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PubMed2006

Confidence intervals for survival quantiles in the Cox regression model.

Median survival times and their associated confidence intervals are often used to summarize the survival outcome of a group of patients in clinical trials with failure-time endpoints. Although there is an extensive literature on this topic for the case in which the patients come from a homogeneous population, few papers have dealt with the case in which covariates are present as in the proportional hazards model. In this paper we propose a new approach to this problem and demonstrate its advantages over existing methods, not only for the proportional hazards model but also for the widely studied cases where covariates are absent and where there is no censoring. As an illustration, we apply it to the Stanford Heart Transplant data. Asymptotic theory and simulation studies show that the proposed method indeed yields confidence intervals and bands with accurate coverage errors.

Biometry↗
PubMed1998

The calculation of a confidence interval on the absolute estimated benefit for an individual patient.

Physicians need a method to predict the individualized absolute therapeutic benefit before deciding which therapy to prescribe to a given patient and the confidence intervals around this estimate. We have derived a method to predict the absolute individual therapeutic benefit in a previous work. In this paper, we present a Monte Carlo simulation to estimate the bias of prediction for an individual with certain characteristics and use a bootstrap method to compute its confidence intervals. Because the bootstrap approach does not depend upon the parametric assumption for the distribution of the prediction, it can be applied to situations where the parametric distribution is unknown. Over 35,000 cases of subjects at risk of cardiovascular events were available for analysis. Our results show the 95% confidence intervals for each individual. In a clinical setting, the use of this approach makes it possible to predict the absolute therapeutic benefit for each patient (the quantity of individual benefit) with sufficient precision.

Cardiovascular Diseases↗
PubMed2004

Simple table for estimating confidence interval of discrepancy frequencies in microbiological safety evaluation.

We provide a simple tool to determine discrepancies confidence interval (CI) in microbiology validation studies such as technical accuracy of a qualitative test result. This tool enables to determine exact confidence interval (binomial CI) from an observed frequency when normal approximation is inadequate, that is, in case of rare events. This tool has daily applications in microbiology and we are presenting an example of its application to antimicrobial susceptibility systems evaluation.

Binomial Distribution↗
PubMed1996

Exact confidence intervals for the common mean of several normal populations.

The problem of interval estimation of the common mean of several normal populations when the variances are unknown and unequal is considered. Some new confidence intervals are proposed. One of them is centered at the well-known Graybill-Deal estimator of the common mean. These new intervals and an existing exact confidence interval are numerically compared with respect to their expected lengths. Based on comparison studies, some recommendations are made regarding the choice of the intervals to be used in applications.

Animals↗
PubMed1983

Evaluation and propagation of confidence intervals in nonlinear, asymmetrical variance spaces. Analysis of ligand-binding data.

Two problems that are often overlooked in studies employing nonlinear least-squares techniques for parameter estimation are confidence-interval estimation and propagation. When the parameters are correlated, the variance space and consequently the confidence intervals are nonlinear and asymmetrical. The presented mathematical method for the evaluation of confidence intervals and error propagation addresses these problems. The examples employed to demonstrate these methods include linear least-squares and the nonlinear least-squares analysis of ligand-binding problems, such as hormone receptor interactions and oxygen binding to human hemoglobin. The mathematical procedures have proven very useful for analyzing the molecular mechanism of cooperativity in human hemoglobin (Johnson, M. L., and G. K. Ackers, 1982. Biochemistry 21:201-211).

Computers↗
PubMed2002

Comparison of populations as a function of confidence intervals of gene probability.

