Main Effect Plans Research Articles

On the [formula omitted]-optimality of orthogonal and nonorthogonal blocked main effects plans

In this paper, we consider experimental situations where a main effects plan is to be used to study m two-level factors using n runs which are partitioned into b blocks of equal size k = n / b . For the case n ≡ 2 ( mod 8 ) and k > 2 where k is even, sufficient conditions are derived for certain designs having all main effects orthogonal to blocks to be D -optimal among all available designs. For n ≡ 2 ( mod 8 ) we also give conditions under which one type of design not having all main effects orthogonal to blocks is D -optimal among all available designs.

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.

Optimal main effect plans in nested row-column set-up of small size

In many experimental situations, there are two blocking factors nested within another blocking factor. This is termed nested row-column set-up. The available factorial designs in such a set-up are large and so may not always be practically useful. We consider factorial designs in these set-ups where the number of levels of each factor, including the blocking factor, is small and obtain main effect plans which are optimal for the estimation of all the treatment factors.

A note on the construction of optimal main effects plans in blocks of size two

In this paper we consider experimental situations in which n factors are to be studied in N runs arranged in N / 2 blocks of size two. A simple method for constructing optimal designs in many such situations is given and examples are used to illustrate the technique.

Partial duplication in two-level fractional factorial designs

Most fractional factorial designs have no replicated points and thus do not provide an estimate for pure error. The construction methods for orthogonal main-effect plan in the literature usually do not produce designs with duplicate points. However, it is possible to combine four fractions to provide a set of duplicate points without sacrificing the orthogonality of main effects. This paper proposes two techniques of this idea to produce designs with replicate points in two-level fractional factorial designs.

Optimal main effect plans in blocks of small size

In this paper we first construct an universally optimal main effect plan (MEP) for an s s experiment on s 2 ( s - 1 ) / 2 nonorthogonal blocks of size two each, s a power of 2. Next we derive another set of sufficient conditions for an MEP on nonorthogonal blocks requiring a smaller number of blocks. These conditions are used to obtain universally optimal saturated MEPs in blocks of size 2 for (i) 4 2 experiment on 6 blocks and (ii) 5 2 × 2 2 on 10 blocks.

Construction of D-optimal s1 × s2 × s3 factorial designs using graph theory

In this paper, the maximum determinant of the associated 0-1 matrix in D-Optimal saturated main effect plans for 3× s_2 × s_3 factorials, is derived by the use of Graph theory and Combinatorics. The present work is related to a problem suggested by Chatterjee and Narasimhan (2002). Using the theoretical results, we also give the designs for s3≥s2 + 1.

Projection Properties of Certain Three-Level Main Effect Plans with Quantitative Factors

ABSTRACT Orthogonal arrays are used as screening designs to identify active main effects, after which the properties of the subdesign for estimating these effects and possibly their interactions become important. Such a subdesign is known as a “projection design”. In this article, we have identified all the geometric non isomorphic projection designs of an OA(27,13,3,2), an OA(18,7,3,2) and an OA(36,13,3,2) into k = 3,4, and 5 factors when they are used for screening out active quantitative experimental factors, with regard to the prior selection of the middle level of factors. We use the popular D-efficiency criterion to evaluate the ability of each design found in estimating the parameters of a second order model.

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.

A comparison between the Gröbner bases approach and hidden projection properties in factorial designs

Screening designs are useful for situations where a large number of factors ( q ) are examined but only few ( k ) of these are expected to be important. Plackett–Burman designs have traditionally been studied for this purpose. Since these designs are only main effects plans and since the number of runs are greater than the number of active factors (main effects), there are plenty of degrees of freedom unused for identifying and estimating interactions of factors. Computational Algebraic Geometry can be used to solve identifiability problems in design of experiments in Statistics. The theory of Gröbner bases allows one to identify the whole set of estimable effects (main or interactions) of the factors of the design. On the other hand, the hidden projection property approach, that deals with the same identification problem, provides a measure of how efficient the identification of effects is. The advantages and disadvantages of both methods are discussed with application to a certain two level (fractional) factorial designs that arise from Plackett–Burman designs.

Optimal main effect plans in blocks and related nested row-column designs

Abstract Optimal main effect plans with nonorthogonal blocks of size two each are constructed. Using these block designs, optimal block designs with nested rows and columns are constructed for some main effect plans.

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.

Adding more observations to saturated D‐optimal resolution III two‐level factorial designs

AbstractWe consider the class of saturated main effect plans for the 2k factorial. With these saturated designs, the overall mean and all main effects can be unbiasedly estimated provided that there are no interactions. However, there is no way to estimate the error variance with such designs. Because of this and other reasons, we like to add some additional runs to the set of (k+1) runs in the D‐optimal design in this class. Our goals here are: (1) to search for s additional runs so that the resulting design based on (k+s+1) runs yields a D‐optimal design in the class of augmented designs; (2) to classify all the runs into equivalent classes so that the runs in the same equivalent class give us the same value of the determinant of the information matrix. This allows us to trade runs for runs if this becomes necessary; (3) to obtain upper bounds for determinant of the information matrices of augmented designs. In this article we shall address these approaches and present some new results. © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 51–77, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10026

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.

