A BAYESIAN METHOD FOR THE CHOICE OF THE SAMPLE SIZE IN EQUIVALENCE TRIALS

In this paper we consider a Bayesian predictive approach to sample size determination in equivalence trials. Equivalence experiments are conducted to show that the unknown difference between two parameters is small. For instance, in clinical practice this kind of experiment aims to determine whether the effects of two medical interventions are therapeutically similar. We declare an experiment successful if an interval estimate of the effects‐difference is included in a set of values of the parameter of interest indicating a negligible difference between treatment effects (equivalence interval). We derive two alternative criteria for the selection of the optimal sample size, one based on the predictive expectation of the interval limits and the other based on the predictive probability that these limits fall in the equivalence interval. Moreover, for both criteria we derive a robust version with respect to the choice of the prior distribution. Numerical results are provided and an application is illustrated when the normal model with conjugate prior distributions is assumed.     

A REVIEW OF BOUNDS FOR THE EFFICIENCY FACTOR OF BLOCK DESIGNS

This paper draws together bounds for the efficiency factor of block designs, starting with the papers of Conniffe & Stone (1974) and Williams & Patterson (1977). By extending the methods of Jarrett (1983), firstly to cover supercomplete block designs and then to cover resolvable designs, a set of bounds is obtained which provides the best current bounds for any block design with equal replication and equal block size, including resolvable designs and two-replicate resolvable designs as special cases. The bounds given for non-resolvable designs apply strictly only to designs which are either regular-graph (John & Mitchell, 1977) or whose duals are regular-graph. It is conjectured (John & Williams, 1982) that they are in fact global bounds. Similar qualifications apply to the bounds for resolvable designs.     

LOWER BOUNDS FOR THE DIVERGENCE OF DIRECTIONAL AND AXIAL ESTIMATORS

Consider estimation of a unit vector parameter a in two classes of distributions. In the first, α is a direction. In the second, α is an axis, so that –α and α are equivalent: the aim is to obtain the projector αα^t. In each case the paper uses first principles to define measures of the divergence of such estimators and derives lower bounds for them. These bounds are computed explicitly for the Fisher-Von Mises and Scheidegger-Watson densities on the g-dimensional sphere, ω_q. In the latter case, the tightness of the bound is established by simulations.     

LOWER BOUNDS FOR THE NORMAL APPROXIMATION OF A SUM OF INDEPENDENT RANDOM VARIABLES

The rates of convergence to the normal distribution are investigated for a sum of independent random variables. Using Stein's method, we derive a lower bound of the uniform distance between two distributions of independent sum and normal.     

THE OPTIMAL TEST FOR SELECTING THE GREATER OF TWO BINOMIAL PROBABILITIES

The efficiency of a sequential test is related to the “importance” of the trials within the test. This relationship is used to find the optimal test for selecting the greater of two binomial probabilities, p_α and p_b, namely, the stopping rule is “gambler's ruin” and the optimal discipline when p_α+p_b≤ 1 (≥ 1) is play-the-winner (loser), i.e. an α-trial which results in a success is followed by an α-trial (b-trial) whereas an α-trial which results in a failure is followed by α b-trid (α-trial) and correspondingly for b-trials.     

SHARP BOUNDS FOR EXPECTATIONS OF kTH RECORD INCREMENTS

This paper presents sharp bounds for expectations of non‐adjacent increments of kth record statistics, measured in various scale units, for a sequence of independent identically distributed random variables with continuous cumulative distribution function. The results for kth record spacings are considered as special cases. The paper also characterizes probability distributions for which the bounds are attained.     

BOUNDS ON THE BIAS OF WINSORIZED MEANS

This paper provides sharp upper bounds on the bias of the Winsorized mean as an estimator of the population mean in a nonparametric setting. The resulting bounds are numerically evaluated. Further, the probability distributions attaining the bounds are determined.     

BOOTSTRAP TESTS FOR THE ERROR DISTRIBUTION IN LINEAR AND NONPARAMETRIC REGRESSION MODELS

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and non‐parametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution‐free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness‐of‐fit testing of assumptions regarding the error distribution in linear and non‐parametric regression models.     

THE GEOMETRY OF SPECIFICATION ERROR

This paper discusses the traditional specification problem from a geometric (or co-ordinate-free) viewpoint. While the traditional emphasis is on the properties of estimators, the geometric approach also allows an easy development of corresponding results for inference. Errors arising from artificial inclusion or exclusion of variables are considered in terms of augmentations or restrictions on a given maintained hypothesis, and this allows a corresponding interpretation of tests based upon the Wald and Lagrange Multiplier Principles. It is demonstrated that biases arising from incorrect exclusion of variables do not invalidate the traditional F-test.     

