A theoretical analysis of the power of biased coin designs

Outlining some recently obtained results of Hu and Rosenberger [2003. Optimality, variability, power: evaluating response-adaptive randomization procedures for treatment comparisons. J. Amer. Statist. Assoc. 98, 671–678] and Chen [2006. The power of Efron's biased coin design. J. Statist. Plann. Inference 136, 1824–1835] on the relationship between sequential randomized designs and the power of the usual statistical procedures for testing the equivalence of two competing treatments, the aim of this paper is to provide theoretical proofs of the numerical results of Chen [2006. The power of Efron's biased coin design. J. Statist. Plann. Inference 136, 1824–1835]. Furthermore, we prove that the Adjustable Biased Coin Design [Baldi Antognini A., Giovagnoli, A., 2004. A new “biased coin design” for the sequential allocation of two treatments. J. Roy. Statist. Soc. Ser. C 53, 651–664] is uniformly more powerful than the other “coin” designs proposed in the literature for any sample size.     

The comparison of designs for sequential clinical trials with covariate information

Summary. The paper develops methods for the comparison of randomized rules of the biased coin type for the sequential allocation of treatments in a clinical trial. One important characteristic is the loss , which measures the increase in the variance of parameter estimates due to the imbalance caused by randomization. The other important characteristic is the selection bias measuring the probability of correctly guessing which treatment is to be allocated next. The combination of these two measures leads to the elucidation of admissible designs. Simulations provide clear plots of the behaviour of the designs and make it possible to distinguish good designs from those which are less good.     

Optimality of random allocation design for the control of accidental bias in sequential experiments

In comparing two treatments, suppose the suitable subjects arrive sequentially and must be treated at once. Known or unknown to the experimenter there may be nuisance factors systematically affecting the subjects. Accidental bias is a measure of the influence of these factors in the analysis of data. We show in this paper that the random allocation design minimizes the accidental bias among all designs that allocate n, out of 2n, subjects to each treatment and do not prefer either treatment in the assignment. When the final imbalance is allowed to be nonzero, optimal and efficient designs are given. In particular the random allocation design is shown to be very efficient in this broader setup.     

Implementing optimal allocation for sequential continuous responses with multiple treatments

In practice, it is important to find optimal allocation strategies for continuous response with multiple treatments under some optimization criteria. In this article, we focus on exponential responses. For a multivariate test of homogeneity, we obtain the optimal allocation strategies to maximize power while (1) fixing sample size and (2) fixing expected total responses. Then the doubly adaptive biased coin design [Hu, F., Zhang, L.-X., 2004. Asymptotic properties of doubly adaptive biased coin designs for multi-treatment clinical trials. The Annals of Statistics 21, 268–301] is used to implement the optimal allocation strategies. Simulation results show that the proposed procedures have advantages over complete randomization with respect to both inferential (power) and ethical standpoints on average. It is important to note that one can usually implement optimal allocation strategies numerically for other continuous responses, though it is usually not easy to get the closed form of the optimal allocation theoretically.     

Renewal theory and the sequential design of experiments with two states of nature

In the sequential design of experiments in experimentel is sequentially performing experiments to help him make an inference about the true state of nature. Using results from renewal twenty at derive approximations for the operations characteristics and average sample numbers for this problem when there are two states of nature.

A critical problem in the sequential design of experiments is finding a good procedure. We investigate a Bayesian formulation of this problem and use our approximations to approximate the Bayes risk. Minimization of this approximate Bayes risk over procedures is discussed as a method of finding a good procedure, but difficulties are encountered due to the discrete time character of the sequential process. To avoid these difficulties, we consider minimization of an approximation related to a diffusion process. This leads to a simple rule for the sequential selection of experiments.     

A comparison of two procedures to select the best binomial population with sequential elimination of inferior populations

We compare the selection procedure of Levin and Robbins [1981. Selecting the highest probability in binomial or multinomial trials. Proc. Nat. Acad. Sci. USA 78, 4663–4666.] with the procedure of Paulson [1994. Sequential procedures for selecting the best one of k Koopman–Darmois populations. Sequential Analysis 13, 207–220.] to identify the best of several binomial populations with sequential elimination of unlikely candidates. We point out situations in which the Levin–Robbins procedure dominates the Paulson procedure in terms of the duration of the experiment, the expected total number of observations, and the expected number of failures. Because the Levin–Robbins procedure is also easier to implement than Paulson's procedure and gives a tighter guarantee for the probability of correct selection, we conclude that it holds a competitive edge over Paulson's procedure.     

