Reporting Mixed-Model and Inflation Diagnostics

Why Reporting Is Part of the Analysis

A mixed-model GWAS report should let another analyst determine what was modeled, how relatedness entered the model, and what the calibration diagnostics do—and do not—support. Naming a method or reporting a genomic inflation factor alone is not enough. The relationship matrix depends on sample and variant choices; the fitted model depends on phenotype and covariate definitions; and summary-statistic diagnostics depend on their own assumptions and reference data.

The goal is not to claim that a model has “removed confounding.” A report should make the analysis auditable: give enough detail to reproduce important choices, show estimates and uncertainty, and describe diagnostic patterns without treating them as definitive tests of model validity.

What to Report for the Relationship Matrix

A genomic relationship matrix, denoted \(K\), summarizes genetic similarity among the analyzed individuals. Since its entries depend on how genotypes are encoded and which variants are used, reporting only “a genetic relationship matrix was included” is insufficient. State the genome build, the sample set, variant inclusion and exclusion criteria, allele coding, missing-genotype handling, and the matrix construction and scaling method. Also report whether the variants used to construct \(K\) were pruned or otherwise adjusted for linkage disequilibrium, and how variants with low frequency or poor imputation quality were handled.

For reproducibility, identify whether the matrix was built from the same individuals as the association analysis, and how any exclusions or ordering differences were reconciled. State whether a leave-one-chromosome-out (LOCO) matrix was used for testing variants on each chromosome, or whether a single genome-wide matrix was used. If \(K\) was normalized, report the normalization convention. For example, scaling so that \(\operatorname{tr}(K)/n=1\) should be stated explicitly; another scaling convention may give variance components on a different numerical scale.

A matrix should be symmetric and positive semidefinite, up to numerical tolerance, for the usual Gaussian random-effect covariance model. Reports should describe any numerical processing that changes it, such as eigenvalue truncation, and explain the reason. Such details matter because a changed matrix can change the fitted covariance structure.

State the Mixed Model and Its Fit

For a continuous phenotype, a common single-variant linear mixed model can be written as

Here \(y\) is the phenotype vector, \(X\) is the fixed-effect design matrix, \(\beta\) contains fixed-effect coefficients, \(g_j\) is the genotype dosage vector for variant \(j\), and \(\alpha_j\) is its tested effect. The random genetic effect \(u\) has covariance \(\sigma_g^2K\); the residual vector \(\varepsilon\) has covariance \(\sigma_e^2I\). The usual model assumes the stated Gaussian distributions, the specified covariance structure, and independence of \(u\) and \(\varepsilon\). It also assumes that the fixed effects and phenotype scale have been specified appropriately for the scientific question. For binary outcomes, a report should name the generalized mixed-model method and its link rather than imply that this Gaussian model was fitted.

Report the phenotype definition and units or transformation, the analyzed sample size, fixed covariates, genotype encoding, missing-data handling, and the association test used. Name the software and version, estimation method, convergence criteria, and any model options that affect the fit. Give estimates of \(\sigma_g^2\) and \(\sigma_e^2\), with uncertainty when available, and state how \(K\) was scaled. If reporting a variance ratio such as \(h^2=\sigma_g^2/(\sigma_g^2+\sigma_e^2)\), define it as the model-based ratio under the stated matrix and phenotype conventions; do not present it as a universal, scale-free measure.

Report Association Calibration in Context

Calibration reporting should include the number of variants tested, the test statistic and degrees of freedom, and how variants with missing or unreliable statistics were handled. A quantile–quantile (QQ) plot displays observed against expected test-statistic quantiles under the null. Describe its overall pattern, including whether departures are broad or concentrated in the extreme tail; do not use the plot as a substitute for model and sample description.

For one-degree-of-freedom association tests, the genomic inflation factor \(\lambda\) is commonly defined as the median observed chi-square statistic divided by the median of a central chi-square distribution with one degree of freedom. The latter median is approximately 0.4549. This definition presumes the relevant test statistics are on a comparable one-degree-of-freedom chi-square scale. For other test degrees of freedom or statistic types, use an appropriate null reference and state it.

LD score regression (LDSC) is a complementary summary-statistic diagnostic. In its usual mean model, the expected association chi-square statistic is related to the variant’s LD score through an intercept and a slope. Report the estimate and standard error of each, the number of variants used, the LD score reference panel and its ancestry relevance, variant filters, regression options, and any treatment of sample overlap. The intercept can reflect confounding-related inflation, but is not a definitive estimate of confounding in every setting; the slope reflects polygenic signal under the model’s assumptions. Earlier material on LDSC assumptions and interpretation explains why neither coefficient should be read in isolation.

Combine Diagnostics Without Overclaiming

Lambda, QQ plots, and LDSC answer related but different questions. Lambda compresses the middle of the test-statistic distribution to one number. A QQ plot shows distributional shape, including tail behavior, but does not by itself separate causes of departure. LDSC uses the relationship between chi-square statistics and LD scores under its assumptions; reference-panel mismatch or sample overlap can complicate interpretation. Report all diagnostics used, with sample and variant definitions, rather than selecting whichever one appears most reassuring.

For a mixed-model GWAS, also state whether calibration diagnostics were calculated from the mixed-model association statistics, and whether genomic control correction was applied. Do not imply that a mixed model makes diagnostic reporting unnecessary. Conversely, do not apply a further correction automatically just because \(\lambda\) exceeds 1: first examine the model, trait, sample structure, and diagnostic pattern, and justify any correction.

Common Reporting Mistakes and Failure Modes

A frequent mistake is to report “\(\lambda=1.12\), therefore the analysis is confounded.” The calculation establishes a median inflation relative to a specified null reference, not its cause. The correction is to report the calculation, show the QQ plot, and interpret these alongside the model and relevant LDSC results.

• Omitting matrix construction. A named software package does not reveal which variants, samples, or scaling were used. Report those choices and any nondefault options.
• Reporting variance components without scale. Their numerical values depend on phenotype and \(K\) scaling. State both, and define any variance ratio.
• Giving LDSC coefficients without their context. Provide standard errors, reference panel, filters, and relevant overlap handling; do not label an intercept as a complete confounding estimate.
• Using one summary to dismiss a visible pattern. A modest lambda does not rule out tail-specific departures, and a QQ plot cannot identify their cause. Describe all the evidence actually examined.
• Claiming calibration from model convergence. Convergence means the fitting procedure met its stated numerical criteria; it does not establish that the model assumptions are correct.

Diagnostics can also fail when the analyzed variants do not match the LDSC reference adequately, when test statistics have been filtered in an undocumented way, or when an incorrect null distribution is used for lambda. A clear report names the data behind each diagnostic, making such limitations visible rather than hiding them behind a single result.