THE CONVERGENCE RATE OF SEQUENTIAL FIXED-WIDTH CONFIDENCE INTERVALS FOR REGULAR FUNCTIONALS

Let X₁X₂,.be i.i.d. random variables and let U_n= (n r)^-1S?^(n,r) h (X_i1,., X_ir,) be a U-statistic with EU_n= v, v unknown. Assume that g(X₁) =E[h(X₁,.,X_r) - v |X₁]has a strictly positive variance s?². Further, let a be such that φ(a) - φ(-a) =α for fixed α, 0 < α < 1, where φ is the standard normal d.f., and let S²_n be the Jackknife estimator of n Var U_n. Consider the stopping times N(d)= min {n: S²_n: + n^-1≤²a^-2},d > 0, and a confidence interval for v of length 2d,of the form I_n,d= [U_n,-d, Un + d]. We assume that Var U_n is unknown, and hence, no fixed sample size method is available for finding a confidence interval for v of prescribed width 2d and prescribed coverage probability α Turning to a sequential procedure, let I_N(d),d be a sequence of sequential confidence intervals for v. The asymptotic consistency of this procedure, i.e. lim_{d → 0}P(v ∈ I_N(d),d)=α follows from Sproule (1969). In this paper, the rate at which |P(v ∈ I_N(d),d) converges to α is investigated. We obtain that |P(v ∈ I_N(d),d) - α| = 0 (d^{1/2-(1+k)/2(1+m)}), d → 0, where K = max {0,4 - m}, under the condition that E|h(X₁, X_r)|^m < ∞m > 2. This improves and extends recent results of Ghosh & DasGupta (1980) and Mukhopadhyay (1981).     

LOCATION OF THE OPTIMAL SAMPLING POINT FOR THE QUALITY ASSESSMENT OF CONTINUOUS STREAMS

The paper makes an appraisal of the most appropriate sampling point for situations where a single sample must be used to estimate the mean flow of a continuous stream during a set time interval. Taking ‘optimal’ to mean the point at which the estimation error variance is minimised, optimal sampling locations are obtained for constant, linear and exponential flow rates when the process variogram is assumed linear or exponential. Numerical results illustrate the significance of failing to sample at the optimal point.     

ON THE EQUIVALENCE OF SOME INDICES OF SIMILARITY: IMPLICATION FOR BINARY PRESENCE/ABSENCE DATA

Cohen’s kappa, a special case of the weighted kappa, is a chance‐corrected index used extensively to quantify inter‐rater agreement in validation and reliability studies. In this paper, it is shown that in inter‐rater agreement for 2 × 2 tables, for two raters having the same number of opposite ratings, the weighted kappa, Cohen’s kappa, Peirce, Yule, Maxwell and Pilliner and Fleiss indices are identical. This implies that the weights in the weighted kappa are less important under such assumptions. Equivalently, it is shown that for two partitions of the same data set, resulting from two clustering algorithms having the same number of clusters with equal cluster sizes, these similarity indices are identical. Hence, an important characterisation is formulated relating equal numbers of clusters with the same cluster sizes to the presence/absence of a trait in a reliability study. Two numerical examples that exemplify the implication of this relationship are presented.     

Dynamic allocation as a balancing act

The CPMP Points to Consider on Adjustment for Baseline Covariates recently came into operation. As well as providing sound guidance, this document states a strong position against dynamic allocation. This paper reviews the most important issues involved and aims to stimulate interest in further investigating alternative allocation methods in clinical trials. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

An elementary approach to naturality,predictability, and the fundamental theorem of local martingales

We give non-technical proofs of the predictability of natural processes and the Fundamental Theorem of Local Martingales. Some arguments are shortened through a novel use of an elementary inequality known as “Chebyshev's Other Inequality”.     

ADAPTIVE GROUP SEQUENTIAL PROCEDURES FOR MULTIPLE COMPARISON OF TREATMENTS

ABSTRACT

We present a flexible group sequential procedure for comparing several treatments to a control. Though longitudinal data corresponding to a two stage mixed effects model are considered, ranges of application include any process with independent increments. The procedure allows the experimenter to drop the inferior treatments from the trial as soon as they are detected. It control strongly the familywise error rate. We also discuss a new error spending function (ESF) and study the performance of the procedure using various ESFs and time scales. Finally, the procedure is illustrated on a real example and implementation considerations are discussed.     

UPPER BOUNDS FOR THE HARMONIC MEAN, WITH AN APPLICATION TO EXPERIMENTAL DESIGN

For positive-valued random variables, the paper provides a sequence of upper bounds for the harmonic mean, the ith of these bounds being exact if and only if the random variable is essentially i-valued. Sufficient conditions for the convergence of the bounds to the harmonic mean are given. The bounds have a number of applications, particularly in experimental design where they may be used to check how close a given design is to A-optimality     

ERROR VARIANCE ESTIMATION FOR THE SINGLE‐INDEX MODEL

Single‐index models provide one way of reducing the dimension in regression analysis. The statistical literature has focused mainly on estimating the index coefficients, the mean function, and their asymptotic properties. For accurate statistical inference it is equally important to estimate the error variance of these models. We examine two estimators of the error variance in a single‐index model and compare them with a few competing estimators with respect to their corresponding asymptotic properties. Using a simulation study, we evaluate the finite‐sample performance of our estimators against their competitors.