Causal modelling of heavy-tailed variables and confounders with application to river flow

Confounding variables are a recurrent challenge for causal discovery and inference. In many situations, complex causal mechanisms only manifest themselves in extreme events, or take simpler forms in the extremes. Stimulated by data on extreme river flows and precipitation, we introduce a new causal discovery methodology for heavy-tailed variables that allows the effect of a known potential confounder to be almost entirely removed when the variables have comparable tails, and also decreases it sufficiently to enable correct causal inference when the confounder has a heavier tail. We also introduce a new parametric estimator for the existing causal tail coefficient and a permutation test. Simulations show that the methods work well and the ideas are applied to the motivating dataset.

Supplementary Information

The online version contains supplementary material available at 10.1007/s10687-022-00456-4.

Human mortality at extreme age

We use a combination of extreme value statistics, survival analysis and computer-intensive methods to analyse the mortality of Italian and French semi-supercentenarians. After accounting for the effects of the sampling frame, extreme-value modelling leads to the conclusion that constant force of mortality beyond 108 years describes the data well and there is no evidence of differences between countries and cohorts. These findings are consistent with use of a Gompertz model and with previous analysis of the International Database on Longevity and suggest that any physical upper bound for the human lifespan is so large that it is unlikely to be approached. Power calculations make it implausible that there is an upper bound below 130 years. There is no evidence of differences in survival between women and men after age 108 in the Italian data and the International Database on Longevity, but survival is lower for men in the French data.

An unethical optimization principle

If an artificial intelligence aims to maximize risk-adjusted return, then under mild conditions it is disproportionately likely to pick an unethical strategy unless the objective function allows sufficiently for this risk. Even if the proportion η of available unethical strategies is small, the probability p _U of picking an unethical strategy can become large; indeed, unless returns are fat-tailed p _U tends to unity as the strategy space becomes large. We define an unethical odds ratio, Υ (capital upsilon), that allows us to calculate p _U from η, and we derive a simple formula for the limit of Υ as the strategy space becomes large. We discuss the estimation of Υ and p _U in finite cases and how to deal with infinite strategy spaces. We show how the principle can be used to help detect unethical strategies and to estimate η. Finally we sketch some policy implications of this work.

A Global-Local Approach for Detecting Hotspots in Multiple-Response Regression

We tackle modelling and inference for variable selection in regression problems with many predictors and many responses. We focus on detecting hotspots, that is, predictors associated with several responses. Such a task is critical in statistical genetics, as hotspot genetic variants shape the architecture of the genome by controlling the expression of many genes and may initiate decisive functional mechanisms underlying disease endpoints. Existing hierarchical regression approaches designed to model hotspots suffer from two limitations: their discrimination of hotspots is sensitive to the choice of top-level scale parameters for the propensity of predictors to be hotspots, and they do not scale to large predictor and response vectors, for example, of dimensions 103-105 in genetic applications. We address these shortcomings by introducing a flexible hierarchical regression framework that is tailored to the detection of hotspots and scalable to the above dimensions. Our proposal implements a fully Bayesian model for hotspots based on the horseshoe shrinkage prior. Its global-local formulation shrinks noise globally and, hence, accommodates the highly sparse nature of genetic analyses while being robust to individual signals, thus leaving the effects of hotspots unshrunk. Inference is carried out using a fast variational algorithm coupled with a novel simulated annealing procedure that allows efficient exploration of multimodal distributions.

Parameter estimation for discretely observed linear birth-and-death processes

Birth-and-death processes are widely used to model the development of biological populations. Although they are relatively simple models, their parameters can be challenging to estimate, as the likelihood can become numerically unstable when data arise from the most common sampling schemes, such as annual population censuses. A further difficulty arises when the discrete observations are not equi-spaced, for example, when census data are unavailable for some years. We present two approaches to estimating the birth, death, and growth rates of a discretely observed linear birth-and-death process: via an embedded Galton-Watson process and by maximizing a saddlepoint approximation to the likelihood. We study asymptotic properties of the estimators, compare them on numerical examples, and apply the methodology to data on monitored populations.

Automatic module selection from several microarray gene expression studies

Independence of genes is commonly but incorrectly assumed in microarray data analysis; rather, genes are activated in co-regulated sets referred to as modules. In this article, we develop an automatic method to define modules common to multiple independent studies. We use an empirical Bayes procedure to estimate a sparse correlation matrix for all studies, identify modules by clustering, and develop an extreme-value-based method to detect so-called scattered genes, which do not belong to any module. The resulting algorithm is very fast and produces accurate modules in simulation studies. Application to real data identifies modules with significant enrichment and results in a huge dimension reduction, which can alleviate the computational burden of further analyses.

