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# Identifying loci affecting trait variability and detecting interactions in genome-wide association studies

## Abstract

Identification of genetic variants with effects on trait variability can provide insights into the biological mechanisms that control variation and can identify potential interactions. We propose a two-degree-of-freedom test for jointly testing mean and variance effects to identify such variants. We implement the test in a linear mixed model, for which we provide an efficient algorithm and software. To focus on biologically interesting settings, we develop a test for dispersion effects, that is, variance effects not driven solely by mean effects when the trait distribution is non-normal. We apply our approach to body mass index in the subsample of the UK Biobank population with British ancestry (n ~408,000) and show that our approach can increase the power to detect associated loci. We identify and replicate novel associations with significant variance effects that cannot be explained by the non-normality of body mass index, and we provide suggestive evidence for a connection between leptin levels and body mass index variability.

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## Data availability

The primary data analyzed in this study come from the UK Biobank. Applications for access can be made on the UK Biobank website.

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## Acknowledgements

This work was supported by Wellcome Trust grant 095552/Z/11/Z to P.D. and grants 090532/Z/09/Z and 20314/Z/16/Z as core support for the Wellcome Trust Centre for Human Genetics. A.Y. was supported by a Wellcome Trust Doctoral Studentship (099670/Z/12/Z) and by the Li Ka Shing Foundation.

## Author information

A.Y. developed the method, led its application to the UK Biobank data, and wrote the paper. F.L.W. was involved in the development and application of the method. P.D. supervised the research and wrote the paper. All work undertaken by F.L.W. was done while F.L.W. was at University of Oxford.

### Competing interests

P.D. is a founder and director of Genomics plc, and a partner of Peptide Groove LLP. The remaining authors declare no competing interests.

Correspondence to Alexander I. Young or Peter Donnelly.

## Integrated supplementary information

1. ### Supplementary Figure 1 Association signal of the additive variance (AV) test for simulated phenotypes with different parameters.

The expected −log10 (P value) of the AV test for different additive and log-linear variance effects of the test locus is indicated by shading. Phenotypes were simulated for 100,000 unrelated individuals (Methods). The test locus had frequency 0.05. To make this plot comparable to Fig. 1, we used the same set of additive effects. As in Fig. 1, the strength of the additive effect is parameterized by the amount of variance explained, h2, if the allele frequency is 0.5. Here the allele frequency is 0.05, so the actual variance explained is 0.19 times the variance explained when the allele frequency is 0.5. The log-linear variance effect is indicated on the y axis and corresponds approximately to the proportional change in phenotypic variance per allele. We have highlighted two regions of parameter space: the area inside the green lines is where the association signal is stronger under the AV test than under the additive test, and the area inside the yellow lines is where the AV test is genome-wide significant (P < 5 × 10−8) but the additive test is not.

2. ### Supplementary Figure 2 Comparison of association signal for the additive variance (AV) and additive tests for different sample sizes.

The association signal when testing for both additive and log-linear variance effects (AV test) compared to testing for only an additive effect (additive test) in simulations. The y axis gives the expected log ratio (base 10) of the P value from the additive test to the AV test for different variance effects of the test SNP (x axis), with values above zero indicating a stronger signal from the AV test. The simulations were performed for sample sizes of 10,000 (red), 50,000 (green), and 100,000 (blue), indicated with the different colored curves. The log ratio is plotted as a crossed box if the expected P value from the additive variance test would pass the standard genome-wide significance threshold of 5 × 10−8, and it is plotted with a triangle if neither of the expected P values from the two tests would pass the significance threshold.

3. ### Supplementary Figure 3 Relationship between additive and variance effects from GIANT meta-analyses.

Estimated additive (x axis) and variance (y axis) effects on BMI are plotted for all genome-wide loci, shaded in proportion to the negative log10 (P value) for an additive effect, up to a maximum of negative log10 (5 × 10−8), the conventional boundary for genome-wide significance. The additive effects are taken from Locke et al. (Nature 518, 197–206, 2015), and the variance effects are taken from Yang et al. (Nature 490, 267–272, 2012). Because of the mean–variance relationship of untransformed BMI, any locus with an additive effect is expected to have a variance effect, even after inverse normal transformation. The red line has slope 0.1071, determined by robust regression of genome-wide variance effects on additive effects, with weights proportional to the inverse square of the standard error of the estimated variance effects.

4. ### Supplementary Figure 4 Relationship between estimated leptin effect and estimated dispersion effect on BMI.

Estimated leptin effect (s.d. change in leptin per allele) (x axis) and dispersion (y axis) effects on BMI are plotted for the top 100 approximately independent SNPs ranked by evidence for a leptin effect (Methods). The leptin effects are taken from Kilpeläinen et al. (Nat. Commun. 7, 2016), and the dispersion effects are taken from our analysis of the UK Biobank. The red line gives the estimated expected dispersion effect for a given leptin effect (Methods).

## Supplementary information

1. ### Supplementary Text and Figures

Supplementary Figures 1–4, Supplementary Tables 1 and 5, and Supplementary Note

3. ### Supplementary Table 2

Genome-wide summary statistics for BMI

4. ### Supplementary Table 3

Summary statistics for genome-wide significant SNPs

5. ### Supplementary Table 4

Summary statistics from the gene-by-environment interaction analysis

## Rights and permissions

• #### DOI

https://doi.org/10.1038/s41588-018-0225-6