Conditional Inference Following Group Sequential Testing

The work of Fisher (1959) and Buehler (1959) discuss the importance of conditioning on recognizable subsets of the sample space. The stopping time yields an easily identifiable partition of the sample space when considering group sequential testing. We first present confidence intervals that are correct when conditioning on the subset of data for which a trial stopped at a particular analysis. These intervals have very desirable properties for observations that are highly unusual (given any value of the mean). In addition, they provide insight into how information about the mean is distributed between the two sufficient statistics. We then use conditional coverage probabilities to compare the sample mean, stagewise, and repeated confidence intervals. We find that none of these intervals outperforms the others when conditioning on stopping time, and no interval is a uniformly acceptable performer.     

Capture-recapture when time and behavioral response affect capture probabilities

We consider a capture-recapture model in which capture probabilities vary with time and with behavioral response. Two inference procedures are developed under the assumption that recapture probabilities bear a constant relationship to initial capture probabilities. These two procedures are the maximum likelihood method (both unconditional and conditional types are discussed) and an approach based on optimal estimating functions. The population size estimators derived from the two procedures are shown to be asymptotically equivalent when population size is large enough. The performance and relative merits of various population size estimators for finite cases are discussed. The bootstrap method is suggested for constructing a variance estimator and confidence interval. An example of the deer mouse analyzed in Otis et al. (1978, Wildlife Monographs 62, 93) is given for illustration.     

Diminished Confidence Levels in the Interval Estimation of an Unknown Regressor in Model I of Linear Regression

For the model y=β₀ + β₁ x + e (model I of linear regression) in the literature confidence estimators of an unknown position x₀ are given at which either the expectation of y is given (see FIELLER, 1944; FINNEY, 1952), or realizations of y are given (see GRAYBILL, 1961). These confidence regions with level 1—α need not be intervals. The occurrence of interval shape is a random event. Its probability is equal to the power of the t test for the examination of the hypothesis H: β₁ = 0. The papers mentioned above claim to provide confidence intervals with level 1 ? α. But because of the restriction of (1 —α)—confidence regions to intervals the true confidence probability is the conditional probability W_c: W_c = P (the confidence region covers x₀| the region has interval shape). Here this conditional probability is shown to be less than 1 —α. Evidence on the possible deviations from 1 —α has been obtained by simulations.     

A comparison of reflected versus test-based confidence intervals for the median survival time, based on censored data

The small-sample performance of some recently proposed nonparametric methods of constructing confidence intervals for the median survival time, based on randomly right-censored data, is compared with that of two new methods. Most of these methods are equivalent for large samples. All proposed intervals are either 'test-based' or 'reflected' intervals, in the sense defined in the paper. Coverage probabilities for the interval estimates were obtained by exact calculation for uncensored data, and by stimulation for three life distributions and four censoring patterns. In the range of situations studied, 'test-based' methods often have less than nominal coverage, while the coverage of the new 'reflected' confidence intervals is closer to nominal (although somewhat conservative), and these intervals are easy to compute.     

Empirical Bayes interval estimates that are conditionally equal to unadjusted confidence intervals or to default prior credibility intervals

Problems involving thousands of null hypotheses have been addressed by estimating the local false discovery rate (LFDR). A previous LFDR approach to reporting point and interval estimates of an effect-size parameter uses an estimate of the prior distribution of the parameter conditional on the alternative hypothesis. That estimated prior is often unreliable, and yet strongly influences the posterior intervals and point estimates, causing the posterior intervals to differ from fixed-parameter confidence intervals, even for arbitrarily small estimates of the LFDR. That influence of the estimated prior manifests the failure of the conditional posterior intervals, given the truth of the alternative hypothesis, to match the confidence intervals. Those problems are overcome by changing the posterior distribution conditional on the alternative hypothesis from a Bayesian posterior to a confidence posterior. Unlike the Bayesian posterior, the confidence posterior equates the posterior probability that the parameter lies in a fixed interval with the coverage rate of the coinciding confidence interval. The resulting confidence-Bayes hybrid posterior supplies interval and point estimates that shrink toward the null hypothesis value. The confidence intervals tend to be much shorter than their fixed-parameter counterparts, as illustrated with gene expression data. Simulations nonetheless confirm that the shrunken confidence intervals cover the parameter more frequently than stated. Generally applicable sufficient conditions for correct coverage are given. In addition to having those frequentist properties, the hybrid posterior can also be motivated from an objective Bayesian perspective by requiring coherence with some default prior conditional on the alternative hypothesis. That requirement generates a new class of approximate posteriors that supplement Bayes factors modified for improper priors and that dampen the influence of proper priors on the credibility intervals. While that class of posteriors intersects the class of confidence-Bayes posteriors, neither class is a subset of the other. In short, two first principles generate both classes of posteriors: a coherence principle and a relevance principle. The coherence principle requires that all effect size estimates comply with the same probability distribution. The relevance principle means effect size estimates given the truth of an alternative hypothesis cannot depend on whether that truth was known prior to observing the data or whether it was learned from the data.     

