Tweedie’s Formula and Selection Bias

High-dimensional prediction of binary outcomes in the presence of between-study heterogeneity

Many prediction methods have been proposed in the literature, but most of them ignore heterogeneity between populations. Either only data from a single study or population is available for model building and evaluation, or when data from multiple studies make up the training dataset, studies are pooled before model building. As a result, prediction models might perform less than expected when applied to new subjects from new study populations. We propose a linear method for building prediction models with high-dimensional data from multiple studies. Our method explicitly addresses between-population variability and tends to select predictors that are predictive in most of the study populations. We employ empirical Bayes estimators and hence avoid selection bias during the variable selection process. Simulation results demonstrate that the new method works better than other linear prediction methods that ignore the between-study variability. Our method is developed for classification into two groups.

Interval estimation of a population mean using existing knowledge or data on effect sizes

Bayes or empirical Bayes methods to improve inferential accuracy for a population mean has been widely adopted in medical research. As the joint prior distribution of both the mean and variance parameters can be difficult to specify or estimate, most of these methods have relied on certain level of simplifications of the joint prior, which could lead to difficulty in the interpretation of the posterior distribution or compromised inferential accuracy. We propose a framework of interval estimation using existing knowledge or data on the effect size to address this difficulty. Our method has two unique characteristics. First, the interpretation of the interval bears the spirit of both Frequentist and Bayesian thinking. For this reason, it will be called FB interval. Second, we define a new quantity, the hybrid effect size, which is a key quantity that mediates the construction of the FB interval when the population variance is unknown. A simulation study and a real data example are presented to evaluate and illustrate our method.

Exploring the predictive power of polygenic scores derived from genome-wide association studies: a study of 10 complex traits

Motivation

It is hoped that advances in our knowledge in disease genomics will contribute to personalized medicine such as individualized preventive strategies or early diagnoses of diseases. With the growth of genome-wide association studies (GWAS) in the past decade, how far have we reached this goal? In this study we explored the predictive ability of polygenic risk scores (PRSs) derived from GWAS for a range of complex disease and traits.

Results

We first proposed a new approach to evaluate predictive performances of PRS at arbitrary P -value thresholds. The method was based on corrected estimates of effect sizes, accounting for possible false positives and selection bias. This approach requires no distributional assumptions and only requires summary statistics as input. The validity of the approach was verified in simulations. We explored the predictive power of PRS for ten complex traits, including type 2 diabetes (DM), coronary artery disease (CAD), triglycerides, high- and low-density lipoprotein, total cholesterol, schizophrenia (SCZ), bipolar disorder (BD), major depressive disorder and anxiety disorders. We found that the predictive ability of PRS for CAD and DM were modest (best AUC = 0.608 and 0.607) while for lipid traits the prediction R-squared ranged from 16.1 to 29.8%. For psychiatric disorders, the predictive power for SCZ was estimated to be the highest (best AUC 0.820), followed by BD. Predictive performance of other psychiatric disorders ranged from 0.543 to 0.585. Psychiatric traits tend to have more gradual rise in AUC when significance thresholds increase and achieve the best predictive power at higher P -values than cardiometabolic traits.

Contact

hcso@cuhk.edu.hk ; pcsham@hku.hk.

Supplementary information

Supplementary data are available at Bioinformatics online.

Using local multiplicity to improve effect estimation from a hypothesis-generating pharmacogenetics study

We propose a multiple estimation adjustment (MEA) method to correct effect overestimation due to selection bias from a hypothesis-generating study (HGS) in pharmacogenetics. MEA uses a hierarchical Bayesian approach to model individual effect estimates from maximal likelihood estimation (MLE) in a region jointly and shrinks them toward the regional effect. Unlike many methods that model a fixed selection scheme, MEA capitalizes on local multiplicity independent of selection. We compared mean square errors (MSEs) in simulated HGSs from naive MLE, MEA and a conditional likelihood adjustment (CLA) method that model threshold selection bias. We observed that MEA effectively reduced MSE from MLE on null effects with or without selection, and had a clear advantage over CLA on extreme MLE estimates from null effects under lenient threshold selection in small samples, which are common among 'top' associations from a pharmacogenetics HGS.