Efficient inference for genetic association studies with multiple outcomes

Combined inference for heterogeneous high-dimensional data is critical in modern biology, where clinical and various kinds of molecular data may be available from a single study. Classical genetic association studies regress a single clinical outcome on many genetic variants one by one, but there is an increasing demand for joint analysis of many molecular outcomes and genetic variants in order to unravel functional interactions. Unfortunately, most existing approaches to joint modeling are either too simplistic to be powerful or are impracticable for computational reasons. Inspired by Richardson and others (2010, Bayesian Statistics 9), we consider a sparse multivariate regression model that allows simultaneous selection of predictors and associated responses. As Markov chain Monte Carlo (MCMC) inference on such models can be prohibitively slow when the number of genetic variants exceeds a few thousand, we propose a variational inference approach which produces posterior information very close to that of MCMC inference, at a much reduced computational cost. Extensive numerical experiments show that our approach outperforms popular variable selection methods and tailored Bayesian procedures, dealing within hours with problems involving hundreds of thousands of genetic variants and tens to hundreds of clinical or molecular outcomes.

Meta-analysis of incomplete microarray studies

Meta-analysis of microarray studies to produce an overall gene list is relatively straightforward when complete data are available. When some studies lack information-providing only a ranked list of genes, for example-it is common to reduce all studies to ranked lists prior to combining them. Since this entails a loss of information, we consider a hierarchical Bayes approach to meta-analysis using different types of information from different studies: the full data matrix, summary statistics, or ranks. The model uses an informative prior for the parameter of interest to aid the detection of differentially expressed genes. Simulations show that the new approach can give substantial power gains compared with classical meta-analysis and list aggregation methods. A meta-analysis of 11 published studies with different data types identifies genes known to be involved in ovarian cancer and shows significant enrichment.

Diabetes imaging-quantitative assessment of islets of Langerhans distribution in murine pancreas using extended-focus optical coherence microscopy

Diabetes is characterized by hyperglycemia that can result from the loss of pancreatic insulin secreting β-cells in the islets of Langerhans. We analyzed ex vivo the entire gastric and duodenal lobes of a murine pancreas using extended-focus Optical Coherence Microscopy (xfOCM). To identify and quantify the islets of Langerhans observed in xfOCM tomograms we implemented an active contour algorithm based on the level set method. We show that xfOCM reveals a three-dimensional islet distribution consistent with Optical Projection Tomography, albeit with a higher resolution that also enables the detection of the smallest islets (≤ 8000 μm(3)). Although this category of the smallest islets represents only a negligible volume compared to the total β-cell volume, a recent study suggests that these islets, located at the periphery, are the first to be destroyed when type I diabetes develops. Our results underline the capability of xfOCM to contribute to the understanding of the development of diabetes, especially when considering islet volume distribution instead of the total β-cell volume only.

Dynamic fragmentation of a ring: predictable fragment mass distributions

We employ a finite element framework, coupled to cohesive elements, to model material decohesion of a uniformly expanding ring. Our study focuses on the average fragment mass, the distribution of fragment masses, and the heaviest fragments. The computed fragment mass distributions are best captured by generalized gamma distributions, regardless of the model parameters. However, the distribution of the heaviest fragments depends on toughness, specimen size, and loading rate.

Reliable confidence intervals in quantitative genetics: narrow-sense heritability

Many quantitative genetic statistics are functions of variance components, for which a large number of replicates is needed for precise estimates and reliable measures of uncertainty, on which sound interpretation depends. Moreover, in large experiments the deaths of some individuals can occur, so methods for analysing such data need to be robust to missing values. We show how confidence intervals for narrow-sense heritability can be calculated in a nested full-sib/half-sib breeding design (males crossed with several females) in the presence of missing values. Simulations indicate that the method provides accurate results, and that estimator uncertainty is lowest for sampling designs with many males relative to the number of females per male, and with more females per male than progenies per female. Missing data generally had little influence on estimator accuracy, thus suggesting that the overall number of observations should be increased even if this results in unbalanced data. We also suggest the use of parametrically simulated data for prior investigation of the accuracy of planned experiments. Together with the proposed confidence intervals an informed decision on the optimal sampling design is possible, which allows efficient allocation of resources.

Rapid classification of phenotypic mutants of Arabidopsis via metabolite fingerprinting

We evaluated the application of gas chromatography-mass spectrometry metabolic fingerprinting to classify forward genetic mutants with similar phenotypes. Mutations affecting distinct metabolic or signaling pathways can result in common phenotypic traits that are used to identify mutants in genetic screens. Measurement of a broad range of metabolites provides information about the underlying processes affected in such mutants. Metabolite profiles of Arabidopsis (Arabidopsis thaliana) mutants defective in starch metabolism and uncharacterized mutants displaying a starch-excess phenotype were compared. Each genotype displayed a unique fingerprint. Statistical methods grouped the mutants robustly into distinct classes. Determining the genes mutated in three uncharacterized mutants confirmed that those clustering with known mutants were genuinely defective in starch metabolism. A mutant that clustered away from the known mutants was defective in the circadian clock and had a pleiotropic starch-excess phenotype. These results indicate that metabolic fingerprinting is a powerful tool that can rapidly classify forward genetic mutants and streamline the process of gene discovery.

