Search

Suppose items can be purchased from one of k-suppliers and it is required to purchase from the one with the smaller failure rate or equivalently from the one with the larger mean-time-to-failure. It is assumed that data d, in the form of the times-to-failure for n1,…,nk items from suppliers 1,…,k, respectively is available. There are two suggested selection criteria studied in this paper and when comparing only two suppliers they reduce to P(θ1 cY2|d), where b and c are prespecified practical constants; θ1 and θ2 are the respective mean failure rates; Y1 and Y2 are the predicted times to failure for individual items purchased from each supplier. In addition partial prior information about the k-suppliers collectively is assumed to have been elicited. This situation is modelled using the hierarchical Bayesian approach, which easily facilitates interpreting the elicited partial prior information as constraints on the hyperpriors, i.e. hyperpriors that are known only to be contained in families with specified properties. In this paper these properties are assumed to be in the form of specifying certain quantiles arising from the elicited information. Minimum and maximum values of the above selection criteria are obtained and are used to indicate whether or not the elicited prior information is useful. Specific examples are given for comparing two suppliers but generalisation to k-suppliers follows easily.

The minimum detectable concentration (MDC) is one of the most important properties of an assay. It is a statement about the smallest physical quantity an assay can reliably measure, and is used in assay design and quality control assessments. A plethora of measures of the MDC have been reported in a widely scattered literature. Many of these were developed at a time when accuracy and relevance had to be sacrificed for computational feasibility. This paper identifies limitations of existing measures and demonstrates how Bayesian inference may be used to overcome these limitations. Several new measures of the MDC are developed. These are conceptually simpler than existing measures, and are free of analytical approximations. The recent advances in Bayesian computation make them efficient to evaluate. A procedure developed in this paper measures the difference in the quality of two assays and shows that the new Bayesian measures perform better than existing measures.

To develop a method of limited sampling that would enable accurate estimation of the area under the concentration time curve (AUC) when using the log trapezoidal method.A series of datasets were simulated. Each dataset comprised 1000 subjects. Each subject was "administered" an intravenous bolus dose of a drug that displays two compartment pharmacokinetics. In the first series of simulations, a variety of combinations of the number of sampling times (K) and number of replicate measurements (R) at each of these times were tested, where K x R = 12 (i.e. N = 12). The times that each of the K samples were taken were chosen to be those that divided the AUC into K - 1 trapezoids of equal area. The concentration-time curves were estimated based on a priori estimates of the population parameters. The best combination of K and R was tested under various conditions of parameter variability and assay variability. The combinations were compared with a conventional sampling strategy, where N = 12, K = 12 (R = 1).The combination K = 4 and R = 3 proved to be the "best". It had similar accuracy to the conventional method. The best limited sampling combination was superior to the conventional method when assay variability was high (CV= 30%), was similar when assay variability was 15%, but the conventional method became statistically superior when assay variability was 7.5% or less. The accuracy of the best limited sampling combination was inversely related to the parameter variability. If K was set to 4 and R allowed to increase to 6 (i.e. N is not equal to 12), there was no further gain in accuracy.The proposed method of limited sampling is at least as accurate as the conventional intensive sampling technique, but more efficient in terms of sampling.

Summary In the context of the data and issues discussed by Goldstein and Spiegelhalter, we suggest refinements which can be used by decision makers when confronted with ranking problems associated with ‘league tables’. Two ranking criteria are defined and their performance illustrated for one of the studies reported by Goldstein and Spiegelhalter.

The reduction of quantitative diagnostic test scores to the dichotomous case is a wasteful and unnecessary simplification in the era of high-speed computing. Physicians could make better use of the information embedded in quantitative test results if modern generalized curve estimation techniques were applied to the likelihood functions of Bayes' theorem. Hand calculations could be completely avoided and computed graphical summaries provided instead. Graphs showing posttest probability of disease as a function of pretest probability with confidence intervals (POD plots) would enhance acceptance of these techniques if they were immediately available at the computer terminal when test results were retrieved. Such constructs would also provide immediate feedback to physicians when a valueless test had been ordered.

Identification, ranking and selecting hazardous traffic accident locations from a group under consideration is a fundamental goal for traffic safety researchers. Few methods exist that can quantitatively, accurately and easily discriminate between sites that commonly have small and variable observation count periods. One method that embodies all these advantages is the hierarchical Bayesian model, the method proposed in this paper. The particular hierarchical Bayesian approach that we use incorporates expert knowledge about accident sites as a group believed a priori to be exchangeable, the Poisson assumption and a conjugate gamma prior. We then propose three natural strategies for ranking and selecting the most hazardous subgroup of accident locations. Also presented is an especially useful procedure that gives the probability that each particular site is worst and by how much it is worst. All proposed strategies are illustrated using previously published fatality accident data from 35 sites in Auckland, New Zealand.