A new method for population comparison has been developed in order to determine the most discriminating systems in relation to diverse populations. The method compares pairs of populations on the basis of confidence intervals for allelic frequencies and does not require any previous assumption, such as normality of the population. A data set of 12 populations studied for Polymarker Kit (Roche) genetic systems, 13 biallelic systems (LDLR, GYPA, D7S8) and two systems with three alleles each (HBGG, GC), was considered. The analysis produced a total of 144 confidence intervals (12 populations x 12 alleles). The raw data were then analyzed comparing pairs of populations, allele by allele, through their respective confidence intervals. The method calculates the matrix of similarity between pairs of populations for each allele and eventually for each genetic system. Finally, the 12 populations were classified depending on the number of nondiscriminating alleles. The results of this method are compared to those produced by such usual techniques as Principal Component Analysis.

Alleles↗
PubMed1998

Regression models: calculating the confidence interval of effects in the presence of interactions.

The main goal of regression analysis (multiple, logistic, Cox) is to assess the relationship of one or more exposure variables to a response variable, in the presence of confounding and interaction. The confidence interval for the regression coefficient of the exposure variable, obtained through the use of a computer statistical package, quantify these relationships for models without interaction. Relationships between variables that present interactions are represented by two or more terms, and the corresponding confidence intervals can be calculated 'manually' from the covariance matrix. This paper suggests an easy procedure for obtaining confidence intervals from any statistical package. This procedure is applicable for modifying variables which are continuous as well as categorical.

Adult↗
PubMed1992

A nonstatistical approach to estimating confidence intervals about model parameters: application to respiratory mechanics.

Estimates of parameters obtained by fitting models to physiologic data are of little use unless accompanied by confidence intervals. The standard methods for estimating confidence intervals are statistical, and make the assumption that the fitted model accounts for all the deterministic variation in the data while the residuals between the fitted model and the data reflect only stochastic noise. In practice, this is frequently not the case, as one often finds the residuals to be systematically distributed about zero. In this paper, we develop an approach for assessing confidence in a parameter estimate when the order of the model is clearly less than that of the system being modeled. Our approach does not require a parameter to have a single value located within a region of confidence. Instead, we let the parameter value vary over the data set in such a way as to provide a good fit to the entire data set. We apply our approach to the estimation of the resistance of the respiratory system in which a simple model is fitted to measurements of tracheal pressure and flow by recursive multiple linear regression. The values of resistance required to achieve a good fit are represented as a modified histogram in which the contribution of a particular resistance value to the histogram is weighted by the amount of information used in its determination. Our approach provides parameter frequency distribution functions that convey the degree of confidence one may have in the parameter, while not being based on erroneous statistical assumptions.

Airway Resistance↗
PubMed1991

Confidence intervals for seroprevalence determined from pooled sera.

In population surveys of seroprevalence, it may not be most efficient to test all individual samples. The laboratory and statistical issues encountered when one first pools individual samples into groups of samples before laboratory analyses were discussed recently in relation to the seroprevalence of human immunodeficiency virus. In particular, point and confidence interval estimates for seroprevalence from pooled sera were derived. A potential problem with these confidence intervals is that they may contain negative values. This problem is most likely to occur in low-prevalence populations where pooling is most efficient. An alternative method of obtaining confidence intervals that cannot contain negative values is proposed and an example provided.

Confidence Intervals↗
PubMed1990

Use and abuse of imprecision profiles: some pitfalls illustrated by computing and plotting confidence intervals.

Published estimates of the imprecision characteristics of immunoassays are often based on quantities of data that seem to be inadequate. The increasing use of imprecision profiles has not necessarily improved the situation. We describe and illustrate a method of computing 95% confidence intervals for profiles estimated directly from replicated assay results. The data used were chosen to mimic the sort of data that are typically available in practice, either within laboratories or from external quality-assessment programs. Use of confidence intervals is an effective pictorial way of incorporating information about the quantity and the distribution of the data used for estimation. This is in keeping with an important property of imprecision profile plots: potentially complex information is summarized in a readily comprehensible way. A computer program is available for estimating and plotting profiles and their 95% confidence intervals.

Computer Simulation↗
PubMed1997

Approximate confidence intervals for heritability from method R estimates.

