Covariance Analysis for Split-Plot and Split-Block Designs

Commentaries are informative essays dealing with viewpoints of statistical practice, statistical education, and other topics considered to be of general interest to the broad readership of The American Statistician. Commentaries are similar in spirit to Letters to the Editor, but they involve longer discussions of background, issues, and perspectives. All commentaries will be refereed for their merit and compatibility with these criteria.

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.     

Regular graph designs

In this survey paper, measures of optimality of experimental designs are discussed from a combinatorial viewpoint. The main vehicle for this discussion is the regular graph design. This family of designs was introduced in the hope that they would be ‘good’ designs in cases where no balanced incomplete block design can exist. Various properties of these designs are discussed.     

On the analysis of long-term experiments

Summary.  Long-term experiments are commonly used tools in agronomy, soil science and other disciplines for comparing the effects of different treatment regimes over an extended length of time. Periodic measurements, typically annual, are taken on experimental units and are often analysed by using customary tools and models for repeated measures. These models contain nothing that accounts for the random environmental variations that typically affect all experimental units simultaneously and can alter treatment effects. This added variability can dominate that from all other sources and can adversely influence the results of a statistical analysis and interfere with its interpretation. The effect that this has on the standard repeated measures analysis is quantified by using an alternative model that allows for random variations over time. This model, however, is not useful for analysis because the random effects are confounded with fixed effects that are already in the repeated measures model. Possible solutions are reviewed and recommendations are made for improving statistical analysis and interpretation in the presence of these extra random variations.     

Embedded experiments in repeated and overlapping surveys

Summary.  Statistical agencies make changes to the data collection methodology of their surveys to improve the quality of the data collected or to improve the efficiency with which they are collected. For reasons of cost it may not be possible to estimate the effect of such a change on survey estimates or response rates reliably, without conducting an experiment that is embedded in the survey which involves enumerating some respondents by using the new method and some under the existing method. Embedded experiments are often designed for repeated and overlapping surveys; however, previous methods use sample data from only one occasion. The paper focuses on estimating the effect of a methodological change on estimates in the case of repeated surveys with overlapping samples from several occasions. Efficient design of an embedded experiment that covers more than one time point is also mentioned. All inference is unbiased over an assumed measurement model, the experimental design and the complex sample design. Other benefits of the approach proposed include the following: it exploits the correlation between the samples on each occasion to improve estimates of treatment effects; treatment effects are allowed to vary over time; it is robust against incorrectly rejecting the null hypothesis of no treatment effect; it allows a wide set of alternative experimental designs. This paper applies the methodology proposed to the Australian Labour Force Survey to measure the effect of replacing pen-and-paper interviewing with computer-assisted interviewing. This application considered alternative experimental designs in terms of their statistical efficiency and their risks to maintaining a consistent series. The approach proposed is significantly more efficient than using only 1 month of sample data in estimation.     

Computer-aided unbalanced supersaturated designs involving interactions

Supersaturated designs (SSDs) are defined as fractional factorial designs whose experimental run size is smaller than the number of main effects to be estimated. While most of the literature on SSDs has focused only on main effects designs, the construction and analysis of such designs involving interactions has not been developed to a great extent. In this paper, we propose a backward elimination design-driven optimization (BEDDO) method, with one main goal in mind, to eliminate the factors which are identified to be fully aliased or highly partially aliased with each other in the design. Under the proposed BEDDO method, we implement and combine correlation-based statistical measures taken from classical test theory and design of experiments field, and we also present an optimality criterion which is a modified form of Cronbach's alpha coefficient. In this way, we provide a new class of computer-aided unbalanced SSDs involving interactions, that derive directly from BEDDO optimization.     

An intelligent system for experimental design-some ideas

There is a growing movement towards the construction of statistical expert systems by statisticians to assist incertain aspects of their work. As one of the most'abused’ areas, experimental design could benefit from a system ofthis type. This paperoutlines some ideas on how a system for intelligently constructing designs could be built.     

A Variable Selection Method for Analyzing Supersaturated Designs

Supersaturated designs are a large class of factorial designs which can be used for screening out the important factors from a large set of potentially active variables. The huge advantage of these designs is that they reduce the experimental cost drastically, but their critical disadvantage is the confounding involved in the statistical analysis. In this article, we propose a method for analyzing data using several types of supersaturated designs. Modifications of widely used information criteria are given and applied to the variable selection procedure for the identification of the active factors. The effectiveness of the proposed method is depicted via simulated experiments and comparisons.     

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.     

