Poisson Populations Research Articles

A Bayesian homogeneity test for comparing Poisson populations

AbstractFor a wide class of daily applications in industrial quality control, there may be interest in comparing several Poisson means. A large catalogue of frequentist procedures for this hypothesis testing problem is available. However, some common drawbacks of them are their low power, interpretation of the ‐values for multiple comparison, among many others. In this paper, we present a unified Bayesian approach to the problem based on a model selection approach developed using a product partition clustering model. The posterior probabilities for models obtained are derived directly from the associated Bayes factors which are calculated by considering a simple hierarchical prior structure which has a quasi–closed form, easily computed by numerical procedures. This approach constitutes a readily implementable alternative to frequentist multiple testing procedures where uncertainty concerning all possible “types of homogeneity” is ignored. The proposed methodology allows for an intuitive interpretation based directly on posterior probabilities of the partitions involved in the testing problem. We illustrate its performance with three real data sets.

Lower bounds for the expected sample size of sequential procedures for selecting and ranking of binomial and Poisson populations

In this paper we consider selection and ranking problems for the cases of binomial and Poisson populations. Our main goal is to construct lower bounds for the expected sample size of sequential selection and ranking procedures for the aforementioned problems in the setting of the indifference zone approach. Established lower bounds are investigated and their explicit estimates are obtained. After we investigate the efficiency of some sequential selection and ranking procedures with respect to our lower bounds.

EXACT TESTS OF THE SUPERIORITY UNDER THE GENERALIZED POISSON DISTRIBUTION

We propose two exact p-values of two commonly-used test statistics for testing the superiority under Generalized Poisson populations. We show that, the computationally-intensive confidence-set p-value involves at most a supremum search over a closed interval of a single argument. On the other hand, the estimated p-value has adequate performances empirically

On Statistical Tests for Homogeneity of Two Bivariate Zero-Inflated Poisson Populations

The problem of testing treatment difference in the occurrence of a study endpoint in a randomized parallel-group comparative clinical trial with repeated responses under the assumption that the responses follow a bivariate zero-inflated Poisson (ZIP) distribution is considered. Likelihood ratio test for homogeneity of two bivariate ZIP populations is derived. Approximate formula for sample size calculation is also obtained, which achieves a desired power for detecting a clinically meaningful difference under an alternative hypothesis. An example concerning the comparison of treatment effect in an addictive clinical trial in terms of the number of days of illicit drug use during a month is given for illustrative purposes.

Exact tests of the superiority under the Poisson distribution

We propose two exact p-values of two commonly-used test statistics for testing the superiority under Poisson populations. We show that, the computationally-intensive confidence-set p-value involves at most a supremum search over a closed interval of a single argument. On the other hand, the estimated p-value has adequate performances empirically.

Bayesian Fixed Sample Size Procedure for Selecting the Better of Two Poisson Populations With General Loss Function

In this paper an optimal (Bayesian) fixed sample size procedure for selecting the better of two Poisson populations is proposed and studied . Bayesian decision-theoretic approach with general loss function and Gamma priors are used to construct this procedure .The numerical result of this procedure are given with different loss functions constant , linear and quadratic , in one equation we can obtain the Bayes risk for the three types of the loss functions : constant , linear and quadratic . in this paper the numerical results are given by using Math Works Matlab ver. 7.10.0 .

Average worth estimation of the selected subset of Poisson populations

Let Π1, …, Π p be p(p≥2) independent Poisson populations with unknown parameters θ1, …, θ p , respectively. Let X i denote an observation from the population Π i , 1≤i≤p. Suppose a subset of random size, which includes the best population corresponding to the largest (smallest) θ i , is selected using Gupta and Huang [On subset selection procedures for Poisson populations and some applications to the multinomial selection problems, in Applied Statistics, R.P. Gupta, ed., North-Holland, Amsterdam, 1975, pp. 97–109] and (Gupta et al. [On subset selection procedures for Poisson populations, Bull. Malaysian Math. Soc. 2 (1979), pp. 89–110]) selection rule. In this paper, the problem of estimating the average worth of the selected subset is considered under the squared error loss function. The natural estimator is shown to be biased and the UMVUE is obtained using Robbins [The UV method of estimation, in Statistical Decision Theory and Related Topics-IV, S.S. Gupta and J.O. Berger, eds., Springer, New York, vol. 1, 1988, pp. 265–270] UV method of estimation. The natural estimator is shown to be inadmissible, by constructing a class of dominating estimators. Using Monte Carlo simulations, the bias and risk of the natural, dominated and UMVU estimators are computed and compared.

