Methodological Issues in Multistage Genome-Wide Association Studies

Because of the high cost of commercial genotyping chip technologies, many investigations have used a two-stage design for genome-wide association studies, using part of the sample for an initial discovery of ``promising'' SNPs at a less stringent significance level and the remainder in a joint analysis of just these SNPs using custom genotyping. Typical cost savings of about 50% are possible with this design to obtain comparable levels of overall type I error and power by using about half the sample for stage I and carrying about 0.1% of SNPs forward to the second stage, the optimal design depending primarily upon the ratio of costs per genotype for stages I and II. However, with the rapidly declining costs of the commercial panels, the generally low observed ORs of current studies, and many studies aiming to test multiple hypotheses and multiple endpoints, many investigators are abandoning the two-stage design in favor of simply genotyping all available subjects using a standard high-density panel. Concern is sometimes raised about the absence of a ``replication'' panel in this approach, as required by some high-profile journals, but it must be appreciated that the two-stage design is not a discovery/replication design but simply a more efficient design for discovery using a joint analysis of the data from both stages. Once a subset of highly-significant associations has been discovered, a truly independent ``exact replication'' study is needed in a similar population of the same promising SNPs using similar methods.Comment: Published in at http://dx.doi.org/10.1214/09-STS288 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Genome-Wide Significance Levels and Weighted Hypothesis Testing

Genetic investigations often involve the testing of vast numbers of related hypotheses simultaneously. To control the overall error rate, a substantial penalty is required, making it difficult to detect signals of moderate strength. To improve the power in this setting, a number of authors have considered using weighted $p$-values, with the motivation often based upon the scientific plausibility of the hypotheses. We review this literature, derive optimal weights and show that the power is remarkably robust to misspecification of these weights. We consider two methods for choosing weights in practice. The first, external weighting, is based on prior information. The second, estimated weighting, uses the data to choose weights.Comment: Published in at http://dx.doi.org/10.1214/09-STS289 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

SLOPE - Adaptive variable selection via convex optimization

We introduce a new estimator for the vector of coefficients $\beta$ in the linear model $y=X\beta+z$, where $X$ has dimensions $n\times p$ with $p$ possibly larger than $n$. SLOPE, short for Sorted L-One Penalized Estimation, is the solution to $$\min_{b\in\mathbb{R}^p}\frac{1}{2}\Vert y-Xb\Vert _{\ell_2}^2+\lambda_1\vert b\vert _{(1)}+\lambda_2\vert b\vert_{(2)}+\cdots+\lambda_p\vert b\vert_{(p)},$$ where $\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_p\ge0$ and $\vert b\vert_{(1)}\ge\vert b\vert_{(2)}\ge\cdots\ge\vert b\vert_{(p)}$ 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 $\ell_1$ procedures such as the Lasso. Here, the regularizer is a sorted $\ell_1$ 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. B 57 (1995) 289-300] procedure (BH) which compares more significant $p$-values with more stringent thresholds. One notable choice of the sequence $\{\lambda_i\}$ is given by the BH critical values $\lambda_{\mathrm {BH}}(i)=z(1-i\cdot q/2p)$, where $q\in(0,1)$ and $z(\alpha)$ 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 $\lambda_{\mathrm{BH}}$ 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.Comment: Published at http://dx.doi.org/10.1214/15-AOAS842 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

False discovery rates in somatic mutation studies of cancer

The purpose of cancer genome sequencing studies is to determine the nature and types of alterations present in a typical cancer and to discover genes mutated at high frequencies. In this article we discuss statistical methods for the analysis of somatic mutation frequency data generated in these studies. We place special emphasis on a two-stage study design introduced by Sj\"{o}blom et al. [Science 314 (2006) 268--274]. In this context, we describe and compare statistical methods for constructing scores that can be used to prioritize candidate genes for further investigation and to assess the statistical significance of the candidates thus identified. Controversy has surrounded the reliability of the false discovery rates estimates provided by the approximations used in early cancer genome studies. To address these, we develop a semiparametric Bayesian model that provides an accurate fit to the data. We use this model to generate a large collection of realistic scenarios, and evaluate alternative approaches on this collection. Our assessment is impartial in that the model used for generating data is not used by any of the approaches compared. And is objective, in that the scenarios are generated by a model that fits data. Our results quantify the conservative control of the false discovery rate with the Benjamini and Hockberg method compared to the empirical Bayes approach and the multiple testing method proposed in Storey [J. R. Stat. Soc. Ser. B Stat. Methodol. 64 (2002) 479--498]. Simulation results also show a negligible departure from the target false discovery rate for the methodology used in Sj\"{o}blom et al. [Science 314 (2006) 268--274].Comment: Published in at http://dx.doi.org/10.1214/10-AOAS438 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org