On Parametric Confidence Intervals for the Cost‐Effectiveness Ratio

When comparing two competing interventions, confidence intervals for cost‐effectiveness ratios (CERs) provide information on the uncertainty in their point estimates. Techniques for constructing these confidence intervals are much debated. We provide a formal comparison of the Fieller, symmetric and Bonferroni methods for constructing confidence intervals for the CER using only the joint asymptotic distribution of the incremental cost and incremental effectiveness of the two interventions being compared. We prove the existence of a finite interval under the Fieller method when the incremental effectiveness is statistically significant. When this difference is not significant the Fieller method yields an unbounded confidence interval. The Fieller interval is always wider than the symmetric interval, but the latter is an approximation to the Fieller interval when the incremental effectiveness is highly significant. The Bonferroni method is shown to produce the widest interval. Because it accounts for the likely correlation between cost and effectiveness measures, and the intuitively appealing relationship between the existence of a bounded interval and the significance of the incremental effectiveness, the Fieller interval is to be preferred in reporting a confidence interval for the CER.     

Contrasting Diversity Values: Statistical Inferences Based on Overlapping Confidence Intervals

Ecologists often contrast diversity (species richness and abundances) using tests for comparing means or indices. However, many popular software applications do not support performing standard inferential statistics for estimates of species richness and/or density. In this study we simulated the behavior of asymmetric log-normal confidence intervals and determined an interval level that mimics statistical tests with P(α) = 0.05 when confidence intervals from two distributions do not overlap. Our results show that 84% confidence intervals robustly mimic 0.05 statistical tests for asymmetric confidence intervals, as has been demonstrated for symmetric ones in the past. Finally, we provide detailed user-guides for calculating 84% confidence intervals in two of the most robust and highly-used freeware related to diversity measurements for wildlife (i.e., EstimateS, Distance).     

Likelihood-Weighted Confidence Intervals for the Difference of Two Binomial Proportions

Two new methods for computing confidence intervals for the difference δ = p₁ — p₂ between two binomial proportions (p₁, p₂) are proposed. Both the Mid-P and Max-P likelihood weighted intervals are constructed by mapping the tail probabilities from the two-dimensional (p₁, p₂)-space into a one-dimensional function of δ based on the likelihood weights. This procedure may be regarded as a natural extension of the CLOPPER-PEARSON (1934) interval to the two-sample case where the weighted tail probability is α/2 at each end on the δ scale. The probability computation is based on the exact distribution rather than a large sample approximation. Extensive computation was carried out to evaluate the coverage probability and expected width of the likelihood-weighted intervals, and of several other methods. The likelihood-weighted intervals compare very favorably with the standard asymptotic interval and with intervals proposed by HAUCK and ANDERSON (1986), COX and SNELL (1989), SANTNER and SNELL (1980), SANTNER and YAMAGAMI (1993), and PESKUN (1993). In particular, the Mid-P likelihood-weighted interval provides a good balance between accurate coverage probability and short interval width in both small and large samples. The Mid-P interval is also comparable to COE and TAMHANE'S (1993) interval, which has the best performance in small samples.     

Confidence Intervals in Qtl Mapping by Bootstrapping

The determination of empirical confidence intervals for the location of quantitative trait loci (QTLs) was investigated using simulation. Empirical confidence intervals were calculated using a bootstrap resampling method for a backcross population derived from inbred lines. Sample sizes were either 200 or 500 individuals, and the QTL explained 1, 5, or 10% of the phenotypic variance. The method worked well in that the proportion of empirical confidence intervals that contained the simulated QTL was close to expectation. In general, the confidence intervals were slightly conservatively biased. Correlations between the test statistic and the width of the confidence interval were strongly negative, so that the stronger the evidence for a QTL segregating, the smaller the empirical confidence interval for its location. The size of the average confidence interval depended heavily on the population size and the effect of the QTL. Marker spacing had only a small effect on the average empirical confidence interval. The LOD drop-off method to calculate empirical support intervals gave confidence intervals that generally were too small, in particular if confidence intervals were calculated only for samples above a certain significance threshold. The bootstrap method is easy to implement and is useful in the analysis of experimental data.     

Confidence Intervals for Microbial Density Using Serial Dilutions with MPN Estimates

An alteration to Woodward's methods is recommended for deriving a 1 — α confidence interval for microbial density using serial dilutions with most-probable-number (MPN) estimates. Outcomes of the serial dilution test are ordered by their MPNs. A lower limit for the confidence interval corresponding to an outcome y is the density for which y and all higher ordered outcomes have total probability α/2. An upper limit is derived in the analogous way. An alteration increases the lowest lower limits and decreases the highest upper limits. For comparison, a method that is optimal in the sense of null hypothesis rejection is described. This method ranks outcomes dependent upon the microbial density in question, using proportional first derivatives of the probabilities. These and currently used methods are compared. The recommended method is shown to be more desirable in certain respects, although resulting in slightly wider confidence intervals than De Man's (1983) method.     

