Cost‐Effectiveness of a Low‐Carbohydrate Diet and a Standard Diet in Severe Obesity

Objective: Low‐carbohydrate diets have become a popular alternative to standard diets for weight loss. Our aim was to compare the cost‐effectiveness of these two diets. Research Methods and Procedures: The patient population included 129 severely obese subjects (BMI = 42.9) from a randomized trial; participants had a high prevalence of diabetes or metabolic syndrome. We compared within‐trial costs, quality‐adjusted life years (QALYs), and the incremental cost‐effectiveness ratio (CER) for the two study groups. We imputed missing values for QALYs. The CER was bootstrapped to derive 95% confidence intervals and to define acceptability cut‐offs. We took a societal perspective for our analysis. Results: Total costs during the one year of the trial were $6742 ± 6675 and $6249 ± 5100 for the low‐carbohydrate and standard groups, respectively (p = 0.78). Participants experienced 0.64 ± 0.02 and 0.61 ± 0.02 QALYs during the one year of the study, respectively (p = 0.17 for difference). The point estimate of the incremental CER was $?1225/QALY (i.e., the low‐carbohydrate diet dominated the standard diet). However, in the bootstrap analysis, the wide spread of CERs caused the 95% confidence interval to be undefined. The probabilities that the low‐carbohydrate diet was acceptable, using cut‐offs of $50, 000/QALY, $100, 000/QALY, and $150, 000/QALY, were 72.4% 78.6%, and 79.8%, respectively. Discussion: The low‐carbohydrate diet was not more cost‐effective for weight loss than the standard diet in the patient population studied. Larger studies are needed to better assess the cost‐effectiveness of dietary therapies for weight loss.     

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.     

Nonparametric statistical methods for cost-effectiveness analyses

Two measures often used in a cost-effectiveness analysis are the incremental cost-effectiveness ratio (ICER) and the net health benefit (NHB). Inferences on these two quantities are often hindered by highly skewed cost data. In this article, we derive the Edgeworth expansions for the studentized t-statistics for the two measures and show how they could be used to guide inferences. In particular, we use the expansions to study the theoretical performance of existing confidence intervals based on normal theory and to derive new confidence intervals for the ICER and the NHB. We conduct a simulation study to compare our new intervals with several existing methods. The methods evaluated include Taylor's interval, Fieller's interval, the bootstrap percentile interval, and the bootstrap bias-corrected acceleration interval. We found that our new intervals give good coverage accuracy and are narrower compared to the current recommended intervals.     

Some Remarks on the Estimation of the Ratio of the Expectation Values of a Two-dimensional Normal Random Variable (Correction of the Theorem of Milliken)

It is possible to show that the theorem of Fieller yields an exact confidence region derivable from the likelihood ratio test. The statement of Milliken (1982) that this confidence region is conservative is refuted. Bounds for the conditional confidence coefficient under the condition that the confidence region is a “meaningful” confidence interval are given.     

On Estimating Medical Cost and Incremental Cost-Effectiveness Ratios with Censored Data

Medical cost estimation is very important to health care organizations and health policy makers. We consider cost-effectiveness analysis for competing treatments in a staggered-entry, survival-analysis-based clinical trial. We propose a method for estimating mean medical cost over patients in such settings. The proposed estimator is shown to be consistent and asymptotically normal, and its asymptotic variance can be obtained. In addition, we propose a method for estimating the incremental cost-effectiveness ratio and for obtaining a confidence interval for it. Simulation experiments are conducted to evaluate our proposed methods. Finally, we apply our methods to a clinical trial comparing the cost effectiveness of implanted cardiac defibrillators with conventional therapy for individuals at high risk for ventricular arrhythmias.     

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.     

Confidence interval construction for relative toxicity endpoints such as LD50 or LD90 ratios

A common measure of the relative toxicity is the ratio of median lethal doses for responses estimated in two bioassays. Robertson and Preisler previously proposed a method for constructing a confidence interval for the ratio. The applicability of this technique in common experimental situations, especially those involving small samples, may be questionable because the sampling distribution of this ratio estimator may be highly skewed. To examine this possibility, we did a computer simulation experiment to evaluate the coverage properties of the Robertson and Preisler method. The simulation showed that the method provided confidence intervals that performed at the nominal confidence level for the range of responses often observed in pesticide bioassays. Results of this study provide empirical support for the continued use this technique.     

