Identification of factors influencing dispersion in split-plot experiments

As split-plot designs are commonly used in robust design it is important to identify factors in these designs that influence the dispersion of the response variable. In this article, the Bergman-Hynén method, developed for identification of dispersion effects in unreplicated experiments, is modified to be used in the context of split-plot experiments. The modification of the Bergman-Hynén method enables identification of factors that influence specific variance components in unreplicated two-level fractional factorial splitplot experiments. An industrial example is used to illustrate the proposed method.     

Split-plot designs for robust product experimentation

Genichi Taguchi has emphasized the use of designed experiments in several novel and important applications. In this paper we focus on the use of statistical experimental designs in designingproducts to be robust to environmental conditions. The engineering concept of robust product design is very important because it is frequently impossible or prohibitively expensive to control or eliminate variation resulting from environmental conditions. Robust product design enablesthe experimenter to discover how to modify the design of the product to minimize the effect dueto variation from environmental sources. In experiments of this kind, Taguchi's total experimental arrangement consists of a cross-product of two experimental designs:an inner array containing the design factors and an outer array containing the environmental factors. Except in situations where both these arrays are small, this arrangement may involve a prohibitively large amount of experimental work. One of the objectives of this paper is to show how this amount of work can be reduced. In this paper we investigate the applicability of split-plot designs for thisparticular experimental situation. Consideration of the efficiency of split-plot designs and anexamination of several variants of split-plot designs indicates that experiments conductedin a split-plot mode can be of tremendous value in robust product design since they not only enable the contrasts of interest to be estimated efficiently but also the experiments can be considerably easier to conduct than the designs proposed by Taguchi.     

Covariance Analysis for Split-Plot and Split-Block Designs

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Proper methodology for the analysis of covariance for experiments designed in a split-plot or split-block design is not found in the statistical literature. Analyses for these designs are often performed incompletely or even incorrectly. This is especially true when popular statistical computer software packages are used for the analysis of these designs. This article provides several appropriate models, ANOVA tables, and standard errors for comparisons from experiments arranged in a standard split-plot, split–split-plot, or split-block design where a covariate has been measured on the smallest size experimental unit.     

Generalized minimum aberration two-level split-plot designs

Split-plot experiments may arise when it is impractical to completely randomize the treatment combinations of a designed experiment. To provide more flexible design choices in the nonregular split-plot setting, we describe an approach for constructing minimum aberration orthogonal two-level split-plot designs having 12, 16, 20 and 24 runs. We consider five design scenarios that may be of importance to practitioners, and then propose an approach for assigning word lengths under these five scenarios. We then use the extended word length patterns to rank both regular and nonregular orthogonal split-plot designs. While most existing papers concerning orthogonal split-plot designs focus on regular orthogonal designs, we find that many minimum aberration split-plot designs are nonregular orthogonal designs.     

A Permutation Approach to Split-Plot Experiments

Split-plot design may be refer to a common experimental setting where a particular type of restricted randomization has occurred during a planned experiment. The aim of this article is to suggest a new method to perform inference on split-plot experiments by combination-based permutation tests. This novel nonparametric approach has been studied and validated using a Monte Carlo simulation study where we compared it with the parametric and nonparametric procedures proposed in the literature. Results suggest that in each experimental situation where normality is hard to justify and especially when errors have heavy-tailed distribution, the proposed nonparametric procedure can be considered as a valid solution.     

Analyzing Small Samples of Repeated Measures Data with the Mixed-Model Adjusted F Test

This research examines the Type I error rates obtained when using the mixed model with the Kenward-Roger correction and compares them with the Between–Within and Satterthwaite approaches in split-plot designs. A simulated study was conducted to generate repeated measures data with small samples under normal distribution conditions. The data were obtained via three covariance matrices (unstructured, heterogeneous first-order auto-regressive, and random coefficients), the one with the best fit being selected according to the Akaike criterion. The results of the simulation study showed the Kenward-Roger test to be more robust, particularly when the population covariance matrices were unstructured or heterogeneous first-order auto-regressive. The main contribution of this study lies in showing that the Kenward–Roger method corrects the liberal Type I error rates obtained with the Between–Within and Satterthwaite approaches, especially with positive pairings between group sizes and covariance matrices.     

