The Per-Variant Regression Scan

What a Per-Variant Scan Does

A genome-wide association study may examine hundreds of thousands or millions of genetic variants for association with a phenotype. A per-variant regression scan organizes that task as a sequence of regression analyses: for each variant, fit a model relating the phenotype to that variant, possibly while accounting for specified covariates. The result is a collection of variant-level estimates and tests, not one model that includes every variant simultaneously.

This scan structure is useful because it produces a consistent set of results across the genome. But the phrase “associated variant” must be read in relation to the phenotype, the regression model, the included participants, and the statistical testing procedure. A small p-value is evidence against that variant’s model-specific null hypothesis; by itself, it does not establish that the variant causes the phenotype.

Inputs and the Per-Variant Model

A scan requires a phenotype for each participant, genetic data for the variants to be tested, and a clear rule for which participants and variables enter each fit. Covariates may be included to represent features such as study site or other measured characteristics. The exact model and predictor coding depend on the phenotype and analysis plan; the later material on additive regression, dosage coding, and covariate adjustment develops those choices in detail.

For notation, let \(Y_i\) be the phenotype for participant \(i\), let \(X_{ij}\) be the value supplied to the regression for variant \(j\), and let \(\mathbf{Z}_i\) denote the included covariates. A schematic quantitative-trait model is

Here, \(\beta_j\) is the variant-specific association parameter under the chosen model, and \(\varepsilon_{ij}\) represents variation not explained by the predictors. This equation illustrates the repeated-fit structure; it does not prescribe how a genotype must be encoded. In a quantitative-trait analysis, a common test is \(H_0:\beta_j=0\) against a two-sided alternative \(H_a:\beta_j\ne 0\). Other phenotype types require a regression model appropriate to their outcome scale.

Classical coefficient-based tests rely on assumptions appropriate to the selected regression procedure. For a standard linear-model t-test, the conditional mean must be represented adequately by the model, observations must be independent under the sampling design, and the errors must have the variance structure required for that test; exact small-sample t inference additionally assumes normally distributed errors. Large-sample or robust procedures can alter some inferential requirements, but they do not repair a wrongly specified mean model or biased sampling. The analyst should also verify that the design is identifiable: the predictors must contain enough independent information to estimate the requested parameters.

What the Scan Reports

A result table typically has one row per tested variant and may include the variant identifier, the estimated association, its standard error, a test statistic, a p-value, and the number of participants used. It may also record quality-control or model-fit information. The exact columns depend on the software and analysis plan. An estimate gives direction and scale under the selected model; its standard error expresses sampling uncertainty under that model; and the p-value summarizes how incompatible the observed test statistic is with the specified null distribution when the null holds.

The p-value is not the probability that the null hypothesis is true, nor is it the probability that the result arose “by chance.” It is calculated under a null model and its assumptions. A per-variant p-value also does not automatically account for the large number of variants in a genome-wide scan. The scan must have a stated approach to interpreting evidence across the tested family, such as an appropriate familywise procedure. The earlier material on max-T testing provides one approach for correlated tests; whatever procedure is used, do not confuse a raw per-variant p-value with a multiplicity-adjusted result.

Why Results Can Differ Across Variants

The scan is a repeated analysis, but the actual inputs need not be identical from one fit to the next. A participant can have a missing phenotype or covariate, or genotype data may be unavailable for a particular variant. If the analysis uses only participants with complete data for each fit, the effective sample size can vary by variant. Consequently, estimates and p-values can differ partly because they are based on different participants, not solely because the underlying associations differ.

Before interpreting a table, confirm how missing data were handled and whether the reported sample size is consistent with that rule. Check that phenotype and genotype records align to the same participant identifiers, that each tested predictor has usable variation, and that each fitted model is estimable. Inspect distributions and residual diagnostics appropriate to the regression model. Extreme values, strong heteroskedasticity, influential observations, or poor fit can make a conventional standard error or p-value unreliable. Genomic structure and related participants can also violate independence assumptions if they are not addressed by the analysis design.

Deriving the Repeated-Testing Consequence

Even if every per-variant test is calibrated at a nominal level \(\alpha\), running many tests changes the chance of seeing at least one false positive. Under the simplifying assumptions that all \(m\) null hypotheses are true and the tests are independent, each test has probability \(1-\alpha\) of not rejecting its null. The probability that none rejects is therefore \((1-\alpha)^m\), so the probability of one or more rejections is

For example, with \(m=20\) and \(\alpha=0.05\), this probability is \(1-0.95^{20}\approx0.642\). Thus, a per-variant threshold alone does not control the chance of any false rejection across a scan. Genomic tests are often correlated, so this independence calculation is not generally an exact scan-wide error rate; it illustrates why the interpretation must state what testing family and error-rate procedure were used.

Alternatives, Diagnostics, and Limitations

A per-variant scan differs from fitting one model that includes many variants simultaneously. The scan asks, variant by variant, whether each predictor is associated with the phenotype under its own specified model. A joint model instead estimates effects conditional on the other included variants, and may address a different scientific question. Joint models can be difficult to fit or interpret when predictors are highly correlated, and they do not make the scan’s per-variant results interchangeable with conditional effects. Resampling approaches can provide reference distributions that reflect dependence among tests, but they require a valid resampling scheme for the design and tested hypotheses.

A scan also has limits beyond multiple testing. Its findings depend on the chosen phenotype definition, participant inclusion rules, predictor encoding, regression family, covariates, and assumptions. A statistically strong association can arise because the variant is correlated with a causal factor or with population structure; it is not proof of a direct biological mechanism. A weak p-value does not prove absence of association, especially when the estimate is imprecise or the variant has limited usable data.

Common warning signs include unexpected sample-size changes, predictors with little or no variation, convergence failures, implausible estimates, unusual standard errors, and p-value patterns inconsistent with the assumed null calibration. These signs call for investigation of data alignment, model specification, and assumptions—not automatic deletion of inconvenient results. A transparent scan report states the phenotype, per-variant model, sample-size rule, tested family, and the distinction between unadjusted and scan-level evidence.