Balanced incomplete Latin square designs

Latin squares have been widely used to design an experiment where the blocking factors and treatment factors have the same number of levels. For some experiments, the size of blocks may be less than the number of treatments. Since not all the treatments can be compared within each block, a new class of designs called balanced incomplete Latin squares (BILS) is proposed. A general method for constructing BILS is proposed by an intelligent selection of certain cells from a complete Latin square via orthogonal Latin squares. The optimality of the proposed BILS designs is investigated. It is shown that the proposed transversal BILS designs are asymptotically optimal for all the row, column and treatment effects. The relative efficiencies of a delete-one-transversal BILS design with respect to the optimal designs for both cases are also derived; it is shown to be close to 100%, as the order becomes large.     

Using partial ranking information in the design of small-sample comparisons

Random assignment of experimental units to treatment and control groups is a conventional device tob create unbiased comparisons. However, when sample sizes are small and the units differ considerably, there is a significant risk that randomization will create seriously unbalanced partitions of the units into treatment and control groups. We develop and evaluate an alternative to complete randomization for small-sample comparisons involving ordinal data with partial information on ranks of units. For instance, we might know that, of eight units, Rank (A) < Rank (C), Rank (A) < Rank (E) and Rank(D) < Rank(H). We develop an efficient computational procedure to use such information as the basis for restricted randomization of units to the treatment group. We compare our methods to complete randomization in the context of the Mann-Whitney test. With sufficient ranking information, the restricted randomization results in more powerful comparisons.     

A test of the mean square error criterion for linear admissible estimators

A test for choosing between a linear admissible estimator and the least squares estimator (LSE) is developed. A characterization of linear admissible estimators useful for comparing estimators is presented and necessary and sufficient conditions for superiority of a linear admissible estimator over the LS estimetor is derived for the test. The test is based on the MSE matrix superiority, but also new resl?!ts concerning covariance matrix comparisons of linear estimators are derived. Further,shown that the test of Toro - Vizcarrondo and Wailace applies iioi only the restricted least squares estimators but also to certain estimators outside this class.     

Hyper-Graeco-Latin Squares and Fractional Factorial Designs

A Latin square of order s is an arrangement of the s letters in an s × s square so that every letter appears exactly once in every row and exactly once in every column. Copeland and Nelson (2000) used two examples to show that a Latin square can be chosen such that it corresponds to a fractional factorial design. In this article, we are going to study this topic more precisely. Furthermore, we will explore the relationship between fractional factorial designs and hyper-Graeco-Latin squares in general, where s is a prime or a power of a prime.     

Properties of a randomization test for multifactor comparisons of groups

Multivariate hypothesis testing in studies of vegetation is likely to be hindered by unrealistic assumptions when based on conventional statistical methods. This can be overcome by randomization tests. In this paper, the accuracy and power of a MANOVA randomization test are evaluated for one and two factors with interaction with simulated data from three distributions. The randomization test is based on the partitioning of sum of squares computed from Euclidean distances. In one-factor designs, sample size and variance inequality were evaluated. The results showed a high level of accuracy. The power curve was higher with normal distribution, lower with uniform, intermediate with lognormal and was sensitive to variance inequality. In two-factor designs, three methods of permutations and two statistics were compared. The results showed that permutation of the residuals with F _pseudo is accurate and can give good power for testing the interaction and restricted permutation for testing main factors.     

Weighted re‐randomization tests for minimization with unbalanced allocation

Re‐randomization test has been considered as a robust alternative to the traditional population model‐based methods for analyzing randomized clinical trials. This is especially so when the clinical trials are randomized according to minimization, which is a popular covariate‐adaptive randomization method for ensuring balance among prognostic factors. Among various re‐randomization tests, fixed‐entry‐order re‐randomization is advocated as an effective strategy when a temporal trend is suspected. Yet when the minimization is applied to trials with unequal allocation, fixed‐entry‐order re‐randomization test is biased and thus compromised in power. We find that the bias is due to non‐uniform re‐allocation probabilities incurred by the re‐randomization in this case. We therefore propose a weighted fixed‐entry‐order re‐randomization test to overcome the bias. The performance of the new test was investigated in simulation studies that mimic the settings of a real clinical trial. The weighted re‐randomization test was found to work well in the scenarios investigated including the presence of a strong temporal trend. Copyright © 2013 John Wiley & Sons, Ltd.     

