Confidence Interval Procedures Research Articles

Testing Superiority and Non‐Inferiority Hypotheses in Active Controlled Clinical Trials

Switching between testing for superiority and non-inferiority has been an important statistical issue in the design and analysis of active controlled clinical trial. In practice, it is often conducted with a two-stage testing procedure. It has been assumed that there is no type I error rate adjustment required when either switching to test for non-inferiority once the data fail to support the superiority claim or switching to test for superiority once the null hypothesis of non-inferiority is rejected with a pre-specified non-inferiority margin in a generalized historical control approach. However, when using a cross-trial comparison approach for non-inferiority testing, controlling the type I error rate sometimes becomes an issue with the conventional two-stage procedure. We propose to adopt a single-stage simultaneous testing concept as proposed by Ng (2003) to test both non-inferiority and superiority hypotheses simultaneously. The proposed procedure is based on Fieller's confidence interval procedure as proposed by Hauschke et al. (1999).

The Chow and Robbins Fixed Width Confidence Interval Procedure When the Second Moment Barely Does Not Exist

We investigate the behavior of the Chow and Robbins asymptotic fixed width confidence interval procedure when the underlying i.i.d. random variables do not necessarily have a finite second moment, however, are in the domain of attraction of a normal law.

TYPE I ERROR PROBABILITIES BASED ON DESIGN-STAGE STRATEGIES WITH APPLICATIONS TO NONINFERIORITY TRIALS#

ABSTRACT When testing the equality of means from two different populations, a t-test or large sample normal test tend to be performed. For these tests, when the sample size or design for the second sample is dependent on the results of the first sample, the type I error probability is altered for each specific possibility in the null hypothesis. We will examine the impact on the type I error probabilities for two confidence interval procedures and procedures using test statistics when the design for the second sample or experiment is dependent on the results from the first sample or experiment (or series of experiments). Ways for controlling a desired maximum type I error probability or a desired type I error rate will be discussed. Results are applied to the setting of noninferiority comparisons in active controlled trials where the use of a placebo is unethical.

Models and Confidence Intervals for True Values in Interlaboratory Trials

We consider the one-way random-effects model with unequal sample sizes and heterogeneous variances. Using the method of generalized confidence intervals, we develop a new confidence interval procedure for the mean. Additionally, we investigate two alternative models based on different sets of assumptions regarding between-group variability and derive generalized confidence interval procedures for the mean. These procedures are applicable to small samples. Statistical simulation is used to demonstrate that the coverage probabilities of these procedures are close enough to the nominal value so that they are useful in practice. Although the methods are quite general, the procedures are explained with the backdrop of interlaboratory studies.

Interval Estimation For The Scale Parameter Of Burr Type X Distribution Based On Grouped Data

The application of some bootstrap type intervals for the scale parameter of the Burr type X distribution with grouped data is proposed. The general asymptotic confidence interval procedure (Chen & Mi, 2001) is studied. The performance of these intervals is investigated and compared. Some of the bootstrap intervals give better performance for situations of small sample size and heavy censoring.

ALTERNATIVE CONFIDENCE INTERVALS FOR THE TOTAL OF A SKEWED BIOLOGICAL POPULATION

In a 1999 Ecology article, T. G. Gregoire and O. Schabenberger addressed the problem of obtaining truly symmetric confidence intervals for the total of a positively skewed biological population under simple random sampling. Their simulation study revealed that the skewness induced a substantial positive correlation between the estimator of the total and the estimator of its variance. This caused the standard nominally symmetric t-based intervals, based on approximate normality of the estimator of the total, to be highly unbalanced, i.e., intervals much more often missed from below than from above. To better cope with this situation I suggest an alternative confidence interval procedure that takes into account and adjusts for the induced correlation. A simulation study based on one of the populations used by Gregoire and Schabenberger shows that the resulting adjusted intervals have more balanced noncoverage probabilities and often higher coverage probability than the standard intervals in cases of substantial correlation. I also provide an example of an unequal probability design using auxiliary information, where there is much less need for an adjustment.

Confidence interval procedures for the probability of disease transmission in Multiple-Vector-Transfer designs

In plant pathology, group testing has been widely used in vector-transfer designs to study factors affecting the spread of disease by insect vectors. In such contexts, vectors are tested in groups rather than individually. However, the goal is still to estimate p, the probability of disease transmissi on by a single vector. The purpose of this article is to provide a thorough comparison of new and established interval estimators for p in terms of coverage probability and mean length. We ill ustrate our methods using data from an Argentinean study in volving the Mal Rio Cuarto virus and its transmission by the Delphacodes kuscheli planthopper.

