Fieller's Method Research Articles

MOVER-R and Penalized MOVER-R Confidence Intervals for the Ratio of Two Quantities

Developing a confidence interval for the ratio of two quantities is an important task in statistics because of its omnipresence in real world applications. For such a problem, the MOVER-R (method of variance recovery for the ratio) technique, which is based on the recovery of variance estimates from confidence limits of the numerator and the denominator separately, was proposed as a useful and efficient approach. However, this method implicitly assumes that the confidence interval for the denominator never includes zero, which might be violated in practice. In this article, we first use a new framework to derive the MOVER-R confidence interval, which does not require the above assumption and covers the whole parameter space. We find that MOVER-R can produce an unbounded confidence interval, just like the well-known Fieller method. To overcome this issue, we further propose the penalized MOVER-R. We prove that the new method differs from MOVER-R only at the second order. It, however, always gives a bounded and analytic confidence interval. Through simulation studies and a real data application, we show that the penalized MOVER-R generally provides a better confidence interval than MOVER-R in terms of controlling the coverage probability and the median width.

Computing Confidence Intervals for the Postal Service’s Cost-Elasticity Estimates

This paper provides methods for assessing the precision of cost elasticity estimates when the underlying regression function is assumed to be polynomial. Specifically, the paper adapts two well-known methods for computing confidential intervals for ratios: the delta-method and the Fieller method. We show that performing the estimation with mean-centered explanatory variables provides a straightforward way to estimate the elasticity and compute a confidence interval for it. A theoretical discussion of the proposed methods is provided, as well as an empirical example based on publicly available postal data. Possible areas of application include postal service providers worldwide, transportation and electricity.

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.

Non-Standard Confidence Sets for Ratios and Tipping Points with Applications to Dynamic Panel Data

We study estimation uncertainty when the object of interest contains one or more ratios of parameters. The ratio of parameters is a discontinuous parameter transformation; it has been shown that traditional confidence intervals often fail to cover this true ratio with very high probability. Constructing confidence sets for ratios using Fieller’s method is a viable solution as the method can avoid the discontinuity problem. This paper proposes an extension of the multivariate Fieller method beyond standard estimators, focusing on asymptotically mixed normal estimators that commonly arise in dynamic panel polynomial regression with persistent covariates. We discuss the cases where the underlying estimators converge to various distri- butions, depending on the persistence level of the covariates. We show that the asymptotic distribution of the pivotal statistic used for constructing a Fieller’s confidence set remains a standard Chi-squared distribution regardless of rates of convergence, thus the rates are being ‘self-normalized’ and can be unknown. A simulation study illustrates the finite sample properties of the proposed method in a dynamic polynomial panel. Our method is demonstrated to work well in small samples, even when the persistence coefficient is unity.

Combination of the weighted kappa coefficients of two binary diagnostic tests

ABSTRACTThe combination of two binary diagnostic tests in order to increase the accuracy of the diagnosis of a disease is a frequent procedure in clinical practice. When considering the losses associated with an erroneous classification with a binary diagnostic test, the parameter that is used to assess the diagnostic test is the weighted kappa coefficient. The weighted kappa coefficient depends on the sensitivity and the specificity of the diagnostic test, on the disease prevalence and on the weighting index. In this article, we study the combination of the weighted kappa coefficients of two binary diagnostic tests, defining the parameters and studying the conditions under which the combination of the two diagnostic tests increases the value of the weighted kappa coefficient of the combination. The Fieller method has been applied to obtain a confidence interval for the ratio between the combined weighted kappa coefficient and the weighted kappa coefficient of each diagnostic test. A program in R has been written to solve the problem studied and it is available as complementary material. The results have been applied to a real example.

Identification-Robust Factor Pricing: Canadian Evidence

We analyze factor models based on the Arbitrage Pricing Theory (APT). using identification-robust inference methods. Such models involve nonlinear reduced-rank restrictions whose identification may raise serious non-regularities and lead to a failure of standard asymptotic theory. We build confidence sets for structural parameters based on inverting Hotelling-type pivotal statistics. These confidence sets provide much more information than the corresponding tests. Our approach may be interpreted as a multivariate extension of the Fieller method for inference on mean ratios. We also introduce a formal definition for a redundant factor linking the presence of such factors to unbounded confidence sets, and we document their perverse effects on minimum-root-based model tests. Results are applied to multifactor asset-pricing models with Canadian data, the Fama-French-Carhart benchmarks and monthly returns of 25 portfolios from 1991 to 2010. Despite evidence of weak identification, several findings deserve notice when data are analyzed over ten-year subperiods. With equally weighted portfolios, the three-factor model is rejected before 2000, but weakly supported thereafter. In contrast, the three-factor model is not rejected with value-weighted portfolios. Interestingly in this case, the market factor is priced before 2000 along with size, while both Fama-French factors are priced thereafter. The momentum factor severely compromises identification, which calls for caution in interpreting existing work documenting momentum effects on the Canadian market. This empirical analysis underscores the practical usefulness of our analytical confidence sets.

