The false discovery rate (FDR) is a multiple hypothesis testing quantity that describes the expected proportion of false positive results among all rejected null hypotheses. Benjamini and Hochberg introduced this quantity and proved that a particular step-up "p"-value method controls the FDR. Storey introduced a point estimate of the FDR for fixed significance regions. The former approach conservatively controls the FDR at a fixed predetermined level, and the latter provides a conservatively biased estimate of the FDR for a fixed predetermined significance region. In this work, we show in both finite sample and asymptotic settings that the goals of the two approaches are essentially equivalent. In particular, the FDR point estimates can be used to define valid FDR controlling procedures. In the asymptotic setting, we also show that the point estimates can be used to estimate the FDR conservatively over all significance regions simultaneously, which is equivalent to controlling the FDR at all levels simultaneously. The main tool that we use is to translate existing FDR methods into procedures involving empirical processes. This simplifies finite sample proofs, provides a framework for asymptotic results and proves that these procedures are valid even under certain forms of dependence. Copyright 2004 Royal Statistical Society.

We develop a general approach to valid inference after model selection. At the core of our framework is a result that characterizes the distribution of a post-selection estimator conditioned on the selection event. We specialize the approach to model selection by the lasso to form valid confidence intervals for the selected coefficients and test whether all relevant variables have been included in the model.Comment: Published at http://dx.doi.org/10.1214/15-AOS1371 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Anticipated reward magnitude and probability comprise dual components of expected value (EV), a cornerstone of economic and psychological theory. However, the neural mechanisms that compute EV have not been characterized. Using event-related functional magnetic resonance imaging, we examined neural activation as subjects anticipated monetary gains and losses that varied in magnitude and probability. Group analyses indicated that, although the subcortical nucleus accumbens (NAcc) activated proportional to anticipated gain magnitude, the cortical mesial prefrontal cortex (MPFC) additionally activated according to anticipated gain probability. Individual difference analyses indicated that, although NAcc activation correlated with self-reported positive arousal, MPFC activation correlated with probability estimates. These findings suggest that mesolimbic brain regions support the computation of EV in an ascending and distributed manner: whereas subcortical regions represent an affective component, cortical regions also represent a probabilistic component, and, furthermore, may integrate the two.

We present a path algorithm for the generalized lasso problem. This problem penalizes the $\ell_1$ norm of a matrix D times the coefficient vector, and has a wide range of applications, dictated by the choice of D. Our algorithm is based on solving the dual of the generalized lasso, which greatly facilitates computation of the path. For $D=I$ (the usual lasso), we draw a connection between our approach and the well-known LARS algorithm. For an arbitrary D, we derive an unbiased estimate of the degrees of freedom of the generalized lasso fit. This estimate turns out to be quite intuitive in many applications.Comment: Published in at http://dx.doi.org/10.1214/11-AOS878 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

We add a set of convex constraints to the lasso to produce sparse interaction models that honor the hierarchy restriction that an interaction only be included in a model if one or both variables are marginally important. We give a precise characterization of the effect of this hierarchy constraint, prove that hierarchy holds with probability one and derive an unbiased estimate for the degrees of freedom of our estimator. A bound on this estimate reveals the amount of fitting “saved” by the hierarchy constraint. We distinguish between parameter sparsity—the number of nonzero coefficients—and practical sparsity—the number of raw variables one must measure to make a new prediction. Hierarchy focuses on the latter, which is more closely tied to important data collection concerns such as cost, time and effort. We develop an algorithm, available in the R package hierNet, and perform an empirical study of our method.