Exact calculation of minimum sample size for estimating a Poisson parameter

Abstract

We develop an exact approach for the determination of the minimum sample size for estimating a Poisson parameter such that the pre-specified levels of relative precision and confidence are guaranteed. The exact computation is made possible by reducing infinitely many evaluations of coverage probability to finitely many evaluations. The theory for supporting such a reduction is that the minimum of coverage probability with respect to the parameter in an interval is attained at a discrete set of finitely many elements. Computational mechanisms have been developed to further reduce the computational complexity. An explicit bound for the minimum sample size is established.     

A new confidence interval for the difference between two binomial proportions of paired data

Motivated by a study on comparing sensitivities and specificities of two diagnostic tests in a paired design when the sample size is small, we first derived an Edgeworth expansion for the studentized difference between two binomial proportions of paired data. The Edgeworth expansion can help us understand why the usual Wald interval for the difference has poor coverage performance in the small sample size. Based on the Edgeworth expansion, we then derived a transformation based confidence interval for the difference. The new interval removes the skewness in the Edgeworth expansion; the new interval is easy to compute, and its coverage probability converges to the nominal level at a rate of O(n^−1/2). Numerical results indicate that the new interval has the average coverage probability that is very close to the nominal level on average even for sample sizes as small as 10. Numerical results also indicate this new interval has better average coverage accuracy than the best existing intervals in finite sample sizes.     

Performance of Interval Estimators for the Inverse Hypergeometric Distribution

The inverse hypergeometric distribution is of interest in applications of inverse sampling without replacement from a finite population where a binary observation is made on each sampling unit. Thus, sampling is performed by randomly choosing units sequentially one at a time until a specified number of one of the two types is selected for the sample. Assuming the total number of units in the population is known but the number of each type is not, we consider the problem of estimating this parameter. We use the Delta method to develop approximations for the variance of three parameter estimators. We then propose three large sample confidence intervals for the parameter. Based on these results, we selected a sampling of parameter values for the inverse hypergeometric distribution to empirically investigate performance of these estimators. We evaluate their performance in terms of expected probability of parameter coverage and confidence interval length calculated as means of possible outcomes weighted by the appropriate outcome probabilities for each parameter value considered. The unbiased estimator of the parameter is the preferred estimator relative to the maximum likelihood estimator and an estimator based on a negative binomial approximation, as evidenced by empirical estimates of closeness to the true parameter value. Confidence intervals based on the unbiased estimator tend to be shorter than the two competitors because of its relatively small variance but at a slight cost in terms of coverage probability.     

The Fate Worse than Death and other Curiosities and Stupidities

The problem of finding confidence intervals for the success parameter of a binomial experiment has a long history, and a myriad of procedures have been developed. Most exploit the duality between hypothesis testing and confidence regions and are typically based on large sample approximations. We instead employ a direct approach that attempts to determine the optimal coverage probability function a binomial confidence procedure can have from the exact underlying binomial distributions, which in turn defines the associated procedure. We show that a graphical perspective provides much insight into the problem. Both procedures whose coverage never falls below the declared confidence level and those that achieve that level only approximately are analyzed. We introduce the Length/Coverage Optimal method, a variant of Sterne's procedure that minimizes average length while maximizing coverage among all length minimizing procedures, and show that it is superior in important ways to existing procedures.     

Bayesian sample size determination for estimating binomial parameters from data subject to misclassification

We investigate the sample size problem when a binomial parameter is to be estimated, but some degree of misclassification is possible. The problem is especially challenging when the degree to which misclassification occurs is not exactly known. Motivated by a Canadian survey of the prevalence of toxoplasmosis infection in pregnant women, we examine the situation where it is desired that a marginal posterior credible interval for the prevalence of width w has coverage 1−α, using a Bayesian sample size criterion. The degree to which the misclassification probabilities are known a priori can have a very large effect on sample size requirements, and in some cases achieving a coverage of 1−α is impossible, even with an infinite sample size. Therefore, investigators must carefully evaluate the degree to which misclassification can occur when estimating sample size requirements.     

A comment on sample size calculations for binomial confidence intervals

In this article we examine sample size calculations for a binomial proportion based on the confidence interval width of the Agresti–Coull, Wald and Wilson Score intervals. We pointed out that the commonly used methods based on known and fixed standard errors cannot guarantee the desired confidence interval width given a hypothesized proportion. Therefore, a new adjusted sample size calculation method was introduced, which is based on the conditional expectation of the width of the confidence interval given the hypothesized proportion. With the reduced sample size, the coverage probability can still maintain at the nominal level and is very competitive to the converge probability for the original sample size.     

