Orthogonal Main-effect Plans Research Articles

Mixed-level designs with orthogonality and relatively optimal run order

Orthogonality and optimality of run order are two important and worthy to be considered criteria in design of experiment. For the mixed-level designs commonly used in the case of unequal levels of factors in the experiment, both orthogonality and optimality of run order are taken into account at the same time in this paper. The construction methods of mixed-level designs with orthogonality and relatively minimum or maximum level changes are respectively proposed. Based on such designs, the orthogonal main-effect plans with relatively minimum or maximum level changes are obtained by the method of collapsing factor. The designs with relatively minimum or maximum level changes constructed in this paper guarantee the cost minimum or the time trend most robust in the actual experiment arrangement.

Multiple doubling: a simple effective construction technique for optimal two-level experimental designs

Design of experiment is an efficient statistical methodology of establishing which input variables are important (have significant effects) in an experiment (process) and the conditions under which these inputs should work to optimize the outputs of that process. Two-level designs are widely used in high-tech industries and manufacturing for productivity and quality improvement experiments. The construction of (nearly) optimal two-level designs for real-life experiments with large number of input variables can be quite challenging. The practice demonstrated that the existing techniques are complex, highly time-consuming, produce limited types of designs, and likely to fail in large experiments (i.e., optimal results are not expected). To overcome these significant problems, this article gives a simple and effective technique for constructing large two-level designs with good statistical properties. To meet practical needs in different fields, the statistical properties of the generated designs by the new technique are investigated from four basic perspectives: minimizing the similarity among the experimental runs, minimizing the aliasing among the input variables, maximizing the resolution, and filling the experimental domain as uniformly as possible. New recommended saturated orthogonal main effect plans and uniform orthogonal arrays of strength three with thousands or even millions of runs and factors are generated via the new technique without recourse to optimization software.

Inter-Class Orthogonal Main Effect Plans for Asymmetrical Experiments

In this paper we construct ‘inter-class orthogonal’ main effect plans (MEPs) for asymmetrical experiments. In such a plan, the factors are partitioned into classes so that any two factors from different classes are orthogonal. We have also defined the concept of “partial orthogonality” between a pair of factors. In many of our plans, partial orthogonality has been achieved when (total) orthogonality is not possible due to divisibility or any other restriction. We present a method of obtaining inter-class orthogonal MEPs. Using this method and also a method of ‘cut and paste’ we have obtained several series of inter-class orthogonal MEPs. One of them happens to be a series of orthogonal MEP (OMEPs) [see Theorem 3.6], which includes an OMEP for a 330 experiment on 64 runs. We have also obtained a series of MEPs which are almost orthogonal in the sense that every contrast is non-orthogonal to at most one more. A member of this series is an MEP for a 310210 experiment on 32 runs in which the only non-orthogonality is between the linear contrasts of pairs of three-level factors. Plans of small size (≤ 15 runs) are also constructed by ad-hoc methods. Among these plans there are MEPs for a 42.32.2 and a 35.2 experiment on 12 runs and a 52.32 experiment on 15 runs.

Orthogonal main-effect plans in row–column designs for two-level factorial experiments

ABSTRACTRow–column designs for two-level factorial experiments are constructed to estimate all the main effects. We give the interactions for row and column blockings. Based on these blockings, independent treatment combinations are proposed to establish the whole design so that practitioners can easily apply it to their experiments. Some examples are given for illustrations. The estimation of two-factor interactions in these designs is discussed.

Minimum breakdown two-level orthogonal main-effect plans

A new class of robust designs against missing data, called the minimum breakdown designs, is proposed for two-level orthogonal main-effect plans. The minimum breakdown designs provide the highest probability to derive a non-singular residual design for estimating the grand mean and all the main effects, when a fraction of experimental data is unavailable. Some theoretical and computational results for characterizing the minimum breakdown designs are developed. In addition, a catalogue of the proposed designs with practical run-sizes is presented for multifactorial experiments.

Mosaic Plots are Useful for Visualizing Low-Order Projections of Factorial Designs

Factorial experiments are widely used in industrial experimentation and other fields. Whenever a factorial experiment is not designed as a full factorial, but as a regular or nonregular fraction thereof, choice between competing designs and interpretation of experimental results should take into consideration how the experimental plan will confound experimental effects. This article proposes mosaic plots of low-order projections of factorial designs for visualizing confounding of low-order effects. Mosaic plots are particularly useful for design and analysis of orthogonal main effect plans. The R code for the creation of the plots in this article is available online in the supplementary material.

Selection of partial replication on two‐level orthogonal arrays

AbstractDue to the lack of a replication‐based estimate for the experimental error variance, unreplicated orthogonal main‐effect plans (OMEPs) usually appear to be unsatisfactory in screening active factors, when there are certain interactions likely to be non‐negligible. In this study, partially replicated designs constructed through two‐level orthogonal arrays are recommended to compensate for this deficiency. A compound criterion called the extended minimum aberration (EMA) is proposed for choosing the twice‐replicated fractions. According to the EMA criterion, sufficient conditions for constructing the minimal two‐level OMEPs with partial replication are proposed, and a series of EMA designs with practical run‐sizes is provided for practical applications. The Canadian Journal of Statistics 42: 168–183; 2014 © 2014 Statistical Society of Canada

Two-level Orthogonal and Saturated Designs for Estimating Main Effects and Certain Important Interactions

In a two-level factorial experiment, we consider orthogonal designs that allow joint estimation of the grand mean, all main effects, and certain classes of two-level interactions, assuming that the remaining effects are all negligible. Based on a judicious allocation of the factorial effects of interest to the columns of a Hadamard matrix, we propose some general classes of orthogonal and saturated designs which include some existing orthogonal main-effect plans of asymmetric factorials as special cases.