Comparing two measures of creatinine clearance: an application of errors-in-variables and bootstrap techniques

Two indices of creatinine clearance (an index of kidney function) are compared on a group of cancer patients who underwent chemotherapy with a potentially nephrotoxic drug. The standard index, measured creatinine clearance MCC, is cumbersome to use, whereas the more convenient alternative, estimated creatinine clearance ECC, has not yet been conclusively evaluated on cancer patients. We conclude that under certain clinical conditions ECC and MCC are identically calibrated for males, but not for females, and we obtain estimated true and false positive rates for assessing the use of ECC instead of MCC as a diagnostic tool. We use a model that is formally equivalent to an errors-in-variables model with (unbalanced) repeated observations and correlated measurement errors. The bootstrap is used to obtain standard errors and confidence limits.     

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.     

Further exploratory analysis of split-plot experiments to study certain stratified effects

Designed experiments are a key component in many companies' improvement strategies. Because completely randomized experiments are not always reasonable from a cost or physical perspective, split-plot experiments are prevalent. The recommended analysis accounts for the different sources of variation affecting whole-plot and split-plot error. However experiments on industrial processes must be run and, consequently analyzed quite differently from ones run in a controlled environment. Such experiments are typically subject to a wide array of uncontrolled, and barely understood, variation. In particular, it is important to examine the experimental results for additional, unanticipated sources of variation. In this paper, we consider how unanticipated, stratified effects may influence a split-plot experiment and discuss further exploratory analysis to indicate the presence of stratified effects. Examples of such experiments are provided, additional tests are suggested and discussed in light of their power, and recommendations given.     

Subset selection in two-factor experiments using randomization restricted designs

This paper studies subset selection procedures for screening in two-factor treatment designs that employ either a split-plot or strip-plot randomization restricted experimental design laid out in blocks. The goal is to select a subset of treatment combinations associated with the largest mean. In the split-plot design, it is assumed that the block effects, the confounding effects (whole-plot error) and the measurement errors are normally distributed. None of the selection procedures developed depend on the block variances. Subset selection procedures are given for both the case of additive and non-additive factors and for a variety of circumstances concerning the confounding effect and measurement error variances. In particular, procedures are given for (1) known confounding effect and measurement error variances (2) unknown measurement error variance but known confounding effect (3) unknown confounding effect and measurement error variances. The constants required to implement the procedures are shown to be obtainable from available FORTRAN programs and tables. Generalization to the case of strip-plot randomization restriction is considered.     

On Non-Normal Invariance Principles for Multi-Response Permutation Procedures

A non-normal invariance principle is established for a restricted class of univariate multi-response permutation procedures whose distance measure is the square of Euclidean distance. For observations from a distribution with finite second moment, the test statistic is found asymptotically to have a centered chi-squared distribution. Spectral expansions are used to determine the asymptotic distribution for more general distance measures d, and it is shown that if d(x, y) = |x — y|^u, u? 2, the asymptotic distribution is not invariant (i.e. it is dependent on the distribution of the observations).     

Basic concepts of group sequential and adaptive group sequential test procedures

Based on the concept of repeated significance tests, an empirical study may be planned in subsequent stages. Group sequential test procedures offer the possibility of performing the study with a fixed number of observations per stage. At least, the number of observations must be chosen independently of the observed data. In adaptive group sequential test procedures, the number of observations can be changed during the course of the study using all results observed so far. In this article, the basic concepts of these two designs are reviewed. Recent developments in adaptive designs are outlined and potential fields of application are given.     