Confidence regions for two proportions from independent negative binomial distributions

The negative binomial distribution offers an alternative view to the binomial distribution for modeling count data. This alternative view is particularly useful when the probability of success is very small, because, unlike the fixed sampling scheme of the binomial distribution, the inverse sampling approach allows one to collect enough data in order to adequately estimate the proportion of success. However, despite work that has been done on the joint estimation of two binomial proportions from independent samples, there is little, if any, similar work for negative binomial proportions. In this paper, we construct and investigate three confidence regions for two negative binomial proportions based on three statistics: the Wald (W), score (S) and likelihood ratio (LR) statistics. For large-to-moderate sample sizes, this paper finds that all three regions have good coverage properties, with comparable average areas for large sample sizes but with the S method producing the smaller regions for moderate sample sizes. In the small sample case, the LR method has good coverage properties, but often at the expense of comparatively larger areas. Finally, we apply these three regions to some real data for the joint estimation of liver damage rates in patients taking one of two drugs.     

A Conway Maxwell Poisson type generalization of the negative hypergeometric distribution

Abstract

Negative hypergeometric distribution arises as a waiting time distribution when we sample without replacement from a finite population. It has applications in many areas such as inspection sampling and estimation of wildlife populations. However, as is well known, the negative hypergeometric distribution is over-dispersed in the sense that its variance is greater than the mean. To make it more flexible and versatile, we propose a modified version of negative hypergeometric distribution called COM-Negative Hypergeometric distribution (COM-NH) by introducing a shape parameter as in the COM-Poisson and COMP-Binomial distributions. It is shown that under some limiting conditions, COM-NH approaches to a distribution that we call the COM-Negative binomial (COMP-NB), which in turn, approaches to the COM Poisson distribution. For the proposed model, we investigate the dispersion characteristics and shape of the probability mass function for different combinations of parameters. We also develop statistical inference for this model including parameter estimation and hypothesis tests. In particular, we investigate some properties such as bias, MSE, and coverage probabilities of the maximum likelihood estimators for its parameters by Monte Carlo simulation and likelihood ratio test to assess shape parameter of the underlying model. We present illustrative data to provide discussion.     

Improved nonparametric tolerance intervals based on interpolated and extrapolated order statistics

The standard approach to construct nonparametric tolerance intervals is to use the appropriate order statistics, provided a minimum sample size requirement is met. However, it is well-known that this traditional approach is conservative with respect to the nominal level. One way to improve the coverage probabilities is to use interpolation. However, the extension to the case of two-sided tolerance intervals, as well as for the case when the minimum sample size requirement is not met, have not been studied. In this paper, an approach using linear interpolation is proposed for improving coverage probabilities for the two-sided setting. In the case when the minimum sample size requirement is not met, coverage probabilities are shown to improve by using linear extrapolation. A discussion about the effect on coverage probabilities and expected lengths when transforming the data is also presented. The applicability of this approach is demonstrated using three real data sets.     

On exact computation of minimum sample size for restricted estimation of a binomial parameter

The problem of determining minimum sample size for the estimation of a binomial parameter with prescribed margin of error and confidence level is considered. It is assumed that available auxiliary information allows to restrict the parameter space to some interval whose left boundary is above zero. A range-preserving estimator resulting from the conditional maximization of the likelihood function is considered. A method for exact computation of minimum sample size controlling for the relative error is proposed. Several tables of minimum sample sizes for typical situations are also presented. The range-preserving estimator achieves the same precision and confidence level as the unrestricted maximum likelihood estimator but with a smaller sample.     

Fixed width confidence intervals for the location parameter of an exponential distribution

We study confidence intervals of prescribed width for the lo-cation parameter of an exponential distribution. Asymptotic expan-sions up to terms tending to zero are obtained for the coverage probability and expected sample size. The limiting distribution of the sample size is given from which an asymptotic expression for the variance of the sample size is deduced. Sequential procedures with non-asymptotic coverage probability are also investigated     

Robustness of confidence intervals for scale parameters based on m-estimators

The robustness of confidence intervals for a scale parameter based on M-esimators is studied, especially in small size samples. The coverage probablity is used as measure of robustness. A theorem for a lower bound of the minimum coverage probability of M-estimators is presented and it is applied in order to examine the behavior of the standard deviation and the median absolute deviation, as interval estimators. This bound can confirm the robustness of any other scale M-estimator in interval estimation. The idea of stretching is used to formulate the family of distributions that are considered as underlying. Critical values for the confidence interval are computed where it is needed, that is for the median absolute deviation in the Normal, Uniform and Cauchy distribution and for the standard deviation in the Uniform and Cauchy distribution. Simulation results have been achieved for the estimation of the coverage probabilities and the critical values.     

