Fieller's Confidence Interval Research Articles

Stepwise Procedure for Toxicological Assessment Based on Ratio of Mean Differences for a Normally Distributed Data

We propose a new stepwise confidence set procedure for toxicity study based on ratio of mean difference. Statistical approaches for evaluating toxicity studies that properly control familywise error rate (FWER) for difference of means between treatments and a control already exist. However, in some therapeutic areas, ratio of mean differences is desirable. Therefore, we construct stepwise confidence procedure based on Fieller's confidence intervals for multiple ratio of mean difference without multiplicity adjustment for toxicological evaluation. Simulation study revealed that the FWER is well controlled at prespecified nominal level α. Also, the power of our approach increases with increasing sample size and ratio of mean differences.

Penalized Fieller's confidence interval for the ratio of bivariate normal means.

Constructing a confidence interval for the ratio of bivariate normal means is a classical problem in statistics. Several methods have been proposed in the literature. The Fieller method is known as an exact method, but can produce an unbounded confidence interval if the denominator of the ratio is not significantly deviated from 0; while the delta and some numeric methods are all bounded, they are only first-order correct. Motivated by a real-world problem, we propose the penalized Fieller method, which employs the same principle as the Fieller method, but adopts a penalized likelihood approach to estimate the denominator. The proposed method has a simple closed form, and can always produce a bounded confidence interval by selecting a suitable penalty parameter. Moreover, the new method is shown to be second-order correct under the bivariate normality assumption, that is, its coverage probability will converge to the nominal level faster than other bounded methods. Simulation results show that our proposed method generally outperforms the existing methods in terms of controlling the coverage probability and the confidence width and is particularly useful when the denominator does not have adequate power to reject being 0. Finally, we apply the proposed approach to the interval estimation of the median response dose in pharmacology studies to show its practicalusefulness.

A Gaussian alternative to using improper confidence intervals

AbstractThe problem posed by exact confidence intervals (CIs) which can be either all‐inclusive or empty for a nonnegligible set of sample points is known to have no solution within CI theory. Confidence belts causing improper CIs can be modified by using margins of error from the renewed theory of errors initiated by J. W. Tukey—briefly described in the article—for which an extended Fraser's frequency interpretation is given. This approach is consistent with Kolmogorov's axiomatization of probability, in which a probability and an error measure obey the same axioms, although the connotation of the two words is different. An algorithm capable of producing a margin of error for any parameter derived from the five parameters of the bivariate normal distribution is provided. Margins of error correcting Fieller's CIs for a ratio of means are obtained, as are margins of error replacing Jolicoeur's CIs for the slope of the major axis. Margins of error using Dempster's conditioning that can correct optimal, but improper, CIs for the noncentrality parameter of a noncentral chi‐square distribution are also given.

Assessing the ratio of means as a causal estimand in clinical endpoint bioequivalence studies in the presence of intercurrent events.

In clinical endpoint bioequivalence studies, the observed per-protocol (PP) population (compliers and completers in general) is usually used in the primary analysis for equivalence assessment. However, intercurrent events, ie, missingness and noncompliance, are not properly handled. The resulting estimand is not causal. Previously, we proposed the first causal framework to assess equivalence in the presence of missing data and noncompliance. We proposed a causal survivor average causal effect (SACE) estimand for the difference of means (DOM). In equivalence assessment, DOM is not as widely used as the ratio of means (ROM). However, no existing formula links the observed PP estimand to the SACE estimand for ROM as exists for DOM. Herein, we propose a similar causal framework for ROM using the principal stratification approach, one of the strategies recommended by the International Conference on Harmonisation (ICH) E9 R1 addendum. We quantify the bias of the observed ROM PP estimand for the SACE estimand, which provides a basis to identify three conditions under which the two estimands are equal. We propose a sensitivity analysis method to evaluate the robustness of the current PP estimator to estimate the SACE estimand. We extend Fieller's confidence interval for the SACE estimand using ROM, which can be applied to many settings. Simulation demonstrates that the PP estimator is biased in either directions and may inflate type 1 error and/or change power when the three identified conditions are violated. Our work can be applied to comparative clinical biosimilar studies.

Development of Defined Microbial Population Standards Using Fluorescence Activated Cell Sorting for the Absolute Quantification of S. aureus Using Real-Time PCR

In this article, four types of standards were assessed in a SYBR Green-based real-time PCR procedure for the quantification of Staphylococcus aureus (S. aureus) in DNA samples. The standards were purified S. aureus genomic DNA (type A), circular plasmid DNA containing a thermonuclease (nuc) gene fragment (type B), DNA extracted from defined populations of S. aureus cells generated by Fluorescence Activated Cell Sorting (FACS) technology with (type C) or without purification of DNA by boiling (type D). The optimal efficiency of 2.016 was obtained on Roche LightCycler(®) 4.1. software for type C standards, whereas the lowest efficiency (1.682) corresponded to type D standards. Type C standards appeared to be more suitable for quantitative real-time PCR because of the use of defined populations for construction of standard curves. Overall, Fieller Confidence Interval algorithm may be improved for replicates having a low standard deviation in Cycle Threshold values such as found for type B and C standards. Stabilities of diluted PCR standards stored at -20°C were compared after 0, 7, 14 and 30 days and were lower for type A or C standards compared with type B standards. However, FACS generated standards may be useful for bacterial quantification in real-time PCR assays once optimal storage and temperature conditions are defined.

