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PubMed2001

Unbiased confidence intervals for the odds ratio of two independent binomial samples with application to case-control data.

The problem of confidence interval construction for the odds ratio of two independent binomial samples is considered. Two methods of eliminating the nuisance parameter from the exact likelihood, conditioning and maximization, are described. A conditionally exact tail method exists by putting together upper and lower bounds. A shorter interval can be obtained by simultaneous consideration of both tails. We present here new methods that extend the tail and simultaneous approaches to the maximized likelihood. The methods are unbiased and applicable to case-control data, for which the odds ratio is important. The confidence interval procedures are compared unconditionally for small sample sizes in terms of their expected length and coverage probability. A Bayesian confidence interval method and a large-sample chi2 procedure are included in the comparisons.

Case-Control Studies↗
PubMed2000

General deming regression for estimating systematic bias and its confidence interval in method-comparison studies.

BACKGROUND: Various forms of least-squares regression analyses are used to estimate average systematic error (bias) and its confidence interval in method-comparison studies. When assumptions that underlie a particular regression method are inappropriate for the data, errors in estimated statistics result. In this report, I present an improved method for regression analysis that is free from the usual simplifying assumptions and is generally applicable to linearly related method-comparison data. METHODS: Theoretical equations based on the Deming approach, further developed by physicists and extended herein, were applied to method-comparison data analysis. Monte Carlo simulations were used to demonstrate the validity of the new procedure and to compare its performance to ordinary linear regression (OLR) and simple Deming regression (SDR) procedures. RESULTS: Simulation studies included three types of data commonly encountered in method-comparison studies: (a) constant within-method SDs for both methods, (b) constant within-method CVs for both methods, and (c) neither SDs nor CVs constant for both methods. For all cases examined, OLR produced unreliable confidence intervals of the estimated bias. However, OLR point estimates of systematic bias were reliable when the correlation coefficient was >0.975. SDR produced reliable estimates of systematic bias for all cases studied, but the confidence intervals of systematic bias were unreliable when SDs of methods varied as a function of analyte concentration. CONCLUSION: Only iteratively reweighted general Deming regression produced statistically unbiased estimates of systematic bias and reliable confidence intervals of bias for all cases.

Albumins↗
PubMed1997

A parametric confidence interval for a moment-based scaled criterion for individual bioequivalence.

Confidence intervals of proposed individual bioequivalence metrics are difficult to determine in closed form because their stochastic distributions are unknown. In this article, it is shown that, with slightly modified weights, the Relative Individual Risks (RIR) moment-based scaled statistic for individual bioequivalence that was presented by Schall and Williams has an exact noncentral Fisher's F distribution with noncentrality parameter give by a scaled squared difference in formulations means. This can be approximated by a central F with adjusted degrees of freedom from which it follows that an upper (1-alpha) confidence bound for RIR is given by [formula: see text] where [formula: see text] is the least square estimate of RIR; dfER is the degrees of freedom associated with the reference intrasubject variance estimate, v is the subject-by-formulation degrees of freedom adjusted for noncentrality and alpha is the significance level. Individual bioequivalence is concluded if UL does not exceed the regulatory limit. The performance of this confidence interval was investigated by comparing its experimental bioequivalence rate to that of the unweighted metric under known parameter situations through simulations of two formulations in a fully replicated study design. Results showed that the proposed metric is slightly less biased and more precise than the unweighted metric.

Computer Simulation↗
PubMed2006

Conditional and unconditional confidence intervals following a group sequential test.

After a group sequential test, the naive confidence interval (CI) is usually biased in the sense that it does not cover the true parameter at the correct nominal level. Furthermore, when the stopping time is taken into account, the actual conditional confidence coverage probability can be much less accurate. In this article, we study the conditional coverage probability and other related properties of the naive CI and different versions of exact CI's. It is demonstrated that only correcting the overall confidence level does not necessarily improve the confidence level at any given stopping stage. Conditional inference can be applied to construct an exact conditional CI but it is not without serious undesirable properties. We propose a two-step restricted conditional confidence interval (RCCI) which considerably improves the conditional confidence level while minimizing the undesirable properties. Numerical comparisons are made between the proposed method and existing methods. The results show that the RCCI not only improves the conditional coverage probability considerably from the exact CI's but also is free of the major undesirable properties displayed by the pure conditional CI. Differences between the conditional and unconditional CI's and their respective strengths are also discussed.

Clinical Trials as Topic↗
PubMed1999

Constructing confidence intervals for cost-effectiveness ratios: an evaluation of parametric and non-parametric techniques using Monte Carlo simulation.