Determining the cost-effectiveness of mass screening for cervical cancer using common analytic models

OBJECTIVE: To estimate the cost per life-year saved (cost-effectiveness ratio [CER]) for cervical cancer and to evaluate the influence of the decreased incidence upon the cost per life-year saved. STUDY DESIGN: We established hypothetical cohorts at 10-year intervals between 30 and 79 years of age, each of which consisted of 100,000 asymptomatic female subjects, and estimated the cost and effect of single mass screening for cervical cancer. To investigate the influence on CER, we performed a sensitivity analysis of each item, including the consultation rate for further examination, prevalence rate and cost of medical treatment. RESULTS: The estimated CER per one expected life-year of survival was lowest for subjects in their 30s and highest for those in their 70s. The difference between the two was more than five-fold. Sensitivity analysis was rarely affected by changes in the cost of medical treatment and the prevalence rate, but the effectiveness rate could be fairly affected by the consultation rate for closer examination. CONCLUSION: Mass screening for cervical cancer is acceptable in terms of economic effectiveness. Moreover, mass screening for cervical cancer could decrease the morbidity rate for scores of years thereafter.     

Testing Equality of Means and Simultaneous Confidence Intervals in Repeated Measures with Missing Data

In this paper, repeated measures with intraclass correlation model is considered when the observations are missing at random. An exact test for the equality of the mean components and simultaneous confidence intervals (Scheffé and Bonferroni inequality types) are given for linear contrasts of the mean components when the missing observations are of a monotone type. When the missing observations are not of the monotone type, the maximum likelihood estimates are obtained numerically by iterative methods given in Srivastava and Carter (1986). These estimators are then used to obtain asymptotic tests and confidence intervals for the equality of mean components and linear contrasts, respectively. An example is given to illustrate the method.     

Simultaneous confidence intervals for comparing binomial parameters

SUMMARY: To compare proportions with several independent binomial samples, we recommend a method of constructing simultaneous confidence intervals that uses the studentized range distribution with a score statistic. It applies to a variety of measures, including the difference of proportions, odds ratio, and relative risk. For the odds ratio, a simulation study suggests that the method has coverage probability closer to the nominal value than ad hoc approaches such as the Bonferroni implementation of Wald or "exact" small-sample pairwise intervals. It performs well even for the problematic but practically common case in which the binomial parameters are relatively small. For the difference of proportions, the proposed method has performance comparable to a method proposed by Piegorsch (1991, Biometrics 47, 45-52).     

A study on confidence intervals for incremental cost-effectiveness ratios

In health policy and economics studies, the incremental cost-effectiveness ratio (ICER) has long been used to compare the economic consequences relative to the health benefits of therapies. Due to the skewed distributions of the costs and ICERs, much research has been done on how to obtain confidence intervals of ICERs, using either parametric or nonparametric methods, with or without the presence of censoring. In this paper, we will examine and compare the finite sample performance of many approaches via simulation studies. For the special situation when the health effect of the treatment is not statistically significant, we will propose a new bootstrapping approach to improve upon the bootstrap percentile method that is currently available. The most efficient way of constructing confidence intervals will be identified and extended to the censored data case. Finally, a data example from a cardiovascular clinical trial is used to demonstrate the application of these methods.     

Estimating incremental cost-effectiveness ratios and their confidence intervals with differentially censored data

With medical costs escalating over recent years, cost analysis is being conducted more and more to assess economic impact of new treatment options. An incremental cost-effectiveness ratio (ICER) is a measure that assesses the additional cost for a new treatment for each additional unit of effectiveness, such as saving 1 year of life. In this article, we consider cost-effectiveness analysis for new treatments evaluated in a randomized clinical trial setting with staggered entries. In particular, the censoring times are different for cost and survival data. We propose a method for estimating the ICER and obtaining its confidence interval when differential censoring exists. Simulation experiments are conducted to evaluate our proposed method. We also apply our methods to a clinical trial example comparing the cost-effectiveness of implanted defibrillators with conventional therapy for individuals with reduced left ventricular function after myocardial infarction.     

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).     

Confidence intervals for the mean of diagnostic test charge data containing zeros

In this paper, we consider the problem of interval estimation for the mean of diagnostic test charges. Diagnostic test charge data may contain zero values, and the nonzero values can often be modeled by a log-normal distribution. Under such a model, we propose three different interval estimation procedures: a percentile-t bootstrap interval based on sufficient statistics and two likelihood-based confidence intervals. For theoretical properties, we show that the two likelihood-based one-sided confidence intervals are only first-order accurate and that the bootstrap-based one-sided confidence interval is second-order accurate. For two-sided confidence intervals, all three proposed methods are second-order accurate. A simulation study in finite-sample sizes suggests all three proposed intervals outperform a widely used minimum variance unbiased estimator (MVUE)-based interval except for the case of one-sided lower end-point intervals when the skewness is very small. Among the proposed one-sided intervals, the bootstrap interval has the best coverage accuracy. For the two-sided intervals, when the sample size is small, the bootstrap method still yields the best coverage accuracy unless the skewness is very small, in which case the bias-corrected ML method has the best accuracy. When the sample size is large, all three proposed intervals have similar coverage accuracy. Finally, we analyze with the proposed methods one real example assessing diagnostic test charges among older adults with depression.     