Identifying Extreme Decision Makers in Group AHP

SYNOPTIC ABSTRACTThe analytic hierarchy process is a popular and effective tool for multi-criteria decision making. It can be applied by a lone decision maker or by a group or team of decision makers. When a group of decision makers (DMs) is involved in an evaluation of alternatives with respect to criteria and sub-criteria, each DM produces a rating for each alternative. Of course, there may exist one or more DMs whose ratings deviate substantially from the other ratings. These outlier or extreme DMs can exert considerable influence on the overall ratings (and hence the ranking) of the alternatives, since the overall ratings are typically based on the arithmetic or geometric mean of the individual ratings. Two recent papers propose using a confidence-interval procedure for identifying extreme DMs. In this paper, we present two alternative procedures, one based on a multivariate test and the other based on visualization of the ratings. We discuss and illustrate how these new procedures improve upon the earlier procedure.

Stepwise multiple tests for successive comparisons of treatment effects

Suppose that the k treatments under comparison are ordered in a certain way. For example, they may be a sequence of increasing dose levels of a drug. It is interesting to look directly at the successive differences between the treatment effects μ i 's, namely the set of differences μ 2− μ 1, μ 3− μ 2,…, μ k − μ k−1 . Lee and Spurrier (J. Statist. Plann. Inference 43 (1995) 323) propose a one-sided and a two-sided simultaneous confidence interval procedures for making successive comparisons between treatments. In this paper we develop step-down and step-up tests for testing the families of hypotheses H i0: μ i+1−μ i=0 vs H ia: μ i+1−μ i>0, i=1,…,k−1 (one-sided alternatives) H i0: μ i+1−μ i=0 vs H ia: μ i+1−μ i≠0, i=1,…,k−1 (two-sided alternatives) These stepwise tests are uniformly more powerful than the simultaneous confidence interval procedures in terms of rejection of the null hypotheses, though the simultaneous confidence interval procedures provide information on the magnitudes of the μ i+1 − μ i 's. The critical constants required for applying these multiple tests are provided, and the tests are illustrated with a numerical example.

On Non-inferiority Analysis Based On Delta-method Confidence Intervals#

For many indications where there is an effective standard therapy, active controlled trials are generally conducted when it is unethical to use a placebo. The efficacy objective of most such trials is the demonstration that the experimental therapy has superior efficacy to the active-control. The efficacy objective of a non-inferiority trial may be to rule out that the experimental treatment loses some prespecified fraction of the active-control effect. The size of the active-control effect may be based on previous trials comparing this active-control with a placebo—for example, through a meta-analysis. Delta-method 95% confidence interval procedures are among the testing procedures that have been proposed to test a non-inferiority hypothesis that an experimental treatment retains more than some prespecified fraction of the active-control effect. For time-to-event endpoints using hazard ratios, we will examine the type I error probability of such testing procedures under the assumption that the current active-control effect has been correctly modeled. Conditions are discussed for when such testing procedures maintain a desired approximate type I error rate and when such testing procedures will not. Two applications (one in Cardiorenalogy and one in Oncology) are given—one maintains the desired approximate type I error probability and the other does not. The delta-method 95% confidence interval procedures will also be contrasted with Fieller 95% confidence intervals. Testing based on Fieller 95% confidence intervals will maintain a desired approximate type I error rate. #The views expressed in this article do not necessarily represent those of the U.S. Food and Drug Administration.

Some Comments on the Robustness of Student t Procedures

AbstractMany business and engineering courses stress the use of confidence interval and hypothesis testing procedures based on the Student t distribution. However, students are often unaware of the underlying assumptions that govern these procedures and of the consequences of misapplying them. In this exegetic account, we discuss the concept of robustness by exploring violations of the Student t assumptions and the effects of those violations on the resulting inferences. We also discuss the use of randomization methods as possible alternative methods. Our goal is to familiarize readers with the underpinnings of the t procedures, to summarize their use in practice, and to offer words of caution as to when they should not be used.

In Vitro Release: Comparative Evaluation of Vertical Diffusion Cell System and Automated Procedure

In vitro release for topical drug products is carried out using a vertical diffusion cell system and a synthetic support membrane. The improvements in vertical diffusion cell design and automated procedures have been carefully studied and evaluated to establish their reproducibility and ruggedness of the experimental procedure. The data were analyzed using 90% confidence interval procedure developed and published by the US Food and Drug Administration in SUPAC-SS (Start Up and Post Approval Changes-Semi Solid) guidance.

Design and analysis of non-inferiority mortality trials in oncology.

