Notes on the two-stage procedure under an incomplete block two-period crossover design

Abstract

Under an incomplete block crossover design with two periods, we derive the least-squares estimators for the period effect, treatment effects and carry-over effects in explicit formulae based on within-patient differences. Using the commonly-used strategy of searching a base model for making inferences in regression analysis, we define a two-stage test procedure in studying treatment effects. On the basis of Monte Carlo simulation, we evaluate the performance of the two-stage procedure for hypothesis testing, point and interval estimation of treatment effects in a variety of situations. We note that use of the two-stage procedure can be potentially misleading and hence one should not apply a test procedure to exclusively determine whether he/she needs to account for the carry-over effect in studying treatment effects. We use the double-blind crossover trial comparing two different doses of formoterol with placebo on the forced expiratory volume in 1 second (FEV₁) readings to illustrate the use of the two-stage procedure, as well as the distinction between use of two-stage procedure and the approach with assuming no carry-over effects based on one's subjective knowledge.     

Some New Bounds on the Probability of Extinction of a Galton–Watson Process with Numerical Comparisons

Some new upper and lower bounds for the extinction probability of a Galton–Watson process are presented. They are very easy to compute and can be used even if the offspring distribution has infinite variance. These new bounds are numerically compared to previously discussed bounds. Some definite guidelines are given concerning when these new bounds are preferable. Some open problems are also discussed.     

A Multiple Comparison Procedure Based on a Variant of the Schwarz Information Criterion in a Mixed Model

Repeated measurements are collected in a variety of situations and are generally characterized by a mixed model where the correlation within the subject is specified by the random effects. In such a mixed model, we propose a multiple comparison procedure based on a variant of the Schwarz information criterion (SIC; Schwarz, 1978 Schwarz , G. ( 1978 ). Estimating the dimension of a model . Ann. Statist. 6 : 461 – 464 .[Crossref], [Web of Science ®] , [Google Scholar]). The derivation of SIC indicates that SIC serves as an asymptotic approximation to a transformation of the Bayesian posterior probability of a candidate model. Therefore, an approximated posterior probability for a candidate model can be calculated based upon SIC. We suggest a variant of SIC which includes the terms which are asymptotically negligible in the derivation of SIC. The variant improves upon the performance of SIC in small and moderate sample-size applications. Based upon the proposed variant, the corresponding posterior probability can be calculated for each candidate model. A hypothesis testing for multiple comparisons involves one or more models in the candidate class, the posterior probability of the hypothesis testing is therefore evaluated as the sum of the posterior probabilities for the models associated with the testing. The approximated posterior probability based on the variant accommodates the effect of the prior on each model in the candidate class, and therefore is more effectively approximated than that based on SIC for conducting multiple comparisons. We derive the computational formula of the approximated posterior probability based on the variant in the mixed model. The applications in two real data sets demonstrate that the proposed procedure based on the SIC variant can perform effectively in multiple comparisons.     

A Bayesian Test for the Mean of the Power Exponential Distribution

In this article, we deal with the problem of testing a point null hypothesis for the mean of a multivariate power exponential distribution. We study the conditions under which Bayesian and frequentist approaches can match. In this comparison it is observed that the tails of the model are the key to explain the reconciliability or irreconciliability between the two approaches.     

Comparison of Confidence Intervals for a Poisson Mean – Further Considerations

I make recommendations in choosing a confidence interval for the Poisson mean, from twelve different methods, that are based on four general principles: actual coverage should closely match the nominal coverage; narrower expected widths of confidence intervals are better; the right and left non-coverage should be fairly balanced; and some investigators may prefer closed-form intervals. The interval chosen depends on the relative importance the investigator places on each of these principles. The confidence intervals are examined through graphs of their coverage probability, interval widths and shapes.     

Empirical Likelihood Confidence Region for the Parameter in a Partially Linear Errors-in-Variables Model

In this article, a partially linear errors-in-variables model is considered, and empirical log-likelihood ratio statistic for the unknown parameter in the model is suggested. It is proved that the proposed statistic is asymptotically standard chi-square distribution under some suitable conditions, and hence it can be used to construct the confidence region of the parameter. A simulation study indicates that, in terms of coverage probabilities and average lengths of the confidence intervals, the proposed method performs better than the least-squares method.     

The Asymptotic Estimate of Ruin Probability Under a Class of Risk Model in the Presence of Heavy Tails

In contrast with the classical Cramér–Lundberg model where the premium process is a linear function of time, we consider the ruin probability under the risk model where the aggregate premium consists of both a compound Poisson process and a linear process of time. Moreover, a constant interest force is also taken into account in our model. We restrict ourselves to the case where the claim size is heavy-tailed, i.e., the equilibrium distribution function of the claim size belongs to a wide subclass of the subexponential distribution. An asymptotic formula for the ruin probability is obtained by using the similar method of Kalashnikov and Konstantinides (2000 Kalashnikov , V. , Konstantinides , D. ( 2000 ). Ruin under interest force and subexponential claims: a simple treament . Insur. Math. Econ. 27 : 145 – 149 .[Crossref], [Web of Science ®] , [Google Scholar]). The asymptotic formula we get here is the same as the one in Asmussen (1998 Asmussen , S. ( 1998 ). Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distribution and first passage probabilities . Ann. Appl. Probab. 8 : 354 – 374 .[Crossref], [Web of Science ®] , [Google Scholar]), Klüppelberg and Stadtmüller (1998 Klüppelberg , C. , Stadtmüller , U. ( 1998 ). Ruin probabilities in the presence of heavy-tails and interest rates . Scand. Actuarial J. 1 : 49 – 58 .[Taylor & Francis Online] , [Google Scholar]), and Kalashnikov and Konstantinides (2000 Kalashnikov , V. , Konstantinides , D. ( 2000 ). Ruin under interest force and subexponential claims: a simple treament . Insur. Math. Econ. 27 : 145 – 149 .[Crossref], [Web of Science ®] , [Google Scholar]) which did not consider the stochastic premium.     

A New Formula for Inclusion Probabilities in Median-Ranked Set Sampling

This article develops a new generalized formula to compute the inclusion probabilities of a median-ranked set sample in a finite population setting. The use of this formula is illustrated in a numerical example. Furthermore, the inclusion probabilities of a median-ranked set sample is compared with the inclusion probabilities of ranked set and simple random samples.     

Optimal Partitioning of Probability Distributions under General Convex Loss Functions in Selective Assembly

Selective assembly is an effective approach for improving the quality of a product which is composed of two mating components. This article studies optimal partitioning of the dimensional distributions of the components in selective assembly. It extends previous results for squared error loss function to cover general convex loss functions, including asymmetric convex loss functions. Equations for the optimal partition are derived. Assuming that the density function of the dimensional distribution is log-concave, uniqueness of solutions is established and some properties of the optimal partition are shown. Some numerical results compare the optimal partition with some heuristic partitioning schemes.