Main-Effect Plans Orthogonal Through the Block Factor

Many industrial experiments require different number of levels for different factors. Orthogonal main-effect plans (OMEPs) for such asymmetrical experiments often require large run size. If blocking is needed, then run size usually becomes even larger. In this article we show that there are situations when use of blocks may in fact be helpful in finding an OMEP with small run size. Explicitly, we obtained saturated OMEPs for a 33n⋅23n experiment in 3n blocks of size 4 each for each Hadamard number n. In each of these plans, the three-level factors are nonorthogonal to the block factor but are pairwise orthogonal through the block factor. The two-level factors are orthogonal to the block factor. The data analysis for such a plan is illustrated with the help of examples. This article has supplementary material online.

Universally optimal structurally balanced row–column designs with some empty nodes

In this paper, we present two methods of construction of a class of universally optimal b 1 × b 2 row–column designs with b 2 - k 1 > 0 empty nodes in each row and b 1 - k 2 > 0 empty nodes in each column. The number of treatments under consideration is of the form s + 1 where s is an odd prime power number. The designs constructed here are structurally balanced in the sense that every pair of columns is used in an equal number of rows. The proposed designs can also be used as universally optimal two-factor orthogonal main-effect plans arranged in incomplete blocks which are also equivalent to orthogonal pairs of balanced incomplete block designs, balanced incomplete block designs for two sets of treatments, and two stage experiments in incomplete blocks with different numbers of treatments for the two stages.

MinPROMEP: Generation of Partially Replicated Minimal Orthogonal Main-Effect Plans Using a Novel Algorithm

MinPROMEP, written in SAS macros, is a program that generates partially replicated minimal orthogonal main-effect plans (minimal PROMEP's) for a given set of specified factor levels. The proposed algorithm is based on a generalization of the construction of Jacroux (1992, 1993). By using specific collapsing schemes, duplicated points are embedded in the design. In some cases, the generated minimal PROMEP has maximum number of duplicate points.

The Idle Column Method: Design Construction, Properties and Comparisons

Taguchi's idle column method for constructing arrays is investigated and generalized. A general construction method is proposed via a necessary and sufficient condition on two-level orthogonal arrays to which the method is applicable. A method is developed for identifying the idle column and pairs of columns in the arrays. Mixed two- and three-level idle column arrays with 64 or fewer runs are tabulated. Some guidelines are offered on when the idle column arrays should be used and when they should be replaced by orthogonal main-effect plans or D-optimal designs.

Orthogonal main effect plans and an extension to resolution V

Orthogonal main effect plans and an extension to resolution <i>V</i>

An interchange algorithm for four factor orthogonal main effect plans

In this paper we give a construction for four factor orthogonal main effect plans (OMEPs) and an interchange algorithm to give four factor OMEPs with various different numbers of repeated runs.

Small orthogonal main effect plans with four factors

In this paper we study orthogonal main effect plans with four factors, A table of such designs, where each factor has at most 10 levels, and there are at most 40 runs, is generated. We determine the spectrum of the degrees of freedom of pure error for these designs.

Minimal orthogonal main-effect plans and partial replication

This work extends Jacroux's results (Jacroux, Technometrics 34 (1992) 92–96) involving minimal orthogonal main-effect plans (OMEP's). Tables of correct minimal numbers of runs of OMEP's are also provided. In addition, a general method is proposed to construct a class of minimal OMEP's. Moreover, a construction method is provided to obtain partially replicated OMEP's (PROMEP's). Our results provide more duplicate points for some plans than Jacroux's results on PROMEP's (Jacroux Technometrics 35 (1993) 32–36). In addition, a method is derived to construct s 1 × ( s 1 − 1) 2∥ s 2 1 minimal PROMEP's with maximal duplicate points.

Tight 4-factor orthogonal main effect plans

Our main result is a determination of those parameter sets for which a tight OMEP with four rows exists. These are used to characterize those n for which an s 1 × s 2 × s 3 × s 4// n OMEP exists. The main application of this is a method for determining the minimal n for which an s 1 × s 2 × s 3 × s 4// n OMEP exists. We use the same techniques to give an algorithm for determining the minimal n for which an s 1 × s 2 × s 3 × s 4// n OMEP with equal replication exists. More importantly, the methods used here can be applied to OMEPs on any number of factors.

Asymptotic Existence of Tight Orthogonal Main Effect Plans

AbstractOur main result is showing the asymptotic existence of tight OMEPs. More precisely, for each fixed number k of rows, and with the exception of OMEPs of the form 2 × 2 × … 2 × 2s // 4s with s odd and with more than three rows, there are only a finite number of tight OMEP parameters for which the tight OMEP does not exist.

A Note on Main Effect Plans

Method of construction of main effect plans (saturated and unsaturated) through BIB designs, using necessary and sufficient condition of proportional frequencies, is proposed. Using this condition, we obtain two new plans 216/18 and 225/27 not reported earlier. Utilising these plans, we have obtained two new asymmetrical orthogonal main effect plans.

Orthogonal main effect plans for two and three factors in small blocks

Orthogonal main-effect plans for two and three factors in small blocks are obtained from the dual of adjusted orthogonal row-column designs. The method for constructing efficient plans is presented, and a relationship between the average efficiency factors of the row-column design and the corresponding main effects is given for the two-factor case.