Modeling stream fish distributions using interval‐censored detection times

Controlling for imperfect detection is important for developing species distribution models (SDMs). Occupancy‐detection models based on the time needed to detect a species can be used to address this problem, but this is hindered when times to detection are not known precisely. Here, we extend the time‐to‐detection model to deal with detections recorded in time intervals and illustrate the method using a case study on stream fish distribution modeling. We collected electrofishing samples of six fish species across a Mediterranean watershed in Northeast Portugal. Based on a Bayesian hierarchical framework, we modeled the probability of water presence in stream channels, and the probability of species occupancy conditional on water presence, in relation to environmental and spatial variables. We also modeled time‐to‐first detection conditional on occupancy in relation to local factors, using modified interval‐censored exponential survival models. Posterior distributions of occupancy probabilities derived from the models were used to produce species distribution maps. Simulations indicated that the modified time‐to‐detection model provided unbiased parameter estimates despite interval‐censoring. There was a tendency for spatial variation in detection rates to be primarily influenced by depth and, to a lesser extent, stream width. Species occupancies were consistently affected by stream order, elevation, and annual precipitation. Bayesian P‐values and AUCs indicated that all models had adequate fit and high discrimination ability, respectively. Mapping of predicted occupancy probabilities showed widespread distribution by most species, but uncertainty was generally higher in tributaries and upper reaches. The interval‐censored time‐to‐detection model provides a practical solution to model occupancy‐detection when detections are recorded in time intervals. This modeling framework is useful for developing SDMs while controlling for variation in detection rates, as it uses simple data that can be readily collected by field ecologists.     

Confidence Intervals for Microbial Density Using Serial Dilutions with MPN Estimates

An alteration to Woodward's methods is recommended for deriving a 1 — α confidence interval for microbial density using serial dilutions with most-probable-number (MPN) estimates. Outcomes of the serial dilution test are ordered by their MPNs. A lower limit for the confidence interval corresponding to an outcome y is the density for which y and all higher ordered outcomes have total probability α/2. An upper limit is derived in the analogous way. An alteration increases the lowest lower limits and decreases the highest upper limits. For comparison, a method that is optimal in the sense of null hypothesis rejection is described. This method ranks outcomes dependent upon the microbial density in question, using proportional first derivatives of the probabilities. These and currently used methods are compared. The recommended method is shown to be more desirable in certain respects, although resulting in slightly wider confidence intervals than De Man's (1983) method.     

Confidence Intervals in Qtl Mapping by Bootstrapping

The determination of empirical confidence intervals for the location of quantitative trait loci (QTLs) was investigated using simulation. Empirical confidence intervals were calculated using a bootstrap resampling method for a backcross population derived from inbred lines. Sample sizes were either 200 or 500 individuals, and the QTL explained 1, 5, or 10% of the phenotypic variance. The method worked well in that the proportion of empirical confidence intervals that contained the simulated QTL was close to expectation. In general, the confidence intervals were slightly conservatively biased. Correlations between the test statistic and the width of the confidence interval were strongly negative, so that the stronger the evidence for a QTL segregating, the smaller the empirical confidence interval for its location. The size of the average confidence interval depended heavily on the population size and the effect of the QTL. Marker spacing had only a small effect on the average empirical confidence interval. The LOD drop-off method to calculate empirical support intervals gave confidence intervals that generally were too small, in particular if confidence intervals were calculated only for samples above a certain significance threshold. The bootstrap method is easy to implement and is useful in the analysis of experimental data.     