Regression analysis of experiments with complex confounding patterns guided by the alias matrix

Resolution III experiments with complex confounding patterns, often called main effect plans. Their main effects are partially confounded with two-factor interactions rather than being either independent or completely aliased, as they are in regular designs that are constructed from a defining relation. It is possible to detect some two-factor interactions, from experiments with complex confounding patterns if only a few of the factors are active. The partial confounding of interactions with main effects in these experiments allows one to estimate interactions with regression analysis. Recent papers have shown how the interactions can be detected using repeated runs of stepwise regression, or computationally intensive Bayesian methods. This paper shows that, with a short list of candidate interactions, the important interactions can be detected efficiently by a single pass of all subsets regression. The short list of candidates interactions for the all subsets regression are selected with regard to their potential for contributing to the value of the large estimated effects calculated from the data. The potential is determined by summing coefficients in the alias matrix. This paper reviews the concept of the alias matrix, and defines the alias plot that can be obtained from the alias matrix. The alias plot provides simple graphic for viewing the potential contribution of interactions to large estimated effects. The interactions identified on the alias plot, along with main effects, are used as candidates in an all subsets regression analysis of data from partially confounded experiments. The resulting strategy allows exploration of a wide range of the potential model space, can reveal several plausible models for the data if they exist, and can be completed using standard statistical software. Examples are presented.

Optimal main effect plans with non-orthogonal blocking

The current literature on fractional factorial plans in block designs centres around orthogonal blocking which may not, however, always be attainable because of practical restrictions on the block size. For general factorials, including asymmetric ones, sufficient conditions are indicated in this paper for a main effect plan to be universally optimal under possibly non-orthogonal blocking. A construction procedure is given using generalised Youden designs in conjunction with orthogonal arrays. We also illustrate how the procedure can be applied to obtain optimal main effect plans in the practically important situation where each factor has two or three levels and the block size is small.

NESTING OPTIMAL MAIN EFFECTS PLANS AND OPTIMAL FOLDOVER DESIGNS

ABSTRACT Optimal main effects plans (MEPs) and optimal foldover designs can often be performed as a series of nested optimal designs. Then, if the experiment cannot be completed due to time or budget constraints, the fraction already performed may still be an optimal design. We show that the optimal MEP for 4t factors in 4t + 4 points does not contain the optimal MEP for 4t factors in 4t + 2 points nested within it. In general, the optimal MEP for 4t factors in 4t + 4 points does not contain the optimal MEPs for 4t factors in 4t + 1, 4t + 2, or 4t + 3 points and the optimal MEP for 4t + 1 factors in 4t + 4 points does not contain the optimal MEPs for 4t + 1 factors in 4t + 2 or 4t + 3 points. We also show that the runs in an orthogonal design for 4t factors in 4t + 4 points, and the optimal foldover designs obtained by folding, should be performed in a certain sequence in order to avoid the possibility of a singular X'X matrix.

Two ge neral classes of search designs for factor screening experiments with factors at three levels

In this paper, we are presenting general classes of factor screening designs for identifying a few important factors from a list of m (≥ 3) factors each at three levels. A design is a subset of 3m possible runs. The problem of finding designs with small number of runs is considered here. A main effect plan requires at least (2m + 1) runs for estimating the general mean, linear and quadratic effects of m factors. An orthogonal main effect plan requires, in addition, the number of runs as a multiple of 9. For example, when m=5, a main effect plan requires at least 11 runs and an orthogonal main effect plan requires 18 runs. Two general factor screening designs presented here are nonorthogonal designs with (2m− 1) runs. These designs, called search designs permit us to search for and identify at most two important factors out of m factors under the search linear model introduced in Srivastava (1975). For example, when m=5, the two new plans given in this paper have 9 runs, which is a significant improvement over an orthogonal main effect plan with 18 runs in terms of the number of runs and an improvement over a main effect plan with at least 11 runs. We compare these designs, for 4≤m≤ 10, using arithmetic and geometric means of the determinants, traces, and maximum characteristic roots of certain matrices. Two designs D1 and D2 are identical for m=3 and this design is an optimal design in the class of all search designs under the six criteria discussed above. Designs D1 and D2 are also identical for m=4 under some row and column permutations. Consequently, D1 and D2 are equally good for searching and identifying one important factor out of m factors when m=4. The design D1 is marginally better than the design D2 for searching and identifying one important factor out of m factors when m=5, … , 10. The design D1 is marginally better than the D2 for searching and identifying two important factors out of m factors when m=5, 7, 9. The design D2 is somewhat better than the design D1 for m=6, 8. For m=10, D1 is marginally better than D2 w.r.t. the geometric mean and D2 is marginally better than D1 w.r.t. the arithmetic mean of the maximum characteristic roots.

Orthogonal main effect plans and an extension to resolution V

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

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.