In this paper we propose tests for individual and joint irrelevance of network inputs. Such tests can be used to determine whether an input or group of inputs "belong" in a particular model, thus permitting valid statistical inference based on estimated feedforward neural network models. The approaches employ well known statistical resampling techniques. We conduct a small Monte Carlo Experiment showing that our tests have reasonable level and power behavior, and we apply our methods to examine whether there are predictable regularities in foreign exchange rates. We nd that exchange rates do appear to contain information that is exploitable for enhanced point prediction, but the nature of the predictive relations evolves through time.

Consider the multiple testing problem of testing null hypotheses H1,...,Hs. A classical approach to dealing with the multiplicity problem is to restrict attention to procedures that control the familywise error rate (FWER), the probability of even one false rejection. But if s is large, control of the FWER is so stringent that the ability of a procedure that controls the FWER to detect false null hypotheses is limited. It is therefore desirable to consider other measures of error control. This article considers two generalizations of the FWER. The first is the k-FWER, in which one is willing to tolerate k or more false rejections for some fixed k ≥ 1. The second is based on the false discovery proportion (FDP), defined to be the number of false rejections divided by the total number of rejections (and defined to be 0 if there are no rejections). Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289–300] proposed control of the false discovery rate (FDR), by which they meant that, for fixed α, E(FDP) ≤ α. Here, we consider control of the FDP in the sense that, for fixed γ and α, P {FDP> γ} ≤ α. Beginning with any nondecreasing sequence of constants and p-values for the individual tests, we derive stepup procedures that control each of these two measures of error control without imposing any assumptions on the dependence structure of the p-values. We use our results to point out a few interesting connections with some closely related stepdown procedures. We then compare and contrast two FDP-controlling procedures obtained using our results with the stepup procedure for control of the FDR of Benjamini and Yekutieli [Ann. Statist. 29 (2001) 1165–1188]. 1. Introduction. In

Three alternative procedures to adjust significance levels for multiplicity are the traditional Bonferroni technique, a sequential Bonferroni technique devel-oped by Hochberg (1988), and a sequential approach for controlling the false discovery rate proposed by Benjamini and Hochberg (1995). These procedures are illustrated and compared using examples from the National Assessment of Educational Progress (NAEP). A prominent advantage of the Benjamini and Hochberg (B-H) procedure, as demonstrated in these examples, is the greater invariance of statistical significance for given comparisons over alternative family sizes. Simulation studies show that all three procedures maintain a false discovery rate bounded above, often grossly, by ct (or c~/2). For both uncorre-lated and pairwise families of comparisons, the B-H technique is shown to have greater power than the Hochberg or Bonferroni procedures, and its power remains relatively stable as the number of comparisons becomes large, giving it an increasing advantage when many comparisons are involved. We recommend that results from NAEP State Assessments be reported using the B-H technique rather than the Bonferroni procedure. Two questions often asked about each of a set of observed comparisons are: (a) should we be confident about the direction or the sign of the corresponding underlying population comparison, and (b) for what interval of values should we be confident that it contains the value for the population comparison? Most

Multivariate kernel density estimation provides information about structure in data. Feature significance is a technique for deciding whether features – such as local extrema – are statistically significant. This paper proposes a framework for feature significance in d-dimensional data which combines kernel density derivative estimators and hypothesis tests for modal regions. For the gradient and curvature estimators distributional properties are given, and pointwise test statistics are derived. The hypothesis tests extend the two-dimensional feature significance ideas of Godtliebsen et al. (2002). The theoretical framework is complemented by novel visualisation for three-dimensional data. Applications to real data sets show that tests based on the kernel curvature estimators perform well in identifying modal regions. These results can be enhanced by corresponding tests with kernel gradient estimators.

We analyze the relations between Granger (G) non-causality and a notion of structural causality arising naturally from a general nonseparable recursive dynamic structural system. Building on classical notions of G non-causality, we introduce interesting and natural extensions, namely weak G non-causality and retrospective weak G non-causality. We show that structural non-causality and certain (retrospective) conditional exogeneity conditions imply (retrospective) (weak) G non-causality. We strengthen structural causality to notions of (retrospective) strong causality and show that (retrospective) strong causality implies (retrospective) weak G causality. We provide practical conditions and straightforward new methods for testing (retrospective) weak G non-causality, (retrospective) conditional exogeneity, and structural non-causality. Finally, we apply our methods to explore structural causality in industrial pricing, macroeconomics, and …nance.

We describe methods used and some results in a study of schizophrenia in a population of affected and unaffected participants, called patients and controls. The subjects are characterized by diagnosis, genotype, brain anatomy (MRI), laboratory tests on blood samples, and basic demographic data. The long term goal is to identify the causal chains of processes leading to disease. Methods used in this preliminary phase are statistical model building with randomization tests for p-value computations, FDR error rate control, Bayesian model comparisons, and conventional visualization. We describe preliminary findings of the study, which confirm earlier results on deviations of brain tissue volumes in schizophrenia patients, and also indicators of other effects that are presently under further investigation.

The ability to express a behavior during the postnatal period may be related to developmental changes in the recruitment of particular neural systems. Here, we show that developmental changes in the functional interactions involving three cortical regions (the medial prefrontal cortex, orbitofrontal cortex, and anterior cingulate cortex) are associated with maturation of extinction behavior in infant rats. Postnatal day 17 (P17) and P12 pups were trained in a straight-alley runway on an alternating schedule of reward and nonreward [patterned single alternation (PSA)] or on a pseudorandom schedule of partial reinforcement (PRF); the pups were then injected with fluorodeoxyglucose (FDG) and shifted to continuous nonreward (extinction). Handled control groups exposed to the same training environment but not trained on a particular schedule were included. Among P17 pups, extinction proceeded faster in PSA

There has been renewed interest in the area of multiple testing because of its importance in many statistical investigations, particularly in experiments where large data sets are generated, such as genetic microarrays. The purpose of this paper is to present some important developments that have taken place recently in this area, focusing mainly on stepwise multiple testing procedures, and to introduce some open problems.

We address the problem of improving the reliability of independence-based causal discovery algorithms that results from the execution of statistical independence tests on small data sets, which typically have low reliability. We model the problem as a knowledge base containing a set of independence facts that are related through Pearl’s well-known axioms. Statistical tests on finite data sets may result in errors in these tests and inconsistencies in the knowledge base. We resolve these inconsistencies through the use of an instance of the class of defeasible logics called argumentation, augmented with a preference function, that is used to reason about and possibly correct errors in these tests. This results in a more robust conditional independence test, called an argumentative independence test. Our experimental evaluation shows clear positive improvements in the accuracy of argumentative over purely statistical tests. We also demonstrate significant improvements on the accuracy of causal structure discovery from the outcomes of independence tests both on sampled data from randomly generated causal models and on real-world data sets.

Haplotypes of the WNK1 gene associate with blood pressure variation in a severely hypertensive population from the British Genetics of Hypertension study