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PubMed1986

Confidence intervals rather than P values: estimation rather than hypothesis testing.

Overemphasis on hypothesis testing--and the use of P values to dichotomise significant or non-significant results--has detracted from more useful approaches to interpreting study results, such as estimation and confidence intervals. In medical studies investigators are usually interested in determining the size of difference of a measured outcome between groups, rather than a simple indication of whether or not it is statistically significant. Confidence intervals present a range of values, on the basis of the sample data, in which the population value for such a difference may lie. Some methods of calculating confidence intervals for means and differences between means are given, with similar information for proportions. The paper also gives suggestions for graphical display. Confidence intervals, if appropriate to the type of study, should be used for major findings in both the main text of a paper and its abstract.

Adult↗
PubMed2006

Duffy-Santner confidence intervals for the two-stage three-outcome design.

Although the three-outcome phase II clinical trial design (Sargent et al., 2001) has seen increasing use, confidence interval methodology for the binary endpoint had not been explicitly defined. Typical phase II clinical trial designs with binomial endpoints use a sequential binomial confidence interval (SBCI) algorithm (Duffy and Santner, 1987). We reviewed the SBCI literature and software in order to describe the construction of confidence intervals for the three-outcome setting. Simulations confirmed that proper application of the SBCI algorithm provides appropriate coverage for the three-outcome design.

Computer Simulation↗
PubMed2007

Confidence intervals for the standardized effect arising in the comparison of two normal populations.

Confidence intervals for a standardized effect are derived after stabilizing the variance of the Welch t-statistic. Simulation studies demonstrate the viability of the resulting intervals for a wide range of parameter values and sample sizes as small as five. The methodology is extended to the combination of results from several studies, so as to obtain a confidence interval for a representative standardized effect for all the studies. The methods are illustrated on a recent meta-analytic study of systolic blood pressure reduction during a weight reducing regime, as well as the classical Mumford data on psychological intervention and hospital length of stay.

Analysis of Variance↗
PubMed2000

Multiple inferences using confidence intervals.

1. In a recent review article, the problem of making false-positive inferences as a result of making multiple comparisons between groups of experimental units or between experimental outcomes was addressed. 2. It was concluded that the most universally applicable solution was to use the Ryan-Holm step-down Bonferroni procedure to control the family-wise (experiment-wise) type 1 error rate. This procedure consists of adjusting the P values resulting from hypothesis testing. It allows for correlation among hypotheses and has been validated by Monte Carlo simulation. It is a simple procedure and can be performed by hand. 3. However, some investigators prefer to estimate effect sizes and make inferences by way of confidence intervals rather than, or in addition to, testing hypotheses by way of P values and it is the policy of some editors of biomedical journals to insist on this. It is not generally recognized that confidence intervals, like P values, must be adjusted if multiple inferences are made from confidence intervals in a single experiment. 4. In the present review, it is shown how confidence intervals can be adjusted for multiplicity by an extension of the Ryan-Holm step-down Bonferroni procedure. This can be done for differences between group means in the case of continuous variables and for odds ratios or relative risks in the case of categorical variables set out as 2 x 2 tables.

Confidence Intervals↗
PubMed1999

Exact 95% confidence intervals for differences in binomial proportions.

A method is described that allows exact 95% confidence intervals to be computed for differences in binomial proportions. The method itself is not iterative, but it uses the iteratively computed 95% confidence intervals for the individual binomial proportions. The method converges with the usual normal approximation method, as denominators become large. It correlates very closely with the Yates-corrected chi-square test. The method works regardless of the definition of success or failure in the proportions and regardless of the order of the proportions.

Algorithms↗
PubMed2005

An alternative to Cohen's standardized mean difference effect size: a robust parameter and confidence interval in the two independent groups case.

The authors argue that a robust version of Cohen's effect size constructed by replacing population means with 20% trimmed means and the population standard deviation with the square root of a 20% Winsorized variance is a better measure of population separation than is Cohen's effect size. The authors investigated coverage probability for confidence intervals for the new effect size measure. The confidence intervals were constructed by using the noncentral t distribution and the percentile bootstrap. Over the range of distributions and effect sizes investigated in the study, coverage probability was better for the percentile bootstrap confidence interval.

Analysis of Variance↗
PubMed2003

Confidence intervals for the binomial parameter: some new considerations.