A Geometric Comparison of the Delta and Fieller Confidence Intervals

The comparison of the Delta and Fieller confidence intervals for the ratio of parameters estimated by normally distributed random variables has long been of interest. Our contribution is the construction of a common geometric representation of both the Delta and Fieller intervals defined by two related constrained extrema problems that are subject to a common constraint. The diagrammatic solution to these problems can be used to examine how earlier comparisons based on alternate analytic relationships and simulations have resulted in the particular conclusions they report. We find that the degree to which the Delta and Fieller intervals coincide depends not only on the univariate statistics for the estimated parameters, but also on the agreement of the sign of their estimated correlation and the ratio of estimated parameters. This article has supplementary material online.

Two geometric representations of confidence intervals for ratios of linear combinations of regression parameters: An application to the NAIRU

We present two geometric representations of the Fieller confidence interval for the ratio of regression parameter estimates using standard statistics packages. These methods are used to interpret the estimation of the NAIRU for Australia.

MAGhaler® versus Dosieraerosol: Wirksamkeit und Sicherheit von Beclomethason-Dipropionat bei Kindern mit Asthma

Objective: The MAGhaler ® is a CFC-free, refillable inhalation device. The objective of this clinical study was to show equivalent efficacy and safety ofbeclomethasone dipropionate (BDP) delivered via MAGhaler ® or via CFC-containing pressurized metered-dose inhaler (pMDI) in children with mild to moderate bronchial asthma. Usability and acceptance of the two devices were also compared. Methods: This was a double-blind, randomised, reference-controlled, 12-week trial in 38 children with mild to moderate asthma receiving BDP (2 x 250 μg/day) delivered via either the MAGhaler ® (N = 19) or the pMDI (N = 19). Equivalence of the devices was assessed on the basis of the ratio (MAG/pMDI) of the mean forced expiratory volume in one second (FEV 1 ) measured during therapy. In addition, the two-sided 90% Fieller confidence interval (CI) for the ratio of means was calculated. Lung function parameters, clinical symptoms, concomitant intake of salbutamol or fenoterol, adverse events, and laboratory values were recorded. Results: In the per-protocol analysis (N = 35), the ratio (MAG/ pMDI) of mean FEV 1 during therapy was 1.07 [90% CI (0.93; 1.24)] indicating that the two devices were comparable. Mean FEV 1 increased from 2.00 ± 0.45 1 (baseline) to 2.16 ± 0.53 1 (week 12; MAG) and from 1.88 ± 0.54 1 to 2.01 ± 0.56 1 (pMDI). A total of 28 (80%) children preferred the MAGhaler ® to the pMDI. The safety profile of BDP was similar in both groups. Conclusion: BDP administered via the MAGhaler ® was as safe and efficacious as that delivered via the pMDI. The MAGhaler ® was very well accepted.

Simultaneous Test for Superiority and Noninferiority Hypotheses in Active-Controlled Clinical Trials*

Two stage switching between testing for superiority (SUP) and noninferiority (NI) has been an important statistical issue in the design and analysis of the active-controlled clinical trials. Tsong and Zhang (2005) has shown that the Type I error rates do not change when switching between SUP and NI with the traditional generalized historical control (GHC) approach, however, they may change when switching with the cross-trial comparison (X-trial) approach. Tsong and Zhang (2005) further proposed a simultaneous test for both hypotheses to avoid the problem. The procedure was based on Fieller's confidence interval proposed by Hauschke et al. (1999). Since with the X-trial approach, using the simultaneous test, superiority is tested using all four treatment arms (current test and active control arms, active control and placebo arms in historical trials), the Type I error rate and power are expected to be somewhat different from the conventional superiority test (using the current test and active control arms only). Through a simulation study, we demonstrate that the Type I error rate and power between simultaneous test and the conventional superiority test are compatible. We also examine the impact of the assumption of equal variances of the current trial and the historical trial.

Use of the Fieller–Hinkley Distribution of the Ratio of Random Variables in Testing for Noninferiority

We address the noninferiority assessment problem defined in terms of the ratio of population means in a parallel group design analysis of variance setting. The sample ratio as a point estimate of the corresponding population ratio has been considered. It has been shown that the Fieller–Hinkley distribution of the ratio of two correlated normally distributed random variables readily provide a technique for constructing confidence intervals comparable to the bootstrap percentile and Fieller's confidence intervals. A finite parameter space based level α test of an inferiority hypothesis formulated in terms of a fixed margin has been derived. We illustrate our approach using the forced vital capacity (FVC) data. We claim that it is easy to construct and straight forward to interpret our bootstrap equivalent confidence intervals that are used to assess noninferiority. We discuss appropriate methods for calculation of sample sizes.