Optimal allocation of time points for the random effects model

In this article the problem of the optimal selection and allocation of time points in repeated measures experiments is considered. D‐ optimal designs for linear regression models with a random intercept and first order auto‐regressive serial correlations are computed numerically and compared with designs having equally spaced time points. When the order of the polynomial is known and the serial correlations are not too small, the comparison shows that for any fixed number of repeated measures, a design with equally spaced time points is almost as efficient as the D‐ optimal design. When, however, there is no prior knowledge about the order of the underlying polynomial, the best choice in terms of efficiency is a D‐ optimal design for the highest possible relevant order of the polynomial. A design with equally‐spaced time points is the second best choice     

On the efficiency of some non-orthogonal split-plot×split-block designs with control treatments

Many split-plot×split-block (SPSB) type experiments used in agriculture, biochemistry or plant protection are designed to study new crop plant cultivars or chemical agents. In these experiments it is usually very important to compare test treatments with the so-called control treatments. It happens yet that experimental material is limited and it does not allow using a complete (orthogonal) SPSB design. In the paper we propose a non-orthogonal SPSB design for consideration. Two cases of the design are presented here, i.e. when its incompleteness is connected with a crossed treatment structure only or with a nested treatment structure only. It is assumed the factors' levels connected with the incompleteness of the design are split into two groups: a set of test treatments and a set of control treatments. The method of constructions involves applying augmented block designs for some factors' levels. In a modelling data obtained from such experiments the structure of experimental material and appropriate randomization scheme of the different kinds of units before they enter the experiment are taken into account. With respect to the analysis of the obtained randomization model the approach typical to the multistratum experiments with orthogonal block structure is adapted. The proposed statistical analysis of linear model obtained includes estimation of parameters, testing general and particular hypotheses defined by the (basic) treatment contrasts with special reference to the notion of general balance.     

Optimal Experiments for Time-Dependent Mineral Processes

Time dependence is an important characteristic of mineral processing plant data. This paper finds that the time dependence in the recovery data for an experiment at Bougainville Copper Limited (BCL) (Napier-Munn, 1995) can be described by an autoregressive-one process. The paper investigates the optimum form of experimental design for such data. Two intuitive approaches for the design of experiments involving time-dependent data have been disproved recently. Cheng & Steinberg (1991) showed that in some circumstances systematic experiments are preferable to replicated randomized block designs, and Saunders & Eccleston (1992) showed that rather than sampling at a frequency which ensures independent data, in some circumstances sampling intervals should be as small as possible. A third issue, raised in this paper, concerns the use of standard statistical tests when the data are serially correlated. It is shown that the simple paired t -test, suitably modified for time dependence, is appropriate and easily adapted to allow for a covariate and a sequential analysis. The results are illustrated using the BCL data and are already being used to design major experiments involving another mineral process.     

A comparison of inference procedures in unbalanced split-plot designs

Several procedures for constructing confidence intervals and testing hypotheses about fixed effects in unbalanced split-plot experiments are described in this paper. These procedures can also be used for unbalanced repeated measures experiments when the repeated measures satisfy the Huyhn-Feldt (1970) conditions. A number of these procedures require that the whole plot error mean square has a distribution proportional to a chi-square distribution and that it be independent of estimators of the parameter functions. Often, neither of these conditions are met in unbalanced split-plot experiments. Simulation studies of a small design of eight observations and larger designs with 34 to 48 observations are used to investigate the performance of the different procedures.     

A pragmatic approach to experimental design in industry

The importance of statistically designed experiments in industry has been well recognized. However, the use of 'design of experiments' is still not pervasive, owing in part to the inefficient learning process experienced by many non-statisticians. In this paper, the nature of design of experiments, in contrast to the usual statistical process control techniques, is discussed. It is then pointed out that for design of experiments to be appreciated and applied, appropriate approaches should be taken in training, learning and application. Perspectives based on the concepts of objective setting and design under constraints can be used to facilitate the experimenters' formulation of plans for collection, analysis and interpretation of empirical information. A review is made of the expanding role of design of experiments in the past several decades, with comparisons made of the various formats and contexts of experimental design applications, such as Taguchi methods and Six Sigma. The trend of development shows that, from the realm of scientific research to business improvement, the competitive advantage offered by design of experiments is being increasingly felt.     

A pragmatic approach to experimental design in industry

The importance of statistically designed experiments in industry has been well recognized. However, the use of 'design of experiments' is still not pervasive, owing in part to the inefficient learning process experienced by many non-statisticians. In this paper, the nature of design of experiments, in contrast to the usual statistical process control techniques, is discussed. It is then pointed out that for design of experiments to be appreciated and applied, appropriate approaches should be taken in training, learning and application. Perspectives based on the concepts of objective setting and design under constraints can be used to facilitate the experimenters' formulation of plans for collection, analysis and interpretation of empirical information. A review is made of the expanding role of design of experiments in the past several decades, with comparisons made of the various formats and contexts of experimental design applications, such as Taguchi methods and Six Sigma. The trend of development shows that, from the realm of scientific research to business improvement, the competitive advantage offered by design of experiments is being increasingly felt.     