R法遗传参数估值置信区间的计算方法和重复估计次数的研究

探讨R法遗传参数估值置信区间的计算方法和重复估计次数(NORE)对参数估值的影响,利用4种模型通过模拟产生数据集。基础群中公、母畜数分别为200和2000头,BLUP育种值选择5个世代。利用多变量乘法迭代(MMI)法,结合先决条件的共扼梯度(PCG)法求解混合模型方程组估计方差组分。用经典方法、Box-Cox变换后的经典方法和自助法计算参数估值的均数、标准误和置信区间。结果表明,重复估计次数较多时,3种方法均可;重复估计次数较少时,建议使用自助法。简单模型下需要较少的重复估计,但对于复杂模型则需要较多的重复估计。随模型中随机效应数的增加,直接遗传力高估。随着PCG和MMI轮次的增大,参数估值表现出低估的趋势。     

Precision Intervals for Estimates of the Difference in Success Rates for Binary Random Variables Based on the Permutation Principle

Fisher's exact test is a very commonly applied test in clinical trials with a binary outcome variable (e.g. success/failure). However confidence statements about the difference of success rates are usually based on the normal approximation. This may sometimes lead to the confusing statement that the test is statistically significant at a prespecified level while the corresponding confidence interval includes the zero difference and vice versa. Here, we construct precision intervals for the difference of success rates from two independent samples based on the permutation principle which are in perfect agreement with the discrete (permutation) test and compare it to examples from the literature. APL programs are provided.     

Confidence Intervals for fMRI Activation Maps

Neuroimaging activation maps typically color voxels to indicate whether the blood oxygen level-dependent (BOLD) signals measured among two or more experimental conditions differ significantly at that location. This data presentation, however, omits information critical for interpretation of experimental results. First, no information is represented about trends at voxels that do not pass the statistical test. Second, no information is given about the range of probable effect sizes at voxels that do pass the statistical test. This leads to a fundamental error in interpreting activation maps by naïve viewers, where it is assumed that colored, “active” voxels are reliably different from uncolored “inactive” voxels. In other domains, confidence intervals have been added to data graphics to reduce such errors. Here, we first document the prevalence of the fundamental error of interpretation, and then present a method for solving it by depicting confidence intervals in fMRI activation maps. Presenting images where the bounds of confidence intervals at each voxel are coded as color allows readers to visually test for differences between “active” and “inactive” voxels, and permits for more proper interpretation of neuroimaging data. Our specific graphical methods are intended as initial proposals to spur broader discussion of how to present confidence intervals for fMRI data.     

Confidence Intervals on Ratios of Variance Components for the Unbalanced Two Factor Nested Model

The model considered in this article is the two-factor nested unbalanced variance component model: for p = 1, 2, …, P; q = 1, 2, …, Q_p; and r = 1, 2, …, R_pq. The random variables Y_pqr are observable. The constant μ is an unknown parameter, and A_p, B_pq and C_pqr are (unobservable) normal and independently distributed random variables with zero means and finite variances σ²_A, σ²_B, and σ²_C, respectively. Approximate confidence intervals on ?_A and ?_B using unweighted means are derived, where The performance of these approximate confidence intervals are evaluated using computer simulation. The simulated results indicate that these proposed confidence intervals perform satisfactorily and can be used in applied problems.     

Intervals for the assessment of measurement agreement: Similarities,differences, and consequences of incorrect interpretations

Several intervals have been proposed to quantify the agreement of two methods intended to measure the same quantity in the situation where only one measurement per method and subject is available. The limits of agreement are probably the most well‐known among these intervals, which are all based on the differences between the two measurement methods. The different meanings of the intervals are not always properly recognized in applications. However, at least for small‐to‐moderate sample sizes, the differences will be substantial. This is illustrated both using the width of the intervals and on probabilistic scales related to the definitions of the intervals. In particular, for small‐to‐moderate sample sizes, it is shown that limits of agreement and prediction intervals should not be used to make statements about the distribution of the differences between the two measurement methods or about a plausible range for all future differences. Care should therefore be taken to ensure the correct choice of the interval for the intended interpretation.     

Confidence Intervals and Tests of Hypotheses on Variance Components in an Unbalanced Two-fold Nested Design

Confidence intervals and tests of hypotheses on variance components are required in studies that employ a random effects design. The unbalanced random two-fold nested design is considered in this paper and confidence intervals are proposed for the variance components σ2/A and σ2/B. Computer simulation is used to show that even in very unbalanced designs, these intervals generally maintain the stated confidence coefficient. The hypothesis test for σ2/A based on the lower bound of the recommended confidence interval is shown to be better than previously proposed approximate tests.     

