Orthogonalization of vectors with minimal adjustment

Two transformations are proposed that give orthogonal components with a one-to-one correspondence between the original vectors and the components. The aim is that each component should be close to the vector with which it is paired, orthogonality imposing a constraint. The transformations lead to a variety of new statistical methods, including a unified approach to the identification and diagnosis of collinearities, a method of setting prior weights for Bayesian model averaging, and a means of calculating an upper bound for a multivariate Chebychev inequality. One transformation has the property that duplicating a vector has no effect on the orthogonal components that correspond to nonduplicated vectors, and is determined using a new algorithm that also provides the decomposition of a positive-definite matrix in terms of a diagonal matrix and a correlation matrix. The algorithm is shown to converge to a global optimum

Eliciting Dirichlet and Gaussian copula prior distributions for multinomial models

In this paper, we propose novel methods of quantifying expert opinion about prior distributions for multinomial models. Two different multivariate priors are elicited using median and quartile assessments of the multinomial probabilities. First, we start by eliciting a univariate beta distribution for the probability of each category. Then we elicit the hyperparameters of the Dirichlet distribution, as a tractable conjugate prior, from those of the univariate betas through various forms of reconciliation using least-squares techniques. However, a multivariate copula function will give a more flexible correlation structure between multinomial parameters if it is used as their multivariate prior distribution. So, second, we use beta marginal distributions to construct a Gaussian copula as a multivariate normal distribution function that binds these marginals and expresses the dependence structure between them. The proposed method elicits a positive-definite correlation matrix of this Gaussian copula. The two proposed methods are designed to be used through interactive graphical software written in Java

How does reviewing the evidence change veterinary surgeons' beliefs regarding the treatment of ovine footrot? A quantitative and qualitative study

Footrot is a widespread, infectious cause of lameness in sheep, with major economic and welfare costs. The aims of this research were: (i) to quantify how veterinary surgeons’ beliefs regarding the efficacy of two treatments for footrot changed following a review of the evidence (ii) to obtain a consensus opinion following group discussions (iii) to capture complementary qualitative data to place their beliefs within a broader clinical context. Grounded in a Bayesian statistical framework, probabilistic elicitation (roulette method) was used to quantify the beliefs of eleven veterinary surgeons during two one-day workshops. There was considerable heterogeneity in veterinary surgeons’ beliefs before they listened to a review of the evidence. After hearing the evidence, seven participants quantifiably changed their beliefs. In particular, two participants who initially believed that foot trimming with topical oxytetracycline was the better treatment, changed to entirely favour systemic and topical oxytetracycline instead. The results suggest that a substantial amount of the variation in beliefs related to differences in veterinary surgeons’ knowledge of the evidence. Although considerable differences in opinion still remained after the evidence review, with several participants having non-overlapping 95% credible intervals, both groups did achieve a consensus opinion. Two key findings from the qualitative data were: (i) veterinary surgeons believed that farmers are unlikely to actively seek advice on lameness, suggesting a proactive veterinary approach is required (ii) more attention could be given to improving the way in which veterinary advice is delivered to farmers. In summary this study has: (i) demonstrated a practical method for probabilistically quantifying how veterinary surgeons’ beliefs change (ii) revealed that the evidence that currently exists is capable of changing veterinary opinion (iii) suggested that improved transfer of research knowledge into veterinary practice is needed (iv) identified some potential obstacles to the implementation of veterinary advice by farmers

A frequentist framework of inductive reasoning

Reacting against the limitation of statistics to decision procedures, R. A. Fisher proposed for inductive reasoning the use of the fiducial distribution, a parameter-space distribution of epistemological probability transferred directly from limiting relative frequencies rather than computed according to the Bayes update rule. The proposal is developed as follows using the confidence measure of a scalar parameter of interest. (With the restriction to one-dimensional parameter space, a confidence measure is essentially a fiducial probability distribution free of complications involving ancillary statistics.) A betting game establishes a sense in which confidence measures are the only reliable inferential probability distributions. The equality between the probabilities encoded in a confidence measure and the coverage rates of the corresponding confidence intervals ensures that the measure's rule for assigning confidence levels to hypotheses is uniquely minimax in the game. Although a confidence measure can be computed without any prior distribution, previous knowledge can be incorporated into confidence-based reasoning. To adjust a p-value or confidence interval for prior information, the confidence measure from the observed data can be combined with one or more independent confidence measures representing previous agent opinion. (The former confidence measure may correspond to a posterior distribution with frequentist matching of coverage probabilities.) The representation of subjective knowledge in terms of confidence measures rather than prior probability distributions preserves approximate frequentist validity.Comment: major revisio

A comparison of two methods for expert elicitation in health technology assessments.

