Optimized multi-stage designs controlling the false discovery or the family-wise error rate

Controlling type I error rates in multi-arm clinical trials: A case for the false discovery rate

Multi-arm trials are an efficient way of simultaneously testing several experimental treatments against a shared control group. As well as reducing the sample size required compared to running each trial separately, they have important administrative and logistical advantages. There has been debate over whether multi-arm trials should correct for the fact that multiple null hypotheses are tested within the same experiment. Previous opinions have ranged from no correction is required, to a stringent correction (controlling the probability of making at least one type I error) being needed, with regulators arguing the latter for confirmatory settings. In this article, we propose that controlling the false-discovery rate (FDR) is a suitable compromise, with an appealing interpretation in multi-arm clinical trials. We investigate the properties of the different correction methods in terms of the positive and negative predictive value (respectively how confident we are that a recommended treatment is effective and that a non-recommended treatment is ineffective). The number of arms and proportion of treatments that are truly effective is varied. Controlling the FDR provides good properties. It retains the high positive predictive value of FWER correction in situations where a low proportion of treatments is effective. It also has a good negative predictive value in situations where a high proportion of treatments is effective. In a multi-arm trial testing distinct treatment arms, we recommend that sponsors and trialists consider use of the FDR.

Optimal design for high-throughput screening via false discovery rate control

High-throughput screening (HTS) is a large-scale hierarchical process in which a large number of chemicals are tested in multiple stages. Conventional statistical analyses of HTS studies often suffer from high testing error rates and soaring costs in large-scale settings. This article develops new methodologies for false discovery rate control and optimal design in HTS studies. We propose a two-stage procedure that determines the optimal numbers of replicates at different screening stages while simultaneously controlling the false discovery rate in the confirmatory stage subject to a constraint on the total budget. The merits of the proposed methods are illustrated using both simulated and real data. We show that, at the expense of a limited budget, the proposed screening procedure effectively controls the error rate and the design leads to improved detection power.

Robin Hood: A cost-efficient two-stage approach to large-scale simultaneous inference with non-homogeneous sparse effects

A classical approach to experimental design in many scientific fields is to first gather all of the data and then analyze it in a single analysis. It has been recognized that in many areas such practice leaves substantial room for improvement in terms of the researcher's ability to identify relevant effects, in terms of cost efficiency, or both. Considerable attention has been paid in recent years to multi-stage designs, in which the user alternates between data collection and analysis and thereby sequentially reduces the size of the problem. However, the focus has generally been towards designs that require a hypothesis be tested in every single stage before it can be declared as rejected by the procedure. Such procedures are well-suited for homogeneous effects, i.e. effects of (almost) equal sizes, however, with effects of varying size a procedure that permits rejection at interim stages is much more suitable. Here we present precisely such multi-stage testing procedure called Robin Hood. We show that with heterogeneous effects our method substantially improves on the existing multi-stage procedures with an essentially zero efficiency trade-off in the homogeneous effect realm, which makes it especially useful in areas such as genetics, where heterogeneous effects are common. Our method improves on existing approaches in a number of ways including a novel way of performing two-sided testing in a multi-stage procedure with increased power for detecting small effects.

A simple and flexible graphical approach for adaptive group-sequential clinical trials

In this article, we introduce a graphical approach to testing multiple hypotheses in group-sequential clinical trials allowing for midterm design modifications. It is intended for structured study objectives in adaptive clinical trials and extends the graphical group-sequential designs from Maurer and Bretz (Statistics in Biopharmaceutical Research 2013; 5: 311-320) to adaptive trial designs. The resulting test strategies can be visualized graphically and performed iteratively. We illustrate the methodology with two examples from our clinical trial practice. First, we consider a three-armed gold-standard trial with the option to reallocate patients to either the test drug or the active control group, while stopping the recruitment of patients to placebo, after having demonstrated superiority of the test drug over placebo at an interim analysis. Second, we consider a confirmatory two-stage adaptive design with treatment selection at interim.

