Comparison of designs for the three-fold nested random model

The quality of estimation of variance components depends on the design used as well as on the unknown values of the variance components. In this article, three designs are compared, namely, the balanced, staggered, and inverted nested designs for the three-fold nested random model. The comparison is based on the so-called quantile dispersion graphs using analysis of variance (ANOVA) and maximum likelihood (ML) estimates of the variance components. It is demonstrated that the staggered nested design gives more stable estimates of the variance component for the highest nesting factor than the balanced design. The reverse, however, is true in case of lower nested factors. A comparison between ANOVA and ML estimation of the variance components is also made using each of the aforementioned designs.     

Search for optimal designs in a three stagenested random model

In this paper we define a class of unbalanced designs, denoted by C_k,s,t, for estimating the components of variance in a k-stage nested random effects linear model. This class contains many of the designs proposed in the literature for nested components of variance models. We focus on the three-state model and discuss the determination of locally optimal designs within this class using a systematic computer search. For large sample sizes we show that approximate optimal designs may be obtained using a limit argument combined with numerical optimization. A comparison of our designs with previously published designs suggests that, in many cases, our designs result in substantial gains in efficiency.     

The three-fold nested random effects model

Estimation of the variance components and the mean of the balanced and unbalanced threefold nested design is considered. The relative merits of the following procedures are evaluated: Analysis of variance (ANOVA), maximum likelihood (ML), restricted maximum likelihood (REML), and minimum variance quadratic unbiased estimator (MIVQUE). A new procedure called the weighted analysis of means (WAM) estimator which utilizes prior information on the variance components is proposed. It is found to have optimum properties similar to the REML and MIVQUE, and it is also computationally simpler. For the mean, the overall sample average, grand mean, unweighted mean, and generalized least-squares (GLS) estimator with its weights obtained from the above estimators for the variance components are considered. Comparisons of the above procedures for the variance components and the mean are made from exact expressions for the biases and mean square errors (MSEs) of the estimators and from empirical investigations.     

Generalized staggered nested designs for variance components estimation

Staggered nested experimental designs are the most popular class of unbalanced nested designs in practical fields. The most important features of the staggered nested design are that it has a very simple open-ended structure and each sum of squares in the analysis of variance has almost the same degrees of freedom. Based on the features, a class of unbalanced nested designs that is a generalization of the staggered nested design is proposed in this paper. Formulae for the estimation of variance components and their sums are provided. Comparing the variances of the estimators to the staggered nested designs, it is found that some of the generalized staggered nested designs are more efficient than the traditional staggered nested design in estimating some of the variance components and their sums. An example is provided for illustration.     

The mad estimator: when is it non-linear? applications to two-way design models

The mean absolute deviation (MAD) estimator has recently received a great deal of attention as applied to full-rank linear regression models. This paper provides a necessary and sufficient condition for the MAD estimator to be a non-linear estimator, in which case conditions for the variance of the MAD estimator to be larger or smaller than those for OLS are, in general, unknown. The non-linearity of the MAD estimator is examined for several two-way designs; in particular (1) randomized block design (2) two-way nested design (3) two-way classification with interaction and (4) partially balanced incomplete block design     

Multiphase experiments with at least one later laboratory phase. II. Nonorthogonal designs

Principles and laws that apply to nonorthogonal multiphase experiments are developed and illustrated using examples that are nonorthogonal but structure‐balanced, not structure, but first‐order, balanced or unbalanced, thus exposing the differences between the different design types. The design of such experiments using standard designs, a catalogue of designs and computer searches is exemplified. Factor–allocation diagrams are employed to depict the allocations in the examples, and used in producing the anatomies of designs or, when possible, the related skeleton‐analysis‐of‐variance tables, to assess the properties of designs. The formulation of mixed models based on them is also described. Tools used for structure‐balanced experiments are also shown to be applicable to those experiments that are not.     

Asymptotic posterior distributions for balanced nested multi-way models with a large number of main effect levels

We derive the explicit form for the asymptotic posterior distribution of the balanced nested multi-way variance components model with the assumption that the number of the main factor levels tends to infinity while the number of any specific effect factor levels remains fixed. Under the multi-way model, we also study two different parameterizations, called the standard and the centering, and the relationship between certain quadratic forms of random effects and the variance component parameters. The asymptotic results are illustrated by a three-way model and by a simulation study under a two-way case.     

