Obtaining Critical Values for Simultaneous Confidence Intervals and Multiple Testing

There are many situations where it is desired to make simultaneous tests or give simultaneous confidence intervals for linear combinations (contrasts) of population or treatment means. Somerville (1997, 1999) developed algorithms for calculating the critical values for a large class of simultaneous tests and simultaneous confidence intervals. Fortran 90 and SAS‐IML batch programs and interactive programs were developed. These programs calculate the critical values for 15 different simultaneous confidence interval procedures (and the corresponding simultaneous tests) and for arbitrary procedures where the user specifies a combination of one and two sided contrasts. The programs can also be used to obtain the constants for “step‐down” testing of multiple hypotheses. This paper gives examples of the use of the algorithms and programs and illustrates their versatility and generality. The designs need not be balanced, multiple covariates may be present and there may be many missing values. The use of multiple regression and dummy variables to obtain the required variance covariance matrix is illustrated. Under weak normality assumptions the methods are “exact” and make the use of approximate methods or “simulation” unnecessary.     

Simultaneous Statistical Inference Concerning the Standardized Mortality Ratios (SMB) of Several Strata in an Epidemiologic Study

In the present paper, simultaneous prediction intervals are constructed for some mortality measures involving real data in a recent retrospective epidemiologic study of a cohort of man-made mineral fiber workers. The object is to compare a test population with a standard population in which the workers are exposed to several levels (say K) of a suspected carcinogen and the number of deaths in the K exposure groups are recorded. The SMR is chosen as a mortality measure for this comparison. If the total number of deaths d. in the entire study is known, the number of deaths in the various strata become dependent random variables. Using univariate statistical procedures to test the significance of the individual SMR's is inappropriate and may give misleading conclusions. The present paper shows how to test the simultaneous statistical significance of the SMR's of the K exposure groups, i.e., to test the simultaneous null hypothesis H_o : E(SMR₁) = E(SMR₂) =…E(SMR_k) = 100. If the null hypothesis H_o is rejected, it is shown how to identify the SMR (or the SMR's) which contributed to the rejection of H_o. The power function of such a test is also constructed for some simple and useful alternative hypotheses.     

Do rats time filled and empty intervals of equal duration differently?

The goal was to determine whether rats time filled and empty intervals of equal duration differently. Each of five rats was trained for 50 sessions on an instrumental appetitive head entry procedure in which food was available (primed) every 120 s. On "empty" cycles, 30s prior to the next food prime, a 0.5-s pulse of white noise was presented. On "filled" cycles, 30s prior to the next food prime, white noise came on and stayed on until food was delivered. The two types of cycles were presented with equal probability. The results showed that the rats timed both the food-to-food interval and the stimulus-to-food interval. A comparison of the response gradients on filled and empty cycles following stimulus presentation showed better temporal discrimination on filled cycles. The results were modeled using a Packet theory of timing, with a linear averaging rule to combine the temporal information provided by the stimulus and food. The model fits to the individual response gradients were evaluated with a Turing test.     

Multiple contrast tests in the presence of heteroscedasticity

This paper proposes a general approach for handling multiple contrast tests for normally distributed data in the presence of heteroscedasticity. Three candidate procedures are described and compared by simulations. Only the procedure with both comparison-specific degrees of freedom and a correlation matrix depending on sample variances maintains the alpha-level over all situations. Other approaches may fail notably as the variances differ more. Furthermore, related approximate simultaneous confidence intervals are given. The approach will be applied to a toxicological experiment.     

Simultaneous confidence intervals for a success probability and intraclass correlation,with an application to screening mammography

This paper provides asymptotic simultaneous confidence intervals for a success probability and intraclass correlation of the beta‐binomial model, based on the maximum likelihood estimator approach. The coverage probabilities of those intervals are evaluated. An application to screening mammography is presented as an example. The individual and simultaneous confidence intervals for sensitivity and specificity and the corresponding intraclass correlations are investigated. Two additional examples using influenza data and sex ratio data among sibships are also considered, where the individual and simultaneous confidence intervals are provided.     

Population size estimation in a two-list surveillance system with a discrete covariate

Summary .   Capture–recapture methods are widely adopted to estimate sizes of populations of public health interest using information from surveillance systems. For a two-list surveillance system with a discrete covariate, a population is divided into several subpopulations. A unified framework is proposed in which the logits of presence probabilities are decomposed into case effects and list effects. The estimators for the whole population and subpopulation sizes, their adjusted versions, and asymptotic standard errors admit closed-form expressions. Asymptotic and bootstrap individual and simultaneous confidence intervals are easily constructed. Conditional likelihood ratio tests are used to select one from three possible models. Real examples are investigated.     

The Highest Confidence Density Region and Its Usage for Joint Inferences about Constrained Parameters

Summary Suppose that we are interested in making joint inferences about a set of constrained parameters. Confidence regions for these parameters are often constructed via a normal approximation of the distribution of a consistent estimator for a transformation of the parameters. In this article, we utilize the confidence distribution, a frequentist counterpart to the posterior distribution in Bayesian statistics, to obtain optimal confidence regions for the parameters. Members of such a region can be generated efficiently via a standard Markov chain Monte Carlo algorithm. We then apply this technique to draw inferences about the temporal profile of the survival function with censored observations. We illustrate the new proposal with the survival data from the well‐known Mayo primary biliary cirrhosis study and show that the volume of the new 0.95 confidence region is only one thirty‐fourth of that of the conventional confidence band.     