Method R estimates of heritability (h2) and associated confidence intervals (CI) were obtained from simulated data using a single trait, direct effects, full animal model, with 50% subsampling. Five hundred data sets were simulated for each of five levels of h2 (.10, .20, .30, .40, and .50) and two types of pedigree structure (random pedigree structure [N = 2,000] that varied over simulations, or the pedigree structure from a real data set [N = 2,644] that was constant for all simulations). The first 10, 20, and all 50 h2 estimates were used to obtain 80, 90, 95, and 99% CI for each data set. The variance of h2 estimates within data sets approximated the sampling variance of the h2 estimates. The Box-Cox transformation was used to normalize the distribution of estimates from each data set. Confidence intervals were computed on the transformed scale as CI = mu +/- (T x sigma), where mu and sigma = the mean and SD of the N transformed h2 estimates, respectively, and T = the critical value from the T distribution for a 1-alpha CI, with df = N-1. Upper and lower CI bounds were converted back to the original scale by reversing the transformation. The percentages of CI containing the true h2 value, pooled across all levels of h2, types of pedigree, and number of estimates used to obtain CI, for 80, 90, 95, and 99% CI were 81.14, 90.96, 95.27, and 98.76%, respectively. These results suggested that Method R h2 estimates can be used to obtain reliable CI.

Analysis of Variance↗
PubMed1988

Confidence interval estimation of a rate and the choice of sample size.

The problem of estimating a rate or proportion is considered. Four methods for constructing an approximate confidence interval are discussed and compared via a simulation study. The most accurate method is found. Also, for each method a sharp upper bound (dependent only on the sample size) is given for the length of the confidence interval. By choosing an appropriate sample size this bound enables the practitioner to achieve a prespecified maximum length for the confidence interval without knowing the population rate. The striking result is that the most accurate method has the smallest bound, thus requiring the least sample units.

Computer Simulation↗
PubMed1995

Euclidean distance matrix analysis: confidence intervals for form and growth differences.

Analysis of biological forms using landmark data has received substantial attention recently. Much of the statistical work in this area has concentrated on the estimation of average form, average form difference, and average growth difference. From the statistical, as well as the scientific point of view, it is important that any estimate of a scientifically relevant quantity be accompanied by a statement regarding its accuracy. Such a statement is contained in a confidence interval. The purpose of this paper is to provide a method to obtain confidence intervals for form difference and growth difference estimators. The estimators are based on Euclidean distance matrix analysis. The confidence intervals are calculated using the model independent bootstrap method. We illustrate the method by using three examples: morphological differences between samples of craniofacial patients and normal controls using two dimensional data from head X-rays, sexual dimorphism of craniofacial morphology in Cebus apella, and sexual dimorphism of facial growth in Cebus apella using three-dimensional data.

Animals↗
PubMed1999

Bootstrapping: estimating confidence intervals for cost-effectiveness ratios.

Economic evaluations are increasingly being conducted alongside clinical trials of health interventions, with resource consequences being estimated from stochastic data. It is, therefore, important that economic evaluation results, like the clinical results, reflect the underlying variance within the sample data. A statistical methodology, known as bootstrapping, has recently been put forward as a potential method for calculating confidence intervals for cost-effectiveness ratios, yet it is still unusual to see economic evaluations reporting confidence intervals. In this paper we demonstrate the practical application of bootstrapping using real data from clinical trials, and conclude that bootstrapping is easily transferable from theory to practice for the estimation of confidence intervals for cost-effectiveness ratios. We encourage further investigation into its applicability and use.

Confidence Intervals↗
PubMed1998

Estimation of confidence intervals for area under the curve from destructively obtained pharmacokinetic data.

