When simultaneously testing multiple hypotheses, the usual approach in the context of confirmatory clinical trials is to control the familywise error rate (FWER), which bounds the probability of making at least one false rejection. In many trial settings, these hypotheses will additionally have a hierarchical structure that reflects the relative importance and links between different clinical objectives. The graphical approach of Bretz et al. (2009) is a flexible and easily communicable way of controlling the FWER while respecting complex trial objectives and multiple structured hypotheses. However, the FWER can be a very stringent criterion that leads to procedures with low power, and may not be appropriate in exploratory trial settings. This motivates controlling generalised error rates, particularly when the number of hypotheses tested is no longer small. We consider the generalised familywise error rate (k-FWER), which is the probability of making k or more false rejections, as well as the tail probability of the false discovery proportion (FDP), which is the probability that the proportion of false rejections is greater than some threshold. We also consider asymptotic control of the false discovery rate (FDR), which is the expectation of the FDP. In this paper, we show how to control these generalised error rates when using the graphical approach and its extensions. We demonstrate the utility of the resulting graphical procedures on three clinical trial case studies.
This paper introduces a method for studying the correlation structure of a range of responses modelled by a multivariate generalised linear mixed model (MGLMM). The methodology requires the existence of clusters of observations and that each of the several responses studied is modelled using a generalised linear mixed models (GLMM) containing random components representing the clusters. We construct a MGLMM by assuming that the distribution of each of the random components representing the clusters is the marginal distribution of a (sufficiently regular) multivariate elliptically contoured dis…
Based on the work of Romano and Shaikh (2006) and Lehmann and Romano (2005) we give a sufficient criterion for controlling generalised error rates for arbitrarily dependent p-values. This criterion is formulated in terms of matrices associated with the corresponding error rates and thus it is possible to view the corresponding critical constants as solutions of sets of certain linear inequalities. This property can in some cases be used to improve the power of existing procedures by finding optimal solutions to an associated linear programming problem.
Primarily motivated by the drug development process, several publications have now presented methodology for the design of multi-arm multi-stage experiments with normally distributed outcome variables of known variance. Here, we extend these past considerations to allow the design of what we refer to as an abcd multi-arm multi-stage experiment. We provide a proof of how strong control of the a-generalised type-I familywise error-rate can be ensured. We then describe how to attain the power to reject at least b out of c false hypotheses, which is related to controlling the b-generalised type-II…
We consider a framework for determining and estimating the conditional pairwise relationships of variables when the observed samples are contaminated with measurement error in high dimensional settings. Assuming the true underlying variables follow a multivariate Gaussian distribution, if no measurement error is present, this problem is often solved by estimating the precision matrix under sparsity constraints. However, when measurement error is present, not correcting for it leads to inconsistent estimates of the precision matrix and poor identification of relationships. We propose a new Baye…
We present a stepwise approach to estimate high dimensional Gaussian graphical models. We exploit the relation between the partial correlation coefficients and the distribution of the prediction errors, and parametrize the model in terms of the Pearson correlation coefficients between the prediction errors of the nodes' best linear predictors. We propose a novel stepwise algorithm for detecting pairs of conditionally dependent variables. We show that the proposed algorithm outperforms existing methods such as the graphical lasso and CLIME in simulation studies and real life applications. In ou…
Recently, a special case of precision matrix estimation based on a distributionally robust optimization (DRO) framework has been shown to be equivalent to the graphical lasso. From this formulation, a method for choosing the regularization term, i.e., for graphical model selection, was proposed. In this work, we establish a theoretical connection between the confidence level of graphical model selection via the DRO formulation and the asymptotic family-wise error rate of estimating false edges. Simulation experiments and real data analyses illustrate the utility of the asymptotic family-wise e…
