LDpred2 @WCPG2023

<br>

# Recent and Future Updates to LDpred2 for Polygenic Scores and Inference

<br>

## WCPG 2023

<br>

## Florian Privé (Aarhus Uni, DK)
### <svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg">  <path d="M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z"></path></svg> <svg viewBox="0 0 496 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg">  <path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"></path></svg> privefl

---

# Nothing to disclose

---

### Overview of what LDpred2-auto can now provide

<br>

---

### Prior model: spike and slab

<br>

LDpred2 assumes the following model for effect sizes,

<div class="math">
\[
\beta_j = S_j \gamma_j \sim \left\{
\begin{array}{ll}
\mathcal N\left(0, \dfrac{h^2}{M p}\right) & \mbox{with probability } p,\\
0 & \mbox{otherwise,}\end{array}
\right.
\]
</div>

where 
- `\(p\)` is the proportion of causal variants (aka polygenicity), 
- `\(M\)` the number of variants, 
- `\(h^2\)` the (SNP) heritability, 
- `\(\boldsymbol{\gamma}\)` the effect sizes on the allele scale,
- `\(\boldsymbol{S}\)` the standard deviations of the genotypes,
- `\(\boldsymbol{\beta}\)` the effects of the scaled genotypes.

<br>

LDpred2 uses a Gibbs sampler to sample causal variants and their effects.    
LDpred2-auto directly estimates `\(h^2\)` and `\(p\)` from the Gibbs sampler.

---

### Extended 3-parameter model (for LDpred2-auto)

<br>

<div class="math">
\[
\beta_j = S_j \gamma_j \sim \left\{
\begin{array}{ll}
\mathcal N\left(0, ~\sigma_\beta^2 \cdot (S_j^2)^{(\alpha + 1)} \right) & \mbox{with probability } p,\\
0 & \mbox{otherwise.}
\end{array}
\right.
\]
</div>

<br>

- similar to the model assumed in SBayesS

- previous 2-parameter model assumes `\(\alpha = -1\)` and `\(\sigma_\beta^2 = \frac{h^2}{M p}\)`

- `\(\sigma_\beta^2\)` and `\(\alpha\)` are estimated using maximum likelihood estimation (MLE)

<br>

---

### Inference with LDpred2-auto

<br>

Recent work on:

- properly validating the inference of `\(h^2\)`, `\(p\)`, and `\(\alpha\)` (and their CIs)    
using extensive simulations

- showing calibrated per-variant probabilities of being causal

- inferring the out-of-sample predictive performance `\(r^2\)`    
directly from the Gibbs sampler

<br>

---

### Inferred `\(r^2\)` estimates vs the ones from a test set

---

### An extended set of variants for LDpred2

<br>

- We have recommended to use a set of 1,054,330 HapMap3 variants
  - good coverage of the genome 
  - generally well imputed and available in most studies

- We now provide an extended set with 37% more variants
  - designed to maximize tagging of 11.5M common variants    
  in diverse genetic ancestries
  - called HapMap3+

- Using this new set of variants, in UK Biobank analyses, *on average*,
  - we capture 12% more SNP heritability `\(h^2\)`
  - obtain 6% more predictive performance `\(r^2\)`

<br>

---

### UKBB results with HapMap3+ (1.4M) vs HapMap3 (1M)

---

### Overview of what LDpred2-auto can now provide

<br>

---

### Future development

<br>

- Design automated decisions for choosing parameters such as `use_MLE`
    
- Provide means for enhanced quality control of GWAS summary statistics

- Extend LDpred2-auto for

- using more variants 
    
    - incorporating functional annotations
    
    - multi-ancestry prediction and inference 
    
- and for (smaller priority):

- using multiple phenotypes and estimating genetic correlation 
    
    - imputing GWAS summary statistics

---

### Acknowledgments

#### Co-authors:

- Bjarni J. Vilhjálmsson (Aarhus Uni, DK)

- Julyan Arbel (INRIA Grenoble, FR)

- Hugues Aschard (Pasteur Institute, FR)

- Bogdan Pasaniuc (UCLA, CA, USA)

- Yi Ding (UCLA)

- Clara Albiñana (Aarhus Uni)

#### Funding:

- Danish National Research Foundation    
(Niels Bohr Professorship to Prof. John McGrath)

- Lundbeck Foundation    
(Fellowship to Prof. B.J. Vilhjálmsson)

---

# Thanks!

<br>

Presentation available at [bit.ly/ldpred2_wcpg2023](https://bit.ly/ldpred2_wcpg2023)

<br>