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PubMed2005

Advanced statistics: bootstrapping confidence intervals for statistics with "difficult" distributions.

The use of confidence intervals in reporting results of research has increased dramatically and is now required or highly recommended by editors of many scientific journals. Many resources describe methods for computing confidence intervals for statistics with mathematically simple distributions. Computing confidence intervals for descriptive statistics with distributions that are difficult to represent mathematically is more challenging. The bootstrap is a computationally intensive statistical technique that allows the researcher to make inferences from data without making strong distributional assumptions about the data or the statistic being calculated. This allows the researcher to estimate confidence intervals for statistics that do not have simple sampling distributions (e.g., the median). The purposes of this article are to describe the concept of bootstrapping, to demonstrate how to estimate confidence intervals for the median and the Spearman rank correlation coefficient for non-normally-distributed data from a recent clinical study using two commonly used statistical software packages (SAS and Stata), and to discuss specific limitations of the bootstrap.

Confidence Intervals↗
PubMed2003

Confidence intervals for the 50 per cent response dose.

Confidence intervals for the 50 per cent response dose are usually computed using either the Delta method or Fieller's procedure. Recently, confidence intervals computed by inverting the asymptotic likelihood ratio test have also been recommended. There is some controversy as to which of these methods should be used. By means of an extensive simulation study we examine these methods as well as confidence intervals obtained by the approximate bootstrap confidence (ABC) procedure and an adjusted form of the likelihood ratio based confidence intervals.

Animals↗
PubMed1993

Chemistry with confidence: should Clinical Chemistry require confidence intervals for analytical and other data?

Confidence intervals are not commonly provided with analytical or other data reported in Clinical Chemistry although P values are. However, confidence intervals provide an explicit demonstration of the direction and magnitude of uncertainty and are intuitively easy to grasp, unlike P values. It is therefore argued that the Journal should adopt a policy requiring the provision of confidence intervals. Such a policy would improve the statistical rigor of Journal reports.

Chemistry, Clinical↗
PubMed1985

Generalization of distribution--free confidence intervals for bioavailability ratios.

The confidence interval approach to bioavailability assessment depends first on selection of the confidence level, usually 95%, and then determination of the confidence limits for the expected bioavailability ratio AUC(Test)/AUC(Reference). In practice, however, it is sometimes of greater interest to know the probability that the expected bioavailability will fall below a critical value, for example 0.75, or within a clinically set bioequivalence range, for example 0.80 to 1.25. Up to now, posterior probability distributions have been suggested, based on classical analysis of variance (ANOVA) with its rather restrictive assumptions, including that of a (logarithmic) normal distribution. In this report, a distribution-free confidence interval based on the Wilcoxon signed-rank statistic has been generalized so that confidence probabilities can be obtained for any given confidence limits. In the case of unimodal and almost symmetrical sampling distributions, the results obtained are very similar to those of the ANOVA-based posterior probability distribution. However, skewed or multimodal sampling distributions are better reflected by the proposed distribution-free method, and more valid information is obtained in these cases, as demonstrated by examples.

Biological Availability↗
PubMed1994

Confidence intervals for weighted proportions.

We investigate methods for the construction of confidence intervals for a proportion in a stratified two-stage sampling design with few events occurring in a small number of large, unequal size strata. The critical aspect is the incorporation of the weighting scheme into the construction of a single overall confidence interval. With small numbers of events, the binomial based methods may be inadequate since the normal approximation is not valid. Computer simulations compare coverage probability and bias for five methods of obtaining confidence intervals for proportions by combining: (1) binomial variances; (2) confidence intervals based on the F-distribution approximation to the cumulative binomial; (3) the binomial variance method with exact confidence limits when a zero prevalence occurs in any stratum; (4) confidence intervals based on the F-distribution using a rescaling factor; and (5) the binomial variance method with exact confidence limits using a rescaling factor. The method that performs best in terms of coverage probability is the combination of stratum specific confidence intervals based on the F-distribution using a rescaling factor. The methods involving the binomial variance tend to be negatively biased and the methods based on the F-distribution tend to be positively biased. Application of these methods with data from a study of adolescent depression that employs a stratified two-stage sampling design is consistent with these results.

