Predicting complex diseases: performance and robustness

# Predicting complex diseases: performance and robustness
### Florian Privé, Hugues Aschard, Michael G.B. Blum
### RECOMB-Genetics 2018

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# Introduction

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## Disease architectures

<div class="figure" style="text-align: center">
<img src="figures/disease-archi.gif" alt="&lt;small&gt;Source: 10.1126/science.338.6110.1016&lt;/small&gt;" width="75%" />
Source: 10.1126/science.338.6110.1016
</div>

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## Disease architectures

.footnote2[How to derive a genetic risk score for common diseases based on common variants with small effects?]

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## Interest in prediction: polygenic risk scores (PRS)

- Wray, Naomi R., Michael E. Goddard, and Peter M. Visscher. "**Prediction of individual genetic risk** to disease from genome-wide association studies." Genome research 17.10 (**2007**): 1520-1528.

- Wray, Naomi R., et al. "Pitfalls of **predicting complex traits** from SNPs." Nature Reviews Genetics 14.7 (**2013**): 507.

- Dudbridge, Frank. "Power and **predictive accuracy of polygenic risk scores**." PLoS genetics 9.3 (**2013**): e1003348.

- Chatterjee, Nilanjan, Jianxin Shi, and Montserrat García-Closas. "Developing and evaluating **polygenic risk prediction** models for stratified disease prevention." Nature Reviews Genetics 17.7 (**2016**): 392.

- Martin, Alicia R., et al. "Human demographic history impacts **genetic risk prediction** across diverse populations." The American Journal of Human Genetics 100.4 (**2017**): 635-649.

.footnote2[Still a gap between current predictions and clinical utility.Need more optimal predictions + larger sample sizes.]

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## Standard PRS - part 1: estimating effects

### Genome-wide association studies (GWAS)

In a GWAS, each single-nucleotide polymorphism (SNP) is tested **independently**, resulting in one **effect size** `$\hat\beta$` and one **p-value** `$p$` for each SNP.

Easy combining: `$PRS_i = \sum \hat\beta_j \cdot G_{i,j}$`

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## Standard PRS - part 2: restricting predictors

### Clumping + Thresholding ("C+T" or just "PRS")

`$$PRS_i = \sum_{\substack{j \in S_\text{clumping} \\ p_j~<~p_T}} \hat\beta_j \cdot G_{i,j}$$`

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## A more optimal approach to computing PRS?

In C+T: weights learned independently and heuristics for correlation and regularization.

#### Statistical learning

- joint models of all SNPs at once

- use regularization to account for correlated and null effects

- already proved useful in the litterature (Abraham et al. 2013; Okser et al. 2014; Spiliopoulou et al. 2015)

#### Our contribution

- a memory- and computation-efficient implementation to be used for biobank-scale data

- an automatic choice of the regularization hyper-parameter

- a comprehensive comparison for different disease architectures

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# Methods

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## Penalized Logistic Regression

$$\arg\!\min_{\beta_0,~\beta}(\lambda, \alpha)\left\{ \underbrace{ -\sum_{i=1}^n \left( y_i \log\left(p_i\right) + (1 - y_i) \log\left(1 - p_i\right) \right) }_\text{Loss function} + \underbrace{ \lambda \left((1-\alpha)\frac{1}{2}\|\beta\|_2^2 + \alpha \|\beta\|_1\right) }_\text{Penalization} \right\}$$

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- `$p_i=1/\left(1+\exp\left(-(\beta_0 + x_i^T\beta)\right)\right)$`

- `$x$` is denoting the genotypes and covariables (e.g. principal components),

- `$y$` is the disease status we want to predict,

- `$\lambda$` is a regularization parameter that needs to be determined and

- `$\alpha$` determines relative parts of the regularization `$0 \le \alpha \le 1$`.