Shift in re‐randomization distribution with conditional randomization test

Proschan, Brittain, and Kammerman made a very interesting observation that for some examples of the unequal allocation minimization, the mean of the unconditional randomization distribution is shifted away from 0. Kuznetsova and Tymofyeyev linked this phenomenon to the variations in the allocation ratio from allocation to allocation in the examples considered in the paper by Proschan et al. and advocated the use of unequal allocation procedures that preserve the allocation ratio at every step. In this paper, we show that the shift phenomenon extends to very common settings: using conditional randomization test in a study with equal allocation. This phenomenon has the same cause: variations in the allocation ratio among the allocation sequences in the conditional reference set, not previously noted. We consider two kinds of conditional randomization tests. The first kind is the often used randomization test that conditions on the treatment group totals; we describe the variations in the conditional allocation ratio with this test on examples of permuted block randomization and biased coin randomization. The second kind is the randomization test proposed by Zheng and Zelen for a multicenter trial with permuted block central allocation that conditions on the within‐center treatment totals. On the basis of the sequence of conditional allocation ratios, we derive the value of the shift in the conditional randomization distribution for specific vector of responses and the expected value of the shift when responses are independent identically distributed random variables. We discuss the asymptotic behavior of the shift for the two types of tests. Copyright © 2013 John Wiley & Sons, Ltd.     

On balanced incomplete Latin square designs

Recently, balanced incomplete Latin square designs are introduced in the literature. We propose three methods of constructions of balanced incomplete Latin square designs. Particular classes of Latin squares namely Knut Vik designs, semi Knut Vik designs, and crisscross Latin squares play a key role in the construction.     

Stratification of randomization is not required for a pre‐specified subgroup analysis

Published literature and regulatory agency guidance documents provide conflicting recommendations as to whether a pre‐specified subgroup analysis also requires for its validity that the study employ randomization that is stratified on subgroup membership. This is an important issue, as subgroup analyses are often required to demonstrate efficacy in the development of drugs with a companion diagnostic. Here, it is shown, for typical randomization methods, that the fraction of patients in the subgroup given experimental treatment matches, on average, the target fraction in the entire study. Also, mean covariate values are balanced, on average, between treatment arms in the subgroup, and it is argued that the variance in covariate imbalance between treatment arms in the subgroup is at worst only slightly increased versus a subgroup‐stratified randomization method. Finally, in an analysis of variance setting, a least‐squares treatment effect estimator within the subgroup is shown to be unbiased whether or not the randomization is stratified on subgroup membership. Thus, a requirement that a study be stratified on subgroup membership would place an artificial roadblock to innovation and the goals of personalized healthcare. Copyright © 2012 John Wiley & Sons, Ltd.     

Admissibility of exact conditional tests of stochastic order

Consider the problem of testing for independence against stochastic order in a 2 × J contingency table, with two treatments and J ordered categories. By conditioning on the margins, the null hypothesis becomes simple. Careful selection of the conditional alternative hypothesis then allows the problem to be formulated as one of a class of problems for which the minimal complete class of admissible tests is known. The exact versions of many common tests, such as t-tests and the Smirnov test, are shown to be inadmissible, and thus the non-randomized versions are overly conservative. The proportional hazards and proportional odds tests are shown to be admissible for a given data set and size. A new result allows a proof of the admissibility of convex hull and adaptive tests for all data sets and sizes.     

Two-sample post-stratified or subgroup analysis with restricted randomization rules

In a clinical trial to compare two treatments, subjects may be allocated sequentially to treatment groups by a restricted randomization rule. Suppose that at the end of the trial, the investigator is interested in a post-stratified or subgroup analysis with respect to a particular demographic or clinical factor which was not selected prior to the trial for stratified randomization. Under a randomization model, large sample theory of two-sample post-stratified permutational tests is developed with a broad class of restricted randomization treatment allocation rules. The test procedures proposed here are illustrated with a real-life example. The results of this example indicate that it is not always possible to ignore the treatment rule used in the trial in the design-based analysis.     

Admissibility of linear estimators with respect to restricted parameter sets

In this paper the admissibility of a linear estimator for a linear regression parameter is characterized for such cases, where the considered parameter varies in an ellipsoid. We obtain a certain subset of the set of all linear estimators which are admissible with respect to the unrestricted parameter set. Furthermore, various linear estimators which have been proposed for improving the least squares estimator in cases of a restricted parameter set are investigated for admissibility. It turns out that only some shrunken estimators and some estimators of ridge type are admissible, whereas the KUKS-OLMAN estimator and all estimators of MARQUARDT type can be improved.     

Nonparametric rank-based test procedures for non-additive models in the two-way layout I. no replications

One of the major unresolved problems in the area of nonparametric statistics is the need for satisfactory rank-based test procedures for non-additive models in the two-way layout, especially when there is only one observation on each combination of the levels of the experimental factors. In this paper we consider an arbitrary non-additive model for the two-way layout with n levels of each factor. We utilize both alignment and ranking of the data together with basic properties of Latin squares to develop rank tests for interaction (non-additivity). Our technique involves first aligning within one of the main effects, ranking within the other main effects (columns and rows) and then adding the resulting ranks within “interaction bands” corresponding to orthogonal partitions of the interaction for the model, as denoted by the letters of an n × n Latin square. A Friedman-type statistic is then computed on the resulting sums. This is repeated for each of (n?1) mutually orthogonal Latin squares (thus accounting for all the interaction degrees of freedom). The resulting (n?1) Friedman-type statistics are finally combined to obtain an overall test statistic. The necessary null distribution tables for applying the proposed test for non-additivity are presented and we discuss the results of a Monte Carlo simulation study of the relative powers of this new procedure and other (parametric and nonparametric) procedures designed to detect interaction in a two-way layout with one observation per cell.     

