Why Use Biological Knowledge to Target Interaction Tests?
A genome-wide search can involve an enormous number of locus pairs. Even with 500,000 variants, there are \(500{,}000 \times 499{,}999 / 2 = 124{,}999{,}750{,}000\) distinct pairs. Testing a much smaller set of biologically motivated pairs can make an interaction analysis more feasible and can concentrate power on hypotheses that have a mechanistic rationale. But biological motivation does not, by itself, make a statistical test valid or remove the need to account for multiple testing.
The central design task is to specify how biological knowledge produces the candidate set, which interaction hypotheses will be tested, and which findings could ultimately be reported. If these rules are settled before examining the study’s association results, the analysis is easier to interpret. If the rules depend on the same results being tested, the selection process is part of the inferential problem.
From Biological Rationale to a Testable Family
A pathway can nominate pairs whose genes participate in a shared process. Functional annotations can nominate variants or genes with evidence of relevant molecular activity. A mechanistic hypothesis can nominate a specific pair—for example, one locus thought to alter a protein and another thought to affect a pathway on which that protein acts. These sources can guide which tests are worth doing; they do not establish that the loci interact in the study population.
For each nominated pair, specify the phenotype, the genotype coding, the covariates, and the interaction hypothesis. For example, in a regression model with genotype codes \(G_{Ai}\) and \(G_{Bi}\) for person \(i\), covariates \(C_i\), and an appropriate outcome model, the linear predictor might include main effects and an interaction term:
For a one-degree-of-freedom interaction test, the null hypothesis is \(H_0:\gamma_{AB}=0\). If a pair is represented by several interaction terms, the null hypothesis is that the whole interaction block is zero. The test must be calibrated under a model suitable for the phenotype and study design. In particular, the analysis must handle relevant population structure, relatedness, and other sources of dependence; genotype and phenotype data must be sufficiently reliable; and the chosen test must have a valid null distribution. A pathway label does not repair a misspecified model or sparse, uninformative data.
The family is not necessarily just the final list of pairs. It can also include alternative phenotypes, genotype codings, pathway definitions, ancestries, or model specifications if the analyst may choose among them after seeing results and report the most favorable findings. Write down the decision rules and the set of reportable claims before testing where possible.
Counting and Controlling Tests in a Prespecified Search
Suppose the analysis tests \(m\) hypotheses and uses valid null p-values. A simple family-wise error control is the Bonferroni procedure: test each hypothesis at level \(\alpha/m\), or report its Bonferroni-adjusted p-value, \(\min(1,mp)\). This controls the probability of one or more false rejections in the family at no more than \(\alpha\). It can be conservative, particularly when tests are correlated, but it does not require independence among them.
Here, \(\mathcal{H}_0\) is the set of true null hypotheses in the tested family, and \(A_j\) is the event of rejecting the \(j\)th true null. The first inequality is the union bound. The argument requires each null test to have rejection probability no greater than its assigned level; it does not require the tests to be independent.
A less conservative step-down option is Holm’s procedure. Sort the \(m\) p-values from smallest to largest. Starting with the smallest, compare the \(i\)th p-value to \(\alpha/(m-i+1)\); continue while each comparison passes, and stop at the first failure. Reject the hypotheses before that failure, and do not reject those at or after it. The procedure controls family-wise error under arbitrary dependence when the individual p-values are valid.
Worked Example: Counting Pairs in Prespecified Pathways
Hypothetical study. Before examining phenotype associations, a team specifies three nonoverlapping pathways containing 8, 6, and 5 genes, respectively. For simplicity, assume there is one selected variant per gene and the team will test every pair within each pathway, but no pairs across pathways. The number of pair tests in a pathway with \(k\) variants is \(k(k-1)/2\). Thus the family contains
interaction tests. At family-wise level \(\alpha=0.05\), the Bonferroni threshold for each test is \(0.05/53=0.0009434\). If a pair has p-value \(0.0007\), its adjusted p-value is \(53(0.0007)=0.0371\), so it passes this criterion. If another pair has p-value \(0.0012\), its adjusted p-value is \(53(0.0012)=0.0636\), so it does not pass. These conclusions apply to the specified 53-test family; they do not show that the first pair has a causal or mechanistic interaction.
The count depends on the search rule. If the team also intended to test between-pathway pairs, those tests would need to be included in the family. A pathway-based restriction is meaningful only when it is clear which pairs the restriction includes.
How the Selection Rule Changes the Interpretation
The same final list of pairs can arise through very different selection procedures. A list fixed using external evidence before outcome testing is not equivalent to a list chosen after inspecting association results. The distinction is not that external annotations guarantee truth; it is that selection based on information independent of the study’s test outcomes can, under suitable conditions, leave the null p-values valid for the selected tests.
