Whole-plot Factors Research Articles

General minimum lower-order confounding split-plot designs with important whole-plot factors

Fractional factorial split-plot (FFSP) designs have been widely used in industrial experiments. Whole-plot (WP) factors and subplot (SP) factors play different roles in FFSP designs. This paper proposes a new optimality criterion for selecting FFSP designs, which is the general minimum lower-order confounding of type WP (WP-GMC), under the circumstances that the experimenter has prior information that the WP factors are more important than the SP factors. Comparison with minimum aberration FFSP designs of type WP is given. Some optimal FFSP designs under the WP-GMC criterion are constructed.

Optimal Designs For the Restricted Maximum Likelihood Estimators in a Random Split-Plot Model

The design effect for the restricted maximum likelihood estimators of variance components in acompletely randomized split-plot model is studied. The model was used to represent the response generated froman experimental scenario where the whole-plot and split-plot factors are random. The work generated groups ofbalanced designs from same number of experimental runs and compared them for optimality using the derived Fisher Information matrix of the restricted maximum likelihood (REML) estimators. The measure for optimalityis the D-optimality criterion; the resulting optimal designs depend on the relative magnitudes of the true values of the variance components. The results show that when the factor variances are larger than the error variances, designs where the absolute difference between the number of whole-plots and the number of levels of the splitplot factor is relatively small show substantial gain in statistical efficiency over other designs.

High-dimensional repeated measures

Recently, new tests for main and simple treatment effects, time effects, and treatment by time interactions in possibly high-dimensional multi-group repeated-measures designs with unequal covariance matrices have been proposed. Technical details for using more than one between-subject and more than one within-subject factor are presented in this article. Furthermore, application to electroencephalo-graphy (EEG) data of a neurological study with two whole-plot factors (diagnosis and sex) and two subplot factors (variable and region) is shown with the R package HRM (high-dimensional repeated measures).

A Study of Two Types of Split-Plot Designs

This paper discusses two types of split-plot designs, split-plot designs with few whole-plot factors and blocked split-plot designs. We provide a list of efficient designs and compare them with the designs available in the literature with respect to a surrogate for the information capacity criterion derived in Cheng and Tsai (2011). In many cases, either better designs are found or additional designs are identified as possible alternatives. In the remaining cases, the optimality of the designs tabulated in design literature is confirmed.

Searching for D-Efficient Equivalent-Estimation Second-Order Split-Plot Designs

Several industrial experiments are set up using second-order split-plot designs (SPDs). These experiments have two types of factors: whole-plot (WP) factors and subplot (SP) factors. WP factors, also called hard-to-change factors, are factors for which levels are hard or expensive to change. SP factors, also called easy-to-change factors, are factors for which levels are easy or less expensive to change. In a split-plot experiment, the WP factors are confounded with blocks. Certain SPDs possess the equivalent-estimation property. For SPDs with this property, ordinary least-squares estimates of the model parameters are equivalent to the generalized least-squares estimates.This paper describes a fast and simple algorithm that produces D-efficient equivalent-estimation SPDs by interchanging the levels of the SP factors within each WP. The performance of this algorithm is evaluated against the 111 SPD scenarios reported in Macharia and Goos (2010) and Jones and Goos (2012).

An Algorithm for Finding D-Efficient Equivalent-Estimation Second-Order Split-Plot Designs

Many industrial experiments involve restricted rather than complete randomization. This often leads to the use of split-plot designs, which limit the number of independent settings of some of the experimental factors. These factors, named whole-plot factors, are often, in some way, hard to change. The remaining factors, called subplot factors, are easier to change. Their levels are therefore independently reset for every run of the experiment. In general, model estimation from data from split-plot experiments requires the use of generalized least squares (GLS). However, for some split-plot designs, the ordinary least squares (OLS) estimator will produce the same factor-effect estimates as the GLS estimator. These designs are called equivalent-estimation split-plot designs and offer the advantage that estimation of the factor effects does not require estimation of the variance components in the split-plot model. While many of the equivalent-estimation second-order response-surface designs presented in the literature do not perform well in terms of estimation efficiency (as measured by the D-optimality criterion), Macharia and Goos (2010) showed that, in many instances, it is possible to generate second-order equivalent-estimation split-plot designs that are highly efficient and, hence, provide precise factor-effect estimates. In this work, we present an algorithm that allows us to (i) identify equivalent-estimation designs for scenarios where Macharia and Goos (2010) did not find equivalent-estimation designs and (ii) find equivalent-estimation designs that outperform those of Macharia and Goos (2010) in terms of the D-optimality criterion. We also study the performance of equivalent-estimation designs when it comes to estimating the variance components in the split-plot model and observe that they outperform D-optimal designs in this respect.