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).

Simultaneous Confidence Intervals for Ratios with Applications to the Comparison of Several Treatments with a Control

In this article, we illustrate and compare exact simultaneous confidence sets with various approximate simultaneous confidence intervals for multiple ratios as applied to many-to-one comparisons. Quite different datasets are analyzed to clarify the points. The methods are based on existing probability inequalities (e.g., Bonferroni, Slepian and Sidak), estimation of nuisance parameters and re-sampling techniques. Exact simultaneous confidence sets based on the multivariate t-distribution are constructed and compared with approximate simultaneous confidence intervals. It is found that the coverage probabilities associated with the various methods of constructing simultaneous confidence intervals (for ratios) in manyto-one comparisons depend on the ratios of the coefficient of variation for the mean of the control group to the coefficient of variation for the mean of the treatments. If the ratios of the coefficients of variations are less than one, the Bonferroni corrected Fieller confidence intervals have almost the same coverage probability as the exact simultaneous confidence sets. Otherwise, the use of Bonferroni intervals leads to conservative results. When the ratio of the coefficient of variation for the mean of the control group to the coefficient of variation for the mean of the treatments are greater than one (e.g., in balanced designs with increasing effects), the Bonferroni simultaneous confidence intervals are too conservative. Therefore, we recommend not using Bonferroni for this kind of data. On the other hand, the plug-in method maintains the intended confidence coefficient quite satisfactorily; therefore, it can serve as the best alternative in any case.

A practical approach for the assessment of bioequivalence under selected higher-order cross-over designs.

The two-period cross-over design with two sequences of drug administration is a standard experimental design when bioequivalence of one test formulation is to be assessed in comparison with a reference formulation. Previously, an approach based on Fieller's confidence interval has been presented for the assessment of average bioequivalence under this particular design. However, the two-sequence two-period cross-over design is not very useful in the presence of unequal carry-over effects. Besides, this basic design does not provide independent estimators for the intra-subject variabilities. To overcome these limitations, it might be of interest to consider a higher-order cross-over design in which the number of periods and/or the number of sequences is greater than the number of formulations to be compared. Because of this, the present communication will concentrate on the generalization of Fieller's confidence interval concept for a particular group of higher-order cross-over designs. In addition to this, since the evaluation of simple average bioequivalence does not guarantee that the two products can be used interchangeably, the assessment of population and individual bioequivalence is addressed through the application of a comprehensible three-step decision rule. An example study with a two-sequence four-period design is also analysed to illustrate the use of the proposed methods.

A practical approach for the assessment of bioequivalence under selected higher‐order cross‐over designs

The two-period cross-over design with two sequences of drug administration is a standard experimental design when bioequivalence of one test formulation is to be assessed in comparison with a reference formulation. Previously, an approach based on Fieller's confidence interval has been presented for the assessment of average bioequivalence under this particular design. However, the two-sequence two-period cross-over design is not very useful in the presence of unequal carry-over effects. Besides, this basic design does not provide independent estimators for the intra-subject variabilities. To overcome these limitations, it might be of interest to consider a higher-order cross-over design in which the number of periods and/or the number of sequences is greater than the number of formulations to be compared. Because of this, the present communication will concentrate on the generalization of Fieller's confidence interval concept for a particular group of higher-order cross-over designs. In addition to this, since the evaluation of simple average bioequivalence does not guarantee that the two products can be used interchangeably, the assessment of population and individual bioequivalence is addressed through the application of a comprehensible three-step decision rule. An example study with a two-sequence four-period design is also analysed to illustrate the use of the proposed methods. © 1997 John Wiley & Sons, Ltd.

Fieller's confidence intervals for the ratio of two means in the assessment of average bioequivalence from crossover data.

The two-period crossover design is the most commonly used study design for bioequivalence of one test formulation to be assessed in comparison to one reference formulation. Consequently, in this paper, all derivation is based on this particular design. It is assumed that for the underlying statistical model the usual assumptions of normality and additivity are satisfied on the original scale of measurement and that it is wanted to base the assessment of average bioavailability on the ratio of the unknown population means for the test and reference formulation. The purpose of this paper is to illustrate that it is reasonable to assume a uniform covariance structure for the two-period crossover design, because the demand of equal variability in bioavailabilities, in addition to equal average bioavailabilities, for the reference and test formulation makes the assumption of uniform covariance structure very realistic, and also because the properties of a decision rule based upon a Fieller's confidence interval under a uniform covariance structure are competitive with those of the corresponding rule based on a general covariance structure.