Exact confidence limits for the probability of response in two-stage designs

In addition to point estimate for the probability of response in a two-stage design (e.g. Simon's two-stage design for binary endpoints), confidence limits should be computed and reported. The current method of inverting the p-value function to compute the confidence interval does not guarantee coverage probability in a two-stage setting. The existing exact approach to calculate one-sided limits is based on the overall number of responses to order the sample space. This approach could be conservative because many sample points have the same limits. We propose a new exact one-sided interval based on p-value for the sample space ordering. Exact intervals are computed by using binomial distributions directly, instead of a normal approximation. Both exact intervals preserve the nominal confidence level. The proposed exact interval based on the p-value generally performs better than the other exact interval with regard to expected length and simple average length of confidence intervals.     

A comparison of two approaches for power and sample size calculations in logistic regression models

Whittemore (1981) proposed an approach for calculating the sample size needed to test hypotheses with specified significance and power against a given alternative for logistic regression with small response probability. Based on the distribution of covariate, which could be either discrete or continuous, this approach first provides a simple closed-form approximation to the asymptotic covariance matrix of the maximum likelihood estimates, and then uses it to calculate the sample size needed to test a hypothesis about the parameter. Self et al. (1992) described a general approach for power and sample size calculations within the framework of generalized linear models, which include logistic regression as a special case. Their approach is based on an approximation to the distribution of the likelihood ratio statistic. Unlike the Whittemore approach, their approach is not limited to situations of small response probability. However, it is restricted to models with a finite number of covariate configurations. This study compares these two approaches to see how accurate they would be for the calculations of power and sample size in logistic regression models with various response probabilities and covariate distributions. The results indicate that the Whittemore approach has a slight advantage in achieving the nominal power only for one case with small response probability. It is outperformed for all other cases with larger response probabilities. In general, the approach proposed in Self et al. (1992) is recommended for all values of the response probability. However, its extension for logistic regression models with an infinite number of covariate configurations involves an arbitrary decision for categorization and leads to a discrete approximation. As shown in this paper, the examined discrete approximations appear to be sufficiently accurate for practical purpose.     

Comparison of Bayesian sample size criteria: ACC, ALC, and WOC

A challenge for implementing performance-based Bayesian sample size determination is selecting which of several methods to use. We compare three Bayesian sample size criteria: the average coverage criterion (ACC) which controls the coverage rate of fixed length credible intervals over the predictive distribution of the data, the average length criterion (ALC) which controls the length of credible intervals with a fixed coverage rate, and the worst outcome criterion (WOC) which ensures the desired coverage rate and interval length over all (or a subset of) possible datasets. For most models, the WOC produces the largest sample size among the three criteria, and sample sizes obtained by the ACC and the ALC are not the same. For Bayesian sample size determination for normal means and differences between normal means, we investigate, for the first time, the direction and magnitude of differences between the ACC and ALC sample sizes. For fixed hyperparameter values, we show that the difference of the ACC and ALC sample size depends on the nominal coverage, and not on the nominal interval length. There exists a threshold value of the nominal coverage level such that below the threshold the ALC sample size is larger than the ACC sample size, and above the threshold the ACC sample size is larger. Furthermore, the ACC sample size is more sensitive to changes in the nominal coverage. We also show that for fixed hyperparameter values, there exists an asymptotic constant ratio between the WOC sample size and the ALC (ACC) sample size. Simulation studies are conducted to show that similar relationships among the ACC, ALC, and WOC may hold for estimating binomial proportions. We provide a heuristic argument that the results can be generalized to a larger class of models.     

The large sample coverage probability of confidence intervals in general regression models after a preliminary hypothesis test

We derive a computationally convenient formula for the large sample coverage probability of a confidence interval for a scalar parameter of interest following a preliminary hypothesis test that a specified vector parameter takes a given value in a general regression model. Previously, this large sample coverage probability could only be estimated by simulation. Our formula only requires the evaluation, by numerical integration, of either a double or a triple integral, irrespective of the dimension of this specified vector parameter. We illustrate the application of this formula to a confidence interval for the odds ratio of myocardial infarction when the exposure is recent oral contraceptive use, following a preliminary test where two specified interactions in a logistic regression model are zero. For this real‐life data, we compare this large sample coverage probability with the actual coverage probability of this confidence interval, obtained by simulation.     