Repeated measurements designs - a review

Any experiment in which one or more of the experimental units is used more than once is called a repeated measurements experiment. The associated design of a repeated measurements experiment is referred to as a repeated measurements design. This review covers some known results on repeated measurements designs. Emphasis is placed on the impact of optimal design theory. Some construction methods for these designs are presented. Hedayat and Afsarinejad (1975) has an extensive bibliography of earlier literature. A bibliography of these designs published after 1974 is provided at the end of this paper.     

Likelihood Ratio Type Statistics for Repeated Measures Designs with Heterogeneous Covariance Matrices

Because of their simplicity, Wald statistics are typically used in complex experimental designs. Likelihood ratio statistics in factorial designs are more flexible than Wald statistics in the sense of adapting to non-saturated designs by fitting only as many parameters as the model calls for. This leads to a significant gain in power. Here we propose likelihood ratio type statistics for testing hypotheses in repeated measures designs with heterogeneous covariance matrices, and derive their asymptotic distribution in one general theorem that does not require normality or even continuity of the responses. Simulation studies demonstrate their advantages over the Wald statistics.     

Sample Size Calculation and Power Analysis of Changes in Mean Response Over Time

Despite tremendous effort on different designs with cross-sectional data, little research has been conducted for sample size calculation and power analyses under repeated measures design. In addition to time-averaged difference, changes in mean response over time (CIMROT) is the primary interest in repeated measures analysis. We generalized sample size calculation and power analysis equations for CIMROT to allow unequal sample size between groups for both continuous and binary measures, through simulation, evaluated the performance of proposed methods, and compared our approach to that of a two-stage model formulization. We also created a software procedure to implement the proposed methods.     

Experimental designs for open pollination in polycross trials

A polycross is the pollination by natural hybridization of a group of genotypes, generally selected, grown in isolation from other compatible genotypes in such a way to promote random open pollination. A particular practical application of the polycross method occurs in the production of a synthetic variety resulting from cross-pollinated plants. Laying out these experiments in appropriate designs, known as polycross designs, would not only save experimental resources but also gather more information from the experiment. Different situations may arise in polycross nurseries where accordingly different polycross designs may be used. For situations in which some genotypes interfere in the growth or production of other genotypes, but have to be grown together, neighbour-restricted design is a better option. Furthermore, when the topography of the nursery is such that a known wind system in a certain direction may prevail, then designs balanced for neighbour effects of genotypes only in the direction of wind are appropriate which may help in saving experimental resources to a great extent. Also, when genotypes are planted in a small area without leaving much space between rows, designs balanced for neighbour effects from all possible eight directions are useful to have equal chance of pollinating and being pollinated by every other genotype. Here, polycross designs have been obtained to match above-mentioned three situations. SAS Macros have also been developed to generate these proposed designs.     

Estimated Data Collection Organization Effect in the National Medical Care Expenditure Survey

The National Medical Care Expenditure Survey, which has a complex survey design, is further complicated by combining two independently drawn national samples of households from the Research Triangle Institute (RTI) and the National Opinion Research Center (NORC). It is assumed that since the structures of these national-area samples are similar, they are thereby compatible and allow for the derivation of statistically equivalent unbiased national estimates of relevant health parameters. In this paper, national parameter estimates of relevant demographic and health measures were separately produced for each of the two survey organizations and a determination was made regarding statistical equivalence across samples. In addition, the data-collection organization effect was estimated and tested within the framework of a multivariate analysis appropriate for complex survey data. The findings consistently revealed a statistically nonsignificant data-collection organization effect when testing for differences in domain estimates of the relevant health parameters under consideration.     

Design‐Based Inference in a Mixture Model for Ordinal Variables for a Two Stage Stratified Design

In this paper we present methods for inference on data selected by a complex sampling design for a class of statistical models for the analysis of ordinal variables. Specifically, assuming that the sampling scheme is not ignorable, we derive for the class of cub models (Combination of discrete Uniform and shifted Binomial distributions) variance estimates for a complex two stage stratified sample. Both Taylor linearization and repeated replication variance estimators are presented. We also provide design‐based test diagnostics and goodness‐of‐fit measures. We illustrate by means of real data analysis the differences between survey‐weighted and unweighted point estimates and inferences for cub model parameters.