Fisher information matrix of the Dirichlet-multinomial distribution

In this paper we derive explicit expressions for the elements of the exact Fisher information matrix of the Dirichlet-multinomial distribution. We show that exact calculation is based on the beta-binomial probability function rather than that of the Dirichlet-multinomial and this makes the exact calculation quite easy. The exact results are expected to be useful for the calculation of standard errors of the maximum likelihood estimates of the beta-binomial parameters and those of the Dirichlet-multinomial parameters for data that arise in practice in toxicology and other similar fields. Standard errors of the maximum likelihood estimates of the beta-binomial parameters and those of the Dirichlet-multinomial parameters, based on the exact and the asymptotic Fisher information matrix based on the Dirichlet distribution, are obtained for a set of data from Haseman and Soares (1976), a dataset from Mosimann (1962) and a more recent dataset from Chen, Kodell, Howe and Gaylor (1991). There is substantial difference between the standard errors of the estimates based on the exact Fisher information matrix and those based on the asymptotic Fisher information matrix.     

A note on modelling the dynamics of budding yeast populations using branching processes

A multitype branching process is proposed as a model for the behaviour of populations of the budding yeast Saccharomyces Cerevisiae. Using the idea of branching processes counted by random characteristics, we are able to obtain explicit expressions describing different aspects of the asymptotic composition of such populations. The main purpose of this note is to show that the branching process approach is an alternative to deterministic population models based on differential equation methods.Supported by the Swedish Natural Science Research Council     

混合位置尺度分布中分量参数的统计推断

本文讨论混合位置尺度分布当混合比已知时分量参数的假设检验和区间估计．本文所提出的方法基于广义枢轴模型．当只有感兴趣的参数未知时,检验的实际水平等于名义水平,且各置信域的实际覆盖率等于名义覆盖率．在更一般的场合,检验是相合的,并且各置信域的实际覆盖率趋于名义覆盖率．模拟显示我们的方法是令人满意的．     

A note on permutation tests for variance components in multilevel generalized linear mixed models

In many applications of generalized linear mixed models to multilevel data, it is of interest to test whether a random effects variance component is zero. It is well known that the usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold. In this note we propose a permutation test, based on randomly permuting the indices associated with a given level of the model, that has the correct Type I error rate under the null. Results from a simulation study suggest that it is more powerful than tests based on mixtures of chi-square distributions. The proposed test is illustrated using data on the familial aggregation of sleep disturbance.     

Generalized maximally selected statistics

SUMMARY: Maximally selected statistics for the estimation of simple cutpoint models are embedded into a generalized conceptual framework based on conditional inference procedures. This powerful framework contains most of the published procedures in this area as special cases, such as maximally selected chi(2) and rank statistics, but also allows for direct construction of new test procedures for less standard test problems. As an application, a novel maximally selected rank statistic is derived from this framework for a censored response partitioned with respect to two ordered categorical covariates and potential interactions. This new test is employed to search for a high-risk group of rectal cancer patients treated with a neo-adjuvant chemoradiotherapy. Moreover, a new efficient algorithm for the evaluation of the asymptotic distribution for a large class of maximally selected statistics is given enabling the fast evaluation of a large number of cutpoints.     

Simultaneous confidence intervals for a success probability and intraclass correlation,with an application to screening mammography

This paper provides asymptotic simultaneous confidence intervals for a success probability and intraclass correlation of the beta‐binomial model, based on the maximum likelihood estimator approach. The coverage probabilities of those intervals are evaluated. An application to screening mammography is presented as an example. The individual and simultaneous confidence intervals for sensitivity and specificity and the corresponding intraclass correlations are investigated. Two additional examples using influenza data and sex ratio data among sibships are also considered, where the individual and simultaneous confidence intervals are provided.     

Sample size calculations for noninferiority trials with Poisson distributed count data

Clinical trials with Poisson distributed count data as the primary outcome are common in various medical areas such as relapse counts in multiple sclerosis trials or the number of attacks in trials for the treatment of migraine. In this article, we present approximate sample size formulae for testing noninferiority using asymptotic tests which are based on restricted or unrestricted maximum likelihood estimators of the Poisson rates. The Poisson outcomes are allowed to be observed for unequal follow‐up schemes, and both the situations that the noninferiority margin is expressed in terms of the difference and the ratio are considered. The exact type I error rates and powers of these tests are evaluated and the accuracy of the approximate sample size formulae is examined. The test statistic using the restricted maximum likelihood estimators (for the difference test problem) and the test statistic that is based on the logarithmic transformation and employs the maximum likelihood estimators (for the ratio test problem) show favorable type I error control and can be recommended for practical application. The approximate sample size formulae show high accuracy even for small sample sizes and provide power values identical or close to the aspired ones. The methods are illustrated by a clinical trial example from anesthesia.     