A class of simple approximate sequential tests for adaptive comparison of two treatments

We consider the problem of sequentially deciding which of two treatments is superior, A class of simple approximate sequential tests is proposed. These have the probabilities of correct selection approximately independent of the sampling rule and depending on unknown parameters only through the function of interest, such as the difference or ratio of mean responses. The tests are obtained by using a normal approximation, and this is employed to derive approximate expressions for the probabilities of correct selection and the expected sample sizes. A class of data-dependent sampling rules is proposed for minimizing any weighted average of the expected sample sizes on the two treatments, with the weights being allowed to depend on unknown parameters. The tests are studied in the particular cases of exponentially.     

Efficient non-parametric sequential procedures for comparing the rates of change of two treatments

Consider a longitudinal experiment where subjects are allocated to one of two treatment arms and are subjected to repeated measurements over time. Two non-parametric group sequential procedures, based on the Wilcoxon rank sum test and fitted with asymptotically efficient allocation rules, are derived to test the equality of the rates of change over time of the two treatments, when the distribution of responses is unknown. The procedures are designed to allow for early stopping to reject the null hypothesis while allocating less subjects to the inferior treatment. Simulations – based on the normal, the logistic and the exponential distributions – showed that the proposed allocation rules substantially reduce allocations to the inferior treatment, but at the expense of a relatively small increase in the total sample size and a moderate decrease in power as compared to the pairwise allocation rule.     

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.     

Inferences on the difference between future observations for comparing two treatments

The comparison of two treatments with normally distributed data is considered. Inferences are considered based upon the difference between single potential future observations from each of the two treatments, which provides a useful and easily interpretable assessment of the difference between the two treatments. These methodologies combine information from a standard confidence interval analysis of the difference between the two treatment means, with information available from standard prediction intervals of future observations. Win-probabilities, which are the probabilities that a future observation from one treatment will be superior to a future observation from the other treatment, are a special case of these methodologies. The theoretical derivation of these methodologies is based upon inferences about the non-centrality parameter of a non-central t-distribution. Equal and unequal variance situations are addressed, and extensions to groups of future observations from the two treatments are also considered. Some examples and discussions of the methodologies are presented.     

Identification of quantitative trait loci for canine hip dysplasia by two sequential multipoint linkage analyses

Canine hip dysplasia (CHD) is characterized by hip laxity and subluxation that can lead to hip osteoarthritis. Studies have shown the involvement of multiple genetic regions in the expression of CHD. Although we have associated some variants in the region of fibrillin 2 with CHD in a subset of dogs, no major disease-associated gene has been identified. The focus of this study is to identify quantitative trait loci (QTL) associated with CHD. Two sequential multipoint linkage analyses based on a reversible jump Markov chain Monte Carlo approach were applied on a cross-breed pedigree of 366 dogs. Hip radiographic trait (Norberg Angle, NA) on both hips of each dog was tested for linkage to 21,455 single nucleotide polymorphisms across 39 chromosomes. Putative QTL for the NA was found on 11 chromosomes (1, 2, 3, 4, 7, 14, 19, 21, 32, 36, and 39). Identification of genes in the QTL region(s) can assist in identification of the aberrant genes and biochemical pathways involving hip dysplasia in both dogs and humans.     

A latent Markov model for detecting patterns of criminal activity

Summary.  The paper investigates the problem of determining patterns of criminal behaviour from official criminal histories, concentrating on the variety and type of offending convictions. The analysis is carried out on the basis of a multivariate latent Markov model which allows for discrete covariates affecting the initial and the transition probabilities of the latent process. We also show some simplifications which reduce the number of parameters substantially; we include a Rasch-like parameterization of the conditional distribution of the response variables given the latent process and a constraint of partial homogeneity of the latent Markov chain. For the maximum likelihood estimation of the model we outline an EM algorithm based on recursions known in the hidden Markov literature, which make the estimation feasible also when the number of time occasions is large. Through this model, we analyse the conviction histories of a cohort of offenders who were born in England and Wales in 1953. The final model identifies five latent classes and specifies common transition probabilities for males and females between 5-year age periods, but with different initial probabilities.     