Multiple Comparison Procedures for Poisson Parameters in Multi-Sample Models

First, we consider multiple comparisons for the differences among parameters in k Poisson populations. We may give the Tukey-Kramer type multiple test procedure based on estimators of k means. However, the degree of conservativeness for the multiple tests depends on unknown rate parameters. Therefore, multiple tests based on estimators of k standard deviations are proposed. It is found that the degree of con-servativeness for the proposed tests is controlled by the sample sizes. Furthermore, the closed testing procedure, more powerful than the REGW (Ryan/Einot-Gabriel/Welsch) tests, is proposed. Simultaneous confidence intervals for the differences of the square roots of the rates are discussed. Next, for the multiple comparisons with a control, we propose the multiple test procedure based on the estimators of k standard deviations. It is shown that the proposed multiple test is superior to the tests based on the Bonferroni inequality asymptotically. A sequentially rejective procedure is derived under unequal sample sizes.

Bayesian analysis of hypothesis testing problems for general population: A Kullback–Leibler alternative

We consider a hypothesis problem with directional alternatives. We approach the problem from a Bayesian decision theoretic point of view and consider a situation when one side of the alternatives is more important or more probable than the other. We develop a general Bayesian framework by specifying a mixture prior structure and a loss function related to the Kullback–Leibler divergence. This Bayesian decision method is applied to Normal and Poisson populations. Simulations are performed to compare the performance of the proposed method with that of a method based on a classical z-test and a Bayesian method based on the “0–1” loss.

Simultaneous estimation of Poisson means of the selected subset

Let π 1 , π 2 , … , π p be p independent Poisson populations with means λ 1 , … , λ p , respectively. Let { X 1,…, X p } denote the set of observations, where X i is from π i . Suppose a subset of populations is selected using Gupta and Huang's (1975) selection rule which selects π i if and only if X i + 1 ⩾ cX ( 1 ) , where X (1)=max{ X 1,…, X p }, and 0 < c < 1 . In this paper, the simultaneous estimation of the Poisson means associated with the selected populations is considered for the k-normalized squared error loss function. It is shown that the natural estimator is positively biased. Also, a class of estimators that are better than the natural estimator is obtained by solving certain difference inequalities over the sample space. A class of estimators which dominate the UMVUE is also obtained. Monte carlo simulations are used to assess the percentage improvements and an application to a real-life example is also discussed.

Homogeneity tests for several Poisson populations

In this paper we compare the size distortions and powers for Pearson’s χ 2 -statistic, likelihood ratio statistic L R , score statistic S C and two statistics, which we call U T and V T here, proposed by [Potthoff, R.F., Whittinghill, M., 1966. Testing for homogeneity: II. The Poisson distribution. Biometrika 53, 183–190] for testing the equality of the rates of K Poisson processes. Asymptotic tests and parametric bootstrap tests are considered. It is found that the asymptotic U T test is too conservative to be recommended, while the other four asymptotic tests perform similarly and their powers are close to those of their parametric bootstrap counterparts when the observed counts are large enough. When the observed counts are not large, Monte Carlo simulation suggested that the asymptotic test using S C , L R and U T statistics are discouraged; none of the parametric bootstrap tests with the five statistics considered here is uniformly best or worst, and the asymptotic tests using Pearson’s χ 2 and V T always have similar powers to their bootstrap counterparts. Thus, the asymptotic Pearson’s χ 2 and V T tests have an advantage over all five parametric bootstrap tests in terms of their computational simplicity and convenience, and over the other four asymptotic tests in terms of their powers and size distortions.

Testing Homogeneity of Two Zero‐inflated Poisson Populations

The problem of testing treatment difference in the occurrence of a safety parameter in a randomized parallel-group comparative clinical trial under the assumption that the number of occurrence follows a zero-inflated Poisson (ZIP) distribution is considered. Likelihood ratio tests (LRT) for homogeneity of two ZIP populations are derived under the hypotheses that (i) there is no difference in inflation parameters, (ii) there is no difference in non-zero means; and (iii) there is no difference in both inflation parameters and non-zero means. Approximate formulas for sample size calculation are also obtained for achieving a desired power for detecting a clinically meaningful difference under the corresponding alternative hypotheses. An example concerning the assessment of the gastrointestinal (GI) safety in terms of the number of erosion counts of a newly developed compound for the treatment of osteoarthritis and rheumatoid arthritis is given for illustration purpose.

A Note on Inference on Multiple Generalized Poisson Populations

SYNOPTIC ABSTRACTWe consider a fully Bayesian approach to estimation of parameters for generalized Poisson data in a multiple population context. The hierarchical model we consider here extends previous single population models. This hierarchical model has applications in industrial, biological and sociological disciplines. We also extend two recently developed subset selection criteria to the generalized Poisson hierarchical model we propose. The procedures are used to analyze two real data sets. Probabilities are approximated via Markov Chain Monte Carlo.