Large Covariance Estimation by Thresholding Principal Orthogonal Complements

This paper deals with the estimation of a high-dimensional covariance with a conditional sparsity structure and fast-diverging eigenvalues. By assuming sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the Principal Orthogonal complEment Thresholding (POET) method to explore such an approximate factor structure with sparsity. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator (Cai and Liu, 2011) as specific examples. We provide mathematical insights when the factor analysis is approximately the same as the principal component analysis for high-dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the impact of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the unobserved factors and their factor loadings are derived. The asymptotic results are also verified by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.

False discovery rate control in two-stage designs

Background

For gene expression or gene association studies with a large number of hypotheses the number of measurements per marker in a conventional single-stage design is often low due to limited resources. Two-stage designs have been proposed where in a first stage promising hypotheses are identified and further investigated in the second stage with larger sample sizes. For two types of two-stage designs proposed in the literature we derive multiple testing procedures controlling the False Discovery Rate (FDR) demonstrating FDR control by simulations: designs where a fixed number of top-ranked hypotheses are selected and designs where the selection in the interim analysis is based on an FDR threshold. In contrast to earlier approaches which use only the second-stage data in the hypothesis tests (pilot approach), the proposed testing procedures are based on the pooled data from both stages (integrated approach).

Results

For both selection rules the multiple testing procedures control the FDR in the considered simulation scenarios. This holds for the case of independent observations across hypotheses as well as for certain correlation structures. Additionally, we show that in scenarios with small effect sizes the testing procedures based on the pooled data from both stages can give a considerable improvement in power compared to tests based on the second-stage data only.

Conclusion

The proposed hypothesis tests provide a tool for FDR control for the considered two-stage designs. Comparing the integrated approaches for both selection rules with the corresponding pilot approaches showed an advantage of the integrated approach in many simulation scenarios.

Effectiveness of strategies to increase the validity of findings from association studies: size vs. replication

Background

The capacity of multiple comparisons to produce false positive findings in genetic association studies is abundantly clear. To address this issue, the concept of false positive report probability (FPRP) measures "the probability of no true association between a genetic variant and disease given a statistically significant finding". This concept involves the notion of prior probability of an association between a genetic variant and a disease, making it difficult to achieve acceptable levels for the FPRP when the prior probability is low. Increasing the sample size is of limited efficiency to improve the situation.

Methods

To further clarify this problem, the concept of true report probability (TRP) is introduced by analogy to the positive predictive value (PPV) of diagnostic testing. The approach is extended to consider the effects of replication studies. The formula for the TRP after k replication studies is mathematically derived and shown to be only dependent on prior probability, alpha, power, and number of replication studies.

Results

Case-control association studies are used to illustrate the TRP concept for replication strategies. Based on power considerations, a relationship is derived between TRP after k replication studies and sample size of each individual study. That relationship enables study designers optimization of study plans. Further, it is demonstrated that replication is efficient in increasing the TRP even in the case of low prior probability of an association and without requiring very large sample sizes for each individual study.

Conclusions

True report probability is a comprehensive and straightforward concept for assessing the validity of positive statistical testing results in association studies. By its extension to replication strategies it can be demonstrated in a transparent manner that replication is highly effective in distinguishing spurious from true associations. Based on the generalized TRP method for replication designs, optimal research strategy and sample size planning become possible.

Post hoc power estimation in large-scale multiple testing problems

BACKGROUND

The statistical power or multiple Type II error rate in large-scale multiple testing problems as, for example, in gene expression microarray experiments, depends on typically unknown parameters and is therefore difficult to assess a priori. However, it has been suggested to estimate the multiple Type II error rate post hoc, based on the observed data.

METHODS

We consider a class of post hoc estimators that are functions of the estimated proportion of true null hypotheses among all hypotheses. Numerous estimators for this proportion have been proposed and we investigate the statistical properties of the derived multiple Type II error rate estimators in an extensive simulation study.

RESULTS

The performance of the estimators in terms of the mean squared error depends sensitively on the distributional scenario. Estimators based on empirical distributions of the null hypotheses are superior in the presence of strongly correlated test statistics.

AVAILABILITY

R-code to compute all considered estimators based on P-values and supplementary material is available on the authors web page http://statistics.msi.meduniwien.ac.at/index.php?page=pageszfnr.