Worked Example With Two Causal Signals

Why analyze two signals separately?

A genomic region can contain more than one causal signal. When two associated variants lie in distinct, approximately independent LD blocks, a useful analysis compares a model with one causal variant in the whole region against a model with one causal variant in each block. The comparison matters because evidence for one strong association does not rule out a second, weaker association. It also matters for interpretation: a single-causal analysis treats variants as competing explanations for one signal, whereas a two-signal analysis asks which variant best explains each of two signals.

The example below is hypothetical. It uses four variants in two blocks, summary-statistic z-scores, and an approximate Bayes factor (BF) for each variant. The blocks are assumed independent of one another, and each block is assumed to contain exactly one causal variant under the two-signal model. Within each block, the variants can be correlated through LD; the z-scores alone do not make them independent.

Model and assumptions: There are two blocks, A and B, with two candidate variants in each. The LD matrix is block diagonal: correlations between blocks are zero, while the two variants within each block have correlation 0.2. For the two-signal model, exactly one candidate variant is causal in each block, and the two candidates within each block have equal prior probability. Evidence is summarized using the same approximate single-variant Bayes factor for every candidate. Thus the block-level evidence factors across blocks. These conclusions are conditional on the specified candidate set, LD, effect-size prior, and one-causal-per-block assumption.

For numerical illustration, use a normal prior on the standardized effect with prior variance equal to the sampling variance, producing the variance ratio \(r=0.5\). Under the usual approximate z-statistic Bayes-factor formula,

BF_j=\sqrt{1-r}\exp\left(\frac{r z_j^2}{2}\right) =\frac{1}{\sqrt{2}}\exp\left(\frac{z_j^2}{4}\right).

A BF greater than 1 means the observed summary statistic is more compatible with the specified causal-variant model than with the null for that candidate, under this approximation. It is not itself a posterior probability; prior probabilities are also required to obtain posterior probabilities.

Worked example: variant-level evidence

Worked Example: Calculating the Four Bayes Factors

Suppose the hypothetical summary statistics are \(z=(5.0,4.2)\) for variants A1 and A2, and \(z=(4.5,3.7)\) for B1 and B2. Apply the stated formula to each candidate.

For A1, \(z^2=25\), so \(BF_{A1}=2^{-1/2}\exp(25/4)=0.707107\exp(6.25)\approx366.29\). For A2, \(z^2=17.64\), so \(BF_{A2}=0.707107\exp(4.41)\approx58.18\).

For B1, \(z^2=20.25\), so \(BF_{B1}=0.707107\exp(5.0625)\approx111.71\). For B2, \(z^2=13.69\), so \(BF_{B2}=0.707107\exp(3.4225)\approx21.67\).

All four BFs exceed 1, but this does not establish that all four variants are causal. Within a model that permits one causal candidate per block, variants compete to explain the signal in their own block. The posterior probabilities depend on the relative BFs and the prior probabilities.

Compare the single-causal and two-signal models

First consider a single-causal model for the entire region: exactly one of the four variants is causal, and each has prior probability \(1/4\). With equal prior probabilities, the posterior probability for a candidate is its BF divided by the sum of all four BFs. Using unrounded BFs gives a denominator of about \(557.85\). The displayed BFs are rounded, so adding those displayed values can differ in the last hundredth.

\begin{aligned} P(A1\text{ causal}\mid Z,\text{single})&\approx\frac{366.29}{557.85}=0.6566,\\ P(A2\text{ causal}\mid Z,\text{single})&\approx\frac{58.18}{557.85}=0.1043,\\ P(B1\text{ causal}\mid Z,\text{single})&\approx\frac{111.71}{557.85}=0.2003,\\ P(B2\text{ causal}\mid Z,\text{single})&\approx\frac{21.67}{557.85}=0.0388. \end{aligned}

These are probabilities conditional on the single-causal model. They sum to approximately 1, apart from rounding. Under that model, A1 is the most probable explanation, but the model cannot represent two causal signals simultaneously.

