Benjamin Reiser

Statistical inference for Pr(Y < X): The normal case

This article examines statistical inference for Pr(Y < X), where X and Y are independent normal variates with unknown means and variances. The case of unequal variances is stressed. X can be interpreted as the strength of a component subjected to a stress Y, and Pr(Y < X) is the components reliability. Two approximate methods for obtaining confidence intervals and an approximate Bayesian probability interval are obtained. The actual coverage probabilities of these intervals are examined by simulation.

Confidence intervals for the generalized ROC criterion.

Receiver operating characteristic (ROC) curves are frequently used to assess the usefulness of diagnostic markers. When several diagnostic markers are available, they can be combined by a best linear combination: that is, when the area under the ROC curve of this combination is maximized among all possible linear combinations. This maximal area is the generalized ROC criterion, which provides a measure of how effective the combination of the markers is. This criterion needs to be estimated from the data, and is usually evaluated against single markers. In the present paper, we provide confidence intervals for the generalized ROC criterion under the assumption of homogeneous covariance matrices, derive an approximation for the heterogeneous covariance matrices case, and evaluate the approximation via a simulation study. Finally, we present an illustrative example.

Bayesian Inference for Masked System Lifetime Data

Estimating component and system reliabilities frequently requires using data from the system level. Because of cost and time constraints, however, the exact cause of system failure may be unknown. Instead, it may only be ascertained that the cause of system failure is due to a component in a subset of components. This paper develops methods for analysing such masked data from a Bayesian perspective. This work was motivated by a data set on a system unit of a particular type of IBM PS/2 computer. This data set is discussed and our methods are applied to it

mROC: a computer program for combining tumour markers in predicting disease states

Receiver operating characteristic (ROC) curves are limited when several diagnostic tests are available, mainly due to the problems of multiplicity and inter-relationships between the different tests. The program presented in this paper uses the generalised ROC criteria, as well as its confidence interval, obtained from the non-central F distribution, as a possible solution to this problem. This criterion corresponds to the best linear combination of the test for which the area under the ROC curve is maximal. Quantified marker values are assumed to follow a multivariate normal distribution but not necessarily with equal variances for two populations. Other options include Box-Cox variable transformations, QQ-plots, interactive graphics associated with changes in sensitivity and specificity as a function of the cut-off. We provide an example to illustrate the usefulness of data transformation and of how linear combination of markers can significantly improve discriminative power. This finding highlights potential difficulties with methods that reject individual markers based on univariate analyses.

Alternative Estimation Procedures for Pr(X < Y) in Categorized Data

SUMMARY Consider two independent random variables X and Y. The functional R = Pr(X < Y) [or X = Pr(X < Y) - Pr(Y < X)] is of practical importance in many situations, including clinical trials, genetics, and reliability. In this paper several approaches to estimation of X when X and Y are presented in discretized (categorical) form are analyzed and compared. Asymptotic formulas for the variances of the estimators are derived; use of the bootstrap to estimate variances is also discussed. Computer simulations indicate that the choice of the best estimator depends on the value of X, the underlying distribution, and the sparseness of the data. It is shown that the bootstrap provides a robust estimate of variance. Several examples are treated.

TBARS and cardiovascular disease in a population-based sample.

Background Oxygen radicals might play a crucial role in the pathogenesis of various diseases, including atherosclerosis. Thiobarbituric acid reaction substances (TBARS), a biomarker of oxidative stress, have been proposed as a summary measure of total circulating oxidation. However, there is no strong indication that circulating levels of TBARS are increased in patients with atherosclerosis. Design We evaluated the relation between TBARS and cardiovascular disease (CVD) in a cross-sectional random sample of white men and women from Buffalo, New York. Methods Logistic regression was used to estimate the risk associated with high levels of TBARS. The area under the ROC curve was used to evaluate the discriminating power of TBARS. Results After adjusting for age and gender, TBARS levels were significantly higher in those with prevalent CVD (OR=1.73, 95% CI=1.32–2.38), compared to those without a CVD diagnosis. These OR were almost 50% higher after correcting for measurement error (ME) (OR=1.93, 95% CI=1.07–3.40). The area under the ROC curve was 0.69 (95% CI=0.62–0.77) and when corrected for ME reached 0.80 (95% CI=0.65–0.89). Conclusions Our results indicate that elevated levels of TBARS were associated with increase risk of the prevalence of CVD, but this effect was no longer significant after adjusting for glucose.

Measuring the effectiveness of diagnostic markers in the presence of measurement error through the use of ROC curves

The area under the receiver operating characteristic curve is frequently used to assess the effectiveness of diagnostic markers in distinguishing between diseased and healthy individuals. These markers are generally subject to measurement error. In this paper confidence intervals for the area under the curve are developed which take measurement error into account. These intervals depend on the availability of replicated observations for the subject. Both equal and unequal numbers of replicates per subject are considered. The practice of averaging over replicates and then ignoring measurement error is examined and found wanting.

Bayesian inference for the power law process

The power law process has been used to model reliability growth, software reliability and the failure times of repairable systems. This article reviews and further develops Bayesian inference for such a process. The Bayesian approach provides a unified methodology for dealing with both time and failure truncated data. As well as looking at the posterior densities of the parameters of the power law process, inference for the expected number of failures and the probability of no failures in some given time interval is discussed. Aspects of the prediction problem are examined. The results are illustrated with two data examples.

Confidence Intervals for the Overlapping Coefficient: the Normal Equal Variance Case

The overlapping coefficient, defined as the common area under two probability density curves, is used as a measure of agreement between two distributions. It has recently been proposed as a measure of bioequivalence under the name proportion of similar responses. Confidence intervals for this measure have been considered for the special case of two normal distributions with equal variances. We review and compare two procedures for this confidence interval based on the non-central t- and F-distributions. Our comparison is based on both theoretical considerations and a simulation study. Data on a marker from a study of recurrence of breast cancer are used to illustrate the methodology.

Modern Statistical Quality Control and Improvement

Basic concepts and terminology The modern approach to qualityuan overview Graphing and summarizing data Graphical techniques Applied probablity and statistics Control chart concepts Variables control charts Process capability analysis Attributes control charts Measurement systems Acceptance sampling Advanced topics Experimental design Implementation.