You May Believe You Are a Bayesian But You Are Probably Wrong

An elementary sketch of some issues in statistical inference and in particular of the central role of likelihood is given. This is followed by brief outlines of what George Barnard considered were the four great systems of statistical inferences. These can be thought of terms of the four combinations of two factors at two levels. The first is fundamental purpose (decision or inference) and the second probability argument (direct or inverse). Of these four systems the 'fully Bayesiani approach of decision- making using inverse probability particularly associated with the Ramsay, De Finetti, Savage and Lindley has some claims to be the most impressive. It is claimed, however, and illustrated by example, that this approach seems to be impossible to follow. It is speculated that there may be some advantage to the practising statistician to follow George Barnardis advice of being familiar with all four systems.philosophy of statistics, decision theory, likelihood, subjective probability, Bayesianism, Bayes theorem, Fisher, Neyman and Pearson, Jeffreys

The Prosecution of Religious Fraud

The first amendment to the United States Constitution protects the religious freedom of individuals through its establishment and free exercise clauses. Should the government\u27s hands-off policy under the free exercise clause provide a protective blanket for fraudulent moneymaking schemes carried out in the name of religion? The author of this Article argues that the protection of religious freedom can comfortably coexist with protection from religious fraud where courts employ a sincerity test. He concludes that with appropriate procedural safeguards, courts can use this test to distinguish sincere religious exercise from criminally fraudulent enterprise

An investigation into the unusual linkage isomerization and nitrite reduction activity of a novel tris(2-pyridyl) copper complex

The copper-containing nitrite reductases (CuNIRs) are a class of enzymes that mediate the reduction of nitrite to nitric oxide in biological systems. Metal–ligand complexes that reproduce the salient features of the active site of CuNIRs are therefore of fundamental interest, both for elucidating the possible mode of action of the enzymes and for developing biomimetic catalysts for nitrite reduction. Herein, we describe the synthesis and characterization of a new tris(2-pyridyl) copper complex ([Cu1(NO2)2]) that binds two molecules of nitrite, and displays all three of the common binding modes for NO2−, with one nitrite bound in an asymmetric quasi-bidentate κ2-ONO manner and the other bound in a monodentate fashion with a linkage isomerism between the κ1-ONO and κ1-NO2 binding modes. We use density functional theory to help rationalize the presence of all three of these linkage isomers in one compound, before assessing the redox activity of [Cu1(NO2)2]. These latter studies show that the complex is not a competent nitrite reduction electrocatalyst in non-aqueous solvent, even in the presence of additional proton donors, a finding which may have implications for the design of biomimetic catalysts for nitrite reduction

On the relevance of prognostic information for clinical trials: A theoretical quantification

The question of how individual patient data from cohort studies or historical clinical trials can be leveraged for designing more powerful, or smaller yet equally powerful, clinical trials becomes increasingly important in the era of digitalization. Today, the traditional statistical analyses approaches may seem questionable to practitioners in light of ubiquitous historical prognostic information. Several methodological developments aim at incorporating historical information in the design and analysis of future clinical trials, most importantly Bayesian information borrowing, propensity score methods, stratification, and covariate adjustment. Adjusting the analysis with respect to a prognostic score, which was obtained from some model applied to historical data, received renewed interest from a machine learning perspective, and we study the potential of this approach for randomized clinical trials. In an idealized situation of a normal outcome in a two-arm trial with 1:1 allocation, we derive a simple sample size reduction formula as a function of two criteria characterizing the prognostic score: (1) the coefficient of determination R2 on historical data and (2) the correlation ρ between the estimated and the true unknown prognostic scores. While maintaining the same power, the original total sample size n planned for the unadjusted analysis reduces to (1−R2ρ2)×n in an adjusted analysis. Robustness in less ideal situations was assessed empirically. We conclude that there is potential for substantially more powerful or smaller trials, but only when prognostic scores can be accurately estimated. Keywords: clinical trials; covariate adjustment; machine learning; prognostic covariates; sample size reduction

The statistical properties of RCTs and a proposal for shrinkage

We abstract the concept of a randomized controlled trial as a triple (β,b,s) , where β is the primary efficacy parameter, b the estimate, and s the standard error ( s>0 ). If the parameter β is either a difference of means, a log odds ratio or a log hazard ratio, then it is reasonable to assume that b is unbiased and normally distributed. This then allows us to estimate the joint distribution of the z-value z=b/s and the signal-to-noise ratio SNR=β/s from a sample of pairs (bi,si) . We have collected 23 551 such pairs from the Cochrane database. We note that there are many statistical quantities that depend on (β,b,s) only through the pair (z,SNR) . We start by determining the estimated distribution of the achieved power. In particular, we estimate the median achieved power to be only 13%. We also consider the exaggeration ratio which is the factor by which the magnitude of β is overestimated. We find that if the estimate is just significant at the 5% level, we would expect it to overestimate the true effect by a factor of 1.7. This exaggeration is sometimes referred to as the winner's curse and it is undoubtedly to a considerable extent responsible for disappointing replication results. For this reason, we believe it is important to shrink the unbiased estimator, and we propose a method for doing so. We show that our shrinkage estimator successfully addresses the exaggeration. As an example, we re-analyze the ANDROMEDA-SHOCK trial. Keywords: Cochrane review; achieved power; exaggeration; randomized controlled trial; type M error

