A comparison of confidence intervals generated by the scheffé and bonferroni methods

Simultaneous confidence intervals for the p means of a multivariate normal random variable with known variances may be generated by the projection method of Scheffé and by the use of Bonferroni's inequality. It has been conjectured that the Bonferroni intervals are shorter than the Scheffé intervals, at least for the usual confidence levels. This conjecture is proved for all p≥2 and all confidence levels above 50%. It is shown, incidentally, that for all p≥2 Scheffé's intervals are shorter for sufficiently small confidence levels. The results are also applicable to the Bonferroni and Scheffé intervals generated for multinomial proportions.     

Simultaneous confidence intervals for one-way layout based on generalized pivotal quantities

In this paper, we consider simultaneous confidence intervals for all-pairwise comparisons of treatment means in a one-way layout under heteroscedasticity. Two kinds of simultaneous intervals are provided based on the fiducial generalized pivotal quantities of the interest parameters. We prove that they both have asymptotically correct coverage. Simulation results and an example are also reported. It is concluded from calculational evidence that the second kind of simultaneous confidence intervals, which we provide, performs better than existing methods.     

Confidence bands for the laplace distribution

Based on a random sample from the Laplace population with unknown shape and scale parameters, one- and two-sided confidence bands on the entire cumulative distribution function and simultaneous confidence intervals for the interval probabilities under the distribution are constructed using Kolmogorov–Smirnov type statistics. Small sample and asymptotic percentiles of the relevant statistics are provided.     

Ellipsoidal confidence regions for a normal covariance matrix

We obtain an asymptotic expansion of the confidence coefficient for an ellipsoidal confidence region on the elements of a normal covariance matrix. This leads to simultaneous confidence intervals on all linear functions of the elements of this matrix, which are compared with those of Roy (1954).     

Asymptotic inference for coefficients of variation

Asymptotic inference results for the coefficients of variation of normal populations are presented in this article. This includes formulas for test statistics, power, confidence intervals, and simultaneous inference. The results are based on the asymptotic normality of the sample coefficient of variation as derived by Miller (1991). An example which compares the homogeneity of bone test samples produced from two different methods is presented.     

Pairwise comparisons of matched proportions

Consider the problem of comparing the success rates of c treatments, each of which induce a Bernoulli response. The comparison is to be made on the basis of n matched samples. We present a method for deriving confidence intervals for the pair-wise difference in success rates which has the desirable quality of providing uniformly shorter intervals than the procedure recently proposed by Bhapkar and Somes (1976). Comparisons of the lengths of the respective intervals are provided. Some observations regarding the assumptions required for the use of Cochran’s Q-test (1950) are also made.     

A quadratic programming approach to the construction of simultaneous confidence intervals

Richmond (1982) uses a linear programming approach to the construction of simultaneous confidence intervals for a set of linear estimable parametric functions of the normal mean vector. We present a quadratic programming approach which constructs narrower confidence intervals than the linear programming approach given by Richmond (1982).     

Some results on simultaneous confidence intervals for monotone contrasts in one-way anova model

The one-way ANOVA model with common variance is considered. Simultaneous confidence Intervals (SCI) for monotone contrasts in the means are derived and compared to alternative intervals gene¬rated by Williams (1977)     

Simultaneous confidence band for the difference of segmented linear models

Consider comparing between two treatments a response variable, whose expectation depends on the value of a continuous covariate in some nonlinear fashion. We fit separate segmented linear models to each treatment to approximate the nonlinear relationship. For this setting, we provide a simultaneous confidence band for the difference between treatments of the expected value functions. The treatments are said to differ significantly on intervals of the covariate where the simultaneous confidence band does not contain zero. We consider segmented linear models where the locations of the changepoints are both known and unknown. The band is obtained from asymptotic results.     

DISTRIBUTION FREE MULTIPLE COMPARISONS FOR MONOTONICALLY ORDERED TREATMENT EFFECTS

A method of calculating simultaneous one-sided confidence intervals for all ordered pairwise differences of the treatment effects_j–_i, 1 i < j k, in a one-way model without any distributional assumptions is discussed. When it is known a priori that the treatment effects satisfy the simple ordering_1k, these simultaneous confidence intervals offer the experimenter a simple way of determining which treatment effects may be declared to be unequal, and is more powerful than the usual two-sided Steel-Dwass procedure. Some exact critical points required by the confidence intervals are presented for k= 3 and small sample sizes, and other methods of critical point determination such as asymptotic approximation and simulation are discussed.     