Algorithm for determining whether various two-level fractional factorial split-plot row–column designs are non-isomorphic

The confounding and aliasing scheme for fractional factorial split-plot designs with the units within each wholeplot arranged in rows and columns is described and illustrated. Isomorphism for this design type is described, together with a procedure which considers extensions of the concepts of wordlength patterns and letter patterns that can be used to test isomorphism between designs. Using in part this isomorphism testing procedure, a construction algorithm that may be used to obtain a complete set of such non-isomorphic two-level designs is described. Software based on this construction algorithm was used to obtain a complete set of non-isomorphic designs for up to five wholeplot factors, five subplot factors and up to 64 runs, which is presented as a table of designs. To aid the experimenter in distinguishing between competing designs, the estimation capacity sequence for each design is presented.

Optimal Experimental Design of Human Appraisals for Modeling Consumer Preferences in Engineering Design

Human appraisals are becoming increasingly important in the design of engineering systems to link engineering design attributes to customer preferences. Human appraisals are used to assess consumers’ opinions of a given product design, and are unique in that the experiment response is a function of both the product attributes and the respondents’ human attributes. The design of a human appraisal is characterized as a split-plot design, in which the respondents’ human attributes form the whole-plot factors while the product attributes form the split-plot factors. The experiments are also characterized by random block effects, in which the design configurations evaluated by a single respondent form a block. An experimental design algorithm is needed for human appraisal experiments because standard experimental designs often do not meet the needs of these experiments. In this work, an algorithmic approach to identify the optimal design for a human appraisal experiment is developed, which considers the effects of respondent fatigue and the blocked and split-plot structures of such a design. The developed algorithm seeks to identify the experimental design, which maximizes the determinant of the Fisher information matrix. The algorithm is derived assuming an ordered logit model will be used to model the rating responses. The advantages of this approach over competing approaches for minimizing the number of appraisal experiments and model-building efficiency are demonstrated using an automotive interior package human appraisal as an example.

Follow-Up Designs to Resolve Confounding in Split-Plot Experiments

Split-plot designs are effective in industry due to time and/or cost constraints, restriction on randomization of the treatment combinations of the hard-to-change factors, and different sizes of experimental units. Some of the results of fractional factorial split-plot experiments can be ambiguous and a need may arise to conduct follow-up experiments to separate effects of potential interest by breaking their alias links with others. For completely randomized fractional factorial experiments, methods have been developed to construct follow-up experiments. In this article, we extend the foldover technique to break the alias chains of split-plot experiments. Because it is impractical or not economically possible to foldover the whole-plot factors, as their levels are often hard or expensive to change, the focus of this article is on folding over only one or more subplot factors in order to de-alias certain effects. Six rules are provided to develop foldovers for minimum aberration resolution III and resolution IV fractional factorial split-plot designs.

Bounds on the maximum numbers of clear two-factor interactions for $$2^{(n_1 + n_2 ) - (k_1 + k_2 )} $$ fractional factorial split-plot designs

Fractional factorial split-plot (FFSP) designs have an important value of investigation for their special structures. There are two types of factors in an FFSP design: the whole-plot (WP) factors and sub-plot (SP) factors, which can form three types of two-factor interactions: WP2fi, WS2fi and SP2fi. This paper considers FFSP designs with resolution III or IV under the clear effects criterion. It derives the upper and lower bounds on the maximum numbers of clear WP2fis and WS2fis for FFSP designs, and gives some methods for constructing the desired FFSP designs. It further examines the performance of the construction methods.

An integer programming approach to the construction of trend-free experimental plans on split-plot designs

In many experimental designs, the standard procedure involves randomization of the factor-level combination run order. There are cases, however, where it is known that a time or position trend that can seriously compromise the results of the experiment may be present. These trends include wear of tooling and equipment, learning curves, change in temperatures, and so on, and the trends may show up as linear, quadratic, or even higher order trends. All previously published work has dealt with various methods of constructing trend-resistant run order plans on full and fractional factorial designs. These previous efforts have not addressed any additional dimensions in the trends that emerge when using hierarchical designs such as split-plot plans. These designs are common in many manufacturing experiments where complete randomization is not possible or is too expensive to be practical. The objective of this work is to establish the foundations of a method for constructing linear and quadratic trend-resistant plans in two-level split-plot designs that addresses the two-dimensional trends that may occur. The methodology involves development of a hybrid approach using the foldover method in each of the dimensions of interest and embedding these in a nonlinear integer programming model in the search for a feasible solution. Feasibility of this approach is shown for the particular case of a split-plot design (2 5 whole-plot factors and 3 1 × 2 1 split-plot factors) performed on abrasive machining. In this case study, an experimental plan that is robust against all linear trends and most quadratic trends was achieved.