The area under the curve (AUC) of the concentration-time curve for a drug or metabolite, and the variation associated with the AUC, are primary results of most pharmacokinetic (PK) studies. In nonclinical PK studies, it is often the case that experimental units contribute data for only a single time point. In such cases, it is straightforward to apply noncompartmental methods to determine an estimate of the AUC. In this report, we investigate noncompartmental estimation of the AUC using the long-trapezoidal rule during the elimination phase of the concentration-time profile, and we account for the underlying distribution of data at each sampling time. For data that follow a normal distribution, the log-trapezoidal rule is applied to arithmetic means at each time point of the elimination phase of the concentration-time profile. For data that follow a lognormal distribution, as is common with PK data, the log-trapezoidal rule is applied to geometric means at each time point during elimination. Since the log-trapezoidal rule incorporates nonlinear combinations of mean concentrations at each sampling time, obtaining an estimate of the corresponding variation about the AUC is not straightforward. Estimation of this variance is further complicated by the occurrence of lognormal data. First-order approximations to the variance of AUC estimates are derived under the assumptions of normality, and lognormality, of concentrations at each sampling time. AUC estimates and variance approximations are utilized to form confidence intervals. Accuracies of confidence intervals are tested using simulation studies.

Algorithms↗
PubMed2005

On confidence intervals for genotype relative risks and attributable risks from case parent trio designs for candidate-gene studies.

Scherag et al. [Hum Hered 2002;54:210-217] recently proposed point estimates and asymptotic as well as exact confidence intervals for genotype relative risks (GRRs) and the attributable risk (AR) in case parent trio designs using single nucleotide polymorphism (SNP) data. The aim of this study was the investigation of coverage probabilities and bias in estimates if the marker locus is not identical to the disease locus. Using a variety of parameter constellations, including marker allele frequencies identical to and different from the SNP at the disease locus, we performed an analytical study to quantify the bias and a Monte-Carlo simulation study for quantifying both bias and coverage probabilities. No bias was observed if marker and trait locus coincided. Two parameters had a strong impact on coverage probabilities of confidence intervals and bias in point estimates if they did not coincide: the linkage disequilibrium (LD) parameter delta and the allele frequency at the marker SNP. If marker allele frequencies were different from the allele frequencies at the functional SNP, substantial biases occurred. Further, if delta between the marker and the disease locus was lower than the maximum possible delta, estimates were also biased. In general, biases were towards the null hypothesis for both GRRs and AR. If one GRR was not increased, as e.g. in a recessive genetic model, biases away from the null could be observed. If both GRRs were in identical directions and if both were substantially larger than 1, the bias always was towards the null. When applying point estimates and confidence intervals for GRRs and AR in candidate gene studies, great care is needed. Effect estimates are substantially biased towards the null if either the allele frequencies at the marker SNP and the true disease locus are different or if the LD between the marker SNP and the disease locus is not at its maximum. A bias away from the null occurs only in uncommon study situations; it is small and can therefore be ignored for applications.

Confidence Intervals↗
PubMed2005

Small proportions: what to report for confidence intervals?

PURPOSE: It is generally agreed that a confidence interval (CI) is usually more informative than a point estimate or p-value, but we rarely encounter small proportions with CI in the pharmaco-epidemiological literature. When a CI is given it is sporadically reported, how it was calculated. This incorrectly suggests one single method to calculate CIs. To identify the method best suited for small proportions, seven approximate methods and the Clopper-Pearson Exact method to calculate CIs were compared. METHODS: In a simulation study for 90-, 95- and 99%CIs, with sample size 1000 and proportions ranging from 0.001 to 0.01, were evaluated systematically. Main quality criteria were coverage and interval width. The methods are illustrated using data from pharmaco-epidemiology studies. RESULTS: Simulations showed that standard Wald methods have insufficient coverage probability regardless of how the desired coverage is perceived. Overall, the Exact method and the Score method with continuity correction (CC) performed best. Real life examples showed the methods to yield different results too. CONCLUSIONS: For CIs for small proportions (pi < or = 0.01), the use of the Exact method and the Score method with CC are advocated based on this study.

Confidence Intervals↗