Fully powered polygenic prediction using summary statistics

Alkes L. Price Harvard T.H. Chan School of Public Health October 7, 2015 To download slides of this talk: google “Alkes HSPH”

Summary statistics are widely available

—Nat Genet editorial, July 2012

Outline 1. A brief history of summary statistic genetics 2. Introduction to polygenic prediction using summary statistics 3. LDpred method for polygenic prediction using summary statistics 4. Application of LDpred to real data sets

Outline 1. A brief history of summary statistic genetics 2. Introduction to polygenic prediction using summary statistics 3. LDpred method for polygenic prediction using summary statistics 4. Application of LDpred to real data sets

Definition of summary statistics Definition: Summary statistics consist of: • GWAS association z-scores for each typed or imputed SNP + • Sample sizes on which z-scores were computed (may vary by SNP) Note: Many applications also require LD information computed from a reference panel (e.g. 1000 Genomes or UK10K) using a population “very similar” to the target sample.

Meta-analysis can be performed using summary statistics

Evangelou & Ioannidis 2013 Nat Rev Genet

Joint and conditional analysis can be performed using summary statistics

Yang et al. 2012 Nat Genet

Imputation can be performed using summary statistics

Lee et al. 2013 Bioinformatics; Pasaniuc et al. 2014 Bioinformatics also see Park et al. 2015 Bioinformatics, Lee et al. 2015 Bioinformatics

Rare variant meta-analysis can be performed using summary statistics

Lee et al. 2013 AJHG; Hu et al. 2013 AJHG; Liu et al. 2014 Nat Genet also see Clarke et al. 2013 PLoS Genet, Tang & Lin 2015 AJHG

Genetic variance and covariance can be inferred using summary statistics

Palla & Dudbridge 2015 AJHG; Bulik-Sullivan et al. 2015 Nat Genet

Functional enrichment can be inferred using summary statistics

Pickrell 2014 AJHG; Kichaev & Pasaniuc 2015 AJHG; Finucane et al. 2015 Nat Genet

Many projects at ASHG 2015 using summary statistics • Invited talks Pickrell, Pasaniuc, Im (this session) • Platform talks 11 Gusev, 77 Cichonska, 220 Golan, 272 Park • Posters 791 Kichaev, 797 Shi, 807 Roytman, 860 Salem, 868 Pare, 1301 Wu, 1334 Zhu, 1357 Chatterjee, 1477 Brown, 1618 Li, 1668 Khawaja, 1686 Lee, 1687 Zhao, 1728 Torres, 1867 O’Connor

Outline 1. A brief history of summary statistic genetics 2. Introduction to polygenic prediction using summary statistics 3. LDpred method for polygenic prediction using summary statistics 4. Application of LDpred to real data sets

Genetic prediction: why care?

Erbe et al. 2012 J Dairy Sci; Goss et al. 2011 New Engl J Med

Using only genome-wide significant SNPs is a Stone Age genetic prediction method How should we conduct genetic prediction, Fred?

ˆ k   ˆi xik i (published SNPs)

φk = phenotype for sample k βi = effect size for SNP i xik = genotype for SNP i, sample k

Prediction r2 is less than half the r2 attained by polygenic prediction PGC-SCZ 2014 Nature; Vilhjalmsson et al. 2015 AJHG

Polygenic prediction can be performed using genome-wide summary statistics

ˆ k   ˆi xik i (all GWAS SNPs)

φk = phenotype for sample k βi = effect size for SNP i xik = genotype for SNP i, sample k

Is polygenic prediction using raw genotypes more accurate than using summary statistics? Answer: slightly.

r 

h h 2 g

2

2 g

h M /N 2 g

using summary statistics: fit each SNP individually

hg2 = heritability explained by SNPs M = number of (unlinked) SNPs N = number of training samples




E (  i | ˆi ) 

hg2

ˆi

hg2  M / N Uniform shrink on estimated effect sizes ˆi is appropriate

Accounting for non-infinitesimal architectures can improve polygenic prediction Non-infinitesimal architecture: (e.g. point-normal mixture, mixture of normals, etc.) Non-uniform shrink on estimated effect sizes ˆi is appropriate

Accounting for non-infinitesimal architectures can improve polygenic prediction 2 Infinitesimal (Gaussian) architecture:  i ~ N 0, hg / M 

ˆi ~  i  N 0,1 / N  =>

E (  i | ˆi ) 

hg2

ˆi

hg2  M / N Uniform shrink on estimated effect sizes ˆi is appropriate

Non-infinitesimal architecture: (e.g. point-normal mixture, mixture of normals, etc.) Non-uniform shrink on estimated effect sizes ˆi is appropriate Standard heuristic approach: P-value thresholding