Properties of simultaneous confidence intervals for multinomial proportions

Inversion of Pearson's chi-square statistic yields a confidence ellipsoid that can be used for simultaneous inference concerning multinomial proportions. Because the ellipsoid is difficult to interpret, methods of simultaneous confidence interval construction have been proposed by Quesenberry and hurst,goodman,fitzpatrick and scott and sison and glaz . Based on simulation results, we discuss the performance of these methods in terms of empirical coverage probabilities and enclosed volume. None of the methods is uniformly better than all others, but the Goodman intervals control the empirical coverage probability with smaller volume than other methods when the sample size supports the large sample theory. If the expected cell counts are small and nearly equal across cells, we recommend the sison and glaz intervals.     

Sidak-type simultaneous confidence intervals for the risk measures rsmr,based on proportional mortality measures in epidemiologic 1 studies involving several competing risks of death

In the present paper, simultaneous confidence interval estimates are constructed for the mortality measures RSMR. based on propor¬tional mortality measures SPMR. in epidemiologic studies for several competing risks of death to which the individuals in the study are exposed. It is demonstrated that, under a reasonable assumption, the joint sampling distribution of the statistics X. = RSMR./SPMR. for M competing risks9 may be approximated by means of a multi-variafe normal distribution, Sidak's (1967, 1968) mulfivariate normal probability inequalities are applied to construct the simultaneous confidence intervals for the measures RSMR., i=l3 2, ..., M. These are valid regardless of the covariance structure among the risks, As a particular case if the risks may be assumed as independent, our confidence intervals reduce to those for a single measure RSMR., which are narrower than those of Kupper et al., (1978), In this sense, our paper generalizes the results presently available in the literature in two directions; first, to obtain narrower confidence limits, and second3 to discuss the case of competing risks of death irrespective of their covariance structure.     

Max-min multiple comparison procedure for comparing several dose levels with a zero dose control

The comparison of increasing doses of a compound to a zero dose control is of interest in medical and toxicological studies. Assume that the mean dose effects are non-decreasing among the non-zero doses of the compound. A simple procedure that modifies Dunnett's procedure is proposed to construct simultaneous confidence intervals for pairwise comparisons of each dose group with the zero dose control by utilizing the ordering of the means. The simultaneous lower bounds and upper bounds by the new procedure are monotone, which is not the case with Dunnett's procedure. This is useful to categorize dose levels. The expected gains of the new procedure over Dunnett's procedure are studied. The procedure is shown by real data to compare well with its predecessor.     

The simultaneous confidence intervals for all distances from the extreme populations for two-parameter exponential populations based on the multiply type II censored samples

Among k independent two-parameter exponential distributions which have the common scale parameter, the lower extreme population (LEP) is the one with the smallest location parameter and the upper extreme population (UEP) is the one with the largest location parameter. Given a multiply type II censored sample from each of these k independent two-parameter exponential distributions, 14 estimators for the unknown location parameters and the common unknown scale parameter are considered. Fourteen simultaneous confidence intervals (SCIs) for all distances from the extreme populations (UEP and LEP) and from the UEP from these k independent exponential distributions under the multiply type II censoring are proposed. The critical values are obtained by the Monte Carlo method. The optimal SCIs among 14 methods are identified based on the criteria of minimum confidence length for various censoring schemes. The subset selection procedures of extreme populations are also proposed and two numerical examples are given for illustration.     

A new approximation to the distribution function of the studentized maximum root

The Studentized maximum root (SMR) distribution is useful for constructing simultaneous confidence intervals around product interaction contrasts in replicated two-way ANOVA. A three-moment approximation to the SMR distribution is proposed. The approximation requires the first three moments of the maximum root of a central Wishart matrix. These values are obtained by means of numerical integration. The accuracy of the approximation is compared to the accuracy of a two-moment approximation for selected two-way table sizes. Both approximations are reasonably accurate. The three-moment approximation is generally superior.     

Some multiple comparison procedures for comparing treatments within several groups

Multiple comparison procedures are extended to designs consisting of several groups, where the treatment means are to be compared within each group. This may arise in two-factor experiments, with a significant interaction term, when one is interested in comparing the levels of one factor at each level of the other factor. A general approach is presented for deriving the distributions and calculating critical points, following three papers which dealt with two specific procedures. These points are used for constructing simultaneous confidence intervals over some restricted set of contrasts among treatment means in each of the groups. Tables of critical values are provided for two procedures and an application is demonstrated. Some extensions are presented for the case of possible different sets of contrasts and also for unequal variances in the various groups.     