Limitations of a two-step moment method for mapping linked multiple QTL

As an alternative to multiple-interval mapping a two-step moment method was recently proposed to map linked multiple quantitative trait loci (QTLs). The advantage of this moment method was supposed to be its simplicity and computational efficiency, especially in detecting closely linked QTLs within a marker bracket, but also in mapping QTLs in different marker intervals. Using simulations it is shown that the two-step moment method may give poor results compared with multiple-interval mapping, irrespective of whether the QTLs are in the same or in different marker intervals, especially if linked QTLs are in repulsion. The criteria of comparison are number of identified QTLs, likelihood ratio test statistics, means and empirical standard errors of the QTL position and QTL effects estimates, and the accuracy of the residual variance estimates. Further, the joint conditional probabilities of QTL genotypes for two putative QTLs within a marker interval were derived and compared with the unmodified approach ignoring the non-independence of the conditional probabilities.     

Improved exact confidence intervals for the odds ratio in two independent binomial samples

For two independent binomial samples, the usual exact confidence interval for the odds ratio based on the conditional approach can be very conservative. Recently, Agresti and Min (2002) showed that the unconditional intervals are preferable to conditional intervals with small sample sizes. We use the unconditional approach to obtain a modified interval, which has shorter length, and its coverage probability is closer to and at least the nominal confidence coefficient.     

Symmetric tests and confidence intervals for survival probabilities and quantiles of censored survival data

We describe existing tests and introduce two new tests concerning the value of a survival function. These tests may be used to construct a confidence interval for the survival probability at a given time or for a quantile of the survival distribution. Simulation studies show that error rates can differ substantially from their nominal values, particularly at survival probabilities close to zero or one. We recommend our new constrained bootstrap test for its good overall performance.     

Conditional Probability Analyses of the Spike Activity of Single Neurons

With the objective of separating stimulus-related effects from refractory effects in neuronal spike data, various conditional probability analyses have been developed. These analyses are introduced and illustrated with examples based on electrophysiological data from auditory nerve fibers. The conditional probability analyses considered here involve the estimation of the conditional probability of a firing in a specified time interval (defined relative to the time of the stimulus presentation), given that the last firing occurred during an earlier specified time interval. This calculation enables study of the stimulus-related effects in the spike data with the time-since-the-last-firing as a controlled variable. These calculations indicate that auditory nerve fibers “recover” from the refractory effects that follow a firing in the following sense: after a “recovery time” of approximately 20 msec, the firing probabilities no longer depend on the time-since-the-last-firing. Probabilities conditional on this minimum time since the last firing are called “recovered probabilities.” The recovered probabilities presented in this paper are contrasted with the corresponding poststimulus time histograms, and the differences are related to the refractory properties of the nerve fibers.     

Confidence intervals for the substitution number in the nucleotide substitution models

In the nucleotide substitution model for molecular evolution, a major task in the exploration of an evolutionary process is to estimate the substitution number per site of a protein or DNA sequence. The usual estimators are based on the observation of the difference proportion of the two nucleotide sequences. However, a more objective approach is to report a confidence interval with precision rather than only providing point estimators. The conventional confidence intervals used in the literature for the substitution number are constructed by the normal approximation. The performance and construction of confidence intervals for evolutionary models have not been much investigated in the literature. In this article, the performance of these conventional confidence intervals for one-parameter and two-parameter models are explored. Results show that the coverage probabilities of these intervals are unsatisfactory when the true substitution number is small. Since the substitution number may be small in many situations for an evolutionary process, the conventional confidence interval cannot provide accurate information for these cases. Improved confidence intervals for the one-parameter model with desirable coverage probability are proposed in this article. A numerical calculation shows the substantial improvement of the new confidence intervals over the conventional confidence intervals.     

一类个体寿命有限的纯生过程

讨论了一类个体的寿命有限且一生分为若干个阶段的纯生过程 （ｐｕｒｅ ｂｉｒｔｈ ｐｒｏ－ｃｅｓｓ），其中每个阶段的存活率，繁殖率，其繁殖符合 Ｐｏｉｓｓｏｎ过程．证明了其小群体的繁衍分布服从分布；ｉ和ｊ两个不同阶段繁殖的时间间隔之比的密度函数为Ｆｉｂｏｎａｃｃｉ过程［４］中总体的期望数和寿命与指数分布有关等性质．     

A measure of explained risk in the proportional hazards model

A measure of explained risk is developed for application with the proportional hazards model. The statistic, which is called the estimated explained relative risk, has a simple analytical form and is unaffected by censoring that is independent of survival time conditional on the covariates. An asymptotic confidence interval for the limiting value of the estimated explained relative risk is derived, and the role of individual factors in the computation of its estimate is established. Simulations are performed to compare the results of the estimated explained relative risk to other known explained risk measures with censored data. Prostate cancer data are used to demonstrate an analysis incorporating the proposed approach.     