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### Efficient algorithm

- sequential strong rules for discarding predictors in lasso-type problems (Tibshirani et al. 2012; Zeng et al. 2017)

- implemented in our R package {bigstatsr}

#### Our implementation

- uses memory-mapping to matrices stored on disk

- functioning choice of the hyper-parameter `$\lambda$`

- early stopping criterion

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### Choice of the hyper-parameter `$\lambda$`

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## Comprehensive simulations: varying many parameters

#### Simulation models

#### Methods

- PRS: 
    - PRS-all (no p-value thresholding), 
    - PRS-stringent (GWAS threshold of significance) and
    - PRS-max (best prediction for all thresholds, considered as an upper-bound)

- logit-simple: penalized logistic regression with automatic of selection `$\lambda$`

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## Predictive performance measures

**AUC** (Area Under the ROC Curve) and partial AUC (FPR < 10%) are used.

`$$\text{AUC} = P(S_\text{case} > S_\text{control})$$`

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# Results

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### Higher predictive performance with logit-simple

.footnote2[Penalized logistic regression provides higher predictive performance in the cases that matter, especially when there are correlated variables.]

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### Predictive performance of C+T method varies with threshold

.footnote2[Recall that prediction of PRS-max is an upper-bound of the prediction provided by the C+T method.]

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### Prediction with logit-simple is improving faster

.footnote2[Performance of methods improve with larger sample size. Yet, penalized logistic regression is improving faster than the C+T method.]

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## Real data

#### Celiac disease

- intolerance to gluten

- only treatment: gluten-free diet

- heritability: 57-87% (Nisticò et al. 2006)

- prevalence: 1-6%

#### Case-control study for the celiac disease (WTCCC, Dubois et al. 2010)

- ~15,000 individuals

- ~280,000 SNPs

- ~30% cases

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### Results: real Celiac phenotypes

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# Discussion

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### Summary of our penalized regression as compared to the C+T method

- A more **optimal** approach for predicting complex diseases

- models that are **linear** and very **sparse**

- very **fast**

- **automatic choice** for the regularization parameter

- can be extended to capture also recessive and dominant effects

### Prospects: future work using the UK Biobank

- use of external summary statistics to improve models

- better generalization to multiple populations

- integration of clinical and environmental data

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# Thanks!

Presentation: https://privefl.github.io/thesis-docs/recomb18.html

R package {bigstatsr}: https://github.com/privefl/bigstatsr

R package {bigsnpr}: https://github.com/privefl/bigsnpr

[privefl](https://twitter.com/privefl) &nbsp;&nbsp;&nbsp;&nbsp; [privefl](https://github.com/privefl) &nbsp;&nbsp;&nbsp;&nbsp; [F. Privé](https://stackoverflow.com/users/6103040/f-priv%c3%a9)

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## Real genotype data

Use real data from a case-control study for the Celiac disease.

Keep only **controls** from the **UK** and **not deviating from the robust Malahanobis distance**.

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## Simulate new phenotypes

### The liability-threshold model

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### Two models of liability

#### A "simple" model

`$$y_i = \underbrace{\sum_{j\in S_\text{causal}} w_j \cdot \widetilde{G_{i,j}}}_\text{genetic effect} + \underbrace{\epsilon_i}_\text{environmental effect}$$`

#### A "fancy" model

`$$y_i = \underbrace{\sum_{j\in S_\text{causal}^{(1)}} w_j \cdot \widetilde{G_{i,j}}}_\text{linear} + \underbrace{\sum_{j\in S_\text{causal}^{(2)}} w_j \cdot \widetilde{D_{i,j}}}_\text{dominant} + \underbrace{\sum_{\substack{k=1 \\ j_1=e_k^{(3.1)} \\ j_2=e_k^{(3.2)}}}^{k=\left|S_\text{causal}^{(3.1)}\right|} w_{j_1} \cdot \widetilde{G_{i,j_1} G_{i,j_2}}}_\text{interaction} + \epsilon_i$$`

***

- `$w_j$` are **weights** (generated with a Gaussian or a Laplace distribution)
- `$G_{i,j}$` is the **allele count** of individual `$i$` for SNP `$j$`
- `$D_{i,j} = \mathbf{1}\left\{G_{i,j} \neq 0\right\}$`

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### Extension via feature engineering

We construct a separate dataset with, for each SNP variable, two more variables coding for recessive and dominant effects.

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### Feature engineering improves prediction

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### Prediction with logit-simple is improving faster