ˆ k   ˆi xik i

(Note: requires optimization of PT threshold in validation samples)

P-value < PT

Purcell et al. 2009 Nature; Chatterjee et al. 2013 Nat Genet; Dudbridge 2013 PLoS Genet

Accounting for linkage disequilibrium can improve polygenic prediction Problem:ˆ k 

ˆx   i ik

does not account for LD b/t SNPs

i

P-value < PT

Standard heuristic approaches: Random LD-pruning: prune SNPs (e.g. r2 < 0.2), removing one of each pair of linked SNPs (decide randomly which SNP to remove) Informed LD-pruning (LD-clumping): prune SNPs, removing one of each pair of linked SNPs (remove SNP with less significant P-value in training data) Purcell et al. 2009 Nature; Stahl et al. 2012 Nat Genet also see Rietveld et al. 2013 Science (COJO)

Pruning + Thresholding is widely used …

Purcell et al. 2009 Nature; Lango Allen et al. 2010 Nature; Ripke et al. 2011 Nat Genet; Stahl et al. 2012 Nat Genet; Deloukas et al. 2013 Nat Genet; Ripke et al. 2013 Nat Genet; Chatterjee et al. 2013 Nat Genet; Dudbridge 2013 PLoS Genet; PGC-SCZ 2014 Nature

Pruning + Thresholding is widely used, but does not attain maximum prediction accuracy Simulations at different proportions p of causal SNPs:

Non-infinitesimal

Non-infinitesimal

Infinitesimal

Infinitesimal

hg2

Vilhjalmsson et al. 2015 AJHG

Outline 1. A brief history of summary statistic genetics 2. Introduction to polygenic prediction using summary statistics 3. LDpred method for polygenic prediction using summary statistics

4. Application of LDpred to real data sets

LDpred computes posterior means under a point-normal prior, accounting for LD ˆ k   E (  i | ˆi ) xik i

(all GWAS SNPs)

φk = phenotype for sample k βi = effect size for SNP i xik = genotype for SNP i, sample k

where E (  i | ˆi ) are posterior mean effect sizes

Vilhjalmsson et al. 2015 AJHG

LDpred computes posterior means under a point-normal prior, accounting for LD ˆ k   E (  i | ˆi ) xik i

(all GWAS SNPs)

φk = phenotype for sample k βi = effect size for SNP i xik = genotype for SNP i, sample k

where E (  i | ˆi ) are posterior mean effect sizes based on • point-normal prior with 2 parameters: hg2 = heritability explained by SNPs (estimated from training data) p = proportion of causal SNPs (optimized in validation samples) • LD from a reference panel Use validation samples as LD reference (restrict to SNPs with validation data) Vilhjalmsson et al. 2015 AJHG

In the special case of no LD between SNPs, posterior means can be computed analytically E (  i | ˆi ) 

hg2 hg2  Mp / N

pi ˆi

hg2 = heritability explained by SNPs

p = proportion of causal SNPs M = number of (unlinked) SNPs N = number of training samples  ˆi 2

p

where

h / Mp  1 / N 2 g

pi  p

h / Mp  1 / N 2 g

e

2 ( h g2 / Mp 1 / N )

 ˆi 2

e

2 ( h g2 / Mp 1 / N )



1 p 1/ N

e

 ˆi 2 2 (1 / N )

is the posterior probability that  i  0 , i.e. SNP i is causal (generalizes uniform shrink when p = 1: infinitesimal prior, no LD)

In the special case of infinitesimal prior (with LD), posterior means can be computed analytically 1

  M ˆ E (  i | ˆi )   D  I   i 2 Nhg  

hg2 = heritability explained by SNPs

M = number of (unlinked) SNPs N = number of training samples

where D is an LD matrix from a reference panel

(generalizes uniform shrink when D = I: infinitesimal prior, no LD)

General case of non-infinitesimal prior with LD: posterior means cannot be computed analytically

General case of non-infinitesimal prior with LD: posterior means cannot be computed analytically Possible solutions: • Assume 1 causal variant per locus

General case of non-infinitesimal prior with LD: posterior means cannot be computed analytically Possible solutions: • Assume 1 causal variant per locus

• Iterative approach

General case of non-infinitesimal prior with LD: posterior means cannot be computed analytically Possible solutions: • Assume 1 causal variant per locus

• Iterative approach • MCMC

General case of non-infinitesimal prior with LD: posterior means cannot be computed analytically Solution: use MCMC. Initialize i = 0 At each big iteration For each SNP i Re-sample  i based on • Point-normal prior on  i • Observed ˆ ~ N ( D , D / N ) f (  i | ˆ ) ~ f (  i )e





N ˆ   D 2



T

D 1 ( ˆ  D )