Now consider the two-signal model: one causal variant in A and one in B. The prior assigns probability \(1/2\) to either candidate within each block, so there are four two-variant configurations, each with prior probability \(1/4\). Under the block-independence assumption, the evidence for a configuration is the product of the BFs for its two variants. Normalizing these products gives posterior probabilities for the four configurations.

\begin{aligned} P(A1,B1\mid Z,\text{two})&\approx0.86295(0.83755)=0.7227,\\ P(A1,B2\mid Z,\text{two})&\approx0.86295(0.16245)=0.1402,\\ P(A2,B1\mid Z,\text{two})&\approx0.13705(0.83755)=0.1148,\\ P(A2,B2\mid Z,\text{two})&\approx0.13705(0.16245)=0.0223. \end{aligned}

For example, the posterior probability for A1 within block A is its BF divided by the sum of the BFs in that block: \(366.29/(366.29+58.18)\approx0.86295\). Similarly, for B1 it is \(111.71/(111.71+21.67)\approx0.83755\). The product rule follows because, with independent blocks and a factorized prior, the joint likelihood and prior each factor into an A part and a B part. Consequently, the posterior configuration probability is the product of the two block-specific posterior probabilities.

Interpretation: Under the two-signal model, the posterior probability that A1 is the causal candidate in block A is about 0.863, and the corresponding probability for B1 in block B is about 0.838. The joint probability that A1 and B1 are the two causal candidates is about 0.723. These are conditional on the two-signal model and the stated candidate set, priors, and evidence approximation.

Worked example: evidence for the model and signal-specific sets

Worked Example: Comparing Evidence and Forming 95% Sets

With equal prior probability for the two candidates in each block, the block-level BF relative to a null block is the average of its two candidate BFs. The two-signal model's BF relative to the model with no causal variants is the product of these block averages:

\left(\frac{366.29+58.18}{2}\right) \left(\frac{111.71+21.67}{2}\right) \approx212.24\times66.69 \approx14154 \approx1.42\times10^4.

Thus the specified two-signal model has an approximate BF of \(1.42\times10^4\) relative to the null, given these priors and assumptions. This BF is not a posterior probability that the two-signal model is true; posterior model odds also require prior model odds. In particular, a large BF against the null does not by itself compare the two-signal model with every alternative model.

To form a 95% credible set for each signal, rank the candidate variants within that signal by posterior probability and include variants until their cumulative probability reaches at least 0.95. In block A, A1 alone has probability about 0.863, below 0.95, so A2 must also be included. In block B, B1 alone has probability about 0.838, also below 0.95, so B2 is included as well. The signal-specific 95% sets are therefore \(\{A1,A2\}\) and \(\{B1,B2\}\). Each set has cumulative posterior probability approximately 1 in this two-candidate example.

If the target were a 90% set instead, A1 alone would still fall short at about 0.863, and B1 alone would fall short at about 0.838. Both sets would still contain both candidates. Credible-set membership is determined by the posterior probabilities within the relevant signal, not by a universal threshold on an individual variant's probability.

Justification for the block calculation

The separation into two block-specific calculations is valid only under the assumptions stated above. Let \(C_A\) denote the causal candidate in block A and \(C_B\) the candidate in block B. If the likelihood for the summary data factors as \(L(Z\mid C_A,C_B)=L_A(Z_A\mid C_A)L_B(Z_B\mid C_B)\), and the prior factors as \(P(C_A,C_B)=P(C_A)P(C_B)\), then Bayes' rule gives:

P(C_A,C_B\mid Z) =\frac{L_A L_B P(C_A)P(C_B)} {\sum_{c_A,c_B}L_A(c_A)L_B(c_B)P(c_A)P(c_B)} =P(C_A\mid Z_A)P(C_B\mid Z_B).