Charge, orbital and magnetic ordering in transition metal oxides

Neutron and x-ray diffraction has been used to study charge, orbital and magnetic ordering in some transition metal oxides. The long standing controversy regarding the nature of the ground state (Verwey structure) of the canonical charge ordered material magnetite (Fe3O4) has been resolved by x-ray single crystal diffraction studies on an almost single domain sample at 90 K. The Verwey structure is confirmed to have Cc symmetry with 56 unique sites in the asymmetric unit. Charge ordering is shown to be a useful first approximation to describe the nature of the ground state, and the conjecture that Verwey made in 1939 has finally been confirmed. However, three-site distortions which couple to the orbital ordering of the Fe2+ ordered states (trimerons) are shown to provide a more complete description of the low temperature structure. Trimerons explain the rather continuous distribution of the valence states observed in magnetite below Tv, anomalous shortening of Fe-Fe distances and the off-centre distortions resulting in ferroelectricity. DFT+U electronic structure calculations on the experimental coordinates support the conclusion of this crystallographic study, with the highest electron densities calculated for those Fe-Fe distances predicated to participate in the trimeron bonds. The 6H-perovskites of the type Ba3ARu2O9 have been reinvestigated by high resolution neutron and x-ray power diffraction. The charge ordered state of Ba3NaRu2O9 has been characterised at 110 K (P2/c, a =5.84001(2) Å, b = 10.22197(4) Å, c = 14.48497(6) Å, β = 90.2627(3) °) and shown to consist of a structure with near integer charge ordering of Ru5+ 2O9 / Ru6+ 2O9 dimers. The ground state has been shown to be very sensitive to external perturbations, with a novel melting of charge ordering observed under x-ray irradiation below 40 K (C2/c, a =5.84470(2) Å, b = 10.17706(3) Å, c = 14.45866(5) Å, β = 90.2151(3)-° at 10 K). High pressure studies reveal that the Ru-Ru intra-dimer distance may dictate the response of the system to pressure. Empirical trends in the Ba3ARu2O9 series of compounds have shown that change in ‘chemical pressure’ in these systems may be rationalised in terms of Coulomb’s law. In A = La and Y the magnetic ordering is shown to be FM within the Ru2O9 dimers (1.4(2) μB and 0.5(1) μB, respectively per Ru), representing the first case of intra dimer FM coupling reported in a system containing face-sharing RuO6 octahedra . The overall AFM coupling of the dimers implies an as yet unobserved breaking of the parent symmetry. In A = Nd, a complex competition between the crystal field effect of Nd3+ and the magnetic ordering of the Ru2O9 FM moments has been observed, leading first vi to FM order of Nd at 25 K (1.56(7) μB) followed by ordering of Ru moments (0.5(1) μB) and a spin reorientation transition of Nd moments at 18 K. In A = Ca, the formation of a singlet ground state is observed in Ru2O9 rather than the expected AFM coupling and below 100 K Ba3CaRu2O9 is diamagnetic. All five systems indicate that the Ru2O9 dimer is the physically significant unit in these systems when considering structural trends and the ordering of charge, spin and orbital degrees of freedom

Overstating the evidence - double counting in meta-analysis and related problems

Background: The problem of missing studies in meta-analysis has received much attention. Less attention has been paid to the more serious problem of double counting of evidence. Methods: Various problems in overstating the precision of results from meta-analyses are described and illustrated with examples, including papers from leading medical journals. These problems include, but are not limited to, simple double-counting of the same studies, double counting of some aspects of the studies, inappropriate imputation of results, and assigning spurious precision to individual studies. Results: Some suggestions are made as to how the quality and reliability of meta-analysis can be improved. It is proposed that the key to quality in meta-analysis lies in the results being transparent and checkable. Conclusions: Existing quality check lists for meta-analysis do little to encourage an appropriate attitude to combining evidence and to statistical analysis. Journals and other relevant organisations should encourage authors to make data available and make methods explicit. They should also act promptly to withdraw meta-analyses when mistakes are found