Variance calculations and confidence intervals for estimates of the attributable risk based on logistic models

The attributable risk (AR), defined as AR = [Pr(disease) - Pr(disease/no exposure)]/Pr(disease), measures the proportion of disease risk that is attributable to an exposure. Recently Bruzzi et al. (1985, American Journal of Epidemiology 122, 904-914) presented point estimates of AR based on logistic models for case-control data to allow for confounding factors and secondary exposures. To produce confidence intervals, we derived variance estimates for AR under the logistic model and for various designs for sampling controls. Calculations for discrete exposure and confounding factors require covariances between estimates of the risk parameters of the logistic model and the proportions of cases with given levels of exposure and confounding factors. These covariances are estimated from Taylor series expansions applied to implicit functions. Similar calculations for continuous exposures are derived using influence functions. Simulations indicate that those asymptotic procedures yield reliable variance estimates and confidence intervals with near nominal coverage. An example illustrates the usefulness of variance calculations in selecting a logistic model that is neither so simplified as to exhibit systematic lack of fit nor so complicated as to inflate the variance of the estimate of AR.     

The empirical coverage of confidence intervals: Point estimates and confidence intervals for confidence levels

Many confidence intervals calculated in practice are potentially not exact, either because the requirements for the interval estimator to be exact are known to be violated, or because the (exact) distribution of the data is unknown. If a confidence interval is approximate, the crucial question is how well its true coverage probability approximates its intended coverage probability. In this paper we propose to use the bootstrap to calculate an empirical estimate for the (true) coverage probability of a confidence interval. In the first instance, the empirical coverage can be used to assess whether a given type of confidence interval is adequate for the data at hand. More generally, when planning the statistical analysis of future trials based on existing data pools, the empirical coverage can be used to study the coverage properties of confidence intervals as a function of type of data, sample size, and analysis scale, and thus inform the statistical analysis plan for the future trial. In this sense, the paper proposes an alternative to the problematic pretest of the data for normality, followed by selection of the analysis method based on the results of the pretest. We apply the methodology to a data pool of bioequivalence studies, and in the selection of covariance patterns for repeated measures data.     

Constructing confidence intervals for selected parameters

In large-scale problems, it is common practice to select important parameters by a procedure such as the Benjamini and Hochberg procedure and construct confidence intervals (CIs) for further investigation while the false coverage-statement rate (FCR) for the CIs is controlled at a desired level. Although the well-known BY CIs control the FCR, they are uniformly inflated. In this paper, we propose two methods to construct shorter selective CIs. The first method produces shorter CIs by allowing a reduced number of selective CIs. The second method produces shorter CIs by allowing a prefixed proportion of CIs containing the values of uninteresting parameters. We theoretically prove that the proposed CIs are uniformly shorter than BY CIs and control the FCR asymptotically for independent data. Numerical results confirm our theoretical results and show that the proposed CIs still work for correlated data. We illustrate the advantage of the proposed procedures by analyzing the microarray data from a HIV study.     

Measurement error and confidence intervals for ROC curves

Measurement error in a continuous test variable may bias estimates of the summary properties of receiver operating characteristics (ROC) curves. Typically, unbiased measurement error will reduce the diagnostic potential of a continuous test variable. This paper explores the effects of possibly heterogenous measurement error on estimated ROC curves for binormal test variables. Corrected estimators for specific points on the curve are derived under the assumption of known or estimated measurement variances for individual test results. These estimators and associated confidence intervals do not depend on normal assumptions for the distribution of the measurement error and are shown to be approximately unbiased for moderate size samples in a simulation study. An application from a study of emerging imaging modalities in breast cancer is used to demonstrate the new techniques.     

Under-smoothed kernel confidence intervals for the hazard ratio based on censored data

In cancer clinical trials, it is often of interest in estimating the ratios of hazard rates at some specific time points during the study from two independent populations. In this paper, we consider nonparametric confidence interval procedures for the hazard ratio based on kernel estimates for the hazard rates with under-smoothing bandwidths. Two methods are used to derive the confidence intervals: one based on the asymptotic normality of the ratio of the kernel estimates for the hazard rates in two populations and another through Fieller's Theorem. The performances of the proposed confidence intervals are evaluated through Monte-Carlo simulations and applied to the analysis of data from a clinical trial on early breast cancer.     

Simultaneous confidence intervals for comparing binomial parameters

SUMMARY: To compare proportions with several independent binomial samples, we recommend a method of constructing simultaneous confidence intervals that uses the studentized range distribution with a score statistic. It applies to a variety of measures, including the difference of proportions, odds ratio, and relative risk. For the odds ratio, a simulation study suggests that the method has coverage probability closer to the nominal value than ad hoc approaches such as the Bonferroni implementation of Wald or "exact" small-sample pairwise intervals. It performs well even for the problematic but practically common case in which the binomial parameters are relatively small. For the difference of proportions, the proposed method has performance comparable to a method proposed by Piegorsch (1991, Biometrics 47, 45-52).     