BACKGROUND: When data needed to inform parameters in decision models are lacking, formal elicitation of expert judgement can be used to characterise parameter uncertainty. Although numerous methods for eliciting expert opinion as probability distributions exist, there is little research to suggest whether one method is more useful than any other method. This study had three objectives: (i) to obtain subjective probability distributions characterising parameter uncertainty in the context of a health technology assessment; (ii) to compare two elicitation methods by eliciting the same parameters in different ways; (iii) to collect subjective preferences of the experts for the different elicitation methods used. METHODS: Twenty-seven clinical experts were invited to participate in an elicitation exercise to inform a published model-based cost-effectiveness analysis of alternative treatments for prostate cancer. Participants were individually asked to express their judgements as probability distributions using two different methods - the histogram and hybrid elicitation methods - presented in a random order. Individual distributions were mathematically aggregated across experts with and without weighting. The resulting combined distributions were used in the probabilistic analysis of the decision model and mean incremental cost-effectiveness ratios and the expected values of perfect information (EVPI) were calculated for each method, and compared with the original cost-effectiveness analysis. Scores on the ease of use of the two methods and the extent to which the probability distributions obtained from each method accurately reflected the expert's opinion were also recorded. RESULTS: Six experts completed the task. Mean ICERs from the probabilistic analysis ranged between £162,600-£175,500 per quality-adjusted life year (QALY) depending on the elicitation and weighting methods used. Compared to having no information, use of expert opinion decreased decision uncertainty: the EVPI value at the £30,000 per QALY threshold decreased by 74-86 % from the original cost-effectiveness analysis. Experts indicated that the histogram method was easier to use, but attributed a perception of more accuracy to the hybrid method. CONCLUSIONS: Inclusion of expert elicitation can decrease decision uncertainty. Here, choice of method did not affect the overall cost-effectiveness conclusions, but researchers intending to use expert elicitation need to be aware of the impact different methods could have.This paper presents independent research funded by the National Institute of Health Research (NIHR) Collaboration for Leadership in Applied Health Research and Care (CLAHRC) for the South West Peninsula

Expert Status and Performance

Expert judgements are essential when time and resources are stretched or we face novel dilemmas requiring fast solutions. Good advice can save lives and large sums of money. Typically, experts are defined by their qualifications, track record and experience [1], [2]. The social expectation hypothesis argues that more highly regarded and more experienced experts will give better advice. We asked experts to predict how they will perform, and how their peers will perform, on sets of questions. The results indicate that the way experts regard each other is consistent, but unfortunately, ranks are a poor guide to actual performance. Expert advice will be more accurate if technical decisions routinely use broadly-defined expert groups, structured question protocols and feedback

Bayesian profiling of molecular signatures to predict event times

BACKGROUND: It is of particular interest to identify cancer-specific molecular signatures for early diagnosis, monitoring effects of treatment and predicting patient survival time. Molecular information about patients is usually generated from high throughput technologies such as microarray and mass spectrometry. Statistically, we are challenged by the large number of candidates but only a small number of patients in the study, and the right-censored clinical data further complicate the analysis. RESULTS: We present a two-stage procedure to profile molecular signatures for survival outcomes. Firstly, we group closely-related molecular features into linkage clusters, each portraying either similar or opposite functions and playing similar roles in prognosis; secondly, a Bayesian approach is developed to rank the centroids of these linkage clusters and provide a list of the main molecular features closely related to the outcome of interest. A simulation study showed the superior performance of our approach. When it was applied to data on diffuse large B-cell lymphoma (DLBCL), we were able to identify some new candidate signatures for disease prognosis. CONCLUSION: This multivariate approach provides researchers with a more reliable list of molecular features profiled in terms of their prognostic relationship to the event times, and generates dependable information for subsequent identification of prognostic molecular signatures through either biological procedures or further data analysis

Protein expression in experimental malignant glioma varies over time and is altered by radiotherapy treatment

Radiotherapy is one of the mainstays of glioblastoma (GBM) treatment. This study aims to investigate and characterise differences in protein expression patterns in brain tumour tissue following radiotherapy, in order to gain a more detailed understanding of the biological effects. Rat BT4C glioma cells were implanted into the brain of two groups of 12 BDIX-rats. One group received radiotherapy (12 Gy single fraction). Protein expression in normal and tumour brain tissue, collected at four different time points after irradiation, were analysed using surface enhanced laser desorption/ionisation – time of flight – mass spectrometry (SELDI-TOF-MS). Mass spectrometric data were analysed by principal component analysis (PCA) and partial least squares (PLS). Using these multivariate projection methods we detected differences between tumours and normal tissue, radiation treatment-induced changes and temporal effects. 77 peaks whose intensity significantly changed after radiotherapy were discovered. The prompt changes in the protein expression following irradiation might help elucidate biological events induced by radiation. The combination of SELDI-TOF-MS with PCA and PLS seems to be well suited for studying these changes. In a further perspective these findings may prove to be useful in the development of new GBM treatment approaches

Calf health from birth to weaning. II. Management of diarrhoea in pre-weaned calves

Calfhood diseases have a major impact on the economic viability of cattle operations. The second of this three part review series considers the management of diarrhoeic diseases in pre-weaned calves. In neonatal calf diarrhoea, oral rehydration therapy is the single most important therapeutic measure to be carried out by the farmer and is usually successful if instigated immediately after diarrhoea has developed. Continued feeding of milk or milk replacer to diarrhoeic calves is important, to prevent malnourishment and weight loss in affected calves. Indiscriminative antibiotic treatment of uncomplicated diarrhoea is discouraged, whereas systemically ill calves can benefit from systemic antibiotic treatment for the prevention of septicaemia or concurrent diseases. Ancillary treatments and specific preventive measures are discussed. Eimeriosis has a high economic impact on the farming industries due to direct cost of treatment and calf losses, but especially due to decreased performance of clinically as well as sub-clinically affected animals. Emphasis lies on prophylactic or metaphylactic treatment, since the degree of damage to the intestinal mucosa once diarrhoea has developed, makes therapeutic intervention unrewarding