Reporting Conditional Results Without Causal Overclaim

Why Reporting Requires More Than Naming a Lead Variant

A locus analysis may identify a variant with the strongest marginal association, then find that its estimate changes after conditioning on another correlated variant. A concise report should preserve that distinction. It should say which variants were tested together, what each estimate means in that model, and how uncertain the estimates are. It should also state what the analysis does not establish: a statistically distinct conditional association signal is not, by itself, identification of a causal variant.

The aim is not to list every model that could have been fitted. It is to make the reported result interpretable and reproducible. Readers need the variant and allele definitions, phenotype scale, sample and covariate context, model-selection choices, and relevant limitations from linkage disequilibrium (LD). Without these details, even a numerically accurate coefficient can be misleading.

Specify the Model Before Interpreting the Coefficients

For a quantitative phenotype \(Y_i\) measured on individual \(i\), a joint linear association model for two variants can be written as

Here \(G_{i1}\) and \(G_{i2}\) are the genotype dosages for the specified effect alleles, \(\mathbf{C}_i\) contains the stated covariates, and \(\varepsilon_i\) is the residual. In this model, \(\beta_1\) is the expected difference in phenotype per additional effect-allele copy at variant 1, holding variant 2 and the covariates fixed; \(\beta_2\) has the corresponding interpretation for variant 2. If the phenotype is transformed or standardized, the units refer to that transformed or standardized scale.

For the usual model-based standard errors and tests, the mean model must be appropriately specified, observations must be independent across individuals, and the residual variance assumptions used for the reported inference must be reasonable. For example, conventional linear-model standard errors rely on constant residual variance; robust standard errors can address heteroskedasticity, but do not repair a misspecified mean model or dependence between individuals. The analysis should also state how ancestry or other relevant structure was handled, since unaddressed structure can distort association estimates.

A marginal estimate comes from a model that tests a variant without the other locus variants included. A conditional estimate comes from the joint model. When variants are correlated, these quantities need not be similar: each describes a different model and answers a different statistical question. Report them as such rather than describing one as the “true” effect and the other as a correction.

What a Complete Result Should Report

For each highlighted variant, report enough information to identify the tested allele and reproduce the interpretation. At minimum, include the genome build and variant identifier or genomic position, the effect allele and other allele, the effect-allele frequency when available, the sample size, and the phenotype scale. State the model covariates and whether the estimate is marginal or conditional. For a quantitative phenotype, report the coefficient with its units, standard error, confidence interval, and p-value. For a binary phenotype, report the effect measure and its scale, such as a log odds ratio or odds ratio, with its uncertainty and the relevant model specification.

A confidence interval conveys the range of parameter values compatible with the data and model at the stated confidence level. For example, a 95% confidence interval is produced by a procedure that, under repeated sampling and the model assumptions, covers the fixed parameter in 95% of repetitions. It is not a 95% probability statement about this one interval containing the parameter. A p-value describes how incompatible the observed test statistic is with the null hypothesis under the test assumptions; it is not the probability that the null hypothesis is true.

When a lead variant was chosen from a scan, say how it was selected. If the variant was selected because it had the smallest p-value among many tested variants, its reported effect and p-value are conditional on a data-driven selection process unless the inference explicitly accounts for that process. A nominal p-value from a selected model should not be presented as if the model had been specified in advance. Distinguish discovery-stage results from estimates obtained in an independent validation sample, when such validation exists.

Short Justification: Why the Interval Uses the Standard Error

For a coefficient estimate \(\hat{\beta}\) that is approximately normally distributed around the model parameter \(\beta\), with estimated standard error \(SE\), the standardized statistic \((\hat{\beta}-\beta)/SE\) is approximately standard normal. The central 95% of a standard normal distribution lies between \(-1.96\) and \(1.96\). Rearranging the inequality

for \(\beta\) gives

This produces the approximate 95% confidence interval used above. The approximation depends on the estimator’s sampling distribution and the standard error being suitable for the model. In a small sample or under a different inferential method, the appropriate critical value or interval construction may differ.

Model Choice, LD, and the Meaning of “Independent”

A report should describe why variants entered the joint model: for example, whether they were selected by a prespecified threshold, a stepwise procedure, or a sensitivity analysis. Include the number of variants considered and the criteria used where these choices materially affect interpretation. Selection can make estimates and nominal p-values optimistic, particularly when the same data are used both to choose variants and to report their associations. A transparent report labels selected-model results and avoids treating a data-dependent p-value as confirmatory evidence.

LD creates an additional interpretive limit. When genotypes are strongly correlated, the data may contain limited information for distinguishing their separate conditional coefficients. Standard errors can increase, and modest changes in variant inclusion, allele coding, sample, or LD reference can change estimates. The lead variant may be a useful statistical representative of a local association pattern, but its rank does not establish that it is functional or causal. If an analysis uses LD from a reference sample, report its ancestry composition and relevance to the study; mismatched LD can affect summary-statistic conditional estimates.

In this setting, “independent association signals” means signals that retain statistical evidence after conditioning on the other variants included under the specified analysis. It does not mean the variants are biologically independent, nor does it prove that each is causal. A conditional test can separate association patterns to some degree while leaving multiple correlated variants statistically compatible with the observed data. The next question of assigning posterior support to candidate causal variants requires a distinct inferential target; conditional regression alone does not provide that probability.

Common Mistakes and a Practical Reporting Sequence

• Calling the lead variant causal. The variant with the strongest scan statistic is selected by an association criterion. That rank alone does not establish a causal effect.
• Calling conditional signals causally independent. Conditional association means association remains in a particular fitted model. It does not establish separate biological mechanisms or causal variants.
• Reporting a coefficient without its scale. A number such as \(0.045\) is uninterpretable without the phenotype units, allele dosage, and whether the result is marginal or conditional.
• Reporting a nominal p-value after selection as confirmatory. If the same data selected variants and estimated their associations, state that selection and the limits of the nominal inference.
• Ignoring LD-related instability. High correlation can make variant-specific estimates imprecise and sensitive to model specification. Report relevant sensitivity rather than presenting one fit as uniquely decisive.

A practical reporting sequence is to identify the lead variant and its selection rule, define the genotype and phenotype scales, state the joint model and covariates, report conditional estimates with uncertainty, and then describe model-selection and LD limitations. This makes the statistical claim clear while keeping it narrower than a causal claim.