, where f (  i ) reflects point-normal prior (based on hg2 and p)

General case of non-infinitesimal prior with LD: posterior means cannot be computed analytically Solution: use MCMC. Initialize i = 0 At each big iteration For each SNP i Re-sample  i based on • Point-normal prior on  i • Observed ˆ ~ N ( D , D / N ) 100 big iterations generally suffice for convergence Rao-Blackwellization: average the posterior means sampled Related MCMC methods for prediction from raw genotypes are described in Erbe et al. 2012 J Dairy Sci, Zhou et al. 2013 PLoS Genet, Moser et al. 2015 PLoS Genet

LDpred performs well in simulations Simulations with real genotypes, 1% of SNPs causal

Understanding polygenic prediction Let’s hide away and dance. -- Freddie K.

Let’s hide away with data. -- Alkes

Outline 1. A brief history of summary statistic genetics 2. Introduction to polygenic prediction using summary statistics 3. LDpred method for polygenic prediction using summary statistics 4. Application of LDpred to real data sets

LDpred performs well on within-cohort prediction of WTCCC traits …

Data from WTCCC 2007 Nature. Results are similar to MCMC-based methods that require raw genotypes: Zhou et al. 2013 PLoS Genet, Moser et al. 2015 PLoS Genet

LDpred performs well on within-cohort prediction of WTCCC traits …

2 2 2 (see Lee et al. 2012 Genet Epidemiol) Rnag  Robs  Rliab

Data from WTCCC 2007 Nature. Results are similar to MCMC-based methods that require raw genotypes: Zhou et al. 2013 PLoS Genet, Moser et al. 2015 PLoS Genet

LDpred performs well on within-cohort prediction of WTCCC traits … Dominated by HLA

Data from WTCCC 2007 Nature. Results are similar to MCMC-based methods that require raw genotypes: Zhou et al. 2013 PLoS Genet, Moser et al. 2015 PLoS Genet

LDpred performs well on within-cohort prediction of WTCCC traits …

Do not validate in new cohort Data from WTCCC 2007 Nature. Results are similar to MCMC-based methods that require raw genotypes: Zhou et al. 2013 PLoS Genet, Moser et al. 2015 PLoS Genet

… but within-cohort prediction accuracy may be too good to be true

2 Rnag CAD Training: WTCCC 0.0451 Validation: WTCCC Training: WTCCC 0.0048 Validation: WGHS

T2D 0.0467 0.0095

Results presented for LDpred; similar relative results for other methods Cryptic relatedness? Population structure? (Wray et al. 2013 Nat Rev Genet)

LDpred performs well on summary statistics with independent validation cohorts

Training N=70K

PGC-SCZ 2014 Nature; MGS replication sample

LDpred performs well on summary statistics with independent validation cohorts

Training N=70K

Training N=30K

Training N=60K

LDpred performs well on summary statistics with independent validation cohorts

Training N=70K

Training N=30K

Training N=70K

Training N=90K

Training N=60K

LDpred performs well on summary statistics with independent validation cohorts Height: complexities due to population stratification. Including PCs can improve prediction accuracy. (Chen et al. 2015 Genet Epidemiol) Training N=130K (Lango Allen et al. 2010 Nature)

Conclusions … • Explicitly modeling both LD and non-infinitesimal architectures improves polygenic prediction from summary statistics. • Polygenic prediction should be evaluated using independent validation cohorts. • Although polygenic predictions are not yet clinically useful, prediction accuracies will increase as sample sizes increase (bounded by heritability explained by SNPs; hg2).

… and Future directions • Polygenic prediction in non-European samples is challenging. How to combine training data from Europeans (large sample size) with training data from target population (small sample size)? (cross-population genetic correlation; Poster 1477 Brown)

• Enrichment of heritability in functional annotation classes could potentially be used to improve polygenic prediction (Poster 1357 Chatterjee) • Methods for large raw genotype data sets (e.g. UK Biobank) should be developed in parallel with summary statistic methods (Platform talk 38 Loh; Platform talk 170 Young)

Acknowledgements Bjarni Vilhjalmsson + Vilhjalmsson et al. 2015 AJHG co-authors

Everyone in alkesgrp. Please check out our other ASHG 2015 talks: • Platform talk 11 Gusev “Large-scale transcriptome-wide association study …” • Platform talk 38 Loh “Contrasting regional architectures of schizophrenia …” • Platform talk 196 Bhatia “Haplotypes of common SNPs explain a large …” • Platform talk 352 Galinsky “Population differentiation analysis of 54,734 …” • Platform talk 346 Hayeck “Mixed model association with family-biased …” • Platform talk 354 Palamara “Leveraging distant relatedness to quantify …”