Combining One‐sided and Two‐sided Confidence Interval Procedures for Successive Comparisons of Ordered Treatment Effects

Lee and Spurrier (1995) present one‐sided and two‐sided confidence interval procedures for making successive comparisons between ordered treatments. Their procedures have important applications for problems where the treatments can be assumed to satisfy a simple ordering, such as for a sequence of increasing dose‐levels of a drug. The two‐sided procedure provides both upper and lower bounds on the differences between successive treatments, whereas the one‐sided procedure only provides lower bounds on these differences. However, the one‐sided procedure allows sharper inferences regarding which treatments can be declared to be better than their previous ones. In this paper we apply the results obtained in Hayter , Miwa , and Liu (2000) to develop a new procedure which combines the good aspects of both the one‐sided and the two‐sided procedures. This new procedure maintains the inferential sensitivity of the one‐sided procedure while also providing both upper and lower bounds on the differences between successive treatments. Some new critical points are needed which are tabulated for the balanced case where the sample sizes are all equal. The application of the new procedure is illustrated with an example.     

Applications and Tabulations of the Multivariate t Distribution with ρ = 0

Quantiles of the multivariate t distribution with ρ = 0 are tabulated. Some applications are discussed.     

An Improvement of the Westlake Symmetric Confidence Interval

A common statistical method for assessing bioequivalence of two formulations of a chemical substance is the symmetric confidence interval of WESTLAKE (1972). As mentioned by WEST -LAKE (1981) and SCHUIRMAN (1981) a more powerful method consists of two one-sided t-tests. An (1-α)-confidence interval consistent with the two one-sided t-tests procedure is given by [min(α, 0), max (0, b)] where [a, b] is the conventional (1–2α)-confidence interval of the t-test. This “central” confidence interval is always a strict subset of the symmetric confidence interval and thus has more power in proving bioequivalence. The central confidence interval has properties comparable with those of the conventional one-sided confidence intervals.     

Simultaneous inference in general parametric models

Simultaneous inference is a common problem in many areas of application. If multiple null hypotheses are tested simultaneously, the probability of rejecting erroneously at least one of them increases beyond the pre-specified significance level. Simultaneous inference procedures have to be used which adjust for multiplicity and thus control the overall type I error rate. In this paper we describe simultaneous inference procedures in general parametric models, where the experimental questions are specified through a linear combination of elemental model parameters. The framework described here is quite general and extends the canonical theory of multiple comparison procedures in ANOVA models to linear regression problems, generalized linear models, linear mixed effects models, the Cox model, robust linear models, etc. Several examples using a variety of different statistical models illustrate the breadth of the results. For the analyses we use the R add-on package multcomp, which provides a convenient interface to the general approach adopted here.     

Rectangular tolerance regions and multivariate normal reference regions in laboratory medicine

Reference intervals are widely used in the interpretation of results of biochemical and physiological tests of patients. When there are multiple biochemical analytes measured from each subject, a multivariate reference region is needed. Because of their greater specificity against false positives, such reference regions are more desirable than separate univariate reference intervals that disregard the cross-correlations between variables. Traditionally, under multivariate normality, reference regions have been constructed as ellipsoidal regions. This approach suffers from a major drawback: it cannot detect component-wise extreme observations. In the present work, procedures are developed to construct rectangular reference regions in the multivariate normal setup. The construction is based on the criteria for tolerance intervals. The problems addressed include the computation of a rectangular tolerance region and simultaneous tolerance intervals. Also addressed is the computation of mixed reference intervals that include both two-sided and one-sided limits, simultaneously. A parametric bootstrap approach is used in the computations, and the accuracy of the proposed methodology is assessed using estimated coverage probabilities. The problem of sample size determination is also addressed, and the results are illustrated using examples that call for the computation of reference regions.     

Inferences on the common mean of several log-normal populations: the generalized variable approach

This paper proposes a novel approach for the confidence interval estimation and hypothesis testing of the common mean of several log-normal populations using the concept of generalized variable. Simulation studies demonstrate that the proposed approach can provide confidence intervals with satisfying coverage probabilities and can perform hypothesis testing with satisfying type-I error control even at small sample sizes. Overall, it is superior to the large sample approach. The proposed method is illustrated using two examples.     

Multiple Comparisons for a Large Number of Parameters

When there are many parameters of interest (finitely large or infinite), standard multiple comparison procedures for a finite number of parameters (called discrete‐domain approaches) may lead to a simultaneous confidence region (SCR) too conservative to be useful. Such cases often arise in locating disease genes, detecting changes in image data and examining shapes and patterns in growth curves; or generally, in quantifying uncertainty in an estimate of a regression function (as one entity). In these cases, procedures designed for a continuous domain must be used. Scheffe's method is a classical example of continuous‐domain approaches. It provides an SCR for a regression function when errors are iid Gaussian and the predictor space is unconstrained, i.e. the domain of interest is the q dimensional Euclidean space. In practice, however, functions defined on finite intervals or other constrained domains are often of interest and data may not be Gaussian. Thus, Scheffe's SCR becomes either too conservative or inadequate. In this paper, we introduce and survey a modern‐type continuous‐domain approach, and explore a connection between some discrete‐ and continuous‐domain multiple comparison procedures. We show that, in some cases, even for a small number of parameters, it is still better to use a continuous‐domain multiple comparison procedure. The main ideas behind the continuous‐domain procedures are shown. A new procedure for comparing a finite number of contrasts about k regression curves is developed. Relevant software is provided.