Several methods have been proposed to construct confidence intervals for the binomial parameter. Some recent papers introduced the "mean coverage" criterion to evaluate the performance of confidence intervals and suggested that exact methods, because of their conservatism, are less useful than asymptotic ones. In these studies, however, exact intervals were always represented by the Clopper-Pearson interval (C-P). Now we focus on Sterne's interval, which is also exact and known to be better than the C-P in the two-sided case. Introducing a computer intensive level-adjustment procedure which allows constructing intervals that are exact in terms of mean coverage, we demonstrate that Sterne's interval performs better than the best asymptotic intervals, even in the mean coverage context. Level adjustment improves the C-P as well, which, with an appropriate level adjustment, becomes equivalent to the mid-P interval. Finally we show that the asymptotic behaviour of the mid-P method is far poorer than is generally expected.

Binomial Distribution↗
PubMed2006

Confidence intervals for an exposure adjusted incidence rate difference with applications to clinical trials.

To summarize safety data such as clinical adverse experiences in clinical trials with a moderate to long-term follow-up, we may use a measurement which accounts for the potential differences in the follow-up duration between treatment groups. The incidence rate, which uses the total person-time follow-up in a treatment group as the denominator, is one of these measures. When treatment comparisons are based on the difference of the incidence rates, it is of interest to construct confidence intervals for the rate differences. In this paper, we first discuss the assumptions and scenarios in which the exposure adjusted incidence rate may be appropriate. Then we review several methods of calculating confidence intervals for the difference of two incidence rates assuming that the number of events come from a Poisson distribution. The methods considered include Wald's method, the two-by-two table method, the Miettinen and Nurminen (MN) method, and the conditional MN method. For all but the MN method, explicit confidence intervals can be obtained. For the MN method, some numerical iterations are required. The properties of these methods are evaluated by simulations. The results show that the MN method outperforms the other three in terms of the coverage of the confidence interval, especially when the rates and therefore the number of events are small. Lastly, data from a clinical study are used to demonstrate the application of the methods.

Aged↗
PubMed2001

The psychometric function: II. Bootstrap-based confidence intervals and sampling.

The psychometric function relates an observer's performance to an independent variable, usually a physical quantity of an experimental stimulus. Even if a model is successfully fit to the data and its goodness of fit is acceptable, experimenters require an estimate of the variability of the parameters to assess whether differences across conditions are significant. Accurate estimates of variability are difficult to obtain, however, given the typically small size of psychophysical data sets: Traditional statistical techniques are only asymptotically correct and can be shown to be unreliable in some common situations. Here and in our companion paper (Wichmann & Hill, 2001), we suggest alternative statistical techniques based on Monte Carlo resampling methods. The present paper's principal topic is the estimation of the variability of fitted parameters and derived quantities, such as thresholds and slopes. First, we outline the basic bootstrap procedure and argue in favor of the parametric, as opposed to the nonparametric, bootstrap. Second, we describe how the bootstrap bridging assumption, on which the validity of the procedure depends, can be tested. Third, we show how one's choice of sampling scheme (the placement of sample points on the stimulus axis) strongly affects the reliability of bootstrap confidence intervals, and we make recommendations on how to sample the psychometric function efficiently. Fourth, we show that, under certain circumstances, the (arbitrary) choice of the distribution function can exert an unwanted influence on the size of the bootstrap confidence intervals obtained, and we make recommendations on how to avoid this influence. Finally, we introduce improved confidence intervals (bias corrected and accelerated) that improve on the parametric and percentile-based bootstrap confidence intervals previously used. Software implementing our methods is available.

Confidence Intervals↗
PubMed2005

Problems with existing procedures to calculate exact unconditional P-values for non-inferiority/superiority and confidence intervals for two binomials and how to resolve them.

Recently several papers have been published that deal with the construction of exact unconditional tests for non-inferiority and confidence intervals based on the approximative unconditional restricted maximum likelihood test for two binomial random variables. Soon after the papers have been published the commercially available software for exact tests StatXact has incorporated the new methods. There are however gaps in the proofs which since have not been resolved adequately. Further it turned out that the methods for testing non-inferiority are not coherent and test for non-inferiority can easily come to different conclusions compared to the confidence interval inclusion rule. In this paper, a proposal is made how to resolve the open problems. Berger and Boos (1994) developed the confidence interval method for testing equality of two proportions. StatXact (Version 5) has extended this method for shifted hypotheses. It is shown that at least for unbalanced designs (i.e. largely different sample sizes) the Berger and Boos method can lead to controversial results.