The statistic of interest in most health economic evaluations is the incremental cost-effectiveness ratio. Since the variance of a ratio estimator is intractable, the health economics literature has suggested a number of alternative approaches to estimating confidence intervals for the cost-effectiveness ratio. In this paper, Monte Carlo simulation techniques are employed to address the question of which of the proposed methods is most appropriate. By repeatedly sampling from a known distribution and applying the different methods of confidence interval estimation, it is possible to calculate the coverage properties of each method to see if these correspond to the chosen confidence level. As the results of a single Monte Carlo experiment would be valid only for that particular set of circumstances, a series of experiments was conducted in order to examine the performance of the different methods under a variety of conditions relating to the sample size, the coefficient of variation of the numerator and denominator of the ratio, and the covariance between costs and effects in the underlying data. Response surface analysis was used to analyse the results and substantial differences between the different methods of confidence interval estimation were identified. The methods, both parametric and non-parametric, which assume a normal sampling distribution performed poorly, as did the approach based on simply combining the separate intervals on costs and effects. The choice of method for confidence interval estimation can lead to large differences in the estimated confidence limits for cost-effectiveness ratios. The importance of such differences is an empirical question and will depend to a large extent on the role of hypothesis testing in economic appraisal. However, where it is suspected that the sampling distribution is skewed, normal approximation methods produce particularly poor results and should be avoided.

Computer Simulation↗
PubMed1990

An improved approximation for calculation of confidence intervals after a sequential clinical trial.

A method for the calculation of confidence intervals following a sequential clinical trial is proposed which is more accurate than methods based solely on continuous monitoring assumptions, but is not as computationally laborious as previous methods suggested for group sequential trials. The calculation takes account of the excess of the sample path over the boundary at the final inspection of the trial, which may be substantial after group sequential monitoring, and is ignored when continuous monitoring is assumed. Accuracy of such confidence intervals after a triangular test is investigated by simulation of coverage probabilities, individual confidence limit probabilities and p-value curves.

Clinical Trials as Topic↗
PubMed2002

Confidence intervals for seasonal relative risk with null boundary values.

In evaluating the relative risk of seasonality, the null value is a boundary value. In this case, tests for the null hypothesis exist, but standard methods for confidence intervals are not appropriate. We provide a method for constructing confidence intervals under the circular normal model. The proposed confidence intervals are valid for all values of the underlying seasonal risk if the model is correct and for the null boundary value of the seasonal risk, regardless of model assumptions and sample size. We apply our method to seasonal suicide data from a recent report.

Algorithms↗
PubMedDate not supplied

A note on the estimation of confidence intervals for cost-effectiveness when costs and effects are censored.

BACKGROUND: The relation between methodological advances in estimation of confidence intervals (CIs) for incremental cost-effectiveness ratios (ICER) and estimation of cost effectiveness in the presence of censoring has not been explored. The authors address the joint problem of estimating ICER precision in the presence of censoring. METHODS: Using patient-level data (n = 168) on cost and survival from a published placebo-controlled trial, the authors compared 2 methods of measuring uncertainty with censored data: 1) Bootstrap with censor adjustment (BCA); 2) Fieller's method with censor adjustment (FCA). The authors estimate the FCA over all possible values for the correlation (rho) between costs and effects (range = -1 to + 1) and also examine the use of the correlation between cases without censoring adjustment (i.e., simple time-on-study) for costs and effects as an approximation for rho. RESULTS: Using time-on-study, which considers all censored observations as responders (deaths), yields 0.64 life-years gained at an additional cost of 87.9 for a cost per life-year of 137 (95% CI by bootstrap -5.9 to 392). Censoring adjustment corrects for the bias in the time-on-study approach and reduces the cost per life-year estimate to 132 (=72/0.54). Confidence intervals with censor adjustment were approximately 40% wider than the base-case without adjustment. Using the Fieller method with an approximation of rho based on the uncensored cost and effect correlation provides a 95% CI of (-48 to 529), which is very close to the BCA interval of (-52 to 504). CONCLUSIONS: Adjustment for censoring is necessary in cost-effectiveness studies to obtain unbiased estimates of ICER with appropriate uncertainty limits. In this study, BCA and FCA methods, the latter with approximated covariance, are simple to compute and give similar confidence intervals.

Confidence Intervals↗
PubMed1997

Quantitative coronary angiography: impact of imaging parameters on accuracy and confidence interval.