Corrigendum: Designing Fractional Factorial Split-Plot Experiments with Few Whole-Plot Factors

[Appl. Statist., 53 (2004), 325339]. Owing to an inadvertent mistake in the enumeration program, some of the entries in Table 2 are in error. We reproduce a corrected version of Table 2 below. We thank Peter Goos, James M. Lucas and Frank T. Anbari, who independently pointed out individual faulty entries in the original table and thus gave us an opportunity for correction.

Formula omitted] Fractional factorial split-plot designs containing clear effects

Whole-plot (WP) factors and sub-plot (SP) factors play different roles in fractional factorial split-plot (FFSP) designs. In this paper, we consider FFSP designs with resolution III or IV from the viewpoint of clear factorial effects, classify two-factor interactions (2FIs) into three types, and give sufficient and necessary conditions for the existence of FFSP designs containing various clear factorial effects, including two types of main effects and three types of 2FIs. The structures of these designs are also shown and illustrated with examples.

A BAYESIAN RANDOM COEFFICIENT NONLINEAR REGRESSION MODEL FOR A SPLIT-PLOT EXPERIMENT

In random coefficients regression, we are often interested in the mean of a certain para-meter particular to the experimental unit (EU). When the mean depends on some treatment regimen, we are then interested in comparing the means among the different treatments. When the EUs are repeatedly measured on a variable containing information about the EU parameter, a standard procedure is to estimate each EU parameter and treat the estimates as the response variables. This is especially true when the regression model for an EU is non-linear. Often, for designed experiments with a factorial treatment structure, the estimated EU parameters are then modeled with an appropriate linear (mixed) model. Here, we consider a split-plot experiment conducted to detect differences in the half-life of a compound between different treatment regimens of the compound, namely compound preparation and temperature (whole-plot factors) and initial compound amount (split-plot factor). Initially, we provide a standard (classical) analysis plan, and then present a Bayes random coefficients regression model to address the researcher’s questions of interest. We finally compare the results from the standard and Bayes analyses.

Designing Fractional Factorial Split-Plot Experiments with Few Whole-Plot Factors

Summary When it is impractical to perform the experimental runs of a fractional factorial design in a completely random order, restrictions on the randomization can be imposed. The resulting design is said to have a split-plot, or nested, error structure. Similarly to fractional factorials, fractional factorial split-plot designs can be ranked by using the aberration criterion. Techniques that generate the required designs systematically presuppose unreplicated settings of the whole-plot factors. We use a cheese-making experiment to demonstrate the practical relevance of designs with replicated settings of these factors. We create such designs by splitting the whole plots according to one or more subplot effects. We develop a systematic method to generate the required designs and we use the method to create a table of designs that is likely to be useful in practice.

Detailed Wordlength Pattern of Regular Fractional Factorial Split-Plot Designs in Terms of Complementary Sets

Abstract With reference to regular fractional factorial split-plot designs, we consider a detailed wordlength pattern taking due cognizance of the distinction between the whole-plot and sub-plot factors. A generalized version of the MacWilliams' identity is employed to express the detailed wordlength pattern in terms of complementary sets. Several special features make this result intrinsically different from the corresponding one in classical fractional factorial designs where all factors have the same status. An application to robust parameter designs is indicated and examples given.

Selecting for Common Bacterial Blight Resistance in Common Bean: Effects of Multiple Leaf Inoculation and Detached Pod Inoculation Test

Breeding for resistance is an important strategy to manage common bacterial blight disease caused by Xanthomonas campestris pv. phaseoli (E. Smith) Dye (Xep) in common bean (Phaseolus vulgaris L.). It is necessary to determine if prior inoculation of the first trifoliolate leaf with Xcp influences subsequent reactions in other plant organs by increasing or decreasing resistance to Xcp. It is difficult to get an accurate estimate of heritability of disease reaction in pods since environment greatly affects the heritability estimate if flowering occurs over extended time periods. Thus, the disease reaction in attached pods versus detached pods was compared. A split-split plot design with two replications (growth chambers as blocks) was used, with bean lines as the whole-plot factors, Xcp strains as subplot factors, and bacterial inoculation treatments for leaf reactions or pod treatments as split-split plot factors. The first trifoliolate leaves, later developed leaves, and attached and detached pods were inoculated. No effects of prior inoculation on the disease reactions of subsequently inoculated leaves and pods were observed, indicating that the different plant organs can be inoculated at different times. The fact that detached and attached pods showed similar disease symptoms would suggest use of the former to reduce environment variance and improve heritability estimates of resistance.