Carboplatin is associated with significantly less nephrotoxicity and neurotoxicity than is cisplatin. The dose-limiting toxicity of carboplatin is myelotoxicity. A number of dosing methods have been described that allow a value for the area under the concentration-time curve to be targeted on the basis of the patient's renal function. Recently a formalised analysis of the pharmacodynamic response to carboplatin revealed a therapeutic window in which the response rate was maximal and toxicity, tolerable. Optimal therapy would result from targeting this window in the individual patient. The aim of this study was to develop a Bayesian dose-individualisation method for carboplatin. The method involved (1) development of a high-performance liquid chromatography (HPLC) method to measure serum concentrations of carboplatin; (2) a pharmacokinetic study in 12 women receiving carboplatin for ovarian cancer to estimate the population pharmacokinetic values for this group of patients; (3) development of population models to describe the concentration-time course of carboplatin in serum along with associated errors; and (4) development of an algorithm that uses a sequential Bayesian design, which enables estimation of future doses of carboplatin on the basis of feedback from serum concentrations. The results of each of the stages were (1) the coefficient of variation of the assay was 6.3% within day and 8.4% between days (r2 = 0.9993), and the limit of detection was 0.25 mg/l; (2) Patients' ages ranged from 49 to 68 years, their weights varied from 46 to 85 kg, and their glomerular filtration rate ranged from 3.2 to 7.4 l/h. A geometric mean clearance (Cl) of 6.8 L/h and a steady-state volume of distribution (Vss) of 221 were estimated, which are similar to previously published data; (3) and a two-compartment model best described the data. Two error models were developed, the first describing the error associated with the assay and the second, the error of the two-compartment model, i.e. error due to individual variation in pharmacokinetics and error due to model mis-specification. Finally, (4) the development of a sequential Bayesian dose-individualisation method for carboplatin is described. To our knowledge, this is the first sequential design that has been used for dose individualisation of chemotherapy. The program is specific for carboplatin and operates independently of commercially available Bayesian software. Doses predicted by this program are being tested prospectively against conventional dosing methods.

The high reliability of modern components requires accelerated testing to compare or predict survival or failure rates in the use condition. If the testing of highly reliable components is done in the use condition, the times to failure are too long to observe. Therefore, it is often required to compare or predict mean times to failure in the use condition when the only available data are from a highly accelerated condition. Comparison of failure rates in this situation is possible using frequentist methods but estimation of the individual failure rates is not. Using Bayes methods, both comparison and prediction results are easily possible and computable. The accelerated model in this paper is similar to the model used in health-related research when the data are from a paired experiment. This use of the model to compare and predict survival rates using paired data is also handled easily by the Bayes approach. Let the two failure rates be /spl lambda//sub 1/, /spl lambda//sub 2/; the results are presented as the posterior probability /spl Pscr/r{/spl lambda//sub 1/

We idenify the invertible coherent functional relation between an array of asserted conditional probabilities and the probability distribution for the sum of events that are regarded exchangeably, in the regular case thatP(N N+1 |S N =a) ∈ (0, 1) for everya=0, 1, ...,N. The result is used to construct a useful algebraic and geometrical representation of all coherent inferences in the regular case, including those that are nonlinear in the sum of the conditioning events. The special case in which conditional probabilities mimic observed frequencies within (0, 1) receives an exact solution, which allows an easy interpretation of its surprising consequences. Finally, we introduce a new direction in research on prior opinion assessment that this approach, inverse to the usual one, suggests.

These rules for terminating testing involve achieving specified levels of credibility in both: (1) the probability of the posterior estimate of the exponential scale-parameter, /spl theta/, after some, m, of the units have been tested, and (2) the mean of the probability of /spl theta/ over the remaining units, when viewed pessimistically. Sample decision tables and a numerical example illustrate both the sequential and batch testing cases. Large savings in test times can be achieved whenever the first m units present strong evidence in favor of either hypothesis (H/sub 0/: /spl theta/=/spl thetasub 0/ vs. H/sub 1/: /spl theta/=/spl thetasub 1/, /spl thetasub 1/ >

The classical allocation formula of Neyman is replaced by a new formula which can accommodate prior information on the ratio of means in two strata. To accomplish this, a Bayesian approach is adopted and in this context partial prior information is quantified over the spectrum from absolute certainty to completely unknown.

