Lecture 10

LD Score regression (LDSD)

can be used to calculate chip heritability, genetic correlation between traits, and to attribute the inflation of summary statistics from GWAS to either polygenicity (true causal effect) or confounding due to population structure or other confounders.

Mixed effects models

Unlike the simple GWAS where only the effect of one SNP is considered, mixed effects modeling allows for join modeling of all SNPs.

\[Y = \text{fixed effects} + \sum_j X_k \cdot \beta_k + \epsilon\] It is not possible to estimate the effect sizes of millions of SNPs using a much smaller number of observations. Recall that you need at least as many equations as unknowns in a traditional setting. If we assume that \(\beta_k\)’s are normally distributed with mean 0, then we only need to estimate one parameter: the variance of the \(\beta_k\), \(\sigma_\beta^2\). That’s a huge advantage of using mixed effects modeling.

Connection between EMMAX random effect and the aggregate effect of all SNPs on the trait

\[u = \sum_k \beta_k X_k\]

We demonstrated that the random effect proposed by EMMAX and \(\sum_k \beta_k X_k\) are the same. This fact can be shown by confirming that the variance of \(u\) is the same as the variance of \(\sum_k \beta_k X_k\) (\(\sigma^2 \cdot \mathbf{K}\) where $ $ is the genetic relatedness matrix \(\approx\) correlation matrix of the genotype data \(\mathbf{X}\cdot\mathbf{X}'/M\))

LOCO (Leave One Chromosome Out)

LOCO improves the power of EMMAX by removing the chromosome where the test SNP is in the calculation of the genetic relatedness matrix.

Biobank scale mixed effects methods

With ever increasing sample sizes, methods that can handle a Million individuals is needed. BOLT-LMM AND fastGWA do that by implementing a computationally more efficient versions of EMMAX. In the case of the UK Biobank, these methods allowed the inclusion of related individuals to the analysis, expanding the sample size by 50%.

EMMAX

Kang et al (2010). Variance component model to account for sample structure in genome-wide association studies. Nature Genetics.

boltLMM

Loh et al (2018). Mixed-model association for biobank-scale datasets. Nature Genetics.

SAIGE

Jiang et al (2019). A resource-efficient tool for mixed model association analysis of large-scale data. Nature Genetics.

BSLMM

Zhou X, Carbonetto P, Stephens M (2013) Polygenic Modeling with Bayesian Sparse Linear Mixed Models. PLOS Genetics 9(2): e1003264. https://doi.org/10.1371/journal.pgen.1003264

PRS methods

• Zhu, X., & Stephens, M. (2017). Bayesian large-scale multiple regression with summary statistics from genome-wide association studies. AOAS
• Vilhjálmsson et al. (2015). Modeling Linkage Disequilibrium Increases Accuracy of Polygenic Risk Scores. AJHG
• Mak,et al (2017). Polygenic scores via penalized regression on summary statistics. Genetic Epidemiology, 41(6), 469–480.
• Luke R. Lloyd-Jones (2019). Improved polygenic prediction by Bayesian multiple regression on summary statistics. BioRxiv.
• Ge, T., Chen, CY., Ni, Y. et al. Polygenic prediction via Bayesian regression and continuous shrinkage priors. Nat Commun 10, 1776 (2019). https://doi.org/10.1038/s41467-019-09718-5

Clinical Utility of PRS

Khera et al (2018) Nature Genetics

Issues with transfer across ancestries

Martin, A.R., Kanai, M., Kamatani, Y. et al. Clinical use of current polygenic risk scores may exacerbate health disparities. Nat Genet 51, 584–591 (2019). https://doi.org/10.1038/s41588-019-0379-x