The impact of patient size, image intensifier (II) dose, focal spot size, and matrix size on accuracy and the confidence interval of a quantitative coronary analysis (QCA) system has been investigated using calibrated tube phantoms in combination with a varying amount of water representing the patient's size. We found good accuracy for the whole range of patients' sizes (15-30 cm H2O) except that a 0.5-mm vessel diameter (phi) can no longer be measured at > or = 25 cm H2O. Moreover, at 1-mm vessel phi gives no reliable confidence interval for > or = 30 cm H2O. A small focus (F = 0.4 mm) is favorable for small objects (< or = 15 cm H2O) only. For larger patient sizes the lack of contrast due to higher kilovoltage compensates the improved resolution. Low digital "cine dose" (e.g., 0.1 microGy/f at 17 cm II) is acceptable except for 0.5 mm vessel phi. Better contrast compensates for the increased noise. There is a small improvement in accuracy and confidence interval for vessels < or = 1.0 mm phi using the matrix 1024(2). The HICOR system allows one to stay within the DICOM card standard (matrix 512(2) using the mode "acquisition zoom" while providing high resolution in a limited field size.

Calibration↗
PubMed2004

Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis.

This article presents confidence interval methods for improving on the standard F tests in the balanced, completely between-subjects, fixed-effects analysis of variance. Exact confidence intervals for omnibus effect size measures, such as or and the root-mean-square standardized effect, provide all the information in the traditional hypothesis test and more. They allow one to test simultaneously whether overall effects are (a) zero (the traditional test), (b) trivial (do not exceed some small value), or (c) nontrivial (definitely exceed some minimal level). For situations in which single-degree-of-freedom contrasts are of primary interest, exact confidence interval methods for contrast effect size measures such as the contrast correlation are also provided.

Analysis of Variance↗
PubMed1998

Confidence intervals or surfaces? Uncertainty on the cost-effectiveness plane.

Although cost-effectiveness analysis is not new, it is only recently that economic analysis has been conducted alongside clinical trials. Whereas in the past economic analysts most often used sensitivity analysis to examine the implications of uncertainty for their results, the existence of patient-level data on costs and effects opens up the possibility of statistical analysis of uncertainty. Unfortunately, ratio statistics can cause problems for standard statistical methods of confidence interval estimation. The recent health economics literature contains a number of suggestions for estimating confidence limits for ratios. In this paper, we begin by reviewing the different methods of confidence interval estimation with a view to providing guidance concerning the most appropriate method. We go on to argue that the focus on confidence interval estimation for cost-effectiveness ratios in the recent literature has been concerned more with problems of estimation than with problems of decision-making. We argue that decision-makers are most likely to be interested in one-sided tests of hypothesis and that confidence surfaces are better suited to such tests than confidence intervals. This approach is consistent with decision-making on the cost-effectiveness plane and with the cost-effectiveness acceptability curve approach to presenting uncertainty due to sampling variation in stochastic cost-effectiveness analyses.

Confidence Intervals↗
PubMed2000

[Confidence intervals and their relevance for the interpretation of results. Audit of the journal "Strahlentherapie und Onkology"].

BACKGROUND: The statistical quality of the contributions to "Strahlentherapie und Onkologie" is assessed, aiming for improvement of the journal and consequently its impact factor. MATERIAL AND METHODS: All 181 articles published during 1998 and 1999 in the categories "review", "original contribution", and "short communication" were analysed concerning the appropriate use of confidence intervals. RESULT: Forty-four publications were excluded from analysis, because they did not contain quantitative data or because the quotation of a confidence interval would not have been meaningful for other reasons. Of the remaining 137 publications only 27 presented all relevant results with clearly defined and correctly interpreted confidence intervals. This corresponds to a fraction of 20% (95% CI: 13-28%). CONCLUSION: Authors, peer reviewers, and editors could contribute to improve the quality of the journal by setting value on the documentation of confidence intervals.

Confidence Intervals↗
PubMed1996

Confidence intervals for median survival times under a piecewise exponential model with proportional hazards covariate effects.

Brookmeyer and Crowley derived a non-parametric confidence interval for the median survival time of a homogeneous population by inverting a generalization of the sign test for censored data. The 1-alpha confidence interval for the median is essentially the set of all values t such that the Kaplan--Meier estimate of the survival function at time t does not differ significantly from one-half at significance level alpha. Here I extend the method to incorporate covariates into the analysis by assuming an underlying piecewise exponential model with proportional hazards covariate effects. Maximum likelihood estimates of the model parameters are obtained via iterative techniques, from which the estimated (log) survival curve is easily constructed. The delta method provides asymptotic standard errors. Following Brookmeyer and Crowley, I find the confidence interval for the median survival time at a specified value of the covariate vector by inverting the sign test. I illustrate the methods using data from a clinical trial conducted by the Radiation Therapy Oncology Group in cancer of the mouth and throat. It is seen that the piecewise exponential model provides considerable flexibility in accommodating to the shape of the underlying survival curve and thus offers advantages to other, more restrictive, parametric models. Simulation studies indicate that the method provides reasonably accurate coverage probabilities.