A new approach called "minimum intervention" has been introduced for restoration of carious lesions to preserve tooth structure. This approach suggests that perhaps caries need not always be removed completely from deeper portions of the cavity. It is, therefore, important to characterize caries-affected dentin structures, because of the potential changes in bonding quality when using different dentinal substrates.Ninety teeth (30 teeth each group) were studied. The first group (CF) consisted of 30 caries-free teeth. The second group (CC) consisted of 30 teeth, for which caries-free dentin teeth was chemically demineralized. The third group (ND) consisted of 30 extracted human molars with coronal carious lesions. After all tooth samples were water-polished with grit #600 SiC paper, they were tested by surface contact angle measurements and the electron-probe microanalyzer to measure Ca/P mol ratio.Contact angles were CF = 60.07 degrees ; CC = 30.8 degrees; ND = 26.11 degrees , p0.05. Ca/P mol ratios were as follows; CF = 1.549 (+/-0.0435); CC = 1.324 (+/-0.2305); ND = 1.568 (+/-0.0523), p0.05. Weibull analyses for Ca/P mol ratio indicated shape parameter (m) of CF was 13.3; it was 12.8 for ND and 11.8 for CC. Above the delta point (=1.65 in Ca/P ratio), for both groups m = 3.4.Caries-affected dentin surfaces (naturally-developed and chemically created) were statistically more chemically active than caries-free dentin surface. Ca/P mol ratio of chemically created caries was less than other two groups.

The purpose of this paper is to illustrate the value of a Bayesian approach in practical problems involving two groups (treatments) which are to be compared. Using practical criteria for comparisons, the advantage of the Bayesian approach will be demonstrated in two situations. Problem I will deal with cholesterol data from two groups of people and it is the aim of the experiment to say which group has the higher cholesterol levels. Problem II will compare two treatments in a clinical trial situation at an interim before the trial is completed. This situation is referred to as interim analysis because only part of the data in the planned trial has been collected. At this interim stage it is desirable to decide whether to stop the experiment now or to continue until the end.

ARE (absolute relative error) and SqRE (squared relative error), are random variables that are suggested as measurements of forecast accuracy of the total number of estimated software failures at the end of a mission time. The purpose is to compare the predictive merit of competing software reliability models, an important concern to software reliability analysts. This technique calculates the Bayes probability of how much better the prediction accuracy is for one method relative to a competitor. This novel approach is more realistic, in the assessment of predictive merit, than (a) comparing merely the average values of ARE and SqRE as conventionally done; and (b) conducting statistical hypothesis tests of pair-wise means of ARE and SqRE, an approach somewhat more sensible than (a), because (b) incorporates variability of predicted values, which (a) does not. To implement this technique, first noninformative (across the border) are used and then informative (specified) priors. For the informative case, half-normal priors are placed on the mean of the ARE or SqRE random variables, because these means are hypothesized to remain peaked around zero relative-error (ideal error percentage). This problem is related to the general problem of ranking usual means discussed in the literature by Berger and Deely (1988), and is a follow-up to an invited research paper presented at ISI-97 by Sahinoglu and Capar (1997).

The problem of ranking and selecting normal means as originally studied by Shanti Gupta is approached herein from a robust Bayesian perspective. This model uses the usual hierarchical Bayesian setup but does not require complete specifications of the hyperpriors. Instead, elicited prior information about the population of means as a group is used to specify a quantile class for the hyperpriors. Two criteria are suggested for ranking and selecting which provide insight not only to which population is best, but in addition give quantitative methods for deciding how much better one population is than another. Using these criteria, minimum and maximum values are calculated for the derived quantile class. Relative sizes of these evaluations and the distance between the max and min give insight as to the quality of the data and the sufficiency of the sample size. These concepts are illustrated with a numerical example.

This article considers the variation of the Secretary Problem where at any stage each of the last m applicants are available for employment. We show that the form of the stopping rule which maximizes the probability of employing the best applicant is: stop at the first stage, from r* onwards, when the relatively best applicant is about to become unavailable. A simple algorithm for evaluating r* and the probability of employing the best applicant is outlined.

