Graphical technique for comparing designs for random models

Methods for comparing designs for a random (or mixed) linear model have focused primarily on criteria based on single-valued functions. In general, these functions are difficult to use, because of their complex forms, in addition to their dependence on the model's unknown variance components. In this paper, a graphical approach is presented for comparing designs for random models. The one-way model is used for illustration. The proposed approach is based on using quantiles of an estimator of a function of the variance components. The dependence of these quantiles on the true values of the variance components is depicted by plotting the so-called quantile dispersion graphs (QDGs), which provide a comprehensive picture of the quality of estimation obtained with a given design. The QDGs can therefore be used to compare several candidate designs. Two methods of estimation of variance components are considered, namely analysis of variance and maximum-likelihood estimation.     

Comparison of designs for generalized linear models under model misspecification

The purpose of this article is to demonstrate the use of the quantile dispersion graphs (QDGs) approach for comparing candidate designs for generalized linear models in the presence of model misspecification in the linear predictor. The proposed design criterion is based on the mean-squared error of prediction which incorporates the prediction variance and the bias caused by fitting the wrong model. The method of kriging is used to estimate the unknown function assumed to be the cause of model misspecification. The QDGs approach is also useful in assessing the robustness of a given design to values of the unknown parameters in the linear predictor. Three numerical examples are presented to illustrate the application of the proposed methodology.     

Comparison of designs for multivariate generalized linear models

The purpose of this paper is to discuss response surface designs for multivariate generalized linear models (GLMs). Such models are considered whenever several response variables can be measured for each setting of a group of control variables, and the response variables are adequately represented by GLMs. The mean-squared error of prediction (MSEP) matrix is used to assess the quality of prediction associated with a given design. The MSEP incorporates both the prediction variance and the prediction bias, which results from using maximum likelihood estimates of the parameters of the fitted linear predictor. For a given design, quantiles of a scalar-valued function of the MSEP are obtained within a certain region of interest. The quantiles depend on the unknown parameters of the linear predictor. The dispersion of these quantiles over the space of the unknown parameters is determined and then depicted by the so-called quantile dispersion graphs. An application of the proposed methodology is presented using the special case of the bivariate binary distribution.     

Quantile dispersion graphs for analysis of variance estimates of variance components

SUMMARY The exact distribution of an analysis of variance estimator of a variance component is obtained by determining its quantiles on the basis of R. B. Davies' algorithm. A plot of these quantiles provides useful information concerning the efficiency of the estimator, including the extent to which it can be negative. Furthermore, the variability in the values of each quantile is assessed by varying the values of the variance components for the model under consideration. The maximum and minimum of such quantile values can then be determined. A plot of the maxima and minima for various selected quantiles produces the so-called 'quantile dispersion graphs'. These graphs can be used to provide a comprehensive picture of the quality of estimation obtained with a particular design. They also provide an effective graphical tool for comparing designs on the basis of their estimation capabilities.     

Adjusted homogeneity tests of odds ratios when data are clustered

Three modified tests for homogeneity of the odds ratio for a series of 2 × 2 tables are studied when the data are clustered. In the case of clustered data, the standard tests for homogeneity of odds ratios ignore the variance inflation caused by positive correlation among responses of subjects within the same cluster, and therefore have inflated Type I error. The modified tests adjust for the variance inflation in the three existing standard tests: Breslow–Day, Tarone and the conditional score test. The degree of clustering effect is measured by the intracluster correlation coefficient, ρ. A variance correction factor derived from ρ is then applied to the variance estimator in the standard tests of homogeneity of the odds ratio. The proposed tests are an application of the variance adjustment method commonly used in correlated data analysis and are shown to maintain the nominal significance level in a simulation study. Copyright © 2004 John Wiley & Sons, Ltd.     

Control of type 1 error in a hybrid complete two‐period vaccine efficacy trial

A complete two‐period experimental design has been defined as one in which subjects are randomized to treatment, observed for the occurrence of an event of interest, re‐randomized, and observed again for the event in a second period. A 4‐year vaccine efficacy trial was planned to compare a high‐dose vaccine with a standard dose vaccine. Subjects would be randomized each year, and subjects who had participated in a previous year would be allowed to re‐enroll in a subsequent year and would be re‐randomized. A question of interest is whether positive correlation between observations on subjects who re‐enrolled would inflate the variance of test statistics. The effect of re‐enrollment and correlation on type 1 error in a 4‐year trial is investigated by simulation. As conducted, the trial met its power requirements after two years. Subjects therefore included some who participated for a single year and others who participated in both years. Those who participated in both years constituted a complete two‐period design. An algebraic expression for the variance of the treatment difference in a complete two‐period design is derived. It is shown that under a ‘no difference’ null, correlation does not result in variance inflation in this design. When there is a treatment difference, there is variance inflation but it is small. In the vaccine efficacy trial, the effect of correlation on the statistical inference was negligible. Copyright © 2014 John Wiley & Sons, Ltd.     

