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Privacy-Preserving Data Sharing for Genome-Wide Association Studies
Caroline Uhler, Aleksandra B. Slavkovic, Stephen E. Fienberg
TL;DR
GWAS aggregate data need privacy protection that remains meaningful despite arbitrary external information, while traditional protections do not scale well. The paper applies differential privacy to release MAFs, chi-square statistics, p-values, and association results, finding that direct chi-square perturbation offers the best reported privacy–utility trade-off among tested summary-statistic releases, while sparse data and epistasis require more complex methods.
Problem
Traditional confidentiality methods do not scale well to GWAS data, especially against linkage with arbitrary external information.
Method
The paper develops differentially private releases for MAFs, chi-square statistics, p-values, and relevant SNPs, plus a two-stage private penalized-logistic-regression approach for association discovery.
Results
Directly adding noise to chi-square statistics achieves the best privacy–utility trade-off versus perturbing p-values or cell entries for small-to-moderate counts and overall sample sizes.
Takeaways & Limitations
Differential privacy can support release of some aggregate GWAS data, but simple summary statistics are problematic for bigger, sparse data and insufficient for models integrating across SNPs to detect epistasis.
Takeaways & Limitations
Perturbing p-values introduces too much noise for inference, and the paper omits simulation results on private aggregate MAF utility.
Abstract
from arXiv · showhide
Traditional statistical methods for confidentiality protection of statistical databases do not scale well to deal with GWAS (genome-wide association studies) databases especially in terms of guarantees regarding protection from linkage to external information. The more recent concept of differential privacy, introduced by the cryptographic community, is an approach which provides a rigorous definition of privacy with meaningful privacy guarantees in the presence of arbitrary external information, although the guarantees come at a serious price in terms of data utility. Building on such notions, we propose new methods to release aggregate GWAS data without compromising an individual's privacy. We present methods for releasing differentially private minor allele frequencies, chi-square statistics and p-values. We compare these approaches on simulated data and on a GWAS study of canine hair length involving 685 dogs. We also propose a privacy-preserving method for finding genome-wide associations based on a differentially-private approach to penalized logistic regression.
1 Introduction
The introduction frames GWAS aggregate releases as vulnerable to privacy attacks and motivates differential privacy as a rigorous alternative. It then outlines private release methods for MAFs, chi-square statistics, p-values, and genome-wide association analysis.
- Privacy problem: Homer et al. showed that aggregate minor allele frequencies could potentially be used to identify whether an individual’s genotype appears in a test population.The approach compared the individual’s MAFs with reference and test-population distributions using a t-test.
- Privacy problem: GWAS aggregate data combine genotype information with disease status, creating a privacy concern when case and control summaries are publicly released.The introduction describes HapMap as a possible reference population and GWAS cases as a test population.
- Prior work: Homer et al.’s attack depended on same-population and SNP-independence assumptions that are usually unmet, producing high false-positive rates.Despite these limitations, NIH removed aggregate MAFs, chi-square statistics, and p-values from open-access databases.
- Proposed approach: The paper proposes differential privacy to release aggregate GWAS data with rigorous guarantees in the presence of arbitrary external information.The stated goal is to avoid compromising an individual’s privacy while bypassing disputes about the attack’s validity.
- Proposed approach: The contributions include private case-control MAF release, ϵ-differentially private chi-square statistics and p-values, and private SNP selection.The paper also adapts penalized logistic regression so both stages of a two-stage epistasis-search procedure satisfy ϵ-differential privacy.
- Evaluation: The methods are evaluated through simulations and a canine hair-length GWAS involving 685 dogs.The paper separately describes methods for releasing private MAFs, chi-square statistics, p-values, and genome-wide associations.
2 Main Definitions and Notation
This section defines the GWAS setting, differential privacy framework, sensitivity, and KL divergence used to analyze private releases. It assumes balanced cases and controls and uses KL divergence to compare original and perturbed chi-square distributions.
- GWAS setting: The GWAS model studies interactions between SNPs and a binary phenotype, with N individuals divided equally into N/2 cases and N/2 controls.The equal group sizes are described as usual in GWAS and necessary for sufficient power to detect disease-associated SNPs.
- Differential privacy: A randomized mechanism is ϵ-differentially private when its output remains controlled for datasets differing in at most one individual.The definition quantifies this condition over datasets and possible outputs.
- Sensitivity: The sensitivity of a function is the smallest bound on how much its output can change when one individual changes in the database.The supplied definition describes functions mapping databases to d-dimensional outputs.
- Laplace mechanism: The Laplace mechanism releases f(D) plus random mean-zero Laplace noise scaled by the function’s sensitivity and privacy parameter ϵ.This release mechanism is stated to satisfy the definition of ϵ-differential privacy.
- Evaluation metric: KL divergence measures the difference between probability distributions, including an original chi-square statistic and its ϵ-differentially private version.The paper uses this divergence to analyze simulation results.
3 Privacy-Preserving Methodology
The methodology releases aggregate GWAS statistics under differential privacy by calibrating noise to statistic sensitivity and accounting for information revealed by ranking significant SNPs. It develops private releases for MAFs, χ2-statistics, and p-values, including an algorithm for the M most relevant SNPs.
- Privacy-Preserving Release of Aggregate MAFs: The averaged MAFs for cases and controls across M SNPs have sensitivity 2M/N, enabling Laplace-noise release with scale 2M/(Nϵ) per table cell.The data are organized as a table of averaged case and control MAFs for M SNPs.
- Privacy-Preserving Release of χ2-Statistics and p-Values: For a 3 × 2 contingency table with N/2 cases and N/2 controls, the χ2-statistic sensitivity is 4N/(N+2).The sensitivity grows with N but approaches a constant, so private χ2-statistics become more accurate as N increases.