The last equality follows because the denominator factors into \(\left[\sum_{c_A}L_A(c_A)P(c_A)\right] \left[\sum_{c_B}L_B(c_B)P(c_B)\right]\). If cross-block LD is appreciable, the likelihood generally does not factor this way. Then multiplying block-level posterior probabilities can be wrong, and joint modeling with the appropriate LD structure is needed. A diagonal or block-diagonal approximation should be checked against the LD information used in the analysis.

Diagnostics, alternatives, and limitations

Several checks help determine whether this calculation is informative. Confirm that variant identities and alleles align between the summary statistics and the LD reference, that the same variants occur in both, and that the estimated cross-block LD is negligible enough for the factorization to be a reasonable approximation. Confirm that the z-statistics and effect-size prior use compatible scaling. If the candidate list omits a relevant variant, or if there are multiple causal variants within a block, the stated candidate configurations may not describe the data adequately.

A single-causal analysis is simpler and can be useful when there is evidence for one signal, but its posterior probabilities force all candidates to explain the same association. A two-signal analysis allows two variants to be causal at once and yields a separate posterior distribution for each signal. Conditional or joint evidence methods can also assess additional signals while accounting for LD; they are preferable when the block-independence simplification is not justified. The approximate BF calculation here is an instructive summary-statistic analysis, not a substitute for checking the assumptions and the LD matrix.

The main limitation is that model certainty and variant uncertainty are different. The example conditions on exactly one causal variant in each block; its signal-specific probabilities do not determine whether that configuration is more probable than a one-signal or null model. Likewise, a 95% credible set is a posterior-probability statement under a model and prior, not a guarantee that the causal variant is among the measured candidates. If the model, prior, or LD reference is misspecified, posterior probabilities and sets can be misleading.

Worked Example: Spotting a Single-Model Misinterpretation

A colleague notes that A1 has posterior probability 0.6566 under the single-causal model and concludes that it is the sole likely causal variant in the region. What does the calculation actually support?

The value 0.6566 is conditional on the model that exactly one of A1, A2, B1, or B2 is causal. It compares these four alternatives, not the possibility of two causal variants. Under the two-signal model, A1 has probability about 0.863 of being the block-A candidate, while B1 has probability about 0.838 of being the block-B candidate. The joint A1-and-B1 configuration has probability about 0.723 conditional on that two-signal model. Therefore, the single-model result cannot establish that A1 is the sole causal variant; model comparison and its prior odds are needed to assess competing signal counts.

Common pitfall

Wrong reasoning: “A1 has the largest posterior probability in the region, so the other variants can be discarded and there is only one signal.” This treats mutually exclusive alternatives from a single-causal model as though they were candidates in a model that allows multiple signals.

Correction: State which model each probability conditions on. In the single-causal model, the four variants compete to explain one signal. In the two-signal model, candidates compete within each block, and one candidate from each block is selected jointly. Report signal-specific probabilities and credible sets only with their conditioning model and assumptions.

Key takeaway: With independent LD blocks, one causal variant per block, and factorized priors, two-signal posterior probabilities can be obtained from block-specific probabilities and multiplied across blocks. The single-causal and two-signal analyses answer different questions, so their posterior probabilities and credible sets should not be interchanged.

Check Your Understanding

Use the hypothetical values and assumptions in this tutorial. Brief solutions follow the questions.

Why does the two-signal posterior probability for configuration (A1, B1) equal the product of the two block-specific probabilities?
Under the single-causal model, what is the approximate posterior probability that B2 is causal?
Why does A1 alone not make a 95% credible set for the block-A signal?
What feature of the LD structure would make the blockwise product calculation questionable?

Solutions. (1) Both the likelihood and prior factor across blocks, so the posterior factors into the block-specific posteriors; \(0.86295\times0.83755\approx0.7227\). (2) It is approximately \(21.67/557.85=0.0388\), conditional on exactly one causal variant in the region. (3) A1's block-A posterior probability is about 0.863, below 0.95, so A2 must be added to reach the target. (4) Appreciable LD between blocks can prevent the likelihood from factoring, making the product of blockwise probabilities inappropriate.