Adjusting for bias in randomized cluster trials

The randomized cluster design is typical in studies where the unit of randomization is a cluster of individuals rather than the individual. Evaluating various intervention strategies across medical care providers at either an institutional level or at a physician group practice level fits the randomized cluster model. Clearly, the analytical approach to such studies must take the unit of randomization and accompanying intraclass correlation into consideration. We review alternative methods to the typical Pearson's chi-square analysis and illustrate these alternatives. We have written and tested a Fortran program that produces the statistics outlined in this paper. The program, in an executable format is available from the author on request.     

Robust Designs in Generalized Linear Models: A Quantile Dispersion Graphs Approach

This article studies design selection for generalized linear models (GLMs) using the quantile dispersion graphs (QDGs) approach in the presence of misspecification in the link and/or linear predictor. The uncertainty in the linear predictor is represented by a unknown function and estimated using kriging. For addressing misspecified link functions, a generalized family of link functions is used. Numerical examples are shown to illustrate the proposed methodology.     

Quantile plots for mean effects in the presence of variance effects for 2k-p designs

In robust parameter design, variance effects and mean effects in a factorial experiment are modelled simultaneously. If variance effects are present in a model, correlations are induced among the naive estimators of the mean effects. A simple normal quantile plot of the mean effects may be misleading because the mean effects are no longer iid under the null hypothesis that they are zero. Adjusted quantiles are computed for the case when one variance effect is significant and examples of 8-run and 16-run fractional factorial designs are examined in detail. We find that the usual normal quantiles are similar to adjusted quantiles for all but the largest and smallest ordered effects for which they are conservative. Graphically, the qualitative difference between the two sets of quantiles is negligible (even in the presence of large variance effects) and we conclude that normal probability plots are robust in the presence of variance effects.     

Efficient design of geographically-defined clusters with spatial autocorrelation

Clusters form the basis of a number of research study designs including survey and experimental studies. Cluster-based designs can be less costly but also less efficient than individual-based designs due to correlation between individuals within the same cluster. Their design typically relies on ad hoc choices of correlation parameters, and is insensitive to variations in cluster design. This article examines how to efficiently design clusters where they are geographically defined by demarcating areas incorporating individuals and households or other units. Using geostatistical models for spatial autocorrelation, we generate approximations to within cluster average covariance in order to estimate the effective sample size given particular cluster design parameters. We show how the number of enumerated locations, cluster area, proportion sampled, and sampling method affect the efficiency of the design and consider the optimization problem of choosing the most efficient design subject to budgetary constraints. We also consider how the parameters from these approximations can be interpreted simply in terms of ‘real-world’ quantities and used in design analysis.     

New improved estimators for overdispersion in models with clustered multinomial data and unequal cluster sizes

It is usual to rely on the quasi-likelihood methods for deriving statistical methods applied to clustered multinomial data with no underlying distribution. Even though extensive literature can be encountered for these kind of data sets, there are few investigations to deal with unequal cluster sizes. This paper aims to contribute to fill this gap by proposing new estimators for the intracluster correlation coefficient.     

Hypothesis Testing in Two-Stage Cluster Sampling

Correlated observations often arise in complex sampling schemes such as two-stage cluster sampling. The resulting observations from this sampling scheme usually exhibit certain positive intracluster correlation, as a result of which the standard statistical procedures for testing hypotheses concerning linear combinations of the parameters may lack some of the optimal properties that these possess when the data are uncorrelated. The aim of this paper is to present exact methods for testing these hypotheses by combining within and between cluster information much as in Zhou & Mathew (1993).     

Selection and screening procedures to determine optimal product designs

To compare several promising product designs, manufacturers must measure their performance under multiple environmental conditions. In many applications, a product design is considered to be seriously flawed if its performance is poor for any level of the environmental factor. For example, if a particular automobile battery design does not function well under temperature extremes, then a manufacturer may not want to put this design into production. Thus, this paper considers the measure of a product's quality to be its worst performance over the levels of the environmental factor. We develop statistical procedures to identify (a near) optimal product design among a given set of product designs, i.e., the manufacturing design that maximizes the worst product performance over the levels of the environmental variable. We accomplish this by intuitive procedures based on the split-plot experimental design (and the randomized complete block design as a special case); split-plot designs have the essential structure of a product array and the practical convenience of local randomization. Two classes of statistical procedures are provided. In the first, the δ-best formulation of selection problems, we determine the number of replications of the basic split-plot design that are needed to guarantee, with a given confidence level, the selection of a product design whose minimum performance is within a specified amount, δ, of the performance of the optimal product design. In particular, if the difference between the quality of the best and second best manufacturing designs is δ or more, then the procedure guarantees that the best design will be selected with specified probability. For applications where a split-plot experiment that involves several product designs has been completed without the planning required of the δ-best formulation, we provide procedures to construct a ‘confidence subset’ of the manufacturing designs; the selected subset contains the optimal product design with a prespecified confidence level. The latter is called the subset selection formulation of selection problems. Examples are provided to illustrate the procedures.     