Adolescent↗
PubMed1989

Statistical significance versus clinical relevance. Part III. Methods for calculating confidence intervals.

Formulae for calculating confidence intervals for different types of studies and conditions are presented. These include means, proportions, difference in means and proportions for paired and unpaired data, more complicated analyses of variance, medians, quantiles, relative risks (odd ratios), standardised ratios and rates, intercepts, slopes and correlation coefficients in regression and correlation analyses, as well as survival proportions calculated in survival analysis. For less frequently used formulae, examples are included.

Clinical Trials as Topic↗
PubMed1994

Constructing confidence intervals for QTL location.

We describe a method for constructing the confidence interval of the QTL location parameter. This method is developed in the local asymptotic framework, leading to a linear model at each position of the putative QTL. The idea is to construct a likelihood ratio test, using statistics whose asymptotic distribution does not depend on the nuisance parameters and in particular on the effect of the QTL. We show theoretical properties of the confidence interval built with this test, and compare it with the classical confidence interval using simulations. We show in particular, that our confidence interval has the correct probability of containing the true map location of the QTL, for almost all QTLs, whereas the classical confidence interval can be very biased for QTLs having small effect.

Chromosome Mapping↗
PubMedDate not supplied

Confidence intervals for cost-effectiveness ratios: a comparison of four methods.

We evaluated four methods for computing confidence intervals for cost-effectiveness ratios developed from randomized controlled trials: the box method, the Taylor series method, the nonparametric bootstrap method and the Fieller theorem method. We performed a Monte Carlo experiment to compare these methods. We investigated the relative performance of each method and assessed whether or not it was affected by differing distributions of costs (normal and log normal) and effects (10% absolute difference in mortality resulting from mortality rates of 25% versus 15% in the two groups as well as from mortality rates of 55% versus 45%) or by differing levels of correlation between the costs and effects (correlations of -0.50, -0.25, 0.0, 0.25 and 0.50). The principal criterion used to evaluate the performance of the methods was the probability of miscoverage. Symmetrical miscoverage of the intervals was used as a secondary criterion for evaluating the four methods. Overall probabilities of miscoverage for the nonparametric bootstrap method and the Fieller theorem method were more accurate than those for the other the methods. The Taylor series method had confidence intervals that asymmetrically underestimated the upper limit of the interval. Confidence intervals for cost-effectiveness ratios resulting from the nonparametric bootstrap method and the Fieller theorem method were more dependably accurate than those estimated using the Taylor series or box methods. Routine reporting of these intervals will allow individuals using cost-effectiveness ratios to make clinical and policy judgments to better identify when an intervention is a good value for its cost.

Computer Simulation↗
PubMed1994

Mid-P confidence intervals for the Poisson expectation.

Conventionally a confidence interval (CI) for the standardized mortality ratio is set using the conservative CI for a Poisson expectation, mu. Employing the mid-P argument we present alternative CIs that are shorter than the conventional ones. The mid-P intervals do not guarantee the nominal confidence level, but the true coverage probability is only lower than the nominal level for a few short ranges of mu. The implications for mid-P confidence intervals of various proposed definitions of two-sided tests for discrete data are discussed.

Confidence Intervals↗
PubMed1992

Two-sample nonparametric estimation and confidence intervals under truncation.

We consider point estimates and confidence intervals for the difference in location or scale between two populations when the observations are subject to truncation. We suggest procedures analogous to those for the complete-sample case. A rigorous justification is presented to support the proposed confidence interval procedure. Finally, some simulations verify the properties of the estimators and confidence intervals. We illustrate the procedure using data on tumor size.

Algorithms↗
PubMedDate not supplied

Confidence intervals for the mean of sound exposure levels.