On the asymptotics of marginal regression splines with longitudinal data

There have been studies on how the asymptotic efficiency ofa nonparametric function estimator depends on the handling ofthe within-cluster correlation when nonparametric regressionmodels are used on longitudinal or cluster data. In particular,methods based on smoothing splines and local polynomial kernelsexhibit different behaviour. We show that the generalized estimationequations based on weighted least squares regression splinesfor the nonparametric function have an interesting property:the asymptotic bias of the estimator does not depend on theworking correlation matrix, but the asymptotic variance, andtherefore the mean squared error, is minimized when the truecorrelation structure is specified. This property of the asymptoticbias distinguishes regression splines from smoothing splines.     

Notes on the Mack-Wolfe and Chen-Wolfe Tests for Umbrella Alternatives

In this paper we find a class of umbrella alternatives for which the MACK -WOLFE (1981) peak known test is optimal in the sense of Pitman efficiency. The asymptotic null distribution of the CHEN -WOLFE (1989) statistic for the peak-unknown umbrella alternatives problem is obtained. Some percentiles of the asymptotic distribution computed by simulation are presented.     

Nonparametric All‐Pairs Multiple Comparisons

Nonparametric all‐pairs multiple comparisons based on pairwise rankings can be performed in the one‐way design with the Steel‐Dwass procedure. To apply this test, Wilcoxon's rank sum statistic is calculated for all pairs of groups; the maximum of the rank sums is the test statistic. We provide exact calculations of the asymptotic critical values (and P‐values, respectively) even for unbalanced designs. We recommend this asymptotic method whenever large sample sizes are present. For small sample sizes we recommend the use of the new statistic according to Baumgartner , Weiss , and Schindler (1998, Biometrics 54 , 1129–1135) instead of Wilcoxon's rank sum for the multiple comparisons. We show that the resultant procedure can be less conservative and, according to simulation results, more powerful than the original Steel‐Dwass procedure. We illustrate the methods with a practical data set.     

A class of multiplicity adjusted tests for spatial clustering based on case-control point data

A class of tests with quadratic forms for detecting spatial clustering of health events based on case-control point data is proposed. It includes Cuzick and Edwards's test statistic (1990, Journal of the Royal Statistical Society, Series B 52, 73-104). Although they used the property of asymptotic normality of the test statistic, we show that such an approximation is generally poor for moderately large sample sizes. Instead, we suggest a central chi-square distribution as a better approximation to the asymptotic distribution of the test statistic. Furthermore, not only to estimate the optimal value of the unknown parameter on the scale of cluster but also to adjust for multiple testing due to repeating the procedure by changing the parameter value, we propose the minimum of the profile p-value of the test statistic for the parameter as an integrated test statistic. We also provide a statistic to estimate the areas or cases which make large contributions to significant clustering. The proposed methods are illustrated with a data set concerning the locations of cases of childhood leukemia and lymphoma and another on early medieval grave site locations consisting of affected and nonaffected grave sites.     

Expected Error Rates for Logistic Regression Versus Normal Discriminant Analysis

The expected error rates associated with using the allocation rule based on logistic regression are derived in the context of two multivariate normal populations with a common covariance matrix and compared with the corresponding error rates of the classical rule based on this normality assumption. It is shown in terms of the actual sizes of the asymptotic expected error rates that the performance of the logistic procedure does not fall far short of the normality based method, even for widely separated populations. This latter result is not obvious from previously available work on the asymptotic relative efficiency of the logistic procedure.     

A Modified Chi-Square Test for Testing Equality of Two Multinomial Populations Against an Ordering Restricted Alternative

A modified chi-square test for testing the equality of two multinomial populations against an ordering restricted alternative in one sample and two sample cases is constructed. The relation between a concept of dependence called dependence by chi-square and stochastic ordering is established. A tabulation of the asymptotic distribution of the test statistic under the null hypothesis is given. Simulations are used to compare the power of this test with the power of the likelihood ratio test of stochastic ordering of the two multinomial populations.     

Modeling late entry bias in survival analysis

In a failure time analysis, we sometimes observe additional study subjects who enter during the study period. These late entries are treated as left-truncated data in the statistical literature. However, with real data, there is a substantial possibility that the delayed entries may have extremely different hazards compared to the other standard subjects. We focus on a situation in which such entry bias might arise in the analysis of survival data. The purpose of the present article is to develop an appropriate methodology for making inference about data including late entries. We construct a model that includes parameters for the effect of delayed entry bias having no specification for the distribution of entry time. We also discuss likelihood inference based on this model and derive the asymptotic behavior of estimates. A simulation study is conducted for a finite sample size in order to compare the analysis results using our method with those using the standard method, where independence between entry time and failure time is assumed. We apply this method to mortality analysis among atomic bomb survivors defined in a geographical study region.     