Bootstrap confidence intervals for smoothing splines and their comparison to bayesian confidence intervals

We construct bootstrap confidence intervals for smoothing spline estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from .exponential families. Several vari- ations of bootstrap confidence intervals are considered and compared. We find that the commonly used ootstrap percentile intervals are inferior to the T intervals and to intervals based on bootstrap estimation of mean squared errors. The best variations of the bootstrap confidence intervals behave similar to the well known Bayesian confidence intervals. These bootstrap confidence intervals have an average coverage probability across the function being estimated, as opposed to a pointwise property.     

Confidence Intervals for Difference Between Two Poisson Rates

In this article, we develop four explicit asymptotic two-sided confidence intervals for the difference between two Poisson rates via a hybrid method. The basic idea of the proposed method is to estimate or recover the variances of the two Poisson rate estimates, which are required for constructing the confidence interval for the rate difference, from the confidence limits for the two individual Poisson rates. The basic building blocks of the approach are reliable confidence limits for the two individual Poisson rates. Four confidence interval estimators that have explicit solutions and good coverage levels are employed: the first normal with continuity correction, Rao score, Freeman and Tukey, and Jeffreys confidence intervals. Using simulation studies, we examine the performance of the four hybrid confidence intervals and compare them with three existing confidence intervals: the non-informative prior Bayes confidence interval, the t confidence interval based on Satterthwait's degrees of freedom, and the Bayes confidence interval based on Student's t confidence coefficient. Simulation results show that the proposed hybrid Freeman and Tukey, and the hybrid Jeffreys confidence intervals can be highly recommended because they outperform the others in terms of coverage probabilities and widths. The other methods tend to be too conservative and produce wider confidence intervals. The application of these confidence intervals are illustrated with three real data sets.     

Multiple comparisons with a control for exponential location parameters under heteroscedasticity

In this paper, a new design-oriented two-stage two-sided simultaneous confidence intervals, for comparing several exponential populations with control population in terms of location parameters under heteroscedasticity, are proposed. If there is a prior information that the location parameter of k exponential populations are not less than the location parameter of control population, one-sided simultaneous confidence intervals provide more inferential sensitivity than two-sided simultaneous confidence intervals. But the two-sided simultaneous confidence intervals have advantages over the one-sided simultaneous confidence intervals as they provide both lower and upper bounds for the parameters of interest. The proposed design-oriented two-stage two-sided simultaneous confidence intervals provide the benefits of both the two-stage one-sided and two-sided simultaneous confidence intervals. When the additional sample at the second stage may not be available due to the experimental budget shortage or other factors in an experiment, one-stage two-sided confidence intervals are proposed, which combine the advantages of one-stage one-sided and two-sided simultaneous confidence intervals. The critical constants are obtained using the techniques given in Lam [9,10]. These critical constant are compared with the critical constants obtained by Bonferroni inequality techniques and found that critical constant obtained by Lam [9,10] are less conservative than critical constants computed from the Bonferroni inequality technique. Implementation of the proposed simultaneous confidence intervals is demonstrated by a numerical example.     

Multiplicity: discussion points from the Statisticians in the Pharmaceutical Industry multiplicity expert group

In May 2012, the Committee of Health and Medicinal Products issued a concept paper on the need to review the points to consider document on multiplicity issues in clinical trials. In preparation for the release of the updated guidance document, Statisticians in the Pharmaceutical Industry held a one‐day expert group meeting in January 2013. Topics debated included multiplicity and the drug development process, the usefulness and limitations of newly developed strategies to deal with multiplicity, multiplicity issues arising from interim decisions and multiregional development, and the need for simultaneous confidence intervals (CIs) corresponding to multiple test procedures. A clear message from the meeting was that multiplicity adjustments need to be considered when the intention is to make a formal statement about efficacy or safety based on hypothesis tests. Statisticians have a key role when designing studies to assess what adjustment really means in the context of the research being conducted. More thought during the planning phase needs to be given to multiplicity adjustments for secondary endpoints given these are increasing in importance in differentiating products in the market place. No consensus was reached on the role of simultaneous CIs in the context of superiority trials. It was argued that unadjusted intervals should be employed as the primary purpose of the intervals is estimation, while the purpose of hypothesis testing is to formally establish an effect. The opposing view was that CIs should correspond to the test decision whenever possible. Copyright © 2013 John Wiley & Sons, Ltd.