GWIS
Summary¶
GWIS (genome-wide inferred statistics) approximates the GWAS summary statistics of a variable V = f(P1, ..., Pk) that is a function of k phenotypes, using only each phenotype's existing GWAS summary statistics, phenotypic means, and covariance matrix — without requiring V, or even all of P1...Pk, to be measured in any single cohort. It works via a first-order (or second-order) Taylor linearization of f around each genotype-conditional phenotype mean, combined with the delta method to derive standard errors, and requires no sample overlap between the constituent GWASs [1].
Method¶
For a SNP with S ~ Bin(2, q) effect alleles, GWIS approximates E[V | S = s] by linearizing f around E(s), the phenotype means conditional on genotype dosage s. This yields closed-form estimators for the GWIS intercept λ0 (f evaluated at the phenotype intercepts) and slope λ1 (a weighted combination of the partial derivatives of f with respect to each constituent phenotype, evaluated using each phenotype's GWAS effect sizes). Standard errors are obtained via the delta method applied to an approximate covariance matrix of the constituent regression coefficients, which under partial or no sample overlap is estimated from each phenotype's standard errors and their (LD-score-regression-estimated, if unknown) phenotypic or genetic correlation. A second-order Taylor approximation (derived for BMI in the paper's Appendix A) trades additional complexity for higher power at moderate sample sizes.
Validation¶
- BMI replication: reconstructing a BMI GWAS from independent height and weight GWAS summary statistics, GWIS replicated 310 of 356 genome-wide significant hits from the true BMI GWAS (87.1%) with 3 false positives, when height and weight samples fully overlapped. Using an independent (female) height sample in place of the (male) height/weight cohort's own height data, replication dropped to 135/356 (37.9%) with no false positives — sample overlap between constituent GWASs materially affects GWIS power but not its type I error control.
- Simulations: type I error rate matched a standard GWAS under correctly and mildly misspecified population parameters. With full sample overlap, GWIS power matched GWAS power; with zero sample overlap between constituent phenotypes, power fell by roughly 15–20% at moderate effect and sample sizes, with no inflation of type I error even when constituent GWAS sample sizes differed by two orders of magnitude.
- Genetic correlation concordance: LD score regression genetic correlations between BMI-GWIS and external traits (rheumatoid arthritis, age at menarche, LDL cholesterol, educational attainment, coronary artery disease) closely tracked the correlations obtained from the true BMI GWAS (e.g., BMI GWAS vs. educational attainment: -0.160 [0.047]; BMI GWIS male-height vs. educational attainment: -0.145 [0.053]).
Application: Decomposing Schizophrenia and Bipolar Disorder Liability¶
As a more complex demonstration, GWIS was used to define "unique SCZ" and "unique BIP" — the residual genetic liability for schizophrenia (SCZ) and bipolar disorder (BIP) after removing the part of each shared with the other, using coheritability and heritability estimated by LD score regression. Genetic correlations from these decomposed GWISs showed that bipolar disorder's unique liability, but not schizophrenia's unique liability, is genetically correlated with educational attainment (unique BIP–EA r = 0.218 [0.102]; unique SCZ–EA r = 0.041 [0.082]), while raw SCZ and BIP GWASs are both correlated with educational attainment. This indicates the previously reported schizophrenia–educational-attainment genetic correlation is likely induced by schizophrenia's genetic overlap with bipolar disorder, rather than a direct relationship.
Use Cases¶
Because a GWIS produces per-SNP effect estimates and standard errors just like a GWAS, its output can feed directly into polygenic risk scoring, LD score regression, Mendelian randomization, and biological annotation pipelines. It is also proposed for boosting meta-analysis power when some cohorts measured only the constituent phenotypes of a composite trait rather than the composite itself (e.g., weight and height but not BMI), and for traits defined by (bio)chemical equations such as steady-state reaction kinetics or membrane transport, where GWAS summary statistics exist for concentrations on either side of a barrier but not for the derived quantity.
Limitations¶
GWIS accuracy depends on the quality of the constituent GWAS summary statistics, degrades as more constituent phenotypes are combined (error accumulates), and is sensitive to the degree of approximation used (second-order improves on first-order but adds complexity). GWIS requires each constituent phenotype's regression intercept, though substituting the population mean works reasonably well in practice; SNPs with low minor allele frequency or poor imputation quality should be removed from constituent GWASs before applying GWIS.
Availability¶
A heuristic implementation is provided at sites.google.com/site/mgnivard/gwis.
See Also¶
- N-GWAMA / MA-GWAMA — a related multivariate summary-statistics method from an overlapping research group: it meta-analyzes several already-measured, genetically correlated phenotypes, whereas GWIS infers the GWAS of a phenotype defined as a function of other phenotypes.