Book Reviews

Books Reviewed:
D.J. Daley and J. Gani, Epidemic Modelling: An Introduction
H. Thorisson, Coupling, Stationarity, and Regeneration
Chris J. Lloyd, Statistical Analysis of Categorical Data
Warren Gilchrist, Statistical Modelling with Quantile Functions
D.R. Cox and N. Reid, The Theory of the Design of Experiments     

Random stopping sets in a sequential analysis of random measures and fields

A random stopping set is defined and some of its properties are proved. Namely, we prove the theorem on the absolute continuity of measures on a σ-algebra F_τ connected with a random stopping set τ, which can be applied to the sequential analysis of random measures and fields.     

Bayesian prediction for two-stage sequential analysis in clinical trials

This article deals with a Bayesian predictive approach for two-stage sequential analyses in clinical trials, applied to both frequentist and Bayesian tests. We propose to make a predictive inference based on the notion of satisfaction index and the data accrued so far together with future data. The computations and the simulation results concern an inferential problem, related to the binomial model.     

The size and power of analytical amd bootstrap corrections to score tests in regression models

The paper invsetigation the size and power of analyticaland computer-based (bootstrap) 8nite=saniple adjustments to score tests in a class of regression models. Our results show that score tests can display substantial size distortions and that small-sample adjustments can be quite effective. Bootstrap corrections typically perform slightly better than analytical ones     

On the sample size of curtailed tests

Formulas are given for the asymptotic distribution, mean, and variance of m^-1N_m,where NN_m is the random sample size of the curtailed version of a fixed-sample most powerful test based on sample size m. The adequacy of the formulas is numerically investigated in some important applications where exact formulas can also be derived     

On Comparison of Stopping Times in Sequential Procedures for Exponential Families of Stochastic Processes

For curved ( k + 1), k -exponential families of stochastic processes a natural and often studied sequential procedure is to stop observation when a linear combination of the coordinates of the canonical process crosses a prescribed level. For such procedures the model is, approximately or exactly, a non-curved exponential family. Subfamilies of these stopping rules defined by having the same Fisher (expected) information are considered. Within a subfamily the Bartlett correction for a point hypothesis is also constant. Methods for comparing the durations of the sampling periods for the stopping rules in such a subfamily are discussed. It turns out that some stopping times tend to be smaller than others. For exponential families of diffusions and of counting processes the probability that one such stopping time is smaller than another can be given explicity. More generally, an Edgeworth expansion of this probability is given     

Randomized confidence intervals of a parameter for a family of discrete exponential type distributions

For a family of one-parameter discrete exponential type distributions, the higher order approximation of randomized confidence intervals derived from the optimum test is discussed. Indeed, it is shown that they can be asymptotically constructed by means of the Edgeworth expansion. The usefulness is seen from the numerical results in the case of Poisson and binomial distributions.     

Approximations of the Distributions of Test Statistics for Homogeneity of a Product Multinomial Model

Statistics R ^a based on power divergence can be used for testing the homogeneity of a product multinomial model. All R ^a have the same chi-square limiting distribution under the null hypothesis of homogeneity. R ⁰ is the log likelihood ratio statistic and R ¹ is Pearson's X ² statistic. In this article, we consider improvement of approximation of the distribution of R ^a under the homogeneity hypothesis. The expression of the asymptotic expansion of distribution of R ^a under the homogeneity hypothesis is investigated. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, a new approximation of the distribution of R ^a is proposed. A moment-corrected type of chi-square approximation is also derived. By numerical comparison, we show that both of the approximations perform much better than that of usual chi-square approximation for the statistics R ^a when a ≤ 0, which include the log likelihood ratio statistic.     

Fixed-width sequential confidence interval for the correlation coefficient of a bivariate normal distribution

A sequential confidence interval of fixed width 2d d > 0, is constructed for the correlation coefficient of a bivariate normal distribution. It is shown that the coverage probability is approximately equal to a preassigned number γ, 0 < γ < as d → 0.     

Asymptotic expansions of the distributions of the chi-square statistic based on the asymptotically distribution-free theory in covariance structures

An asymptotic expansion of the null distribution of the chi-square statistic based on the asymptotically distribution-free theory for general covariance structures is derived under non-normality. The added higher-order term in the approximate density is given by a weighted sum of those of the chi-square distributed variables with different degrees of freedom. A formula for the corresponding Bartlett correction is also shown without using the above asymptotic expansion. Under a fixed alternative hypothesis, the Edgeworth expansion of the distribution of the standardized chi-square statistic is given up to order O(1/n). From the intermediate results of the asymptotic expansions for the chi-square statistics, asymptotic expansions of the joint distributions of the parameter estimators both under the null and fixed alternative hypotheses are derived up to order O(1/n).     

A Bayesian approach to sequential analysis in post‐licensure vaccine safety surveillance

With rapid development of computing technology, Bayesian statistics have increasingly gained more attention in various areas of public health. However, the full potential of Bayesian sequential methods applied to vaccine safety surveillance has not yet been realized, despite acknowledged practical benefits and philosophical advantages of Bayesian statistics. In this paper, we describe how sequential analysis can be performed in a Bayesian paradigm in the field of vaccine safety. We compared the performance of the frequentist sequential method, specifically, Maximized Sequential Probability Ratio Test (MaxSPRT), and a Bayesian sequential method using simulations and a real world vaccine safety example. The performance is evaluated using three metrics: false positive rate, false negative rate, and average earliest time to signal. Depending on the background rate of adverse events, the Bayesian sequential method could significantly improve the false negative rate and decrease the earliest time to signal. We consider the proposed Bayesian sequential approach to be a promising alternative for vaccine safety surveillance.     