This text transposes standard statistical estimators of the mean and its confidence intervals to the field of occupational noise exposure, assuming that the samples are independent and lognormally distributed. The hypothesis of lognormality is applied to the values of A-weighted sound exposure as defined in ISO 1999 and expressed in Pa(2).h, which is equivalent to the currently accepted hypothesis of normality applied to the values of equivalent continuous A-weighted sound pressure level L(Aeq,T) expressed in dB(A). By expressing the initial data and results in dB(A), the text provides an unbiased estimator of the mean sound exposure level and tables of confidence intervals according to the sample size n and the standard deviation S(L) of the L(Aeq,T) measured values. The values of the following confidence intervals are given: exact one-sided upper and lower 95 and 97.5% confidence intervals and exact two-sided symmetrical 95% confidence interval. These results are compared with the approximate two-sided symmetrical 95% confidence interval proposed in standard, ISO 9612. This comparison demonstrates that the use of the approximate confidence interval can markedly underestimate the upper limit of the confidence interval when n is small and if S(L) is above 3 dB(A).

Humans↗
PubMed1989

Sample sizes for constructing confidence intervals and testing hypotheses.

Although estimation and confidence intervals have become popular alternatives to hypothesis testing and p-values, statisticians usually determine sample sizes for randomized clinical trials by controlling the power of a statistical test at an appropriate alternative, even those statisticians who recommend the use of confidence intervals for inference. There is merit in achieving consistency in the techniques for data analysis and sample size determination. To that end, this paper compares sample size determination with use of the length of the confidence interval with that obtained by control of power.

Angina Pectoris↗
PubMed2005

Computations of confidence intervals for estimates in the United States National Hospital Discharge Survey, 1979-2000.

INTRODUCTION: The National Hospital Discharge Survey is a primary data source for epidemiology research in the United States. To ensure that estimates are reliable, confidence intervals need to be calculated. The original survey data source is not available to the public, and the usual statistical methods are unsuitable for calculating confidence intervals. Instead, calculating confidence intervals requires using the statistical methods and relative standard errors that the U.S. National Center for Health Statistics has provided. However, the relative standard error parameters differ by hospital, patient category, and group. They also change yearly with sampling and are expressed differently before and during or after 1988. Consequently, manual computations of confidence intervals with multiple groups, diseases, and years are inefficient and prone to error. We developed a SAS program to compute confidence intervals for National Hospital Discharge Survey data from 1979 through 2000, newborns excluded. METHODS: We transposed 22 tables of relative standard error parameters (one for each year) into two new parameter tables that maintain the sampling designs before 1988 and during and after 1988 but are similar in overall structure. We unified all values to make each set of relative standard error parameters unique. We developed a program, COMPURSE, to search for relative standard error parameters for inputted estimates and to calculate confidence intervals. We set up an interface program for users to enter data, time period, confidence interval level, and output location; to read the relative standard error parameter tables; and to run the COMPURSE program. RESULTS: For different sets of National Hospital Discharge Survey data, COMPURSE efficiently and correctly retrieved relevant relative standard error parameters for estimates and accurately calculated relative standard errors, standard errors, and confidence intervals for annual estimates, multiple-year summaries, and average annual estimates. CONCLUSION: The program COMPURSE helps users analyze National Hospital Discharge Survey data efficiently.

Confidence Intervals↗
PubMed2004

Prediction of the confidence interval of quantitative trait Loci location.

In 1997, Darvasi and Soller presented empirical predictions of the confidence interval of quantitative trait loci (QTL) location for dense marker maps in experimental crosses. They showed from simulation results for backcross and F2 populations from inbred lines that the 95% confidence interval was a simple function of sample size and the effect of the QTL. In this study, we derive by theory simple equations that can be used to predict any confidence interval and show that for the 95% confidence interval, they are in good agreement with the empirical results given by Darvasi and Soller. A general form of the confidence interval is given that also applies to other population structures (e.g., collections of sib pairs). Furthermore, the expected shape of the likelihood-ratio-test around the true QTL location is derived, which is shown to be extremely leptokurtic. It is shown that this shape explains why confidence intervals from the Log of Odds (LOD) drop-off method and bootstrap results frequently differ for real data sets.

Animals↗
PubMed1997

Accurate confidence intervals for measures of test performance.