The recent revision of the Declaration of Helsinki and the existence of many new therapies that affect survival or serious morbidity, and that therefore cannot be denied patients, have generated increased interest in active-control trials, particularly those intended to show equivalence or non-inferiority to the active-control. A non-inferiority hypothesis has historically been formulated in terms of a fixed margin. This margin was historically designed to exclude a 'clinically meaningful difference', but has become recognized that the margin must also be no larger than the assured effect of the control in the new study. Depending on how this 'assured effect' is determined or estimated, the selected margin may be very small, leading to very large sample sizes, especially when there is an added requirement that a loss of some specified fraction of the assured effect must be ruled out. In cases where it is appropriate, this paper proposes non-inferiority analyses that do not involve a fixed margin, but can be described as a two confidence interval procedure that compares the 95 per cent two-sided CI for the difference between the treatment and the control to a confidence interval for the control effect (based on a meta-analysis of historical data comparing the control to placebo) that is chosen to preserve a study-wide type I error rate of about 0.025 (similar to the usual standard for a superiority trial) for testing for retention of a prespecified fraction of the control effect. The approach assumes that the estimate of the historical active-control effect size is applicable in the current study. If there is reason to believe that this effect size is diminished (for example, improved concomitant therapies) the estimate of this historical effect could be reduced appropriately. The statistical methodology for testing this non-inferiority hypothesis is developed for a hazard ratio (rather than an absolute difference between treatments, because a hazard ratio seems likely to be less population dependent than the absolute difference). In the case of oncology, the hazard ratio is the usual way of comparing treatments with respect to time to event (time to progression or survival) endpoints. The proportional hazards assumption is regarded as reasonable (approximately holding). The testing procedures proposed are conditionally equivalent to two confidence interval procedures that relax the conservatism of two 95 per cent confidence interval testing procedures and preserve the type I error rate at a one-sided 0.025 level. An application of this methodology to Xeloda, a recently approved drug for the treatment of metastatic colorectal cancers, is illustrated. Other methodologies are also described and assessed - including a point estimate procedure, a Bayesian procedure and two delta-method confidence interval procedures. Published in 2003 by John Wiley & Sons, Ltd.

Non-inferiority trials: the 'at least as good as' criterion.

To demonstrate in a clinical trial that a new or experimental therapy (et) is 'at least as good as' a standard therapy (st), a statistical test or confidence interval procedure must rule out clinical inferiority with a high probability. The term 'at least as good as' implies equivalent but not necessarily superior efficacy. As it is statistically impossible to demonstrate equivalence (that is, prove the null hypothesis of no difference), Blackwelder proposed a one-sided significance test to reject the null hypothesis that standard therapy is better than experimental therapy by a clinically acceptable amount, delta(BW). In this paper, Blackwelder's approach is redefined in terms of the ratio of two means (R(True)= mu(et)/mu(st)) based on a continuous variate with higher values denoting greater improvement. The ratio-based equivalents to Blackwelder's hypotheses will be shown. The ratio parameter has the benefit of being available as a dimensionless percentage, not tied to a specified difference in means. Thus, a study can be sized to assure, with high probability, that the experimental therapy is 'at least' (R(LB)x100) per cent 'as effective as' the standard therapy, where R(LB) is the selected lower bound on the percentage effectiveness. A practical rationale is given for defining non-inferiority as a high fraction or percentage of the standard drug's efficacy, both in terms of statistical efficiency and medical relevance. For most typical 'at least as good as' applications (when R(LB)<R(True) < or =1), the ratio formatted test of H(0):R(True)< or =R(LB) is shown to be more efficient than Blackwelder's test of H(0): mu(st) - mu(et) > or = delta(BW), thereby requiring smaller sample sizes to detect the directionally based non-null alternatives contained in H(1): mu(et)/mu(st)>R(LB) or, equivalently, mu(st) - mu(et)<delta(BW). Further, when R(True)= 1.0, tests of Blackwelder's hypotheses, their ratio-based equivalents and conventional superiority can be evaluated for comparative efficiency. Testing H(0):R(True)< or =R(LB) with single-sided critical region of size alpha, versus H(1):R(True)>R(LB), is shown to be more efficient than excluding R(LB) from the lower limit of a 100(1-2alpha) per cent two-sided symmetric confidence interval centred by R. Relevant examples will be presented.

A multiple directional decision procedure for successive comparisons of treatment effects

Suppose that the k treatments under comparison are ordered in a certain way. For example, there may be a sequence of increasing dose levels of a drug. It is interesting to look directly at the successive differences between the treatment effects μ i 's, namely the set of differences μ 2− μ 1, μ 3− μ 2,…, μ k − μ k−1 . In particular, directional inferences on whether μ i < μ i+1 or μ i > μ i+1 for i=1,…, k−1 are useful. Lee and Spurrier (J. Statist. Plann. Inference 43 (1995) 323) present a one- and a two-sided confidence interval procedures for making successive comparisons between treatments. In this paper, we develop a new procedure which is sharper than both the one- and two-sided procedures of Lee and Spurrier in terms of directional inferences. This new procedure is able to make more directional inferences than the two-sided procedure and maintains the inferential sensitivity of the one-sided procedure. Note however this new procedure controls only type III error, but not type I error. The critical point of the new procedure is the same as that of Lee and Spurrier's one-sided procedure. We also propose a power function for the new procedure and determine the sample size necessary for a guaranteed power level. The application of the procedure is illustrated with an example.