Elicitation threshold of cobalt chloride: analysis of patch test dose-response studies.

Cobalt is a strong skin sensitizer (grade 5 of 5 in the guinea-pig maximization test) that is used in various industrial and consumer applications. To prevent sensitization to cobalt and elicitation of allergic cobalt dermatitis, information about the elicitation threshold level of cobalt is important. To identify the dermatitis elicitation threshold levels in cobalt-allergic individuals. Published patch test dose-response studies were reviewed to determine the elicitation dose (ED) levels in dermatitis patients with a previous positive patch test reaction to cobalt. A logistic dose-response model was applied to data collected from the published literature to estimate ED values. The 95% confidence interval (CI) for the ratio of mean doses that can elicit a reaction in 10% (ED(10)) of a population was calculated with Fieller's method. On the basis of five included studies, the ED10 values of aqueous cobalt chloride ranged between 0.0663 and 1.95 µg cobalt/cm(2), corresponding to 30.8-259 ppm. Our analysis provides an overview of the doses of cobalt that are required to elicit allergic cobalt contactdermatitis in sensitized individuals, and thereby the basis for future prevention of cobalt allergy.

Constructing Confidence Intervals for Effect Size Measures of an Indirect Effect

Confidence intervals for an effect size can provide the information about the magnitude of an effect and its precision as well as the binary decision about the existence of an effect. In this study, the performances of five different methods for constructing confidence intervals for ratio effect size measures of an indirect effect were compared in terms of power, coverage rates, Type I error rates, and widths of confidence intervals. The five methods include the percentile bootstrap method, the bias-corrected and accelerated (BCa) bootstrap method, the delta method, the Fieller method, and the Monte Carlo method. The results were discussed with respect to the adequacy of the distributional assumptions and the nature of ratio quantity. The confidence intervals from the five methods showed similar results for samples of more than 500, whereas, for samples of less than 500, the confidence intervals were sufficiently narrow to convey the information about the population effect sizes only when the effect sizes of regression coefficients defining the indirect effect are large.

On finite sample performance of confidence intervals methods for willingness to pay measures

This paper systematically compares finite sample performances of methods to build confidence intervals for willingness to pay measures in a choice modeling context. It contributes to the field by also considering methods developed in other research fields. Various scenarios are evaluated under an extensive Monte Carlo study. Results show that the commonly used Delta method, producing symmetric intervals around the point estimate, often fails to account for skewness in the estimated willingness to pay distribution. Both the Fieller method and the likelihood ratio test inversion method produce more realistic confidence intervals for small samples. Some bootstrap methods also perform reasonably well, in terms of effective coverage. Finally, empirical data are used to illustrate an application of the methods considered.

A cluster randomised controlled trial to determine the clinical effectiveness and cost-effectiveness of classroom-based cognitive–behavioural therapy (CBT) in reducing symptoms of depression in high-risk adolescents