The use of confidence intervals to estimate population parameters is briefly reviewed. Exact binomial confidence intervals can be calculated through the use of tables or statistical software packages. As an alternative, a microcomputer program to calculate sensitivity and specificity, point estimates and binomial confidence intervals for false-negative and -positive rate, positive and negative predictive power, prevalence of cases and non-cases, correct classification rate, and misclassification rate has also been developed. Characteristics of the computer program, 'AccuCon', which is available from the authors, are described.

Confidence Intervals↗
PubMed2006

Confidence interval calculation methods are infrequently reported in emergency-medicine literature.

BACKGROUND: There are many different confidence interval calculation methods, each providing different as well as in some cases inadequate interval estimates. Readers who know which method is used are better able to understand potentially significant limitations in study reports. OBJECTIVES: To quantify how often confidence interval calculation methods are disclosed by authors in four peer-reviewed North American emergency-medicine journals. METHODS: The authors independently performed searches of four journals for all studies in which comparisons were made between means, medians, proportions, odds ratios, or relative risks. Case reports, editorials, subject reviews, and letters were excluded. Using a standardized abstraction form developed on a spreadsheet, the authors evaluated each article for the reporting of confidence intervals and evaluated the description of methodology used to calculate the confidence intervals. RESULTS: A total of 212 articles met the inclusion criteria. Confidence intervals were reported in 123 articles (58%; 95% CI = 51% to 64%); of these, a description of methodology was reported in 12 (9.8%; 95% CI = 5.7% to 16%). CONCLUSIONS: Confidence interval methods of calculation are disclosed infrequently in emergency medicine literature.

Confidence Intervals↗
PubMed2005

Confidence intervals in procedural dermatology: an intuitive approach to interpreting data.

BACKGROUND: Significant differences observed in therapeutic trials in procedural dermatology are typically denoted by p values of less than .05. Alternatively, significance can be conveyed by use of confidence intervals. OBJECTIVES: The purpose of this article is to clarify how confidence intervals convey the same information about outcomes as p values, albeit in a slightly different manner. METHODS: (1) Selective review of textbooks and other relevant literature and (2) presentation of a brief tutorial describing confidence interval determination for therapeutic clinical trials comparing differences between means of two groups. RESULTS: Routine use of confidence intervals is an intuitively satisfying means for conveying the statistical significance of results and can be used in combination with p values for understanding these results. Specifically, confidence intervals are a useful tool for indicating the size, spread, and direction of the observed differences. Unfortunately, dermatologic surgery trials tend to have low sample sizes, which frequently result in outcomes below the threshold of statistical significance (p > .05, or confidence intervals including 1.00). In the absence of statistical significance, neither p values nor confidence intervals yield definitive results. CONCLUSION: Confidence intervals can complement p values as a means for explaining statistical significant differences. When differences are not statistically significant but are clinically significant and approach statistical significance, neither p values nor confidence intervals can definitively establish whether the observed trends are indicative of an underlying difference. In these cases, common in procedural dermatology, larger, better designed, randomized prospective trials are needed.

Confidence Intervals↗
PubMed2002

Improved confidence intervals in quantitative trait loci mapping by permutation bootstrapping.

The nonparametric bootstrap approach is known to be suitable for calculating central confidence intervals for the locations of quantitative trait loci (QTL). However, the distribution of the bootstrap QTL position estimates along the chromosome is peaked at the positions of the markers and is not tailed equally. This results in conservativeness and large width of the confidence intervals. In this study three modified methods are proposed to calculate nonparametric bootstrap confidence intervals for QTL locations, which compute noncentral confidence intervals (uncorrected method I), correct for the impact of the markers (weighted method I), or both (weighted method II). Noncentral confidence intervals were computed with an analog of the highest posterior density method. The correction for the markers is based on the distribution of QTL estimates along the chromosome when the QTL is not linked with any marker, and it can be obtained with a permutation approach. In a simulation study the three methods were compared with the original bootstrap method. The results showed that it is useful, first, to compute noncentral confidence intervals and, second, to correct the bootstrap distribution of the QTL estimates for the impact of the markers. The weighted method II, combining these two properties, produced the shortest and less biased confidence intervals in a large number of simulated configurations.

Chromosome Mapping↗