Bootstrap confidence intervals for the contributions of individual variables to a Mahalanobis distance

Hotelling's T 2 and Mahalanobis distance are widely used in the statistical analysis of multivariate data. When either of these quantities is large, a natural question is: How do individual variables contribute to its size? The Garthwaite–Koch partition has been proposed as a means of assessing the contribution of each variable. This yields point estimates of each variable's contribution and here we consider bootstrap methods for forming interval estimates of these contributions. New bootstrap methods are proposed and compared with the percentile, bias-corrected percentile, non-studentized pivotal and studentized pivotal methods via a large simulation study. The new methods enable use of a broader range of pivotal quantities than with standard pivotal methods, including vector pivotal quantities. In the context considered here, this obviates the need for transformations and leads to intervals that have higher coverage, and yet are narrower, than intervals given by the standard pivotal methods. These results held both when the population distributions were multivariate normal and when they were skew with heavy tails. Both equal-tailed intervals and shortest intervals are constructed; the latter are particularly attractive when (as here) squared quantities are of interest

Adaptive Optimal Scaling of Metropolis-Hastings Algorithms Using the Robbins-Monro Process

We present an adaptive method for the automatic scaling of Random-Walk Metropolis-Hastings algorithms, which quickly and robustly identifies the scaling factor that yields a specified overall sampler acceptance probability. Our method relies on the use of the Robbins-Monro search process, whose performance is determined by an unknown steplength constant. We give a very simple estimator of this constant for proposal distributions that are univariate or multivariate normal, together with a sampling algorithm for automating the method. The effectiveness of the algorithm is demonstrated with both simulated and real data examples. This approach could be implemented as a useful component in more complex adaptive Markov chain Monte Carlo algorithms, or as part of automated software packages

Modified confidence intervals for the Mahalanobis distance

Reiser (2001) proposes a method of forming confidence interval for a Mahalanobis distance that yields intervals which have exactly the nominal coverage, but sometimes the interval is View the MathML source (0,0). We consider the case where Mahalanobis distance quantifies the difference between an individual and a population mean, and suggest a modification that avoids implausible intervals

Running an international survey in a small country : challenges and opportunities

Background: National and international authorities recognize that health surveys are major sources of information on health conditions. Smaller states may prefer using health surveys to registries because they are cheaper to maintain. Nevertheless, smaller states carry out far fewer national health surveys than larger states. One reason could be that the value of surveys depends on the number of people interviewed rather than the proportion of the population. Therefore, survey costs per capita are substantially higher in smaller states. Methods: Malta is a small state with a population of under half a million. It forms part of the European Union, which has provided financial assistance and external expertise in performing international health surveys. We present the European Health Interview Survey in Malta as a case study to review the challenges for small states and the typical adaptations necessary for implementing national health surveys and meeting international health data obligations. Results: We identified the lack of health survey infrastructure, difficulties in recruiting the large samples recommended by international organizations, survey fatigue, and a lack of resources for marketing, incentivization, analysis and dissemination. Low-cost solutions have been devised to address some issues, such as marketing and incentives, which exploit specific characteristics of small states. Conclusion: In the absence of administrative data or epidemiological registers, surveys are important tools for evidence-based policy-making in small states. The experience of Malta could help other small states to minimize the resources required to run national health surveys.peer-reviewe

Contribution of individual variables to the regression sum of squares

In applications of multiple regression, one of the most common goals is to measure the relative importance of each predictor variable. If the predictors are uncorrelated, quantification of relative importance is simple and unique. However, in practice, predictor variables are typically correlated and there is no unique measure of a predictor variable’s relative importance. Using a transformation to orthogonality, new measures are constructed for evaluating the contribution of individual variables to a regression sum of squares. The transformation yields an orthogonal approximation of the columns of the predictor scores matrix and it maximizes the sum of the covariances between the cross-product of individual regressors and the response variable and the cross-product of the transformed orthogonal regressors and the response variable. The new measures are compared with three previously proposed measures through examples and the properties of the measures are examined