GENERALIZED SIGNED-RANK ESTIMATORS FOR AUTOREGRESSION PARAMETERS

ABSTRACT A new class of weighted signed-rank-based estimates for estimating the parameter vector of an autoregressive time series is considered. The Wilcoxon signed-rank estimate and the GR-estimates of Terpstra et al. (GR-Estimates for an Autoregressive Time Series. Statistics and Probability Letters 2001, 51, 165–172; Generalized Rank Estimates for an Autoregressive Time Series: A U-Statistic Approach. Statistical Inference for Stochastic Processes 2001, 4, 155–179) can be viewed as special cases of the so-called GSR-estimates. Asymptotic linearity properties are derived for the GSR-estimates. Based on these properties, and a symmetry assumption, the GSR-estimates are shown to be asymptotically normal at rate n 1/2. The theory of U-Statistics along with a characterization of weak dependence that is inherent in stationary AR(p) models are the primary tools used to obtain the results. Tests of hypotheses as well as standard errors for confidence interval procedures can be based on such results. An efficiency study indicates that, for an appropriately chosen set of weights, the GSR-estimate is more efficient than the GR-estimate. Furthermore, the GSR-estimate has an added advantage in that an intercept term can be estimated simultaneously; unlike the GR-estimate. Two examples and a small simulation study are used to illustrate the computational and robust aspects of the GSR-estimates.

Confidence Intervals for Effect Sizes in Analysis of Variance

Although confidence interval procedures for analysis of variance (ANOVA) have been available for some time, they are not well known and are often difficult to implement with statistical packages. This article discusses procedures for constructing individual and simultaneous confidence intervals on contrasts on parameters of a number of fixed-effects ANOVA models, including multivariate analysis of variance (MANOVA) models for the analysis of repeated measures data. Examples show how these procedures can be implemented with accessible software. Confidence interval inference on parameters of random-effects models is also discussed.

Stepwise procedures for the identification of minimum effective dose with unknown variances

Hsu and Berger (J. Amer. Statist. Assoc. 50 (1999) 468) proposed a stepwise confidence interval procedure for identifying the minimum effective dose in dose-response studies under homoscedasticity. In practice, homogeneity of variance is often in doubt. In this paper, we extend Hsu and Berger's stepwise confidence interval procedure to the case of heteroscedasticity (i.e., inequality of variances across dosages). In addition, we show that the new procedure can properly control the family-wise error rate. We illustrate our procedure with a real example. Finally, we investigate the performance of our procedure and a step-down test procedure based on Welch's approximate t-test via a simulation study. Our procedure is shown, empirically, to perform similar to Welch's procedure.

Stepwise Procedures under Unknown Variances for Toxicological Evaluation

Toxicological study is of practical importance in modern drug development. Proper statistical methodologies for toxicological evaluation of new developed drugs are undoubtedly necessary. In toxicological studies, it is practically desirable for a method to not declare the safety of a developed drug at a higher dosage prior to the declaration of the safety at lower dosages. Hsu and Berger's stepwise confidence interval method was recently proposed for this purpose. Unfortunately, their procedure necessitates the homogeneity of variances among dosages, which is seldom satisfied in practice. In this article, via the application of the Stein's two-stage sampling method, we propose a stepwise confidence interval procedure for the same task without the homoscedasticity restriction. In addition, our procedure is shown to control its family-wise type I error rate at the pre-chosen nominal level. A simulation study will be conducted to compare our method, Hsu and Berger's stepwise confidence interval method, and a single stage stepwise testing procedure based on Welch's approximation. Our procedure is empirically shown to outperform Hsu and Berger's procedure under heteroscedasticity and perform similarly with Welch's procedure. An example will be used to illustrate our method.

A Bootstrap Confidence Interval Procedure for the Treatment Effect Using Propensity Score Subclassification

In the analysis of observational studies, propensity score subclassification has been shown to be a powerful method for adjusting unbalanced covariates for the purpose of causal inferences. One practical difficulty in carrying out such an analysis is to obtain a correct variance estimate for inference, while reducing bias in the estimate of the treatment effect due to an imbalance in the measured covariates. In this paper, we propose a bootstrap procedure for the inferences concerning the average treatment effect; our bootstrap method is based on an extension of Efron's bias-corrected accelerated (BCa) bootstrap confidence interval to a two-sample problem. Unlike the currently available inference procedures based on propensity score subclassifications, the validity of the proposed method does not rely on a particular form of variance estimation. A brief simulation study is included to evaluate the operating characteristics of the proposed procedure. We conclude the paper by illustrating the new procedure through a clinical application comparing the renal effects of two non-steroidal anti-inammatory drugs (NSAIDs).