A Generalized Diagonal Ridge-type Estimator in Linear Regression

This article introduces a general class of biased estimator, namely a generalized diagonal ridge-type (GDR) estimator, for the linear regression model when multicollinearity occurs. The estimator represents different kinds of biased estimators when different parameters are obtained. Some properties of this estimator are discussed and an iterative procedure is provided for selecting the parameters. A Monte Carlo simulation study and an application show that the GDR estimator performs much better than the ordinary least squares (OLS) estimator under the mean square error (MSE) criterion when severe multicollinearity is present.     

Effects of unstratified and centre‐stratified randomization in multi‐centre clinical trials

This paper deals with the analysis of randomization effects in multi‐centre clinical trials. The two randomization schemes most often used in clinical trials are considered: unstratified and centre‐stratified block‐permuted randomization. The prediction of the number of patients randomized to different treatment arms in different regions during the recruitment period accounting for the stochastic nature of the recruitment and effects of multiple centres is investigated. A new analytic approach using a Poisson‐gamma patient recruitment model (patients arrive at different centres according to Poisson processes with rates sampled from a gamma distributed population) and its further extensions is proposed. Closed‐form expressions for corresponding distributions of the predicted number of the patients randomized in different regions are derived. In the case of two treatments, the properties of the total imbalance in the number of patients on treatment arms caused by using centre‐stratified randomization are investigated and for a large number of centres a normal approximation of imbalance is proved. The impact of imbalance on the power of the study is considered. It is shown that the loss of statistical power is practically negligible and can be compensated by a minor increase in sample size. The influence of patient dropout is also investigated. The impact of randomization on predicted drug supply overage is discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

On finite-sample properties of adaptive least squares regression estimates

This paper is mainly concerned with deriving finite-sample properties of least squares estimators for the regression function in a nonparametric regression situation under some simplifying assumptions such as normally distributed errors with a common known variance. The selection of basis functions to be used for the construction of an estimator may be regarded as a smoothing problem, and will usually be done in a data-dependent way, A straightforward application of a result by P. J. Kernpthorne yields that, under a squared error loss, all selection procedures are admissible. Furthermore, the minimax approach provides an interpolating estimator, which is often impractical, Therefore, within a certain class of selection procedures an optimal one is determined using the minimax regret principle. It can be seen to behave similarly to the procedure minimizing either an unbiased risk estimator or, equivalently, the C_p-criterion.     

Use of simulation to compare the performance of minimization with stratified blocked randomization

Minimization is an alternative method to stratified permuted block randomization, which may be more effective at balancing treatments when there are many strata. However, its use in the regulatory setting for industry trials remains controversial, primarily due to the difficulty in interpreting conventional asymptotic statistical tests under restricted methods of treatment allocation. We argue that the use of minimization should be critically evaluated when designing the study for which it is proposed. We demonstrate by example how simulation can be used to investigate whether minimization improves treatment balance compared with stratified randomization, and how much randomness can be incorporated into the minimization before any balance advantage is no longer retained. We also illustrate by example how the performance of the traditional model-based analysis can be assessed, by comparing the nominal test size with the observed test size over a large number of simulations. We recommend that the assignment probability for the minimization be selected using such simulations.     

Localized classification

The main problem with localized discriminant techniques is the curse of dimensionality, which seems to restrict their use to the case of few variables. However, if localization is combined with a reduction of dimension the initial number of variables is less restricted. In particular it is shown that localization yields powerful classifiers even in higher dimensions if localization is combined with locally adaptive selection of predictors. A robust localized logistic regression (LLR) method is developed for which all tuning parameters are chosen data-adaptively. In an extended simulation study we evaluate the potential of the proposed procedure for various types of data and compare it to other classification procedures. In addition we demonstrate that automatic choice of localization, predictor selection and penalty parameters based on cross validation is working well. Finally the method is applied to real data sets and its real world performance is compared to alternative procedures.     

Low-order spatially-distinct latin squares

The order three to five spatially-distinct Latin squares, and the order three to six spatially-distinct Latin square treatment designs are listed. Some statistical results are given. Designs for 4, 5 and 6 treatments that were found previously to be robust to a linear by linear interacrion are shown to be optimal. Designs with good neighbour balanced are also considered.     

Multilevel maximum likelihood estimation with application to covariance matrices

The asymptotic variance of the maximum likelihood estimate is proved to decrease when the maximization is restricted to a subspace that contains the true parameter value. Maximum likelihood estimation allows a systematic fitting of covariance models to the sample, which is important in data assimilation. The hierarchical maximum likelihood approach is applied to the spectral diagonal covariance model with different parameterizations of eigenvalue decay, and to the sparse inverse covariance model with specified parameter values on different sets of nonzero entries. It is shown computationally that using smaller sets of parameters can decrease the sampling noise in high dimension substantially.