Confidence Intervals for Concentration and Brightness from Fluorescence Fluctuation Measurements

The theory of photon count histogram (PCH) analysis describes the distribution of fluorescence fluctuation amplitudes due to populations of fluorophores diffusing through a focused laser beam and provides a rigorous framework through which the brightnesses and concentrations of the fluorophores can be determined. In practice, however, the brightnesses and concentrations of only a few components can be identified. Brightnesses and concentrations are determined by a nonlinear least-squares fit of a theoretical model to the experimental PCH derived from a record of fluorescence intensity fluctuations. The χ² hypersurface in the neighborhood of the optimum parameter set can have varying degrees of curvature, due to the intrinsic curvature of the model, the specific parameter values of the system under study, and the relative noise in the data. Because of this varying curvature, parameters estimated from the least-squares analysis have varying degrees of uncertainty associated with them. There are several methods for assigning confidence intervals to the parameters, but these methods have different efficacies for PCH data. Here, we evaluate several approaches to confidence interval estimation for PCH data, including asymptotic standard error, likelihood joint-confidence region, likelihood confidence intervals, skew-corrected and accelerated bootstrap (BCa), and Monte Carlo residual resampling methods. We study these with a model two-dimensional membrane system for simplicity, but the principles are applicable as well to fluorophores diffusing in three-dimensional solution. Using simulated fluorescence fluctuation data, we find the BCa method to be particularly well-suited for estimating confidence intervals in PCH analysis, and several other methods to be less so. Using the BCa method and additional simulated fluctuation data, we find that confidence intervals can be reduced dramatically for a specific non-Gaussian beam profile.     

Scheffy's Confidence Intervals in the Models of Analysis of Covariance

Scheffe's confidence intervals for linear functions of some subvectors of a vector of parameters are presented. The considered subvectors are such that covariance matrices of their estimators are known non-negative definite matrices multiplied by unknown positive constants. This property is characteristic of the least squares estimators of vectors of main and interaction effects in the analysis of covariance models of the following experimental designs: split-block, split-plot, completely randomized two-factor design and randomized complete block design. The formulas for confidence intervals for linear functions of vectors of main or interaction effects in the designs mentioned above are given in the paper. The practical example is given as an illustration.     

Confidence Intervals on the Total Variance in an Unbalanced Two-fold Nested Design

Confidence intervals on the total variance in an unbalanced random two-fold nested design are constructed and compared. Computer simulation indicates the proposed intervals provide confidence coefficients that are generally close to the stated level.     

Simple Methods of Determining Confidence Intervals for Functions of Estimates in Published Results

Often, the reader of a published paper is interested in a comparison of parameters that has not been presented. It is not possible to make inferences beyond point estimation since the standard error for the contrast of the estimated parameters depends upon the (unreported) correlation. This study explores approaches to obtain valid confidence intervals when the correlation is unknown. We illustrate three proposed approaches using data from the National Health Interview Survey. The three approaches include the Bonferroni method and the standard confidence interval assuming (most conservative) or (when the correlation is known to be non-negative). The Bonferroni approach is found to be the most conservative. For the difference in two estimated parameter, the standard confidence interval assuming yields a 95% confidence interval that is approximately 12.5% narrower than the Bonferroni confidence interval; when the correlation is known to be positive, the standard 95% confidence interval assuming is approximately 38% narrower than the Bonferroni. In summary, this article demonstrates simple methods to determine confidence intervals for unreported comparisons. We suggest use of the standard confidence interval assuming if no information is available or if the correlation is known to be non-negative.     

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.     

Some Results on the Tukey-Mclaughlin and Yuen Methods for Trimmed Means when Distributions are Skewed

In applied work, distributions are often highly skewed with heavy tails, and this can have disastrous consequences in terms of power when comparing groups based on means. One solution to this problem in the one-sample case is to use the TUKEY and MCLAUGHLIN (1963) method for trimmed means, while in the two-group case YUEN's (1974) method can be used. Published simulations indicate that they yield accurate confidence intervals when distributions are symmetric. Using a Cornish-Fisher expansion, this paper extends these results by describing general circumstances under which methods based on trimmed means can be expected to give more accurate confidence intervals than those based on means. The results cover both symmetric and asymmetric distributions. Simulations are also used to illustrate the accuracy of confidence intervals using trimmed means versus means.     

Accurate Confidence Intervals for the Ratio of Specific Occurrence/Exposure Rates in Risk and Survival Analysis

Transformation and computer intensive methods such as the jackknife and bootstrap are applied to construct accurate confidence intervals for the ratio of specific occurrence/exposure rates, which are used to compare the mortality (or survival) experience of individuals in two study populations. Monte Carlo simulations are employed to compare the performances of the proposed confidence intervals when sample sizes are small or moderate.     

Generalized Confidence Intervals for Ratios of Regression Coefficients with Applications to Bioassays

The problem of constructing a confidence interval for the ratio of two regression coefficients is addressed in the context of multiple regression. The concept of a Generalized Confidence Interval is used, and the resulting confidence interval is shown to perform well in terms of coverage probability. The proposed methodology always results in an interval, unlike the confidence region generated from Fieller's theorem. The procedure can easily be implemented for parallel‐line assays, slope‐ratio assays, and quantal assays under a probit model. Furthermore, this approach can also be extended to compute confidence intervals based on data from multiple bioassays. The results are illustrated using several examples.     

Confidence Intervals for the Mean of a Poisson Distribution: A Review

The paper provides a comprehensive review of methodology for setting confidence intervals for the parameter of a Poisson distribution. The results are illustrated by a numerical example.