Test‐compatible confidence intervals for adaptive two‐stage single‐arm designs with binary endpoint

Inference after two‐stage single‐arm designs with binary endpoint is challenging due to the nonunique ordering of the sampling space in multistage designs. We illustrate the problem of specifying test‐compatible confidence intervals for designs with nonconstant second‐stage sample size and present two approaches that guarantee confidence intervals consistent with the test decision. Firstly, we extend the well‐known Clopper–Pearson approach of inverting a family of two‐sided hypothesis tests from the group‐sequential case to designs with fully adaptive sample size. Test compatibility is achieved by using a sample space ordering that is derived from a test‐compatible estimator. The resulting confidence intervals tend to be conservative but assure the nominal coverage probability. In order to assess the possibility of further improving these confidence intervals, we pursue a direct optimization approach minimizing the mean width of the confidence intervals. While the latter approach produces more stable coverage probabilities, it is also slightly anti‐conservative and yields only negligible improvements in mean width. We conclude that the Clopper–Pearson‐type confidence intervals based on a test‐compatible estimator are the best choice if the nominal coverage probability is not to be undershot and compatibility of test decision and confidence interval is to be preserved.     

Comparison of different methods for decision-making in bioequivalence assessment

If the regulatory requirements are symmetrical, the use of symmetrical confidence intervals as a decision rule for bioequivalence assessment leads, as shown by simulations, to better level properties and an inferior power compared to a rule based on shortest confidence intervals. A choice between these two approaches will have to depend on a loss function. For asymmetric regulatory requirements, symmetrical confidence intervals should not be used; however, a decision can still be based on posterior probabilities, pr (theta epsilon [r1, r2]/x), or shortest confidence intervals. For purposes of inference, presentation and interpretation of results, we think that the use of symmetrical confidence intervals alone can be misleading and we therefore recommend that the posterior probabilities and densities, or at least the shortest confidence intervals, be given.     

Some properties of the point processes generated by integral density modulation of renewal processes

Integral density modulation of point processes is defined, and the properties of the modulated point processes are described. When a homogeneous renewal process is modulated by a step random signal, the mathematical expressions are derived of the probability density functions of the sums of r-successive inter point intervals, the intensity functions and the first order correlation coefficient of intervals. These quantities are calculated and illustrated for several parameter values. Modulated point sequences are generated by computer simulation method. The interval histograms and the serial correlation coefficients of counts and of intervals of the sequences are obtained. The results are compared with the theoretical results on the point processes modulated by the step random signal.Abbreviations IDM Integral density modulation - SRS Step random signal - PP Point process - PS Point sequence - PDF Probability density function - CPDF Conditional probability density function - JCPDF Joint conditional probability density function - SED Special Earlangian distribution - SCC Serial correlation coefficients - SIC Serial intensity coefficients     

Optimal approximate conversions of odds ratios and hazard ratios to risk ratios

Odds ratios approximate risk ratios when the outcome under consideration is rare but can diverge substantially from risk ratios when the outcome is common. In this paper, we derive optimal analytic conversions of odds ratios and hazard ratios to risk ratios that are minimax for the bias ratio when outcome probabilities are specified to fall in any fixed interval. The results for hazard ratios are derived under a proportional hazard assumption for the exposure. For outcome probabilities specified to lie in symmetric intervals centered around 0.5, it is shown that the square-root transformation of the odds ratio is the optimal minimax conversion for the risk ratio. General results for any nonsymmetric interval are given both for odds ratio and for hazard ratio conversions. The results are principally useful when odds ratios or hazard ratios are reported in papers, and the reader does not have access to the data or to information about the overall outcome prevalence.     

Kernel estimates of dose response

A nonparametric method for analyzing quantal response data from an indirect bioassay experiment is proposed. Kernel estimates of the dose-response curve are used to develop approximate confidence intervals for (i) the optimal combination dose of a drug with therapeutic effects at low doses and toxic effects at high doses, and (ii) the lethal dose levels of a toxic chemical. This nonparametric procedure was implemented on real and simulated data. The confidence interval for problem (i) has high coverage probabilities when the dose-response curve is symmetric about the optima. However, the coverage probabilities are adversely affected by asymmetry about the optima and consequently are not reliable unless the sample sizes are large. The use of kernel estimators with higher-order kernels may alleviate this sensitivity to asymmetry. The confidence interval for problem (ii) has high coverage probabilities robust with respect to the shape or symmetry of the underlying dose-response curve.