Biopharmaceutics↗
PubMed2006

Multiple comparisons and multiple contrasts in randomized dose-response trials--confidence interval oriented approaches.

According to the ICH E9 recommendation, the evaluation of randomized dose-finding trials focuses on the graphical presentation of different kinds of simultaneous confidence intervals: i) superiority of at least one dose vs. placebo with and without the assumption of order restriction, ii) noninferiority of at least one dose vs. active control, iii) identification of the minimum effective dose, iv) identification of the peak dose, v) identification of the maximum safe dose for a safety endpoint, and vi) estimation of simultaneous confidence intervals for "many-to-one-by-condition interaction contrasts." Moreover, global tests for a monotone trend or a trend with a possible downturn effect are discussed. The basic approach involved obtaining multiple contrasts for different problem-related contrast definitions. For all approaches, definitions of relevance margins for superiority or noninferiority are needed. Because consensus on margins only exists for selected therapeutic areas and the definition of absolute thresholds may be difficult, simultaneous confidence intervals for ratio to placebo were also used. All approaches are demonstrated in an example-based manner using the R-packages multcomp (difference), for hypotheses based on difference, and mratios (ratio), for hypotheses based on ratios.

Algorithms↗
PubMed1999

Sample size determination for multiple comparison studies treating confidence interval width as random.

Methods for optimal sample size determination are developed using four popular multiple comparison procedures (Scheffe's, Bonferroni's, Tukey's and Dunnett's procedures), where random samples of the same size n are to be selected from k (>/=2) normal populations with common variance sigma2, and where primary interest concerns inferences about a family of L linear contrasts among the k population means. For a simultaneous coverage probability of (1-alpha), the optimal sample size is defined to be the smallest integer value n*m such that, simultaneously for all L confidence intervals, the width of the lth confidence interval will be no greater than tolerance 2deltal (l=1,2,...,L) with tolerance probability at least (1-gamma), treating the pooled sample variance S2p as a random variable. Using Scheffe's procedure as an illustration, comparisons are made to usual sample size methods that incorrectly ignore the stochastic nature of S2p. The latter approach can lead to serious underestimation of required sample sizes and hence to unacceptably low values of the actually tolerance probability (1-gamma'). Our approach guarantees a lower bound of [1-(alpha+gamma)] for the probability that the L confidence intervals will both cover the parametric functions of interest and also be sufficiently narrow. Recommendations are provided regarding the choices among the four multiple comparison procedures for sample size determination and inference-making.

Confidence Intervals↗
PubMed2001

Bootstrap-type confidence intervals for quantiles of the survival distribution.

In this paper we outline and illustrate an easy to program method for analytically calculating both parametric and non-parametric bootstrap-type confidence intervals for quantiles of the survival distribution based on right censored data. This new approach allows for the incorporation of covariates within the framework of parametric models. The procedure is based upon the notion of fractional order statistics and is carried forth using a simple beta transformation of the estimated survival function (parametric or non-parametric). It is the only direct method currently available in the sense that all other methods are based on inverting test statistics or employing confidence intervals for other survival quantities. We illustrate that the new method has favourable coverage probabilities for median confidence intervals as compared to six other competing methods.

Bile Duct Neoplasms↗
PubMed2005

Binomial distribution sample confidence intervals estimation for positive and negative likelihood ratio medical key parameters.

Likelihood Ratio medical key parameters calculated on categorical results from diagnostic tests are usually express accompanied with their confidence intervals, computed using the normal distribution approximation of binomial distribution. The approximation creates known anomalies,especially for limit cases. In order to improve the quality of estimation, four new methods (called here RPAC, RPAC0, RPAC1, and RPAC2) were developed and compared with the classical method (called here RPWald), using an exact probability calculation algorithm.Computer implementations of the methods use the PHP language. We defined and implemented the functions of the four new methods and the five criterions of confidence interval assessment. The experiments run for samples sizes which vary in 14 - 34 range, 90 - 100 range (0 < X < m, 0< Y < n), as well as for random numbers for samples sizes (4m, n </= 1000) and binomial variables (1 </= X, Y < m, n). The experiment run shows that the new proposed RPAC2 method obtains the best overall performance of computing confidence interval for positive and negative likelihood ratios.