Comparison of confidence intervals for adjusted attributable risk estimates under multinomial sampling

The epidemiologic concept of the adjusted attributable risk is a useful approach to quantitatively describe the importance of risk factors on the population level. It measures the proportional reduction in disease probability when a risk factor is eliminated from the population, accounting for effects of confounding and effect-modification by nuisance variables. The computation of asymptotic variance estimates for estimates of the adjusted attributable risk is often done by applying the delta method. Investigations on the delta method have shown, however, that the delta method generally tends to underestimate the standard error, leading to biased confidence intervals. We compare confidence intervals for the adjusted attributable risk derived by applying computer intensive methods like the bootstrap or jackknife to confidence intervals based on asymptotic variance estimates using an extensive Monte Carlo simulation and within a real data example from a cohort study in cardiovascular disease epidemiology. Our results show that confidence intervals based on bootstrap and jackknife methods outperform intervals based on asymptotic theory. Best variants of computer intensive confidence intervals are indicated for different situations.     

Exact and Asymptotic Power Determination For Dose‐Response Studies With Ordinal Response

In this article, we describe a conditional score test for detecting a monotone dose‐response relationship with ordinal response data. We consider three different versions of this test: asymptotic, conditional exact, and mid‐P conditional score test. Exact and asymptotic power formulae based on these tests will be studied. Asymptotic sample size formulae based on the asymptotic conditional score test will be derived. The proposed formulae are applied to a vaccination study and a developmental toxicity study for illustrative purposes. Actual significance level and exact power properties of these tests are compared in a small empirical study. The mid‐P conditional score test is observed to be the most powerful test with actual significance level close to the pre‐specified nominal level.     

Size Matters: Individual Variation in Ectotherm Growth and Asymptotic Size

Body size, and, by extension, growth has impacts on physiology, survival, attainment of sexual maturity, fecundity, generation time, and population dynamics, especially in ectotherm animals that often exhibit extensive growth following attainment of sexual maturity. Frequently, growth is analyzed at the population level, providing useful population mean growth parameters but ignoring individual variation that is also of ecological and evolutionary significance. Our long-term study of Lake Erie Watersnakes, Nerodia sipedon insularum, provides data sufficient for a detailed analysis of population and individual growth. We describe population mean growth separately for males and females based on size of known age individuals (847 captures of 769 males, 748 captures of 684 females) and annual growth increments of individuals of unknown age (1,152 males, 730 females). We characterize individual variation in asymptotic size based on repeated measurements of 69 males and 71 females that were each captured in five to nine different years. The most striking result of our analyses is that asymptotic size varies dramatically among individuals, ranging from 631–820 mm snout-vent length in males and from 835–1125 mm in females. Because female fecundity increases with increasing body size, we explore the impact of individual variation in asymptotic size on lifetime reproductive success using a range of realistic estimates of annual survival. When all females commence reproduction at the same age, lifetime reproductive success is greatest for females with greater asymptotic size regardless of annual survival. But when reproduction is delayed in females with greater asymptotic size, lifetime reproductive success is greatest for females with lower asymptotic size when annual survival is low. Possible causes of individual variation in asymptotic size, including individual- and cohort-specific variation in size at birth and early growth, warrant further investigation.     

An asymptotic maximum principle for essentially linear evolution models

Recent work on mutation-selection models has revealed that, under specific assumptions on the fitness function and the mutation rates, asymptotic estimates for the leading eigenvalue of the mutation-reproduction matrix may be obtained through a low-dimensional maximum principle in the limit N-->infinity (where N, or N(d) with d> or =1, is proportional to the number of types). In order to extend this variational principle to a larger class of models, we consider here a family of reversible matrices of asymptotic dimension N(d) and identify conditions under which the high-dimensional Rayleigh-Ritz variational problem may be reduced to a low-dimensional one that yields the leading eigenvalue up to an error term of order 1/N. For a large class of mutation-selection models, this implies estimates for the mean fitness, as well as a concentration result for the ancestral distribution of types.     

Maximum Likelihood Methods for Fitting the Burr Type XII Distribution to Life Test Data

This paper describes mathematical and computational methodology for estimating the parameters of the Burr Type XII distribution by the method of maximum likelihood. Expressions for the asymptotic variances and covariances of the parameter estimates are given, and the modality of the log-likelihood and conditional log-likelihood functions is analyzed. As a result of this analysis for various a priori known and unknown parameter combinations, conditions are given which guarantee that the parameter estimates obtained will, indeed, be maximum likelihood estimates. An efficient numerical method for maximizing the conditional log-likelihood function is described, and mathematical expressions are given for the various numerical approximations needed to evaluate the expressions given for the asymptotic variances and covariances of the parameter estimates. The methodology discussed is applied in a numerical example to life test data arising in a clinical setting.     

A stochastic gompertz model with hereditary effect

Summary In this paper we have studied a stochastic version of the Gompertz model for population growth of a single species after incorporating the aspect of heredity. Various statistical characteristics-the mean-value function, covariance-kernel, etc.-are evaluated for a delta-correlated process and their asymptotic values obtained. The effect of the hereditary kernel on the various statistics is discussed and it is found that it is to shift the distribution towards the origin.