Depression in adolescents is a significant problem that impairs everyday functioning and increases the risk of severe mental health disorders in adulthood. Although this is a major problem, relatively few adolescents with, or at risk of developing, depression are identified and referred for treatment. This suggests the need to investigate alternative approaches whereby preventative interventions are made widely available in schools. To investigate the clinical effectiveness and cost-effectiveness of classroom-based cognitive-behavioural therapy (CBT) in reducing symptoms of depression in high-risk adolescents. Cluster randomised controlled trial. Year groups ( n = 28) randomly allocated on a 1 : 1 : 1 basis to one of three trial arms once all schools were recruited and balanced for number of classes, number of students, Personal, Social and Health Education (PSHE) lesson frequency, and scheduling of PSHE. Year groups 8 to 11 (ages 12-16 years) in mixed-sex secondary schools in the UK. Data were collected between 2009 and 2011. Young people who attended PSHE at participating schools were eligible ( n = 5503). Of the 5030 who agreed to participate, 1064 (21.2%) were classified as 'high risk': 392 in the classroom-based CBT arm, 374 in the attention control PSHE arm and 298 in the usual PSHE arm. Primary outcome data on the high-risk group at 12 months were available for classroom-based CBT ( n = 296), attention control PSHE ( n = 308) and usual PSHE ( n = 242). The Resourceful Adolescent Programme (RAP) is a focused CBT-based intervention adapted for the UK (RAP-UK) and delivered by two facilitators external to the school. Control groups were usual PSHE (usual school curriculum delivered by teachers) and attention control (usual school PSHE with additional support from two facilitators). Interventions were delivered universally to whole classes. Clinical effectiveness: symptoms of depression [Short Mood and Feelings Questionnaire (SMFQ)] in adolescents at high risk of depression 12 months from baseline. Cost-effectiveness: incremental cost-effectiveness ratios (ICERs) based on SMFQ score and quality-adjusted life-years (from European Quality of Life-5 Dimensions scores) between baseline and 12 months. Process evaluation: reach, attrition and qualitative feedback from service recipients and providers. SMFQ scores had decreased for high-risk adolescents in all trial arms at 12 months, but there was no difference between arms [classroom-based CBT vs. usual PSHE adjusted difference in means 0.97, 95% confidence interval (CI) -0.34 to 2.28; classroom-based CBT vs. attention control PSHE -0.63, 95% CI -1.99 to 0.73]. Costs of interventions per child were estimated at £41.96 for classroom-based CBT and £34.45 for attention control PSHE. Fieller's method was used to obtain a parametric estimate of the 95% CI for the ICERs and construct the cost-effectiveness acceptability curve, confirming that classroom-based CBT was not cost-effective relative to the controls. Reach of classroom-based CBT was good and attrition was low (median 80% attending ≥ 60% of sessions), but feedback indicated some difficulties with acceptability and sustainability. Classroom-based CBT, attention control PSHE and usual PSHE produced similar outcomes. Classroom-based CBT may result in increased self-awareness and reporting of depressive symptoms. Classroom-based CBT was not shown to be cost-effective. While schools are a convenient way of reaching a wide range of young people, implementing classroom-based CBT within schools is challenging, particularly with regard to fitting programmes into a busy timetable, the lack of value placed on PSHE, and difficulties engaging with teachers and young people. Wider use of classroom-based depression prevention programmes should not be undertaken without further research. If universal preventative approaches are to be pursued, their clinical effectiveness and cost-effectiveness with younger children (aged 10-11 years), before the incidence of depression increases, should be investigated. Alternatively, the clinical effectiveness of indicated school-based programmes targeting those already displaying symptoms of depression should be investigated. Current Controlled Trials ISRCTN19083628. This project was funded by the NIHR Health Technology Assessment programme and will be published in full in Health Technology Assessment; Vol. 17, No. 47. See the HTA programme website for further project information.

Identification-Robust Estimation and Testing of the Zero-Beta CAPM

We propose exact simulation-based procedures for: (i) testing mean-variance efficiency when the zero-beta rate is unknown; (ii) building confidence intervals for the zero-beta rate. On observing that this parameter may be weakly identified, we propose likelihood-ratio-type tests as well as Fieller-type procedures based on a Hotelling-HAC statistic, which are robust to weak identification and allow for non-Gaussian distributions including parametric GARCH structures. The Fieller-Hotelling-HAC procedure also accounts (asymptotically) for general forms of heteroskedasticity and autocorrelation. We propose confidence sets for the zero-beta rate based on inverting exact tests for this parameter; for both procedures proposed, these sets can be interpreted as multivariate extensions of the classic Fieller method for inference on ratios. The exact distribution of likelihood-ratio-type statistics for testing efficiency is studied under both the null and the alternative hypotheses. The relevant nuisance parameter structure is established and finite-sample bound procedures are proposed, which extend and improve available Gaussian-specific bounds. Finite-sample distributional invariance results are also demonstrated analytically for the HAC statistic proposed by MacKinlay and Richardson (1991) . We study invariance to portfolio repacking for the tests and confidence sets proposed. The statistical properties of the proposed methods are analysed through a Monte Carlo study and compared with alternative available methods. Empirical results on NYSE returns show that exact confidence sets are very different from asymptotic ones, and allowing for non-Gaussian distributions affects inference results. Simulation and empirical evidence suggests that likelihood-ratio-type statistics--with p-values corrected using the Maximized Monte Carlo test method--are generally preferable to their multivariate Fieller-Hotelling-HAC counterparts from the viewpoints of size control and power. Copyright 2013, Oxford University Press.