The robustness of response surface designs with errors in factor levels

ABSRTACT

Since errors in factor levels affect the traditional statistical properties of response surface designs, an important question to consider is robustness of design to errors. However, when the actual design could be observed in the experimental settings, its optimality and prediction are of interest. Various numerical and graphical methods are useful tools for understanding the behavior of the designs. The D- and G-efficiencies and the fraction of design space plot are adapted to assess second-order response surface designs where the predictor variables are disturbed by a random error. Our study shows that the D-efficiencies of the competing designs are considerably low for big variance of the error, while the G-efficiencies are quite good. Fraction of design space plots display the distribution of the scaled prediction variance through the design space with and without errors in factor levels. The robustness of experimental designs against factor errors is explored through comparative study. The construction and use of the D- and G-efficiencies and the fraction of design space plots are demonstrated with several examples of different designs with errors.     

Adaptive interval estimation in one-way random effects models

Two-stage procedures are introduced to control the width and coverage (validity) of confidence intervals for the estimation of the mean, the between groups variance component and certain ratios of the variance components in one-way random effects models. The procedures use the pilot sample data to estimate an “optimal” group size and then proceed to determine the number of groups by a stopping rule. Such sampling plans give rise to unbalanced data, which are consequently analyzed by the harmonic mean method. Several asymptotic results concerning the proposed procedures are given along with simulation results to assess their performance in moderate sample size situations. The proposed procedures were found to effectively control the width and probability of coverage of the resulting confidence intervals in all cases and were also found to be robust in the presence of missing observations. From a practical point of view, the procedures are illustrated using a real data set and it is shown that the resulting unbalanced designs tend to require smaller sample sizes than is needed in a corresponding balanced design where the group size is arbitrarily pre-specified.     

Cancer immunotherapy trial design with delayed treatment effect

A challenge arising in cancer immunotherapy trial design is the presence of a delayed treatment effect wherein the proportional hazard assumption no longer holds true. As a result, a traditional survival trial design based on the standard log‐rank test, which ignores the delayed treatment effect, will lead to substantial loss of statistical power. Recently, a piecewise weighted log‐rank test is proposed to incorporate the delayed treatment effect into consideration of the trial design. However, because the sample size formula was derived under a sequence of local alternative hypotheses, it results in an underestimated sample size when the hazard ratio is relatively small for a balanced trial design and an inaccurate sample size estimation for an unbalanced design. In this article, we derived a new sample size formula under a fixed alternative hypothesis for the delayed treatment effect model. Simulation results show that the new formula provides accurate sample size estimation for both balanced and unbalanced designs.     

Outcome-adaptive allocation with natural lead-in for three-group trials with binary outcomes

ABSTRACT

Just as Bayes extensions of the frequentist optimal allocation design have been developed for the two-group case, we provide a Bayes extension of optimal allocation in the three-group case. We use the optimal allocations derived by Jeon and Hu [Optimal adaptive designs for binary response trials with three treatments. Statist Biopharm Res. 2010;2(3):310–318] and estimate success probabilities for each treatment arm using a Bayes estimator. We also introduce a natural lead-in design that allows adaptation to begin as early in the trial as possible. Simulation studies show that the Bayesian adaptive designs simultaneously increase the power and expected number of successfully treated patients compared to the balanced design. And compared to the standard adaptive design, the natural lead-in design introduced in this study produces a higher expected number of successes whilst preserving power.     

The present status of confidence interval estimation on variance components in balanced and unbalanced random models

Much research has been conducted to develop confidence Intervals on linear combinations and ratios of variance components in balanced and unbalanced random models.This paper first presents confidence intervals on functions of variance components in balanced designs.These results assume that classical analysis of variance sums of squares are independent and have exact scaled chi-squared distributions.In unbalanced designs, either one or both of these assumptions are violated, and modifications to the balanced model intervals are required.We report results of some recent work that examines various modifications for some particular unbalanced designs.     

Constructions of Nested Pairwise Efficiency and Variance Balanced Designs

ABSTRACT

Nested pairwise efficiency and variance balanced designs form a new class of block designs. In this article, two methods of constructing such designs from a symmetric balanced incomplete block design are proposed with some illustrations.     

A modified bootstrap procedure for cluster sampling variance estimation of species richness

Variance estimators for probability sample-based predictions of species richness (S) are typically conditional on the sample (expected variance). In practical applications, sample sizes are typically small, and the variance of input parameters to a richness estimator should not be ignored. We propose a modified bootstrap variance estimator that attempts to capture the sampling variance by generating B replications of the richness prediction from stochastically resampled data of species incidence. The variance estimator is demonstrated for the observed richness (SO), five richness estimators, and with simulated cluster sampling (without replacement) in 11 finite populations of forest tree species. A key feature of the bootstrap procedure is a probabilistic augmentation of a species incidence matrix by the number of species expected to be ‘lost’ in a conventional bootstrap resampling scheme. In Monte-Carlo (MC) simulations, the modified bootstrap procedure performed well in terms of tracking the average MC estimates of richness and standard errors. Bootstrap-based estimates of standard errors were as a rule conservative. Extensions to other sampling designs, estimators of species richness and diversity, and estimates of change are possible.     

