Coverage probability

Abstract: Simple interval estimate methods for proportions exhibit poor coverage and can produce evidently inappropriate intervals. Criteria appropriate to the evaluation of various proposed methods include: closeness of the achieved coverage probability to its nominal value; whether intervals are located too close to or too distant from the middle of the scale; expected interval width; avoidance of aberrations such as limits outside [0,1] or zero width intervals; and ease of use, whether by tables, software or formulae. Seven methods for the single proportion are evaluated on 96,000 parameter space points. Intervals based on tail areas and the simpler score methods are recommended for use. In each case, methods are available that aim to align either the minimum or the mean coverage with the nominal 1 -alpha.

Abstract: We revisit the problem of interval estimation of a binomial proportion. The erratic behavior of the coverage probability of the stan- d ardWaldconfid ence interval has previously been remarkedon in the literature (Blyth andStill, Agresti andCoull, Santner andothers). We begin by showing that the chaotic coverage properties of the Waldinter- val are far more persistent than is appreciated. Furthermore, common textbook prescriptions regarding its safety are misleading and defective in several respects andcannot be trusted . This leads us to consideration of alternative intervals. A number of natural alternatives are presented, each with its motivation and con- text. Each interval is examinedfor its coverage probability andits length. Basedon this analysis, we recommendthe Wilson interval or the equal- tailedJeffreys prior interval for small n andthe interval suggestedin Agresti andCoull for larger n. We also provide an additional frequentist justification for use of the Jeffreys interval.

Abstract: The rule of thumb that logistic and Cox models should be used with a minimum of 10 outcome events per predictor variable (EPV), based on two simulation studies, may be too conservative. The authors conducted a large simulation study of other influences on confidence interval coverage, type I error, relative bias, and other model performance measures. They found a range of circumstances in which coverage and bias were within acceptable levels despite less than 10 EPV, as well as other factors that were as influential as or more influential than EPV. They conclude that this rule can be relaxed, in particular for sensitivity analyses undertaken to demonstrate adequate control of confounding.

Abstract: Fitting a line to a bivariate dataset can be a deceptively complex problem, and there has been much debate on this issue in the literature. In this review, we describe for the practitioner the essential features of line-fitting methods for estimating the relationship between two variables: what methods are commonly used, which method should be used when, and how to make inferences from these lines to answer common research questions. A particularly important point for line-fitting in allometry is that usually, two sources of error are present (which we call measurement and equation error), and these have quite different implications for choice of line-fitting method. As a consequence, the approach in this review and the methods presented have subtle but important differences from previous reviews in the biology literature. Linear regression, major axis and standardised major axis are alternative methods that can be appropriate when there is no measurement error. When there is measurement error, this often needs to be estimated and used to adjust the variance terms in formulae for line-fitting. We also review line-fitting methods for phylogenetic analyses. Methods of inference are described for the line-fitting techniques discussed in this paper. The types of inference considered here are testing if the slope or elevation equals a given value, constructing confidence intervals for the slope or elevation, comparing several slopes or elevations, and testing for shift along the axis amongst several groups. In some cases several methods have been proposed in the literature. These are discussed and compared. In other cases there is little or no previous guidance available in the literature. Simulations were conducted to check whether the methods of inference proposed have the intended coverage probability or Type I error. We identified the methods of inference that perform well and recommend the techniques that should be adopted in future work.

Abstract: The assessment of the discrimination ability of a survival analysis model is a problem of considerable theoretical interest and important practical applications. This issue is, however, more complex than evaluating the performance of a linear or logistic regression. Several different measures have been proposed in the biostatistical literature. In this paper we investigate the properties of the overall C index introduced by Harrell as a natural extension of the ROC curve area to survival analysis. We develop the overall C index as a parameter describing the performance of a given model applied to the population under consideration and discuss the statistic used as its sample estimate. We discover a relationship between the overall C and the modified Kendall's tau and construct a confidence interval for our measure based on the asymptotic normality of its estimate. Then we investigate via simulations the length and coverage probability of this interval. Finally, we present a real life example evaluating the performance of a Framingham Heart Study model.