Inferences for the ratio: Fieller’s interval, log ratio, and large sample based confidence intervals

In sample surveys and many other areas of application, the ratio of variables is often of great importance. This often occurs when one variable is available at the population level while another variable of interest is available for sample data only. In this case, using the sample ratio, we can often gather valuable information on the variable of interest for the unsampled observations. In many other studies, the ratio itself is of interest, for example when estimating proportions from a random number of observations. In this note we compare three confidence intervals for the population ratio: A large sample interval, a log based version of the large sample interval, and Fieller’s interval. This is done through data analysis and through a small simulation experiment. The Fieller method has often been proposed as a superior interval for small sample sizes. We show through a data example and simulation experiments that Fieller’s method often gives nonsensical and uninformative intervals when the observations are noisy relative to the mean of the data. The large sample interval does not similarly suffer and thus can be a more reliable method for small and large samples.

Métodos de análisis de la incertidumbre

Current budgetary restrictions in health systems have increased the influence of economic evaluation studies on decision making. Nevertheless, uncertainty about the parameters used in pharmacoeconomic models is inevitable and may affect the conclusions drawn. The present article aims to review the main methods proposed to quantify the uncertainty inherent in pharmacoeconomic evaluations applied to health technologies. The most accurate pharmacoeconomic estimations are obtained by probabilistic methods of uncertainty analysis such as Monte Carlo simulation, repetitive sampling and the Fieller method. Alternatives to these methods are calculation of the cost-effectiveness acceptability curve or net health benefit. Equally, the economic impact of uncertainty in the pharmacoeconomic models used in the decision-making process can be estimated by obtaining what is known as the value of perfect information.

Inferential methods for elasticity estimates

Elasticities are often estimated from the results of demand analysis. However, drawing inferences from them may involve assumptions that could influence the outcome. In this paper we investigate one of the most common forms of elasticity which is defined as a ratio of estimated relationships and demonstrate how the Fieller method for the construction of confidence intervals can be used to draw inferences. We estimate the elasticities of expenditure from Engel curves using a variety of estimation models. Parametric Engel curves are modelled using OLS, MM robust regression, and Tobit. Semiparametric Engel curves are estimated using a penalized spline regression. We demonstrate the construction of confidence intervals of the expenditure elasticities for a series of expenditure levels as well as the estimated cumulative density function for the elasticity evaluated for a particular household.

A geometric confidence ellipse approach to the estimation of the ratio of two variables

One is often interested in the ratio of two variables, for example in genetics, assessing drug effectiveness, and in health economics. In this paper, we derive an explicit geometric solution to the general problem of identifying the two tangents from an arbitrary external point to an ellipse. This solution permits numerical integration of a bivariate normal distribution over a wedge-shaped region bounded by the tangents, which yields an evaluation of the tangent slopes as confidence limits on the ratio of the component variables. After suitable adjustment of the confidence coverage of the ellipse, these confidence limits are shown to be equivalent to those from Fieller's method. However, the geometric approach allows additional interpretation of the data through identification of the points of tangency, the ellipse itself, and expressions for the coverage probability of the confidence interval. Numerical evaluations using the theoretical expressions for the geometric confidence intervals (but ignoring sample variation in the underlying parameters) suggested that they perform well overall and are slightly conservative. Simulations that do take account of sample variation in the underlying parameters again suggested that the intervals perform well overall, although here they are slightly anti-conservative. Coverage probabilities for the confidence intervals were only weakly dependent on the distance and correlation of the ellipse, but there were asymmetries in the failure rates of the upper and lower confidence limits in some configurations. The probability of no real solution existing was also evaluated. These ideas are illustrated by a practical example.

Cost‐effectiveness of temozolomide for the treatment of newly diagnosed glioblastoma multiforme

The study aimed to compare the cost-effectiveness of concomitant and adjuvant temozolomide (TMZ) for the treatment of newly diagnosed glioblastoma multiforme versus initial radiotherapy alone from a public health care perspective. The economic evaluation was performed alongside a randomized, multicenter, phase 3 trial. The primary endpoint of the trial was overall survival. Costs included all direct medical costs. Economic data were collected prospectively for a subgroup of 219 patients (38%). Unit costs for drugs, procedures, laboratory and imaging, radiotherapy, and hospital costs per day were collected from the official national reimbursement lists based on 2004. For the cost-effectiveness analysis, survival was expressed as 2.5 years restricted mean estimates. The incremental cost-effectiveness ratio (ICER) was constructed. Confidence intervals for the ICER were calculated using the Fieller method and bootstrapping. The difference in 2.5 years restricted mean survival between the treatment arms was 0.25 life-years and the ICER was euro37,361 per life-year gained with a 95% confidence interval (CI) ranging from euro19,544 to euro123,616. The area between the survival curves of the treatment arms suggests an increase of the overall survival gain for a longer follow-up. An extrapolation of the overall survival per treatment arm and imputation of costs for the extrapolated survival showed a substantial reduction in ICER. The ICER of euro37,361 per life-year gained is a conservative estimate. We concluded that despite the high TMZ acquisition costs, the costs per life-year gained are comparable to accepted first-line treatment with chemotherapy in patients with cancer.