Longitudinal models for polytomous responses

Recent work by Miller and Landis (1991) discusses generalized variance component models for polytomous responses. This work is adapted to longitudinal models for repeated measures of individuals having polytomous responses. In this setting, individuals are considered to be “clusters”. The resulting simplifications are discussed. First, each response has a multinomial distribution with N=l. Second, observed cluster proportions in the variance component estimates must be replaced by their expectations. This technique accommodates patients with missing data in a sequence of repeated observations.     

CHOLESKY DECOMPOSITION OF A VARIANCE MATRIX IN REPEATED MEASURES ANALYSIS

The Cholesky decomposition is given for the inverse of a variance matrix occurring in repeated measures problems where observations have a correlation structure both within and between experimental units. The use of this decomposition is outlined for ML and REML estimation procedures.     

Bayesian quantile regression for skew-normal linear mixed models

Linear mixed models have been widely used to analyze repeated measures data which arise in many studies. In most applications, it is assumed that both the random effects and the within-subjects errors are normally distributed. This can be extremely restrictive, obscuring important features of within-and among-subject variations. Here, quantile regression in the Bayesian framework for the linear mixed models is described to carry out the robust inferences. We also relax the normality assumption for the random effects by using a multivariate skew-normal distribution, which includes the normal ones as a special case and provides robust estimation in the linear mixed models. For posterior inference, we propose a Gibbs sampling algorithm based on a mixture representation of the asymmetric Laplace distribution and multivariate skew-normal distribution. The procedures are demonstrated by both simulated and real data examples.     

Detecting randomization restrictions caused by factors

In practice, randomization in factorial experiments has generally meant the sequential application of treatment combinations to experimental units determined by a random run order and the resetting of levels of factors only when levels change from one run to the next. Whenever factor levels are not reset for consecutive runs requiring the same level of a factor, the experiment is inadvertently split-plotted. The number and the size of the whole plots are formed randomly. The unbalanced split-plot experiment is assumed by experimenters to be completely randomized and analysis usually proceeds by the method of ordinary least squares. We show that once the experiment has been run it is often difficult, sometimes impossible, to determine the effect of such inadvertent split-plotting. Retrospectively, therefore, the correct analysis often cannot be performed. The aims of the paper are: (1) to urge experimenters and statisticians to recognize prospectively the difficulty in resetting factor levels so that split-plotting is deliberate but designed in a manner beneficial to both the conduct of the experiment and analysis of the data; (2) to make recommendations regarding more comprehensive reporting of data from randomized experiments.     

Bayesian assessment of assumptions of regression analysis

Many statistical procedures are based on the models which specify the conditions under which the data are generated. Many applications of linear regression, for example, assume that:(i) the observations are independent; (ii) the errors in the observations are identically distributed; (iii) each error has a normal distribution with mean zero and unknown variance σ²> 0. Previous works have examined individual departures from these assumptions. Here we examine composite departures. It is assumed that the error distribution in a linear model is power-exponential and that the observations are generated via a first order autoregressive model with the possibility of spurious observations. The consequences are illustrated via an example.     

Orthogonal blocking of response surface split-plot designs

When all experimental runs cannot be performed under homogeneous conditions, blocking can be used to increase the power for testing the treatment effects. Orthogonal blocking provides the same estimator of the polynomial effects as the one that would be obtained by ignoring the blocks. In many real-life design scenarios, there is at least one factor that is hard to change, leading to a split-plot structure. This paper shows that for a balanced ordinary least square–generalized least square equivalent split-plot design, orthogonal blocking can be achieved. Orthogonally blocked split-plot central composite designs are constructed and a catalog is provided.     

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.     

An empirical study of eight tests of partial correlation coefficients

Partial correlations can be used to statistically control for the effects of unwanted variables.Perhaps the most frequently used test of a partial correlation is the parametric F test,which requires normality of the joint distribution of observations.The possibility that this assumption may not be met in practice suggests a need for procedures that do not require normality.Unfortunately,the statistical literature provides little guidance for choosing other tests when the normalityassumption is not satisfied.Several nonparametric tests of partial correlations are investigated using a computer simulation study.Recommendations are made for selecting certain tests under particular conditions