Second-Order Approximations to Two Classes of Sequential Estimation Procedures

For the model considered by Chaturvedi, Pandey and Gupta (1991), two classes of sequential procedures are developed to construct confidence regions (which may be interval, ellipsoidal or spherical) of ‘pre-assigned width and coverage probability’ for the parameters of interest and for the minimum risk point estimation (taking loss to be quadratic plus linear cost of sampling) of the nuisance parameter. Second-Order approximations are derived for the expected sample size, coverage probability and ‘regret’ associated with the two classes of sequential procedures. A simple and direct method of obtaining the asymptotic distribution of the stopping time is provided. By means of examples, it is illustrated that several estimation problems can be tackled with the help of proposed classes of sequential procedures.     

Confidence intervals for the difference between independent binomial proportions: comparison using a graphical approach and moving averages

This paper uses graphical methods to illustrate and compare the coverage properties of a number of methods for calculating confidence intervals for the difference between two independent binomial proportions. We investigate both small‐sample and large‐sample properties of both two‐sided and one‐sided coverage, with an emphasis on asymptotic methods. In terms of aligning the smoothed coverage probability surface with the nominal confidence level, we find that the score‐based methods on the whole have the best two‐sided coverage, although they have slight deficiencies for confidence levels of 90% or lower. For an easily taught, hand‐calculated method, the Brown‐Li ‘Jeffreys’ method appears to perform reasonably well, and in most situations, it has better one‐sided coverage than the widely recommended alternatives. In general, we find that the one‐sided properties of many of the available methods are surprisingly poor. In fact, almost none of the existing asymptotic methods achieve equal coverage on both sides of the interval, even with large sample sizes, and consequently if used as a non‐inferiority test, the type I error rate (which is equal to the one‐sided non‐coverage probability) can be inflated. The only exception is the Gart‐Nam ‘skewness‐corrected’ method, which we express using modified notation in order to include a bias correction for improved small‐sample performance, and an optional continuity correction for those seeking more conservative coverage. Using a weighted average of two complementary methods, we also define a new hybrid method that almost matches the performance of the Gart‐Nam interval. Copyright © 2014 John Wiley & Sons, Ltd.     

A flexible multi-stage design for phase II oncology trials

Phase II trials evaluate whether a new drug or a new therapy is worth further pursuing or certain treatments are feasible or not. A typical phase II is a single arm (open label) trial with a binary clinical endpoint (response to therapy). Although many oncology Phase II clinical trials are designed with a two-stage procedure, multi-stage design for phase II cancer clinical trials are now feasible due to increased capability of data capture. Such design adjusts for multiple analyses and variations in analysis time, and provides greater flexibility such as minimizing the number of patients treated on an ineffective therapy and identifying the minimum number of patients needed to evaluate whether the trial would warrant further development. In most of the NIH sponsored studies, the early stopping rule is determined so that the number of patients treated on an ineffective therapy is minimized. In pharmaceutical trials, it is also of importance to know as early as possible if the trial is highly promising and what is the likelihood the early conclusion can sustain. Although various methods are available to address these issues, practitioners often use disparate methods for addressing different issues and do not realize a single unified method exists. This article shows how to utilize a unified approach via a fully sequential procedure, the sequential conditional probability ratio test, to address the multiple needs of a phase II trial. We show the fully sequential program can be used to derive an optimized efficient multi-stage design for either a low activity or a high activity, to identify the minimum number of patients required to assess whether a new drug warrants further study and to adjust for unplanned interim analyses. In addition, we calculate a probability of discordance that the statistical test will conclude otherwise should the trial continue to the planned end that is usually at the sample size of a fixed sample design. This probability can be used to aid in decision making in a drug development program. All computations are based on exact binomial distribution.     

一种简单有效的二项分布比例参数似然比置信区间的求法

对二项分布比例参数p的似然比置信区间,提出一种简便求解方法。在平均覆盖率、平均区间长度及区间长度的95%置信区间准则下与WScore、Plus4、Jeffreys置信区间进行模拟比较。试验表明,在二项分布b(n,p)的参数n≥20且p∈(0.1,0.9)时,该方法获取的似然比置信区间性能优良。当点估计p值不是接近于0或1且n≥20时,推荐使用本方法获取p的置信区间。