Cluster designs to assess the prevalence of acute malnutrition by lot quality assurance sampling: a validation study by computer simulation

Summary.  Traditional lot quality assurance sampling (LQAS) methods require simple random sampling to guarantee valid results. However, cluster sampling has been proposed to reduce the number of random starting points. This study uses simulations to examine the classification error of two such designs, a 67×3 (67 clusters of three observations) and a 33×6 (33 clusters of six observations) sampling scheme to assess the prevalence of global acute malnutrition (GAM). Further, we explore the use of a 67×3 sequential sampling scheme for LQAS classification of GAM prevalence. Results indicate that, for independent clusters with moderate intracluster correlation for the GAM outcome, the three sampling designs maintain approximate validity for LQAS analysis. Sequential sampling can substantially reduce the average sample size that is required for data collection. The presence of intercluster correlation can impact dramatically the classification error that is associated with LQAS analysis.     

Estimating standard error of intra-class correlation coefficients up to three level unbalanced nested clinical trials

ABSTRACT

Very often researchers plan a balanced design for cluster randomization clinical trials in conducting medical research, but unavoidable circumstances lead to unbalanced data. By adopting three or more levels of nested designs, they usually ignore the higher level of nesting and consider only two levels, this situation leads to underestimation of variance at higher levels. While calculating the sample size for three-level nested designs, in order to achieve desired power, intra-class correlation coefficients (ICCs) at individual level as well as higher levels need to be considered and must be provided along with respective standard errors. In the present paper, the standard errors of analysis of variance (ANOVA) estimates of ICCs for three-level unbalanced nested design are derived. To conquer the strong appeal of distributional assumptions, balanced design, equality of variances between clusters and large sample, general expressions for standard errors of ICCs which can be deployed in unbalanced cluster randomization trials are postulated. The expressions are evaluated on real data as well as highly unbalanced simulated data.     

Optimal design of clinical trials comparing several treatments with a control

Clinical trials are often designed to compare several treatments with a common control arm in pairwise fashion. In this paper we study optimal designs for such studies, based on minimizing the total number of patients required to achieve a given level of power. A common approach when designing studies to compare several treatments with a control is to achieve the desired power for each individual pairwise treatment comparison. However, it is often more appropriate to characterize power in terms of the family of null hypotheses being tested, and to control the probability of rejecting all, or alternatively any, of these individual hypotheses. While all approaches lead to unbalanced designs with more patients allocated to the control arm, it is found that the optimal design and required number of patients can vary substantially depending on the chosen characterization of power. The methods make allowance for both continuous and binary outcomes and are illustrated with reference to two clinical trials, one involving multiple doses compared to placebo and the other involving combination therapy compared to mono-therapies. In one example a 55% reduction in sample size is achieved through an optimal design combined with the appropriate characterization of power.     

Simultaneous Confidence Intervals for Ratios of Fixed Effect Parameters in Linear Mixed Models

In multiple comparisons of fixed effect parameters in linear mixed models, treatment effects can be reported as relative changes or ratios. Simultaneous confidence intervals for such ratios had been previously proposed based on Bonferroni adjustments or multivariate normal quantiles accounting for the correlation among the multiple contrasts. We propose Fieller-type intervals using multivariate t quantiles and the application of Markov chain Monte Carlo techniques to sample from the joint posterior distribution and construct percentile-based simultaneous intervals. The methods are compared in a simulation study including bioassay problems with random intercepts and slopes, repeated measurements designs, and multicenter clinical trials.     

Optimal design for the proportional odds model

The authors construct locally optimal designs for the proportional odds model for ordinal data. While they investigate the standard D‐optimal design, they also investigate optimality criteria for the simultaneous estimation of multiple quantiles, namely D_A ‐optimality and the omnibus criterion. The design of experiments for the simultaneous estimation of multiple quantiles is important in both toxic and effective dose studies in medicine. As with c‐optimality in the binary response problem, the authors find that there are distinct phase changes when exploring extreme quantiles that require additional design points. The authors also investigate relative efficiencies of the criteria.     

An extension of the standardized randomized response technique to a multi-stage setup

If nonresponse and/or untruthful answering mechanisms occur, analyzing only the available cases may substantially weaken the validity of sample results. The paper starts with a reference to strategies of empirical social researchers related to respondent cooperation in surveys embedding the statistical techniques of randomized response in this framework. Further, multi-stage randomized response techniques are incorporated into the standardized randomized response technique for estimating proportions. In addition to already existing questioning designs of this family of methods, this generalization includes also several (in particular: two-stage) techniques that have not been published before. The statistical properties of this generalized design are discussed for all probability sampling designs. Further, the efficiency of the model is presented as a function of privacy protection. Hence, it can be shown that not one multi-stage design of this family at the same level of privacy protection can theoretically be more efficient than its one-stage basic version.