Monitoring gasoline blending rates for compliance

AbstractWe consider the problem of obtaining a confidence region for the blending coefficient in the blended underground storage tank (UST) leak model. Under this model, the blending coefficient estimator is a ratio of two correlated normal random variables. Fieller's method provides a simple way of obtaining limits with exact coverage probability. This confidence interval can be inverted to test the hypothesis that the advertised blending ratio is correct. This method is applied to the Cary blended site data, and several variations are included to account for outliers and a change point. Copyright © 2008 John Wiley & Sons, Ltd.

Cost-Effectiveness of Physical Therapy and General Practitioner Care for Sciatica

An economic evaluation alongside a randomized clinical trial in primary care. A total of 135 patients were randomly allocated to physical therapy added to general practitioners' care (n = 67) or to general practitioners' care alone (n = 68). To evaluate the cost-effectiveness of physical therapy and general practitioner care for patients with an acute lumbosacral radicular syndrome (LRS, also called sciatica) compared with general practitioner care only. There is a lack of knowledge concerning the cost-effectiveness of physical therapy in patients with sciatica. The clinical outcomes were global perceived effect and quality of life. The direct and indirect costs were measured by means of questionnaires. The follow-up period was 1 year. The Incremental Cost-effectiveness Ratio (ICER) between both study arms was constructed. Confidence intervals for the ICER were calculated using Fieller's method and using bootstrapping. There was a significant difference on perceived recovery at 1-year follow-up in favor of the physical therapy group. The additional physical therapy did not have an incremental effect on quality of life. At 1-year follow-up, the ICER for the total costs was 6224 euros (95% confidence interval, -10,419, 27,551) per improved patient gained. For direct costs only, the ICER was 837 euros (95% confidence interval, -731, 3186). The treatment of patients with LRS with physical therapy and general practitioners'care is not more cost-effective than general practitioners'care alone.

Providing Intuition to the Fieller Method with Two Geometric Representations using STATA and EVIEWS

The Fieller Method for the construction of confidence intervals for ratios of the expected value of two normally distributed random variables has been shown by a number of authors to be a superior method to the delta approximation. However, it is not widely used due in part, to the tendency to present the intervals only in a formula context. In addition, potential users have been deterred by the potential difficulty in interpreting non-finite confidence intervals when the confidence level is less than 100%. In this paper we present two graphical methods which can be easily constructed using two widely used statistical software packages (Eviews and Stata) for the representation of the Fieller intervals. An application is presented to assess the results of a model of the non-accelerating inflation rate of unemployment (NAIRU).

On Confidence Bounds for the Ratio of Net Differences in the “Gold Standard” Design with Reference, Experimental, and Placebo Treatment

The three-arm clinical study including a placebo has been recommended to be the preferred study design for the comparison of an experimental treatment relative to a reference treatment. In a confirmatory three-arm study multiplicity issues arise that are not present in two-arm studies. In the past a successful demonstration of superiority of the reference over placebo has been regarded a prerequisite validation step for the demonstration of superiority of the experimental treatment over placebo. However, for an investigator this last comparison is the most critical one. In a clinical study the demonstration of superiority of the experimental treatment over placebo is a result of its own value and this should therefore not be made dependent on tests that are of higher priority in a hierarchical test procedure. This can be achieved through a symmetrical formulation of Fieller's method for constructing confidence intervals for ratio of the expected values from normally distributed variables. In the symmetrical formulation the different meanings of nominator and denominator disappear, and simultaneous statements for comparisons between the experimental treatment and placebo, reference treatment and placebo, and reference and experimental treatment can be made. This is accomplished by moving the discussion on confidence sets from the line of real numbers to the unit circle which allows representing confidence sectors always as connected sets, gives always simple geometrical interpretations, and is easy to be transformed back to the real numbers if the traditional calculations behave well. The proposed procedures provides additional insight in existing methods but does obviously not answer the clinical question whether or not the demonstration of superiority of the reference treatment over placebo is a necessary validation step for further comparisons.