A set of unknown normal means (treatment effects, say) {θ1, θ2, …, θk } is to be investigated. Two common questions in analysis of variance and ranking and selection are as follows: (a) What is the strength of evidence against the hypothesis H 0 of equality of means? (b) If H 0 is false, which mean is the largest (or smallest)? A Bayesian approach to the problem is taken, leading to calculation of the posterior probability of H 0 and the posterior probabilities that each mean is the largest, conditional on H 0 being false. A variety of exchangeable, nonexchangeable, informative, and noninformative prior assumptions are considered. Calculations involve, at worst, only low-dimensional numerical integration, in spite of the fact that the dimension k can be arbitrarily large. As an example, Table 1 presents, for each baseball team in the National League in 1984, the highest batting average obtained by any player on the team with at least 150 at bats. The observed batting averages are treated as sampl...

A Bayesian approach is given for various kinds of empirical Bayes problems. In particular it is shown that empirical Bayes procedures are really non-Bayesian, asymptotically optimal, classical procedures for mixtures. In some situations these procedures are Bayes with respect to some prior and in other situations, there is no prior for which they are Bayes. Several examples of these concepts are given as well as a general theory showing the difference between an empirical Bayes model and a Bayes empirical Bayes model.

The purpose of this paper is to make the reader aware of the applicability and advantage of a particular mathematical model. The application is typified by an example and the advantage is via confidence intervals; that is, shorter confidence intervals are possible using the model than if one ignores it, providing the applicability is valid. It is also shown that an improved estimate can be obtained through use of the model. Let f(y|μ,σ) be a normal density with mean μ and variance σ2 and let g(μ|λ, β) be a normal density with mean λ and variance β2. A sequence y 1, y 2, …, y n+1 of independent observations from the mixture of f and g can be considered as follows: An unobservable μi is first drawn from g(μ|λ, β) and then yi which can be observed is drawn from f(y|μi, σ). Confidence intervals on μ n+1 are obtained which are based on the observations y 1, …, y n+1 and which are shorter than the standard interval based on y n+1 only for any n. Shorter intervals are obtained for two cases: (i) λ unkno...

The purpose of this paper is to present an easily applied test useful in determining whether or not a supplier's data have been truncated. The proposed test has the following desirable properties: (i) it is the uniformly most powerful rank test, (ii) it is asymptotically uniformly most powerful, (iii) power computations can easily be made for arbitrary sample sizes, formulas for such computations being given in the paper. Although formulated in the context of verifying a supplier's data, the test can be applied to other situations in which false representation of data in the form of truncation is important. Such is the case, for example, in reliability demonstrations or legal suits involving physical measurements.

The use of the maximum entropy principle for determining prior distribution is compared with other techniques in statistical-decision theory for estimating reliability. The comparison is made in the context of estimating the parameter representing the probability of success in a binomial model and the parameter representing the mean time to failure in a simple exponential model. The existence of partial prior information of varying degrees is also assumed. When a quadratic loss function is used, the techniques based on the maximum entropy principle lose much of their appeal.

SUMMARY It is shown that, for quadratic loss and members of the exponential family with an unbiased sufficient statistic having a variance which is quadratic in the parameter, a C2-minimax estimator exists. In addition, the estimator is of simple form and appears to satisfy intuition. Also, it is shown that the 62-minimax estimator is admissible in the usual sense, is 92-admissible and makes the unbiased sufficient statistic inadmissible in S2. Finally, in this situation, the natural conjugate prior distribution is always a member of the Pearson family and is always least favourable. Asymptotic 92-minimax estimators are discussed briefly.

We consider modified empirical Bayes problems in which the prior distribution of $\Theta$ at stage $n + 1$ is $G^{(n+1)}(\theta)$. The Bayes optimality criterion is now given by the sequence of functionals $R(G^{(n+1)}$. The observations $X_1, \cdots, X_n$ are no longer i.i.d so decision procedures are constructed based on modified empirical density estimates for $f_G^{(n+1)}(x)$. Asymptotic optimality together with asymptotic convergence rates is established for two action and estimation problems when the observations are drawn from a member of the one-parameter exponential family.

The purpose of this paper is to obtain weaker sufficient conditions than those given by Robbins [6] for the asymptotic optimality of empirical Bayes procedures. In so doing a general method for providing asymptotic optimality is given which because of its simplicity should prove quite useful not only for the empirical Bayes problem but for other related asymptotic problems as well. Three areas of application are indicated herein.

(1969). Joint Continuity of Monotonic Functions. The American Mathematical Monthly: Vol. 76, No. 1, pp. 74-76.