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Applied statistical problems often come with pre-specified groupings to predictors. It is natural to test for the presence of simultaneous group-wide signal for groups in isolation, or for multiple groups together. Classical tests for the presence of such signals rely either on tests for the omission of the entire block of variables (the classical F-test) or on the creation of an unsupervised prototype for the group (either a group centroid or first principal component) and subsequent t-tests on these prototypes. In this paper, we propose test statistics that aim for power improvements over these classical approaches. In particular, we first create group prototypes, with reference to the response, hopefully improving on the unsupervised prototypes, and then testing with likelihood ratio statistics incorporating only these prototypes. We propose a (potentially) novel model, called the "prototype model", which naturally models the two-step prototype-then-test procedure. Furthermore, we introduce an inferential schema detailing the unique considerations for different combinations of prototype formation and univariate/multivariate testing models. The prototype model also suggests new applications to estimation and prediction. Prototype formation often relies on variable selection, which invalidates classical Gaussian test theory. We use recent advances in selective inference to account for selection in the prototyping step and retain test validity. Simulation experiments suggest that our testing procedure enjoys more power than do classical approaches., Comment: 27 pages, 6 figures

We propose a new approach for sparse regression and marginal testing, for data with correlated features. Our procedure first clusters the features, and then chooses as the cluster prototype the most informative feature in that cluster. Then we apply either sparse regression (lasso) or marginal significance testing to these prototypes. While this kind of strategy is not entirely new, a key feature of our proposal is its use of the post-selection inference theory of Taylor et al. (2014) and Lee et al. (2014) to compute exact p-values and confidence intervals that properly account for the selection of prototypes. We also apply the recent "knockoff" idea of Barber and Cand\`es to provide exact finite sample control of the FDR of our regression procedure. We illustrate our proposals on both real and simulated data., Comment: 43 pages, 19 figures

We apply the cyclic coordinate descent algorithm of Friedman, Hastie and Tibshirani (2010) to the fitting of a conditional logistic regression model with lasso ($\ell_1$) and elastic net penalties. The sequential strong rules of Tibshirani et al (2012) are also used in the algorithm and it is shown that these offer a considerable speed up over the standard coordinate descent algorithm with warm starts. Once implemented, the algorithm is used in simulation studies to compare the variable selection and prediction performance of the conditional logistic regression model against that of its unconditional (standard) counterpart. We find that the conditional model performs admirably on datasets drawn from a suitable conditional distribution, outperforming its unconditional counterpart at variable selection. The conditional model is also fit to a small real world dataset, demonstrating how we obtain regularisation paths for the parameters of the model and how we apply cross validation for this method where natural unconditional prediction rules are hard to come by.

We tackle the problem of the estimation of a vector of means from a single vector-valued observation $y$. Whereas previous work reduces the size of the estimates for the largest (absolute) sample elements via shrinkage (like James-Stein) or biases estimated via empirical Bayes methodology, we take a novel approach. We adapt recent developments by Lee et al (2013) in post selection inference for the Lasso to the orthogonal setting, where sample elements have different underlying signal sizes. This is exactly the setup encountered when estimating many means. It is shown that other selection procedures, like selecting the $K$ largest (absolute) sample elements and the Benjamini-Hochberg procedure, can be cast into their framework, allowing us to leverage their results. Point and interval estimates for signal sizes are proposed. These seem to perform quite well against competitors, both recent and more tenured. Furthermore, we prove an upper bound to the worst case risk of our estimator, when combined with the Benjamini-Hochberg procedure, and show that it is within a constant multiple of the minimax risk over a rich set of parameter spaces meant to evoke sparsity., Comment: 27 pages, 13 figures

Variance estimation in the linear model when $p > n$ is a difficult problem. Standard least squares estimation techniques do not apply. Several variance estimators have been proposed in the literature, all with accompanying asymptotic results proving consistency and asymptotic normality under a variety of assumptions. It is found, however, that most of these estimators suffer large biases in finite samples when true underlying signals become less sparse with larger per element signal strength. One estimator seems to be largely neglected in the literature: a residual sum of squares based estimator using Lasso coefficients with regularisation parameter selected adaptively (via cross-validation). In this paper, we review several variance estimators and perform a reasonably extensive simulation study in an attempt to compare their finite sample performance. It would seem from the results that variance estimators with adaptively chosen regularisation parameters perform admirably over a broad range of sparsity and signal strength settings. Finally, some intial theoretical analyses pertaining to these types of estimators are proposed and developed., Comment: 30 pages, 9 figures

The study of networks has grown into a substantial interdisciplinary endeavour that encompasses myriad disciplines in the natural, social, and information sciences. Here we introduce a framework for constructing taxonomies of networks based on their structural similarities. These networks can arise from any of numerous sources: they can be empirical or synthetic, they can arise from multiple realizations of a single process, empirical or synthetic, or they can represent entirely different systems in different disciplines. Since the mesoscopic properties of networks are hypothesized to be important for network function, we base our comparisons on summaries of network community structures. While we use a specific method for uncovering network communities, much of the introduced framework is independent of that choice. After introducing the framework, we apply it to construct a taxonomy for 746 individual networks and demonstrate that our approach usefully identifies similar networks. We also construct taxonomies within individual categories of networks, and in each case we expose non-trivial structure. For example we create taxonomies for similarity networks constructed from both political voting data and financial data. We also construct network taxonomies to compare the social structures of 100 Facebook networks and the growth structures produced by different types of fungi., Comment: Main text & supplementary material; revised version