That qualification matters. An annotation score may be associated with features—such as allele frequency or linkage disequilibrium—that affect calibration or power. Analysts should verify that the testing procedure remains valid for the selected candidates. If the selection uses study phenotypes, study association statistics, or repeated choices among biological definitions based on those results, a correction that counts only the final tests can miss part of the search.
Worked Example: Applying a Prespecified Annotation Rule
Hypothetical study. Before testing interactions with the study phenotype, investigators use functional evidence from an independent source and a fixed scoring rule to choose 12 pairs from 1,000 eligible pairs. Assume the selection does not use this study’s phenotype or association results, and that the pairwise p-values are valid conditional on this selection. They test the 12 pairs and obtain p-values of \(0.0008\) and \(0.0030\) for two candidates.
The prespecified testing family has \(m=12\) hypotheses. The Bonferroni threshold is \(0.05/12=0.004167\). The first p-value has adjusted value \(12(0.0008)=0.0096\); the second has adjusted value \(12(0.0030)=0.0360\). Both meet the family-wise criterion. A p-value of \(0.0060\), if observed for another pair, would have adjusted value \(0.0720\) and would not meet it.
This conclusion depends on the stated selection and calibration conditions. If investigators instead chose the 12 pairs because they looked promising in the same phenotype data, the calculation based only on 12 tests would not generally account for the screening. Nor does independent functional evidence make a statistically detected interaction causal: it supplies a rationale for testing, not proof of mechanism.
Worked Example: Selection After Screening Pathways
Hypothetical study. A team examines 40 pathways using the study data, chooses the pathway with the strongest screening result, and then tests 10 locus pairs in that pathway. One pair has a nominal p-value of \(0.010\). It would be incorrect to argue that this finding is adequately adjusted simply because the selected pathway contains 10 pairs.
If the reportable claim is a pair-level interaction and the full search comprised 40 pathways with 10 candidate pair tests each, a Bonferroni bound over the 400 possible pair claims uses threshold \(0.05/400=0.000125\). The pair’s adjusted p-value is \(\min(1,400(0.010))=1\), so it does not pass that conservative family-wise criterion. This calculation is illustrative: the actual family must match the real search, including any additional choices or tests. If the report also claims that a pathway is associated, pathway-level screening claims need appropriate control too.
A more efficient design is to use one sample to screen and select candidates, then test the fixed candidates in an independent validation sample. In that validation sample, the team can adjust over the prespecified validation family. The validation p-values should not be treated as independent confirmation if the same outcomes or test results influenced which candidates were chosen.
A Practical Design Sequence
A disciplined biology-guided analysis makes the path from rationale to inference explicit. The following sequence separates candidate generation from statistical testing and reporting.
Name the pathway, annotation source, or mechanism and explain what kind of pair it motivates. Avoid treating a broad biological story as evidence that a particular interaction exists.
Specify how genes, variants, and pairs enter the search, including pathway membership, annotation thresholds, and whether between-pathway pairs are eligible. Record whether the rule uses external information or data from the association study.
Specify the phenotype, genetic coding, model, and interaction test for each candidate. Count all hypotheses that could contribute to the claims, including the consequences of any data-dependent selection.
Choose an error-control procedure suited to the family and specify what will be reported. If candidates are selected using study results, use a valid selection-aware procedure or reserve independent data for confirmation.
Common Mistake: “Biologically Plausible” Means “No Correction Needed”
The mistaken reasoning is: “We tested only pairs in a well-motivated pathway, so a nominal p-value below \(0.05\) is sufficient.” The correction is that biological plausibility helps define a focused hypothesis set, but does not change the probability of false positives across that set. If 53 pairs were prespecified, for example, the analysis still needs an inferential strategy that accounts for those 53 tests when making family-level claims. If the pathway itself was selected after looking at study results, the selection process may enlarge the relevant family further.
A targeted search is most useful when its restrictions are scientifically defensible, operationally precise, and fixed independently of the results used for testing. This can reduce the number of tests and improve interpretability, but it does not turn association into causation or establish a molecular mechanism. A strong report states how candidates were selected, how many hypotheses were tested, how multiplicity was handled, and whether the selection used the same data as the interaction tests.
Check Your Understanding
Answer each question by identifying the selection rule and the hypotheses that could contribute to the reported claim.
- A team prespecifies two pathways with 7 and 9 variants and tests all within-pathway pairs. How many pair tests are in the family?
- Why does a prespecified pathway restriction not, by itself, make every nominal p-value below \(0.05\) convincing?
- What is the key inferential difference between selecting pairs using external functional evidence and selecting them using the same study’s phenotype associations?
- A pathway is selected after screening 30 pathways in the study data. Why might adjusting only for the pairs in the selected pathway understate the search?
- What role can an independent validation sample play after a data-driven interaction screen?