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Register a new userCombining independent, arbitrarily weighted P-values: a new solution to an old problem using a novel expansion with controllable accuracy
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Goods formula and Fishers method are frequently used for combining independent P-values. Interestingly, the equivalent of Goods formula already emerged in 1910 and mathematical expressions relevant to even more general situations have been repeatedly derived, albeit in different context. We provide here a novel derivation and show how the analytic formula obtained reduces to the two aforementioned ones as special cases. The main novelty of this paper, however, is the explicit treatment of nearly degenerate weights, which are known to cause numerical instabilities. We derive a controlled expansion, in powers of differences in inverse weights, that provides both accurate statistics and stable numerics.
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Given a family of null hypotheses $H_{1},ldots,H_{s}$, we are interested in the hypothesis $H_{s}^{gamma}$ that at most $gamma-1$ of these null hypotheses are false. Assuming that the corresponding $p$-values are independent, we are investigating combined $p$-values that are valid for testing $H_{s}^{gamma}$. In various settings in which $H_{s}^{gamma}$ is false, we determine which combined $p$-value works well in which setting. Via simulations, we find that the Stouffer method works well if the null $p$-values are uniformly distributed and the signal strength is low, and the Fisher method works better if the null $p$-values are conservative, i.e. stochastically larger than the uniform distribution. The minimum method works well if the evidence for the rejection of $H_{s}^{gamma}$ is focused on only a few non-null $p$-values, especially if the null $p$-values are conservative. Methods that incorporate the combination of $e$-values work well if the null hypotheses $H_{1},ldots,H_{s}$ are simple.
Let $(X,Y)$ be a random variable consisting of an observed feature vector $Xin mathcal{X}$ and an unobserved class label $Yin {1,2,...,L}$ with unknown joint distribution. In addition, let $mathcal{D}$ be a training data set consisting of $n$ completely observed independent copies of $(X,Y)$. Usual classification procedures provide point predictors (classifiers) $widehat{Y}(X,mathcal{D})$ of $Y$ or estimate the conditional distribution of $Y$ given $X$. In order to quantify the certainty of classifying $X$ we propose to construct for each $theta =1,2,...,L$ a p-value $pi_{theta}(X,mathcal{D})$ for the null hypothesis that $Y=theta$, treating $Y$ temporarily as a fixed parameter. In other words, the point predictor $widehat{Y}(X,mathcal{D})$ is replaced with a prediction region for $Y$ with a certain confidence. We argue that (i) this approach is advantageous over traditional approaches and (ii) any reasonable classifier can be modified to yield nonparametric p-values. We discuss issues such as optimality, single use and multiple use validity, as well as computational and graphical aspects.
Membrane computing is a branch of natural computingwhich abstracts fromthe structure and the functioning of living cells. The computation models obtained in the field of membrane computing are usually called P systems. P systems have been used to solve computationally hard problems efficiently on the assumption that the execution of each rule is completed in exactly one time-unit (a global clock is assumed for timing and synchronizing the execution of rules). However, in biological reality, different biological processes take different times to be completed, which can also be influenced by many environmental factors. In this work, with this biological reality, we give a time-free solution to independent set problemusing P systems with active membranes, which solve the problem independent of the execution time of the involved rules.
As a concrete setting where stochastic partial differential equations (SPDEs) are able to model real phenomena, we propose a stochastic Meinhardt model for cell repolarisation and study how parameter estimation techniques developed for simple linear SPDE models apply in this situation. We establish the existence of mild SPDE solutions and we investigate the impact of the driving noise process on pattern formation in the solution. We then pursue estimation of the diffusion term and show asymptotic normality for our estimator as the space resolution becomes finer. The finite sample performance is investigated for synthetic and real data.
We review the methods to combine several measurements, in the form of parameter values or $p$-values.
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