Parametric bootstrap tests for unbalanced nested designs under heteroscedasticity

In this article, we consider the two-factor unbalanced nested design model without the assumption of equal error variance. For the problem of testing ‘main effects’ of both factors, we propose a parametric bootstrap (PB) approach and compare it with the existing generalized F (GF) test. The Type I error rates of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the GF test. The PB test performs very satisfactorily even for small samples while the GF test exhibit poor Type I error properties when the number of factorial combinations or treatments goes up. It is also noted that the same tests can be used to test the significance of the random effect variance component in a two-factor mixed effects nested model under unequal error variances.     

Balanced and partially balanced incomplete block designs with autocorrelation errors

The paper aims to find variance balanced and variance partially balanced incomplete block designs when observations within blocks are autocorrelated and we call them BIBAC and PBIBAC designs. Orthogonal arrays of type I and type II when used as BIBAC designs have smaller average variance of elementary contrasts of treatment effects compared to the corresponding Balanced Incomplete Block (BIB) designs with homoscedastic, uncorrelated errors. The relative efficiency of BIB designs compared to BIBAC designs depends on the block size k and the autocorrelation ρ and is independent of the number of treatments. Further this relative efficiency increases with increasing k. Partially balanced incomplete block designs with autocorrelated errors are introduced using partially balanced incomplete block designs and orthogonal arrays of type I and type II.     

General formulae for expectations, variances and covariances of the mean squares for staggered nested designs

Staggered nested experimental designs are the most popular class of unbalanced nested designs. Using a special notation which covers the particular structure of the staggered nested design, this paper systematically derives the canonical form for the arbitrary m-factors. Under the normality assumption for every random variable, a vector comprising m canonical variables from each experimental unit is normally independently and identically distributed. Every sum of squares used in the analysis of variance (ANOVA) can be expressed as the sum of squares of the corresponding canonical variables. Hence, general formulae for the expectations, variances and covariances of the mean squares are directly obtained from the canonical form. Applying the formulae, the explicit forms of the ANOVA estimators of the variance components and unbiased estimators of the ratios of the variance components are introduced in this paper. The formulae are easily applied to obtain the variances and covariances of any linear combinations of the mean squares, especially the ANOVA estimators of the variance components. These results are eff ectively applied for the standardization of measurement methods.     

Comparison of the estimators of the intra-cluster correlation for the nested error regression model

The intra-cluster correlation is insisted on nested error regression model that, in practice, is rarely known. This article demonstrates the size in generalized least squares (GLS) F-test using Fuller–Battese transformation and modification F-test. For the balanced case, the former using strictly positive, analysis of covariance (ANCOVA) and analysis of variance (ANOVA) estimators of intra-cluster correlation can control the size for moderate intra-cluster correlations. For small intra-cluster correlation, they perform well when the numbers of cluster are large. The latter using the ANOVA estimator performs well except for small numbers of cluster. When intra-cluster correlation is large, it cannot control the size. For the unbalanced case, the GLS F-test using the Fuller–Battese transformation and the modification F-test using the strictly positive, the ANCOVA and the ANOVA estimators maintain the significance level for small total sample size and small intra-cluster correlations when there is a large variation in cluster sizes, but they perform well in controlling the size for large total sample size and small different variation in cluster sizes. Besides, Henderson’s method 3 estimator maintains the significance level for a few situations.     

Optimal partially balanced nested row-column designs

A partially balanced nested row-column design, referred to as PBNRC, is defined as an arrangement of v treatments in b p × q blocks for which, with the convention that p q, the information matrix for the estimation of treatment parameters is equal to that of the column component design which is itself a partially balanced incomplete block design. In this paper, previously known optimal incomplete block designs, and row-column and nested row-column designs are utilized to develop some methods of constructing optimal PBNRC designs. In particular, it is shown that an optimal group divisible PBNRC design for v = mn kn treatments in p × q blocks can be constructed whenever a balanced incomplete block design for m treatments in blocks of size k each and a group divisible PBNRC design for kn treatments in p × q blocks exist. A simple sufficient condition is given under which a group divisible PBNRC is Ψ_f-better for all f> 0 than the corresponding balanced nested row-column designs having binary blocks. It is also shown that the construction techniques developed particularly for group divisible designs can be generalized to obtain PBNRC designs based on rectangular association schemes.