Elicitation of prior distributions of variable-selection problems in regression

This paper addresses the problem of quantifying expert opinion about a normal linear regression model when there is uncertainty as to which independent variables should be included in the model. Opinion is modeled as a mixture of natural conjugate prior distributions with each distribution in the mixture corresponding to a different subset of the independent variables. It is shown that for certain values of the independent variables, the predictive distribution of the dependent variable simplifies from a mixture of $t$-distributions to a single $t$-distribution. Using this result, a method of eliciting the conjugate distributions of the mixture is developed. The method is illustrated in an example

Assessment of the learning curve in health technologies: a systematic review

Objective: We reviewed and appraised the methods by which the issue of the learning curve has been addressed during health technology assessment in the past. Method: We performed a systematic review of papers in clinical databases (BIOSIS, CINAHL, Cochrane Library, EMBASE, HealthSTAR, MEDLINE, Science Citation Index, and Social Science Citation Index) using the search term "learning curve:" Results: The clinical search retrieved 4,571 abstracts for assessment, of which 559 (12%) published articles were eligible for review. Of these, 272 were judged to have formally assessed a learning curve. The procedures assessed were minimal access (51%), other surgical (41%), and diagnostic (8%). The majority of the studies were case series (95%). Some 47% of studies addressed only individual operator performance and 52% addressed institutional performance. The data were collected prospectively in 40%, retrospectively in 26%, and the method was unclear for 31%. The statistical methods used were simple graphs (44%), splitting the data chronologically and performing a t test or chi-squared test (60%), curve fitting (12%), and other model fitting (5%). Conclusions: Learning curves are rarely considered formally in health technology assessment. Where they are, the reporting of the studies and the statistical methods used are weak. As a minimum, reporting of learning should include the number and experience of the operators and a detailed description of data collection. Improved statistical methods would enhance the assessment of health technologies that require learning

Assessing the learning curve effect in health technologies: Lessons from the non-clinical literature

Introduction: Many health technologies exhibit some form of learning effect, and this represents a barrier to rigorous assessment. It has been shown that the statistical methods used are relatively crude. Methods to describe learning curves in fields outside medicine, for example, psychology and engineering, may be better. Methods: To systematically search non–health technology assessment literature (for example, PsycLit and Econlit databases) to identify novel statistical techniques applied to learning curves. Results: The search retrieved 9,431 abstracts for assessment, of which 18 used a statistical technique for analyzing learning effects that had not previously been identified in the clinical literature. The newly identified methods were combined with those previously used in health technology assessment, and categorized into four groups of increasing complexity: a) exploratory data analysis; b) simple data analysis; c) complex data analysis; and d) generic methods. All the complex structured data techniques for analyzing learning effects were identified in the nonclinical literature, and these emphasized the importance of estimating intra- and interindividual learning effects. Conclusion: A good dividend of more sophisticated methods was obtained by searching in nonclinical fields. These methods now require formal testing on health technology data sets

Use of expert knowledge in evaluating costs and benefits of alternative service provisions: A case study

Objectives: A treatment pathway model was developed to examine the costs and benefits of the current bowel cancer service in England and to evaluate potential alternatives in service provision. To use the pathway model, various parameters and probability distributions had to be specified. They could not all be determined from empirical evidence and, instead, expert opinion was elicited in the form of statistical quantities that gave the required information. The purpose of this study is to describe the procedures used to quantify expert opinion and note examples of good practice contained in the case study. Methods: The required information was identified and preparatory discussion with four experts refined the questions they would be asked. In individual elicitation sessions they quantified their opinions, mainly in the form of point and interval estimates for specified variables. New methods have been developed for quantifying expert opinion and these were implemented in specialized software that uses interactive graphics. This software was used to elicit opinion about quantities related to measurable covariates. Results: Assessments for thirty-four quantities were elicited and available checks supported their validity. Eight points of good practice in eliciting and using expert judgment were evident. Parameters and probability distributions needed for the pathway model were determined from the elicited assessments. Simulation results from the pathway model were used to inform policy on bowel cancer service provision. Conclusions: The study illustrates that quantifying and using expert judgment can be acceptable in real problems of practical importance. For full benefit to be gained from expert knowledge, elicitation must be conducted carefully and should be reported in detail

An Elicitation Method for Multiple Linear Regression Models

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