Algorithms↗
PubMed1987

A note on confidence intervals with extended least squares parameter estimates.

It has previously been shown that the extended least squares (ELS) method for fitting pharmacokinetic models behaves better than other methods when there is possible heteroscedasticity (unequal error variance) in the data. Confidence intervals for pharmacokinetic parameters, at the target confidence level of 95%, computed in simulations with several pharmacokinetic and error variance models, using a theoretically reasonable approximation to the asymptotic covariance matrix of the ELS parameter estimator, are found to include the true parameter values considerably less than 95% of the time. Intervals with the ordinary least squares method perform better. Two adjustments to the ELS confidence intervals, taken together, result in better performance. These are: (i) apply a bias correction to the ELS estimate of variance, which results in wider confidence intervals, and (ii) use confidence intervals with a target level of 99% to obtain confidence intervals with actual level closer to 95%. Kineticists wishing to use the ELS method may wish to use these adjustments.

Kinetics↗
PubMed1995

Simple confidence intervals for standardized rates based on the approximate bootstrap method.

This paper presents simple expressions for the confidence intervals of indirect and direct standardized rates, based on the approximate bootstrap confidence (ABC) method of DiCiccio and Efron. For indirect rates, the ABC method compares favourably with the exact methods. An exact method is not available for direct standardized rates. I, therefore, compared the ABC to other simple procedures for confidence intervals, namely: the standard normal, log-normal, and a recent method proposed by Dobson et al. Simulation studies of the coverage properties of these methods for direct standardized rates show that the ABC method performs best, with balanced tail probabilities close to their expected values.

Adolescent↗
PubMed2006

Generalized P-values and confidence intervals: a novel approach for analyzing lognormally distributed exposure data.

The problem of assessing occupational exposure using the mean of a lognormal distribution is addressed. The novel concepts of generalized p-values and generalized confidence intervals are applied for testing hypotheses and computing confidence intervals for a lognormal mean. The proposed methods perform well, they are applicable to small sample sizes, and they are easy to implement. Power studies and sample size calculation are also discussed. Computational details and a source for the computer program are given. The procedures are also extended to compare two lognormal means and to make inference about a lognormal variance. In fact, our approach based on generalized p-values and generalized confidence intervals is easily adapted to deal with any parametric function involving one or two lognormal distributions. Several examples involving industrial exposure data are used to illustrate the methods. An added advantage of the generalized variables approach is the ease of computation and implementation. In fact, the procedures can be easily coded in a programming language for implementation. Furthermore, extensive numerical computations by the authors show that the results based on the generalized p-value approach are essentially equivalent to those based on the Land's method. We want to draw the attention of the industrial hygiene community to this accurate and unified methodology to deal with any parameter associated with the lognormal distribution.

Air Pollutants, Occupational↗
PubMed1997

[Ratio of proportions and the use of the delta method for confidence intervals in logistic regression].

OBJECTIVE: To present an application of logistic regression modelling to estimate ratios of proportions, such as prevalence ratio or relative risk, and the Delta Method to estimate confidence intervals. METHOD: The Delta Method was used because it is appropriate for the estimation of variance of non-linear functions of random variables. The method is based on Taylor's series expansion and provides a good approximation of variance estimates. A computer program, utilizing the matrix module of SAS, was developed to compute the variance estimates. A practical demonstration is presented with data from a cross-sectional study carried out on a sample of 611 women, to test the hypothesis that the lack of housework sharing is associated with high scores of psychological symptoms as measured by a validated questionnaire. RESULTS: Crude and adjusted prevalence ratio estimated by logistic regression were similar to those estimated by tabular analysis. Also, ranges of the confidence intervals of the prevalence ratio according to the Delta Method were nearly equal to those obtained by the Mantel-Haenszel approach. CONCLUSIONS: The results give support to the use of the Delta Method for the estimation of confidence intervals for ratios of proportions. The method should be seen as an alternative for situations in which the need to control a large number of potential confounders limits the use of stratified analysis.

Confidence Intervals↗