Age Factors↗
PubMed2000

Bootstrap confidence intervals for relative risk parameters in affected-sib-pair data.

In affected-sib-pair (ASP) studies, parameters such as the locus-specific sibling relative risk, lambda(s), may be estimated and used to decide whether or not to continue the search for susceptibility genes. Typically, a maximum likelihood point estimate of lambda(s) is given, but since this estimate may have substantial variability, it is of interest to obtain confidence limits for the true value of lambda(s). While a variety of methods for doing this exist, there is considerable uncertainty over their reliability. This is because the discrete nature of ASP data and the imposition of genetic "possible triangle" constraints during the likelihood maximization mean that asymptotic results may not apply. In this paper, we use simulation to evaluate the reliability of various asymptotic and simulation-based confidence intervals, the latter being based on a resampling, or bootstrap approach. We seek to identify, from the large pool of methods available, those methods that yield short intervals with accurate coverage probabilities for ASP data. Our results show that many of the most popular bootstrap confidence interval methods perform poorly for ASP data, giving coverage probabilities much lower than claimed. The test-inversion, profile-likelihood, and asymptotic methods, however, perform well, although some care is needed in choice of nuisance parameter. Overall, in simulations under a variety of different genetic hypotheses, we find that the asymptotic methods of confidence interval evaluation are the most reliable, even in small samples. We illustrate our results with a practical application to a real data set, obtaining confidence intervals for the sibling relative risks associated with several loci involved in type 1 diabetes.

Confidence Intervals↗
PubMed2003

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.

Confidence Intervals↗
PubMed1995

Basic statistics for clinicians: 2. Interpreting study results: confidence intervals.

In the second of four articles, the authors discuss the "estimation" approach to interpreting study results. Whereas, in hypothesis testing, study results lead the reader to reject or accept a null hypothesis, in estimation the reader can assess whether a result is strong or weak, definitive or not. A confidence interval, based on the observed result and the size of the sample, is calculated. It provides a range of probabilities within which the true probability would lie 95% or 90% of the time, depending on the precision desired. It also provides a way of determining whether the sample is large enough to make the trial definitive. If the lower boundary of a confidence interval is above the threshold considered clinically significant, then the trial is positive and definitive, if the lower boundary is somewhat below the threshold, the trial is positive, but studies with larger samples are needed. Similarly, if the upper boundary of a confidence interval is below the threshold considered significant, the trial is negative and definitive. However, a negative result with a confidence interval that crosses the threshold means that trials with larger samples are needed to make a definitive determination of clinical importance.

Clinical Trials as Topic↗
PubMedDate not supplied

Can't miss: conquer any number task by making important statistics simple. Part 5. Comparing two confidence intervals, standard error of the difference between two means and between two proportions.

Healthcare quality professionals need to understand and use inferential statistics to interpret sample data from their organizations. Since in quality improvement and healthcare research studies all the data from a population often are not available, investigators take samples and make inferences about that population using inferential statistics. This series of six articles will give readers an understanding of the concepts of inferential statistics as well as the specific tools for calculating confidence intervals and tests of statistical significance for samples of data. This article, Part 5, demonstrates the comparison of two confidence intervals as a method for estimating the difference between two population means. The concept of the standard error of the difference between two sample means is presented along with the confidence interval for estimating the difference between two population means. The article concludes with the standard error and confidence interval for estimating the difference between two population proportions from a binomial variable.

Confidence Intervals↗
PubMed2006

Poor performance of bootstrap confidence intervals for the location of a quantitative trait locus.

The aim of many genetic studies is to locate the genomic regions (called quantitative trait loci, QTL) that contribute to variation in a quantitative trait (such as body weight). Confidence intervals for the locations of QTL are particularly important for the design of further experiments to identify the gene or genes responsible for the effect. Likelihood support intervals are the most widely used method to obtain confidence intervals for QTL location, but the nonparametric bootstrap has also been recommended. Through extensive computer simulation, we show that bootstrap confidence intervals behave poorly and so should not be used in this context. The profile likelihood (or LOD curve) for QTL location has a tendency to peak at genetic markers, and so the distribution of the maximum-likelihood estimate (MLE) of QTL location has the unusual feature of point masses at genetic markers; this contributes to the poor behavior of the bootstrap. Likelihood support intervals and approximate Bayes credible intervals, on the other hand, are shown to behave appropriately.

Bayes Theorem↗