SLOPE-ADAPTIVE VARIABLE SELECTION VIA CONVEX OPTIMIZATION

We introduce a new estimator for the vector of coefficients β in the linear model y = Xβ + z, where X has dimensions n × p with p possibly larger than n. SLOPE, short for Sorted L-One Penalized Estimation, is the solution to [Formula: see text]where λ₁ ≥ λ₂ ≥ … ≥ λ _p ≥ 0 and [Formula: see text] are the decreasing absolute values of the entries of b. This is a convex program and we demonstrate a solution algorithm whose computational complexity is roughly comparable to that of classical ℓ₁ procedures such as the Lasso. Here, the regularizer is a sorted ℓ₁ norm, which penalizes the regression coefficients according to their rank: the higher the rank-that is, stronger the signal-the larger the penalty. This is similar to the Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B57 (1995) 289-300] procedure (BH) which compares more significant p-values with more stringent thresholds. One notable choice of the sequence {λ _i } is given by the BH critical values [Formula: see text], where q ∈ (0, 1) and z(α) is the quantile of a standard normal distribution. SLOPE aims to provide finite sample guarantees on the selected model; of special interest is the false discovery rate (FDR), defined as the expected proportion of irrelevant regressors among all selected predictors. Under orthogonal designs, SLOPE with λ_BH provably controls FDR at level q. Moreover, it also appears to have appreciable inferential properties under more general designs X while having substantial power, as demonstrated in a series of experiments running on both simulated and real data.

Frequentist accuracy of Bayesian estimates

In the absence of relevant prior experience, popular Bayesian estimation techniques usually begin with some form of "uninformative" prior distribution intended to have minimal inferential influence. Bayes rule will still produce nice-looking estimates and credible intervals, but these lack the logical force attached to experience-based priors and require further justification. This paper concerns the frequentist assessment of Bayes estimates. A simple formula is shown to give the frequentist standard deviation of a Bayesian point estimate. The same simulations required for the point estimate also produce the standard deviation. Exponential family models make the calculations particularly simple, and bring in a connection to the parametric bootstrap.

Estimation of treatment effect in a subpopulation: An empirical Bayes approach

It is well recognized that the benefit of a medical intervention may not be distributed evenly in the target population due to patient heterogeneity, and conclusions based on conventional randomized clinical trials may not apply to every person. Given the increasing cost of randomized trials and difficulties in recruiting patients, there is a strong need to develop analytical approaches to estimate treatment effect in subpopulations. In particular, due to limited sample size for subpopulations and the need for multiple comparisons, standard analysis tends to yield wide confidence intervals of the treatment effect that are often noninformative. We propose an empirical Bayes approach to combine both information embedded in a target subpopulation and information from other subjects to construct confidence intervals of the treatment effect. The method is appealing in its simplicity and tangibility in characterizing the uncertainty about the true treatment effect. Simulation studies and a real data analysis are presented.

Two modeling strategies for empirical Bayes estimation

Empirical Bayes methods use the data from parallel experiments, for instance observations Xk ~ 𝒩 (Θ k , 1) for k = 1, 2, …, N, to estimate the conditional distributions Θ k |Xk . There are two main estimation strategies: modeling on the θ space, called "g-modeling" here, and modeling on the×space, called "f-modeling." The two approaches are de- scribed and compared. A series of computational formulas are developed to assess their frequentist accuracy. Several examples, both contrived and genuine, show the strengths and limitations of the two strategies.

Empirical Bayes correction for the Winner's Curse in genetic association studies

We consider an Empirical Bayes method to correct for the Winner's Curse phenomenon in genome-wide association studies. Our method utilizes the collective distribution of all odds ratios (ORs) to determine the appropriate correction for a particular single-nucleotide polymorphism (SNP). We can show that this approach is squared error optimal provided that this collective distribution is accurately estimated in its tails. To improve the performance when correcting the OR estimates for the most highly associated SNPs, we develop a second estimator that adaptively combines the Empirical Bayes estimator with a previously considered Conditional Likelihood estimator. The applications of these methods to both simulated and real data suggest improved performance in reducing selection bias.