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Register a new userCombining parameter values or $p$-values
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We review the methods to combine several measurements, in the form of parameter values or $p$-values.
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We propose a novel method for computing $p$-values based on nested sampling (NS) applied to the sampling space rather than the parameter space of the problem, in contrast to its usage in Bayesian computation. The computational cost of NS scales as $log^2{1/p}$, which compares favorably to the $1/p$ scaling for Monte Carlo (MC) simulations. For significances greater than about $4sigma$ in both a toy problem and a simplified resonance search, we show that NS requires orders of magnitude fewer simulations than ordinary MC estimates. This is particularly relevant for high-energy physics, which adopts a $5sigma$ gold standard for discovery. We conclude with remarks on new connections between Bayesian and frequentist computation and possibilities for tuning NS implementations for still better performance in this setting.
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.
We approach the problem of combining top-ranking association statistics or P-value from a new perspective which leads to a remarkably simple and powerful method. Statistical methods, such as the Rank Truncated Product (RTP), have been developed for combining top-ranking associations and this general strategy proved to be useful in applications for detecting combined effects of multiple disease components. To increase power, these methods aggregate signals across top ranking SNPs, while adjusting for their total number assessed in a study. Analytic expressions for combined top statistics or P-values tend to be unwieldy, which complicates interpretation, practical implementation, and hinders further developments. Here, we propose the Augmented Rank Truncation (ART) method that retains main characteristics of the RTP but is substantially simpler to implement. ART leads to an efficient form of the adaptive algorithm, an approach where the number of top ranking SNPs is varied to optimize power. We illustrate our methods by strengthening previously reported associations of $mu$-opioid receptor variants with sensitivity to pain.
When the data do not conform to the hypothesis of a known sampling-variance, the fitting of a constant to a set of measured values is a long debated problem. Given the data, fitting would require to find what measurand value is the most trustworthy. Bayesian inference is here reviewed, to assign probabilities to the possible measurand values. Different hypothesis about the data variance are tested by Bayesian model comparison. Eventually, model selection is exemplified in deriving an estimate of the Planck constant.
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