Reading QQ Plots After Mixed-Model Adjustment

What a QQ Plot Shows After Mixed-Model Adjustment

A mixed-model association scan produces one test result for each genetic variant. A QQ plot puts those results in rank order and compares their observed p-values with the p-values expected if the tests followed their null distributions. The plot is useful because its shape shows whether departures are spread across much of the distribution or concentrated among the most extreme results.

In this setting, association p-values come from tests that account for relatedness and other modeled structure. A correctly specified mixed model can improve calibration, but the label “mixed-model adjusted” does not guarantee that every p-value is valid. The QQ plot is a diagnostic of the resulting distribution, not a proof that the model assumptions hold or a test that identifies the cause of any departure.

If all tests are null and their p-values are correctly calibrated and continuous, each p-value has a Uniform(0,1) distribution. This means that, before seeing the data, p-values are expected to be spread evenly across the interval from zero to one. For example, about one percent of valid null p-values are expected to be below 0.01. The statement concerns a repeated-sampling expectation; it does not require the observed proportion in one scan to equal exactly one percent.

The tests in a genomic scan are generally not independent: nearby variants can be correlated through linkage disequilibrium. Dependence changes how much the whole plot fluctuates from one data set to another. It does not, by itself, change the marginal uniform null distribution of each correctly calibrated p-value. Consequently, uncertainty bands based on independent tests can be too narrow for a genome-wide scan.

Why the Null Reference Has This Shape

The expected ranks under the null give a precise reference for the plot. Let \(p_{(i)}\) be the \(i\)th smallest of \(m\) independent Uniform(0,1) p-values. Its distribution is Beta with parameters \(i\) and \(m+1-i\), so its expected value is \(i/(m+1)\). This is why that quantity is often used as the expected p-value at rank \(i\).

A short justification follows from counting how many p-values fall below a proposed value \(u\). For \(p_{(i)}\) to be at most \(u\), at least \(i\) of the \(m\) p-values must be at most \(u\). Under independence, that count has a Binomial\((m,u)\) distribution. The resulting order-statistic density is proportional to \(u^{i-1}(1-u)^{m-i}\), the Beta\((i,m+1-i)\) density. Its mean is \(i/(m+1)\).

The variance describes the spread of the p-value at an individual rank under the independent null model. A plotted reference usually transforms the expected p-value, rather than calculating the exact expectation of the transformed value: in general, \(E[-\log_{10}(p_{(i)})]\) is not equal to \(-\log_{10}\{E[p_{(i)}]\}\). The distinction is usually unimportant for a basic visual diagnostic, but it matters if exact expected transformed values or formal bands are required. For genome-wide data, dependence must also be considered when constructing bands; a simple independent-test band may give a misleading impression of how unusual the overall curve is.

Reading Broad Shifts and Tail Enrichment

A broad upward departure occurs when much of the observed curve lies above the null reference. On the usual \(-\log_{10}(p)\) axes, this means observed p-values are smaller than their corresponding expected values across a wide range of ranks, not just at the very end. A broad departure is a reason to examine calibration and model fit. It is not, by itself, evidence for any one explanation: the plot does not determine whether the pattern reflects residual structure, widespread genuine association, an issue in the test implementation, or some combination.

A curve that follows the reference through most of its range and bends upward only among the smallest p-values has tail enrichment. This describes where the departure occurs; it does not establish that every extreme result is a true association. A few extreme points can also arise by chance, and technical or model problems can affect particular tests. The important visual distinction is between a shift through much of the distribution and a departure concentrated near its extreme end.

Interpretation is most reliable when the plot is read as a curve rather than reduced to one summary. The genomic inflation factor \(\lambda\), introduced in the earlier material on genomic inflation, compresses aspects of test-statistic inflation into a scalar. It does not show which ranks depart from expectation. A QQ plot retains that rank information, but does not provide a single calibrated measure of the cause or severity of a departure.

Assumptions, Diagnostics, and Failure Modes

The null reference is meaningful only if the p-values being compared are calculated for the relevant null hypotheses and are calibrated under those hypotheses. For a mixed-model GWAS, this requires that the association test’s stated null distribution be appropriate and that the model used to account for relatedness and fixed covariates adequately represent the data for that test. For example, a one-degree-of-freedom test based on a chi-square reference requires the test statistic to follow that reference distribution under the null, at least to the accuracy needed for the analysis. Misestimated covariance structure, omitted covariates, problematic genotype data, or an incorrect test reference can all undermine calibration.

Useful checks include confirming that the p-values are finite, lie between zero and one, and correspond to the intended test; checking that the plotted number of tests matches the scan; and reviewing whether the axes and ranking convention are as expected. Inspect results by chromosome, allele frequency, imputation quality, or test type when those groupings are relevant. A discrepancy confined to one group may point to a localized data or analysis issue, whereas a curve shifted throughout the distribution warrants a broader calibration review. These checks identify where to investigate; they do not, on their own, diagnose a cause.

A QQ plot can also mislead when viewed without its scale or without attention to the number of tests. With very many tests, small differences from the reference can become visually apparent. At the other extreme, a sparse tail may contain few points, so its apparent shape can change substantially with individual results. Pointwise bands and global bands answer different questions: a pointwise band concerns a particular rank, while a global band concerns the curve as a whole. Under linkage disequilibrium, bands that assume independent tests may not correctly describe either type of uncertainty.

The main alternatives answer different questions. The inflation factor \(\lambda\) provides a compact scalar summary but can conceal whether departure is broad or tail-specific. A QQ plot shows the rank pattern but is mainly descriptive and depends on the validity of its reference. A histogram can show an overall excess of small p-values, but it discards rank-by-rank comparison with the expected quantiles. None of these displays replaces checking the model assumptions, examining the data, or interpreting individual associations with appropriate context.

Common Mistake

A common mistake is to say, “The QQ plot bends upward, so the mixed model failed,” or, conversely, “The model included relatedness, so the tail must be genuine association.” Both conclusions overstate what the display establishes. An upward departure means that observed p-values are smaller than their null reference at the affected ranks. A broad shift is a reason to investigate calibration and other explanations; tail enrichment describes a departure concentrated among extreme results. Neither shape alone distinguishes residual confounding from genuine signal, nor does it prove that the mixed-model assumptions are satisfied.