Central Limit Theorem approximations for the number of runs in Markov-dependent binary sequences

We consider Markov-dependent binary sequences and study various types of success runs (overlapping, non-overlapping, exact, etc.) by examining additive functionals based on state visits and transitions in an appropriate Markov chain. We establish a multivariate Central Limit Theorem for the number of these types of runs and obtain its covariance matrix by means of the recurrent potential matrix of the Markov chain. Explicit expressions for the covariance matrix are given in the Bernoulli and a simple Markov-dependent case by expressing the recurrent potential matrix in terms of the stationary distribution and the mean transition times in the chain. We also obtain a multivariate Central Limit Theorem for the joint number of non-overlapping runs of various sizes and give its covariance matrix in explicit form for Markov dependent trials.     

A note on new confidence intervals for the difference between two proportions based on an Edgeworth expansion

Zhou and Qin [2004. New intervals for the difference between two independent binomial proportions. J. Statist. Plann. Inference 123, 97–115; 2005. A new confidence interval for the difference between two binomial proportions of paired data. J. Statist. Plann. Inference 128, 527–542] “new confidence intervals” for the difference between two treatment proportions exhibit a severe lack of invariance property that is a compelling reason not to use them.     

To split or not to split the US Ninth Circuit Court of Appeals: A simple statistical argument,counterargument, and critique

A statistical argument was presented quite recently, as an Op-Ed in The Los Angeles Times, purporting to conclude that the US Ninth Circuit Court of Appeals should be split. A corresponding statistical counterargument and critique are introduced in the present paper, by making use of some combinatorial formulas and by proposing an alternative empirical approach. In addition, with this issue of the Journal of Statistical Planning and Inference dedicated to Ted Anderson, I also include here comments on his contributions to both statistical inference about Markov chains and to statistical inference about latent classes.     

A class of confidence intervals for sequential phase II clinical trials with binary outcome

The phase II clinical trials often use the binary outcome. Thus, accessing the success rate of the treatment is a primary objective for the phase II clinical trials. Reporting confidence intervals is a common practice for clinical trials. Due to the group sequence design and relatively small sample size, many existing confidence intervals for phase II trials are much conservative. In this paper, we propose a class of confidence intervals for binary outcomes. We also provide a general theory to assess the coverage of confidence intervals for discrete distributions, and hence make recommendations for choosing the parameter in calculating the confidence interval. The proposed method is applied to Simon's [14] optimal two-stage design with numerical studies. The proposed method can be viewed as an alternative approach for the confidence interval for discrete distributions in general.     

A Fixed Point Approach to the Classification of Markov Chains with a Tree Structure

In this paper, we study the classification problem of discrete time and continuous time Markov processes with a tree structure. We first show some useful properties associated with the fixed points of a nondecreasing mapping. Mainly we find the conditions for a fixed point to be the minimal fixed point by using fixed point theory and degree theory. We then use these results to identify conditions for Markov chains of M/G/1 type or GI/M/1 type with a tree structure to be positive recurrent, null recurrent, or transient. The results are generalized to Markov chains of matrix M/G/1 type with a tree structure. For all these cases, a relationship between a certain fixed point, the matrix of partial differentiation (Jacobian) associated with the fixed point, and the classification of the Markov chain with a tree structure is established. More specifically, we show that the Perron-Frobenius eigenvalue of the matrix of partial differentiation associated with a certain fixed point provides information for a complete classification of the Markov chains of interest.     

Optimal allocation of time points for the random effects model

In this article the problem of the optimal selection and allocation of time points in repeated measures experiments is considered. D‐ optimal designs for linear regression models with a random intercept and first order auto‐regressive serial correlations are computed numerically and compared with designs having equally spaced time points. When the order of the polynomial is known and the serial correlations are not too small, the comparison shows that for any fixed number of repeated measures, a design with equally spaced time points is almost as efficient as the D‐ optimal design. When, however, there is no prior knowledge about the order of the underlying polynomial, the best choice in terms of efficiency is a D‐ optimal design for the highest possible relevant order of the polynomial. A design with equally‐spaced time points is the second best choice     

A class of small deviation theorems for the random variables associated with mth-order asymptotic circular Markov chains

ABSTRACT

In this article, we study a class of small deviation theorems for the random variables associated with mth-order asymptotic circular Markov chains. First, the definition of mth-order asymptotic circular Markov chain is introduced, then by applying the known results of the limit theorem for mth-order non homogeneous Markov chain, the small deviation theorem on the frequencies of occurrence of states for mth-order asymptotic circular Markov chains is established. Next, the strong law of large numbers and asymptotic equipartition property for this Markov chains are obtained. Finally, some results of mth-order nonhomogeneous Markov chains are given.