We introduce a new test for conditional independence which is based on what we call the weighted generalised covariance measure (WGCM). It is an extension of the recently introduced generalised covariance measure (GCM). To test the null hypothesis of X and Y being conditionally independent given Z, our test statistic is a weighted form of the sample covariance between the residuals of nonlinearly regressing X and Y on Z. We propose different variants of the test for both univariate and multivariate X and Y . We give conditions under which the tests yield the correct type I error rate. Finally,…
Friedman test is a nonparametric method that proposed for analyzing data from a randomized complete block design as a robust alternative to parametric method and widely applied in many fields such as agriculture, biology, business, education, and medicine. After the null hypothesis of no treatment effects is rejected, the post-hoc pairwise comparisons must be applied to identify where the differences occur. As the number of groups increases, the number of required comparisons becomes large and this may increase the type I error. The aim of this study is twofold. The main aim is to suggest expr…
Despite major methodological developments, Bayesian inference for Gaussian graphical models remains challenging in high dimension due to the tremendous size of the model space. This article proposes a method to infer the marginal and conditional independence structures between variables by multiple testing of hypotheses. Specifically, we introduce closed-form Bayes factors under the Gaussian conjugate model to evaluate the null hypotheses of marginal and conditional independence between variables. Their computation for all pairs of variables is shown to be extremely efficient, thereby allowing…
The Fisher Information Matrix formalism is extended to cases where the data is divided into two parts (X,Y), where the expectation value of Y depends on X according to some theoretical model, and X and Y both have errors with arbitrary covariance. In the simplest case, (X,Y) represent data pairs of abscissa and ordinate, in which case the analysis deals with the case of data pairs with errors in both coordinates, but X can be any measured quantities on which Y depends. The analysis applies for arbitrary covariance, provided all errors are gaussian, and provided the errors in X are small, both …
Many real-world objects can be modeled as a stream of events on the nodes of a graph. In this paper, we propose a class of graphical event models named temporal point process graphical models for representing the temporal dependencies among different components of a multivariate point process. In our model, the intensity of an event stream can depend on the historical events in a nonlinear way. We provide a procedure that allows us to estimate the parameters in the model with a convex loss function in the high-dimensional setting. For the approximation error introduced during the implementatio…
In many large scale multiple testing applications, the hypotheses often have a known graphical structure, such as gene ontology in gene expression data. Exploiting this graphical structure in multiple testing procedures can improve power as well as aid in interpretation. However, incorporating the structure into large scale testing procedures and proving that an error rate, such as the false discovery rate (FDR), is controlled can be challenging. In this paper, we introduce a new general approach for large scale multiple testing, which can aid in developing new procedures under various setting…
We describe an improved statistical downscaling method for Earth science applications using multivariate Basis Graphical Lasso (BGL). We demonstrate our method using a case study of sea surface temperature (SST) projections from CMIP6 Earth system models, which has direct applications for studies of multi-decadal projections of coral reef bleaching. We find that the BGL downscaling method is computationally tractable for large data sets, and that mean squared predictive error is roughly 8% lower than the current state-of-the-art interpolation-based statistical downscaling method. Finally, unli…
Transfer learning for high-dimensional Gaussian graphical models (GGMs) is studied with the goal of estimating the target GGM by utilizing the data from similar and related auxiliary studies. The similarity between the target graph and each auxiliary graph is characterized by the sparsity of a divergence matrix. An estimation algorithm, Trans-CLIME, is proposed and shown to attain a faster convergence rate than the minimax rate in the single study setting. Furthermore, a debiased Trans-CLIME estimator is introduced and shown to be element-wise asymptotically normal. It is used to construct a m…
Differential graphical models are designed to represent the difference between the conditional dependence structures of two groups, thus are of particular interest for scientific investigation. Motivated by modern applications, this manuscript considers an extended setting where each group is generated by a latent variable Gaussian graphical model. Due to the existence of latent factors, the differential network is decomposed into sparse and low-rank components, both of which are symmetric indefinite matrices. We estimate these two components simultaneously using a two-stage procedure: (i) an …