Bootstrap confidence intervals for adaptive cluster sampling

Consider a collection of spatially clustered objects where the clusters are geographically rare. Of interest is estimation of the total number of objects on the site from a sample of plots of equal size. Under these spatial conditions, adaptive cluster sampling of plots is generally useful in improving efficiency in estimation over simple random sampling without replacement (SRSWOR). In adaptive cluster sampling, when a sampled plot meets some predefined condition, neighboring plots are added to the sample. When populations are rare and clustered, the usual unbiased estimators based on small samples are often highly skewed and discrete in distribution. Thus, confidence intervals based on asymptotic normal theory may not be appropriate. We investigated several nonparametric bootstrap methods for constructing confidence intervals under adaptive cluster sampling. To perform bootstrapping, we transformed the initial sample in order to include the information from the adaptive portion of the sample yet maintain a fixed sample size. In general, coverages of bootstrap percentile methods were closer to nominal coverage than the normal approximation.     

Stepwise confidence intervals for monotone dose-response studies

Summary .   In dose–response studies, one of the most important issues is the identification of the minimum effective dose (MED), where the MED is defined as the lowest dose such that the mean response is better than the mean response of a zero-dose control by a clinically significant difference. Dose–response curves are sometimes monotonic in nature. To find the MED, various authors have proposed step-down test procedures based on contrasts among the sample means. In this article, we improve upon the method of Marcus and Peritz (1976, Journal of the Royal Statistical Society, Series B 38 , 157–165) and implement the dose–response method of Hsu and Berger (1999, Journal of the American Statistical Association 94 , 468–482) to construct the lower confidence bound for the difference between the mean response of any nonzero-dose level and that of the control under the monotonicity assumption to identify the MED. The proposed method is illustrated by numerical examples, and simulation studies on power comparisons are presented.     

Step-up related simultaneous confidence intervals for MCC and MCB

The derivation of compatible confidence bounds related to stepwise decision procedures is a serious issue. Especially the derivation of step-up related bounds is rather complex. In this article we consider (one-sided) multiple comparisons with a control (MCC) and multiple comparisons with the best (MCB) with the aim of establishing delta-equivalence to the best and derive step-up related confidence bounds by applying the projection method proposed in Finner and Strassburger (2006). Some examples illustrate the resulting procedures.     

Sensitivity analysis for m-estimates, tests, and confidence intervals in matched observational studies

Huber's m-estimates use an estimating equation in which observations are permitted a controlled level of influence. The family of m-estimates includes least squares and maximum likelihood, but typical applications give extreme observations limited weight. Maritz proposed methods of exact and approximate permutation inference for m-tests, confidence intervals, and estimators, which can be derived from random assignment of paired subjects to treatment or control. In contrast, in observational studies, where treatments are not randomly assigned, subjects matched for observed covariates may differ in terms of unobserved covariates, so differing outcomes may not be treatment effects. In observational studies, a method of sensitivity analysis is developed for m-tests, m-intervals, and m-estimates: it shows the extent to which inferences would be altered by biases of various magnitudes due to nonrandom treatment assignment. The method is developed for both matched pairs, with one treated subject matched to one control, and for matched sets, with one treated subject matched to one or more controls. The method is illustrated using two studies: (i) a paired study of damage to DNA from exposure to chromium and nickel and (ii) a study with one or two matched controls comparing side effects of two drug regimes to treat tuberculosis. The approach yields sensitivity analyses for: (i) m-tests with Huber's weight function and other robust weight functions, (ii) the permutational t-test which uses the observations directly, and (iii) various other procedures such as the sign test, Noether's test, and the permutation distribution of the efficient score test for a location family of distributions. Permutation inference with covariance adjustment is briefly discussed.     

Nonparametric confidence intervals for the one- and two-sample problems

Confidence intervals for the mean of one sample and the difference in means of two independent samples based on the ordinary-t statistic suffer deficiencies when samples come from skewed families. In this article we evaluate several existing techniques and propose new methods to improve coverage accuracy. The methods examined include the ordinary-t, the bootstrap-t, the biased-corrected acceleration and three new intervals based on transformation of the t-statistic. Our study shows that our new transformation intervals and the bootstrap-t intervals give best coverage accuracy for a variety of skewed distributions, and that our new transformation intervals have shorter interval lengths.     

Unconditional small-sample confidence intervals for the odds ratio

The traditional approach to 'exact' small-sample interval estimation of the odds ratio for binomial, Poisson, or multinomial samples uses the conditional distribution to eliminate nuisance parameters. This approach can be very conservative. For two independent binomial samples, we study an unconditional approach with overall confidence level guaranteed to equal at least the nominal level. With small samples this interval tends to be shorter and have coverage probabilities nearer the nominal level.