A note regarding meta‐analysis of sequential trials with stopping for efficacy

It is shown that fixed‐effect meta‐analyses of naïve treatment estimates from sequentially run trials with the possibility of stopping for efficacy based on a single interim look are unbiassed (or at the very least consistent, depending on the point of view) provided that the trials are weighted by information provided. A simple proof of this is given. An argument is given suggesting that this also applies in the case of multiple looks. The implications for this are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Strategies for a sequential selection of the better of two trinomial populations

A sequential procedure for a selection of the better of two trinomial populations has been proposed by ?idók (1988). The present paper shows some Monte Carlo results for 4 different strategies of sequential experimentation in this procedure, on this basis compares the strategies, and gives some practical recommendations for choosing the strategy.     

Discrimination analysis when the variates are grouped and observed in sequential order

Suppose that measurements Math', i = l,....,k, are consecutively taken on an individual at the prescribed costs C_i, i = l,....,k. the individual comes from one of the two populations H₁ and H₂, and it is desired to detect which population the individual belongs to. Given the loss incurreed in selecting population H_r when in fact it belongs to H_s, the prior probability P_r of H_r (r = 1,2), and assuming that H_r has the normal distribution N(µ_r, V), r = 1,2, we derive the sequential Bayesian solution of the discrimination problem when µ₁, µ₂ and V are known. When µ_r, V are unknown and must be estimated, we propose a solution which is asymptotic Bayesian with exponential convergence rate.     

Incorporating scientific,ethical and economic considerations into the design of clinical trials in the pharmaceutical industry: a sequential approach

A flexible sequential approach to the design of clinical trials is discussed herein. This approach is based on a “confidence sequence” viewpoint instead of the rigid stopping and terminal decision rules in conventional sequential testing theory. By using an appropriate confidence sequence, one can always ensure a prescribed degree of scientific rigor (confidence) in establishing the drug to be effective. Moreover, one also has the option of terminating the trial early when there is already enough statistical evidence for concluding that the drug is effective, or when the drug shows uniorseen harmful effects, or when the data predict that there is little chance of arriving at a definitive conclusion in favor of the drug by the scheduled end of the trial. We discuss how these and other ethical and economic considerations can be readily incorporated into the stopping criteria of the trial.     

Use of asymmetric loss functions in sequential estimation problems for multiple linear regression

When estimating in a practical situation, asymmetric loss functions are preferred over squared error loss functions, as the former is more appropriate than the latter in many estimation problems. We consider here the problem of fixed precision point estimation of a linear parametric function in beta for the multiple linear regression model using asymmetric loss functions. Due to the presence of nuissance parameters, the sample size for the estimation problem is not known beforehand and hence we take the recourse of adaptive multistage sampling methodologies. We discuss here some multistage sampling techniques and compare the performances of these methodologies using simulation runs. The implementation of the codes for our proposed models is accomplished utilizing MATLAB 7.0.1 program run on a Pentium IV machine. Finally, we highlight the significance of such asymmetric loss functions with few practical examples.     

Minimum average partial correlation and parallel analysis: The influence of oblique structures

ABSTRACT

Parallel analysis (Horn 1965) and the minimum average partial correlation (MAP; Velicer 1976) have been widely spread as optimal solutions to identify the correct number of axes in principal component analysis. Previous results showed, however, that they become inefficient when variables belonging to different components strongly correlate. Simulations are used to assess their power to detect the dimensionality of data sets with oblique structures. Overall, MAP had the best performances as it was more powerful and accurate than PA when the component structure was modestly oblique. However, both stopping rules performed poorly in the presence of highly oblique factors.     

The Cornish-Fisher expansion for a class of statistics in first order autoregression

ABSTRACT

This article is concerned with the derivation and study of the Cornish-Fisher expansion for a wide class of estimators of the parameter in the first order autoregressive process. Second and third order Cornish-Fisher approximations to the quantile of the distribution of the corresponding asymptotically normal standardized statistic are stated explicitly and their accuracy is examined, both theoretically and numerically, by comparing them with the exact value of the quantile obtained by Monte Carlo simulation.     

Symbolic computation for approximating distributions of some families of one and two-sample nonparametric test statistics

A conditional saddlepoint approximation was provided by Gatto and Jammalamadaka (1999) for computing the distribution function of many test statistics based on dependent quantities like multinomial frequencies, spacing frequencies, etc. The considerable complexity of the formulas involved can be bypassed by symbolic computation. This article illustrates the effectiveness of symbolic computation to evaluate the saddlepoint approximation for the likelihood ratio, the exponential score, and the Wald-Wolfowitz test statistics. The case of composite hypotheses is also discussed.     

New Confidence Intervals for the Proportion of Interest in One-Sample Correlated Binary Data

Correlated binary data is obtained in many fields of biomedical research. When constructing a confidence interval for the proportion of interest, asymptotic confidence intervals have already been developed. However, such asymptotic confidence intervals are unreliable in small samples. To improve the performance of asymptotic confidence intervals in small samples, we obtain the Edgeworth expansion of the distribution of the studentized mean of beta-binomial random variables. Then, we propose new asymptotic confidence intervals by correcting the skewness in the Edgeworth expansion in one direct and two indirect ways. New confidence intervals are compared with the existing confidence intervals in simulation studies.