A Laplace mixture model for identification of differential expression in microarray experiments

Microarrays have become an important tool for studying the molecular basis of complex disease traits and fundamental biological processes. A common purpose of microarray experiments is the detection of genes that are differentially expressed under two conditions, such as treatment versus control or wild type versus knockout. We introduce a Laplace mixture model as a long-tailed alternative to the normal distribution when identifying differentially expressed genes in microarray experiments, and provide an extension to asymmetric over- or underexpression. This model permits greater flexibility than models in current use as it has the potential, at least with sufficient data, to accommodate both whole genome and restricted coverage arrays. We also propose likelihood approaches to hyperparameter estimation which are equally applicable in the Normal mixture case. The Laplace model appears to give some improvement in fit to data, though simulation studies show that our method performs similarly to several other statistical approaches to the problem of identification of differential expression.

Assessment and analysis of mechanical allodynia-like behavior induced by spared nerve injury (SNI) in the mouse

Experimental models of peripheral nerve injury have been developed to study mechanisms of neuropathic pain. In the spared nerve injury (SNI) model in rats, the common peroneal and tibial nerves are injured, producing consistent and reproducible pain hypersensitivity in the territory of the spared sural nerve. In this study, we investigated whether SNI in mice is also a valid model system for neuropathic pain. SNI results in a significant decrease in withdrawal threshold in SNI-operated mice. The effect is very consistent between animals and persists for the four weeks of the study. We also determined the relative frequency of paw withdrawal for each of a series of 11 von Frey hairs. Analysis of response frequency using a mixed-effects model that integrates all variables (nerve injury, paw, gender, and time) shows a very stable effect of SNI over time and also reveals subtle divergences between variables, including gender-based differences in mechanical sensitivity. We tested two variants of the SNI model and found that injuring the tibial nerve alone induces mechanical hypersensitivity, while injuring the common peroneal and sural nerves together does not induce any significant increase in mechanical sensitivity in the territory of the spared tibial nerve. SNI induces a mechanical allodynia-like response in mice and we believe that our improved method of assessment and data analysis will reveal additional internal and external variability factors in models of persistent pain. Use of this model in genetically altered mice should be very effective for determining the mechanisms involved in neuropathic pain.

The evaluation of evidence in the forensic investigation of fire incidents (Part I): an approach using Bayesian networks

The forensic investigation of the origin and cause of a fire incident is a particularly demanding area of expertise. As the available evidence is often incomplete or vague, uncertainty is a key element. The present study is an attempt to approach this through the use of Bayesian networks, which have been found useful in assisting human reasoning in a variety of disciplines in which uncertainty plays a central role. The present paper describes the construction of a Bayesian network (BN) and its use for drawing inferences about propositions of interest, based upon a single, possibly non replicable item of evidence: detected residual quantities of a flammable liquid in fire debris.

The evaluation of evidence in the forensic investigation of fire incidents. Part II. Practical examples of the use of Bayesian networks

This paper extends a previous discussion of the use of Bayesian networks for evaluating evidence in the forensic investigation of fire incidents. Bayesian networks are proposed for two casework examples and the practical implications studied in detail. Such networks were found to provide precious support in addressing some of the wide range of issues that affect the coherent evaluation of evidence.

Non-parametric bootstrap confidence intervals for the intraclass correlation coefficient

The intraclass correlation coefficient rho plays a key role in the design of cluster randomized trials. Estimates of rho obtained from previous cluster trials and used to inform sample size calculation in planned trials may be imprecise due to the typically small numbers of clusters in such studies. It may be useful to quantify this imprecision. This study used simulation to compare different methods for assigning bootstrap confidence intervals to rho for continuous outcomes from a balanced design. Data were simulated for combinations of numbers of clusters (10, 30, 50), intraclass correlation coefficients (0.001, 0.01, 0.05, 0.3) and outcome distributions (normal, non-normal continuous). The basic, bootstrap-t, percentile, bias corrected and bias corrected accelerated bootstrap intervals were compared with new methods using the basic and bootstrap-t intervals applied to a variance stabilizing transformation of rho. The standard bootstrap methods provided coverage levels for 95 per cent intervals that were markedly lower than the nominal level for data sets with only 10 clusters, and only provided close to 95 per cent coverage when there were 50 clusters. Application of the bootstrap-t method to the variance stabilizing transformation of rho improved upon the performance of the standard bootstrap methods, providing close to nominal coverage.

Prediction for small subgroups

Prediction limits are calculated for the number of events likely to occur in a specified time period in an exponentially growing epidemic. The basis for the prediction is the total number of events observed in the past.