- Privacy-Preserving Release of χ2-Statistics and p-Values: The perturbed χ2-statistic has an asymptotic distribution obtained by convolving the χ2 sampling distribution with the Laplace perturbation.The finite-sample distribution is reported to be well approximated by this asymptotic distribution even for low counts.
- Privacy-Preserving Release of χ2-Statistics and p-Values: The p-value sensitivity is exp(−2/3) for the χ2 null distribution with 2 degrees of freedom, and projecting larger p-values to p*=exp(−N/c) reduces sensitivity by a factor of N/2.The projection applies for any fixed constant c ≥ 3.
- Privacy-Preserving Release of χ2-Statistics and p-Values: Because ranking the M most significant SNPs reveals additional information, the method adapts a frequent-itemset release procedure to privately release their χ2-statistics or p-values.Algorithm 1 is stated to satisfy ϵ-differential privacy.
4 Evaluation of Methodology and Results
Evaluation on simulated and canine GWAS data examines the utility of private χ2-statistics and p-values, including asymptotic accuracy, distributional divergence, and association detection. Private χ2-statistics retain useful utility as sample size increases, whereas perturbed p-values are dominated by noise and become uninformative for testing.
- Evaluation data: 685-dog canine GWAS data contained 319 long-haired cases, 364 short-haired controls, and 40,842 SNPs.Simulations used 400 cases, 400 controls, and about 10,000 SNPs per individual.
- Evaluation scope: The evaluation omitted simulation results for private aggregate MAF release and focused instead on private χ2-statistics, p-values, and relevant-SNP release.
- Asymptotic approximation: The private χ2-statistic’s finite-sample distribution was well approximated by its asymptotic distribution even with very small cell or marginal counts.This contrasts with the poor small-sample approximation of the unperturbed χ2-statistic.
- Asymptotic approximation: At total count 10,000, the asymptotic approximation remained inadequate, while at 100,000 it appeared accurate subject to individual-cell-count conditions.
- Private χ2-statistics: KL divergence between original and private χ2-statistics decreased with increasing sample size, supporting useful private χ2-statistic release in a realistic GWAS setting.The divergence also depended on the underlying frequency table and privacy level.
- Private p-values: For private p-values, smaller ϵ increased KL divergence, while larger χ2-statistics reduced the signal-to-noise ratio because p-values became smaller.Projecting perturbed p-values outside [0, 1] to 0 or 1 produced jumps in the divergence plots.
- Private p-values: Perturbed p-values were dominated by projected 0s and 1s, making independence tests not much better than random and preventing reliable Type I and Type II error control.The paper therefore characterizes inference from these perturbed p-values as uninformative.
- Association detection: At ϵ = 0.4, detecting causative SNPs required about 7,500 individuals at MAF=0.4 versus about 10,000 at MAF=0.25.Lower MAF produces sparser tables and requires more individuals at fixed privacy.
5 Extended Work: Differentially-Private Algorithm for Detecting Epistasis
The paper develops a differentially private approach to detecting epistasis that combines private SNP filtering with penalized logistic regression. It perturbs the optimization objective to preserve privacy while enabling an integrated analysis across SNPs.
- Approach: The method addresses epistasis detection through a differentially private version of the standard two-stage pipeline: SNP filtering followed by interaction analysis.Both stages satisfy differential privacy, and the second stage uses penalized logistic regression.
- Model representation: The model encodes disease status and SNP main effects and interactions as feature vectors for penalized logistic regression.SNP genotypes use one-hot encodings, and interaction terms are encoded across genotype combinations.
- Private optimization: The privacy mechanism perturbs the logistic-regression objective with noise drawn from a specified density, replacing the original optimization step.Forward selection and backward deletion based on AIC or BIC remain part of model-size selection.
- Privacy guarantee: Algorithm 2 is ϵ-differentially private under the bounded-feature assumption used for the application.The proof relies on the condition ||x_i||2 ≤ K.
- Algorithm: Algorithm 2 takes individual data vectors and privacy and regularization parameters as input and outputs noisy effects.The algorithm draws the perturbation vector from the density defined in the method.
- Scope: The proposed result supports moving from SNP-by-SNP analysis to an integrated approach without relaxing privacy, but application to actual GWAS data remains ongoing.The paper presents this as a method-level result rather than an empirical GWAS evaluation.
6 Conclusion
The paper shows that differential privacy can support releasing several aggregate GWAS statistics and a private epistasis-detection method. Its evaluation also identifies a privacy–utility trade-off: simple summary-statistic releases become problematic for larger, sparse data, motivating more complex private models.
- Contributions: The paper releases aggregate minor allele frequencies, differentially private χ2-statistics and p-values, and statistics for the most relevant SNPs.These releases are intended to provide genetic data needed by practitioners while protecting individuals.
- Privacy–utility trade-off: For larger and sparse data, releasing simple summary statistics is problematic from both privacy and utility perspectives.The paper therefore turns to more complex methodology for sparse data and models integrating information across SNPs.
- χ2-statistics: Adding noise directly to the χ2-statistic achieves the best privacy–utility trade-off among the compared mechanisms for small-to-moderate counts and overall sample sizes.The comparison is against adding noise to p-values or cell entries.
- Integrated analysis: The paper outlines an ϵ-differentially private penalized logistic-regression algorithm to address sparse data and integrated epistasis detection.The authors suggest that the general strategy may be adaptable to other statistical methods.
- Future work: Future work should consider alternate differential-privacy mechanisms to improve privacy–utility trade-offs for GWAS.The paper identifies these mechanisms as a direction for further research.