Miticide bioassays with spider mites (Acari: Tetranychidae): effect of test design and sample size on the precision of lethal concentration estimates

The effect of interval space between the concentration (symmetric or geometric), number of concentrations and overall sample sizes were investigated using a Petri dish bioassay, the miticide propargite and two spider mites species. The precision (width of the 95% confidence intervals) of the LC₅₀ and LC₉₀ estimates was shown to be influenced by the test design and total sample size The results suggest that a symmetric five-concentration design using a minimum total sample size of 480 mites is required to produce reliable LC estimates using the Petri dish method. Asymmetric designs did not increase the reliability of LC estimates. When less than 480 test subjects are available, a symmetric three-concentration design would give adequate LC estimates, but the precision would be less than with a five-concentration design.     

Comparison of family-based association tests in chromosome regions selected by linkage-based confidence intervals

We use the Genetic Analysis Workshop 14 simulated data to explore the effectiveness of a two-stage strategy for mapping complex disease loci consisting of an initial genome scan with confidence interval construction for gene location, followed by fine mapping with family-based tests of association on a dense set of single-nucleotide polymorphisms. We considered four types of intervals: the 1-LOD interval, a basic percentile bootstrap confidence interval based on the position of the maximum Zlr score, and asymptotic and bootstrap confidence intervals based on a generalized estimating equations method. For fine mapping we considered two family-based tests of association: a test based on a likelihood ratio statistic and a transmission-disequilibrium-type test implemented in the software FBAT. In two of the simulation replicates, we found that the bootstrap confidence intervals based on the peak Zlr and the 1-LOD support interval always contained the true disease loci and that the likelihood ratio test provided further strong confirmatory evidence of the presence of disease loci in these regions.     

Calculating confidence intervals for prediction error in microarray classification using resampling

Cross-validation based point estimates of prediction accuracy are frequently reported in microarray class prediction problems. However these point estimates can be highly variable, particularly for small sample numbers, and it would be useful to provide confidence intervals of prediction accuracy. We performed an extensive study of existing confidence interval methods and compared their performance in terms of empirical coverage and width. We developed a bootstrap case cross-validation (BCCV) resampling scheme and defined several confidence interval methods using BCCV with and without bias-correction. The widely used approach of basing confidence intervals on an independent binomial assumption of the leave-one-out cross-validation errors results in serious under-coverage of the true prediction error. Two split-sample based methods previously proposed in the literature tend to give overly conservative confidence intervals. Using BCCV resampling, the percentile confidence interval method was also found to be overly conservative without bias-correction, while the bias corrected accelerated (BCa) interval method of Efron returns substantially anti-conservative confidence intervals. We propose a simple bias reduction on the BCCV percentile interval. The method provides mildly conservative inference under all circumstances studied and outperforms the other methods in microarray applications with small to moderate sample sizes.     

A New Distribution-Free Approach to Constructing the Confidence Region for Multiple Parameters

Construction of confidence intervals or regions is an important part of statistical inference. The usual approach to constructing a confidence interval for a single parameter or confidence region for two or more parameters requires that the distribution of estimated parameters is known or can be assumed. In reality, the sampling distributions of parameters of biological importance are often unknown or difficult to be characterized. Distribution-free nonparametric resampling methods such as bootstrapping and permutation have been widely used to construct the confidence interval for a single parameter. There are also several parametric (ellipse) and nonparametric (convex hull peeling, bagplot and HPDregionplot) methods available for constructing confidence regions for two or more parameters. However, these methods have some key deficiencies including biased estimation of the true coverage rate, failure to account for the shape of the distribution inherent in the data and difficulty to implement. The purpose of this paper is to develop a new distribution-free method for constructing the confidence region that is based only on a few basic geometrical principles and accounts for the actual shape of the distribution inherent in the real data. The new method is implemented in an R package, distfree.cr/R. The statistical properties of the new method are evaluated and compared with those of the other methods through Monte Carlo simulation. Our new method outperforms the other methods regardless of whether the samples are taken from normal or non-normal bivariate distributions. In addition, the superiority of our method is consistent across different sample sizes and different levels of correlation between the two variables. We also analyze three biological data sets to illustrate the use of our new method for genomics and other biological researches.     

Interval estimation for a difference between intraclass kappa statistics

Model-based inference procedures for the kappa statistic have developed rapidly over the last decade. However, no method has yet been developed for constructing a confidence interval about a difference between independent kappa statistics that is valid in samples of small to moderate size. In this article, we propose and evaluate two such methods based on an idea proposed by Newcombe (1998, Statistics in Medicine, 17, 873-890) for constructing a confidence interval for a difference between independent proportions. The methods are shown to provide very satisfactory results in sample sizes as small as 25 subjects per group. Sample size requirements that achieve a prespecified expected width for a confidence interval about a difference of kappa statistic are also presented.