Sample size calculation for testing non-inferiority and equivalence under Poisson distribution

When counting the number of chemical parts in air pollution studies or when comparing the occurrence of congenital malformations between a uranium mining town and a control population, we often assume Poisson distribution for the number of these rare events. Some discussions on sample size calculation under Poisson model appear elsewhere, but all these focus on the case of testing equality rather than testing equivalence. We discuss sample size and power calculation on the basis of exact distribution under Poisson models for testing non-inferiority and equivalence with respect to the mean incidence rate ratio. On the basis of large sample theory, we further develop an approximate sample size calculation formula using the normal approximation of a proposed test statistic for testing non-inferiority and an approximate power calculation formula for testing equivalence. We find that using these approximation formulae tends to produce an underestimate of the minimum required sample size calculated from using the exact test procedure. On the other hand, we find that the power corresponding to the approximate sample sizes can be actually accurate (with respect to Type I error and power) when we apply the asymptotic test procedure based on the normal distribution. We tabulate in a variety of situations the minimum mean incidence needed in the standard (or the control) population, that can easily be employed to calculate the minimum required sample size from each comparison group for testing non-inferiority and equivalence between two Poisson populations.

Selecting the most reliable Poisson population provided it is better than a control: a nonparametric empirical Bayes approach

We study the problem of selecting the most reliable Poisson population from among k competitors provided it is better than a control using the nonparametric empirical Bayes approach. An empirical Bayes selection procedure is constructed based on the isotonic regression estimators of the posterior means of failure rates associated with the k Poisson populations. The asymptotic optimality of the empirical Bayes selection procedure is investigated. Under certain regularity conditions, we have shown that the proposed empirical Bayes selection procedure is asymptotically optimal and the associated Bayes risk converges to the minimum Bayes risk at a rate of order O(exp(− cn)) for some c>0, where n denotes the number of historical data at hand when the present selection problem is considered.

Hierarchical bayes simultaneous estimation of poisson means

Equal-size samples are observed from k Poisson populations having means Assuming an exchangeable two-stage prior, conjugate at the first stage and noninformative at the second stage, formulas for the posterior means, variances and covariances of the A.λ’s are expressed as the ratio of one-dimensional integrals. Approximations to the hierarchical Bayes posterior means and variances are developed, which do not require numerical integration. The performance of the Bayes estimator compared with other estimators is studied through computer simulation, using the frequentist criterion of percentage risk reduction and the Bayes risk criterion of relative savings loss.

Combining poisson means

The problem of estimating the Poisson mean is considered based on the two samples in the presence of uncertain prior information (not in the form of distribution) that two independent random samples taken from two possibly identical Poisson populations. The parameter of interest is λ1 from population I. Three estimators, i.e. the unrestricted estimator, restricted estimator and preliminary test estimator are proposed. Their asymptotic mean squared errors are derived and compared; parameter regions have been found for which restricted and preliminary test estimators are always asymptotically more efficient than the classical estimator. The relative dominance picture of the estimators is presented. Maximum and minimum asymptotic efficiencies of the estimators relative to the classical estimator are tabulated. A max-min rule for the size of the preliminary test is also discussed. A Monte Carlo study is presented to compare the performance of the estimator with that of Kale and Bancroft (1967).

ON THE ASYMPTOTIC 0PTIMALITY OF CERTAIN EMPIRICAL BAYES SIMULTANEOUS TESTING PROCEDURES FOR POISSON POPULATIONS

Article ON THE ASYMPTOTIC 0PTIMALITY OF CERTAIN EMPIRICAL BAYES SIMULTANEOUS TESTING PROCEDURES FOR POISSON POPULATIONS was published on March 1, 1991 in the journal Statistics & Risk Modeling (volume 9, issue 3).

On the admissibility of estimators of the zero class in poisson populations

Let T = x1+ x2 + … + xn where x1,…, xn are independent and identically distributed randon variables each having a poission distribution with mean θ. It is proved that each of the estimators ((n+k−1)/(n+k))T, k = 0,1,2,… is admissible for estimating exp(−θ) when the loss function is squared error.

Optimality Of Some Two-Stage Sampling Methods

Two-stage sampling plans in which the estimator is computed from the second sample only, and has a guaranteed precision irrespective of the parameter value, are considered. We prove the following optimality property for existing strategies in binomial and Poisson populations: there exists no other strategy, of the same form, that guarantees the same precision, without increasing the expected sample size for some parameter valucs. The method of proof is illustrnted by proving a similar result for normal populations