Takeaway

A frequent mistake is to see \(PP.H4\) exceed \(PP.H3\) and conclude that the result proves the traits share a causal variant. The correction is to describe the result as posterior support under the model, compare \(H_4\) and \(H_3\) quantitatively, and report the total posterior mass on the both-associated hypotheses and on \(H_0\). For example, \(PP.H4=0.55\) and \(PP.H3=0.05\) yield a conditional shared probability of \(0.55/0.60\approx0.9167\), but the remaining 0.40 posterior mass still matters.