Optimal design in average for inference in generalized linear models

This paper considers the problem of optimal design for inference in Generalized Linear Models, when prior information about the parameters is available. The general theory of optimum design usually requires knowledge of the parameter values. These are usually unknown and optimal design can, therefore, not be used in practice. However, one way to circumvent this problem is through so-called “optimal design in average”, or shortly, “ave optimal”. The ave optimal design is chosen to minimize the expected value of some criterion function over a prior distribution. We focus our interest on the aveD _A-optimality, including aveD- and avec-optimality and show the appropriate equivalence theorems for these optimality criterions, which give necessary conditions for an optimal design. Ave optimal designs are of interest when e.g. a factorial experiment with a binary or a Poisson response in to be conducted. The results are applied to factorial experiments, including a control group experiment and a 2×2 experiment.     

Existence conditions for balanced fractional 3m factorial designs of resolution R({00, 10, 01, 20, 11})

We consider a fractional 3^m factorial design derived from a simple array (SA) such that the non negligible factorial effects are the general mean, the linear and the quadratic components of the main effect, and the linear-by-linear and the linear-by-quadratic components of the two-factor interaction. If these effects are estimable, then a design is said to be of resolution R({00, 10, 01, 20, 11}). In this paper, we give a necessary and sufficient condition for an SA to be a balanced fractional 3^m factorial design of resolution R({00, 10, 01, 20, 11}). Such a design is concretely characterized by the suffixes of the indices of an SA.     

Economic and economic statistical design of T 2 control chart with two adaptive sample sizes

The Hotelling's T ² control chart, a direct analogue of the univariate Shewhart X¯ chart, is perhaps the most commonly used tool in industry for simultaneous monitoring of several quality characteristics. Recent studies have shown that using variable sampling size (VSS) schemes results in charts with more statistical power when detecting small to moderate shifts in the process mean vector. In this paper, we build a cost model of a VSS T ² control chart for the economic and economic statistical design using the general model of Lorenzen and Vance [The economic design of control charts: A unified approach, Technometrics 28 (1986), pp. 3–11]. We optimize this model using a genetic algorithm approach. We also study the effects of the costs and operating parameters on the VSS T ² parameters, and show, through an example, the advantage of economic design over statistical design for VSS T ² charts, and measure the economic advantage of VSS sampling versus fixed sample size sampling.     

COMPUTER AIDED-CONSTRUCTION OF D-OPTIMAL 2m FRACTIONAL DESIGNS OF RESOLUTION V

A new exchange algorithm for construction of 2^mD-optimal fractional factorial design (FFD) is devised. This exchange algorithm is a modification of the one due to Fedorov (1969, 1972) and is an improvement over similar algorithm due to Mitchell (1974) and Galil & Kiefer (1980). This exchange algorithm is then used to construct 54 D-optimal 2^m-FFD's of resolution V for m = 4,5,6.     

Minimum Aberration Regular Two-Level Designs in the Presence of Dispersion Factors

In practice, the variance of the response variable may change as some specific factors change from one setting to another in a factorial experiment. These factors affecting the variation of the response are called dispersion factors, which can violate the usual assumption of variance homogeneity. In this study, we modify the conventional minimum aberration criterion to take the impact of dispersion factors into account. The situations of one or two dispersion factors are investigated. As a result, we present regular 2^{n ? p} designs with run sizes equal to 16 and 32 using the modified minimum aberration criterion.     

Hyper-Graeco-Latin Squares and Fractional Factorial Designs

A Latin square of order s is an arrangement of the s letters in an s × s square so that every letter appears exactly once in every row and exactly once in every column. Copeland and Nelson (2000) used two examples to show that a Latin square can be chosen such that it corresponds to a fractional factorial design. In this article, we are going to study this topic more precisely. Furthermore, we will explore the relationship between fractional factorial designs and hyper-Graeco-Latin squares in general, where s is a prime or a power of a prime.     

Sensitivity Analyses of the Economic and Economic Statistical Designs of the Multivariate Synthetic T2 Chart

This article performs a sensitivity analyses of the synthetic T² chart using fractional factorial design, which integrates the interaction effects. We are interested in the effects of the input parameters on the optimal cost, chart's parameters, and average run lengths. We also look at the input parameters responsible for the increase in cost and improvement in statistical performance under statistical constraints, and investigate how the input parameters influence the binding effect of the statistical constraints. The sensitivity analyses of the synthetic T² chart are compared with that of the Hotelling's T² chart, and parameters responsible for the cost advantage of the synthetic T² chart are identified.     

A comparison study of effectiveness and robustness of robust economic designs of T2 chart using genetic algorithm

ABSTRACT

Recently, researchers have tried to design the T² chart economically to achieve the minimum possible quality cost; however, when T² chart is designed, it is important to consider multiple scenarios. This research presents the robust economic designs of the T² chart where there is more than one scenario. An illustrative example is used to demonstrate the effect of the model parameters on the optimal designs. The genetic algorithm optimization method is employed to obtain the optimal designs. Simulation studies show that the robust economic designs of T² chart are more effective than traditional economic design in practice.     

Characterization of Balanced Fractional 3 m Factorial Designs of Resolution III

We consider a fractional 3 ^m factorial design derived from a simple array (SA), which is a balanced array of full strength, where the non negligible factorial effects are the general mean and the linear and quadratic components of the main effect, and m ≥ 2. In this article, we give a necessary and sufficient condition for an SA to be a balanced fractional 3 ^m factorial design of resolution III. Such a design is characterized by the suffixes of indices of an SA.     

Theory of optimal blocking of nonregular factorial designs

The authors introduce the notion of split generalized wordlength pattern (GWP), i.e., treatment GWP and block GWP, for a blocked nonregular factorial design. They generalize the minimum aberration criterion to suit this type of design. Connections between factorial design theory and coding theory allow them to obtain combinatorial identities that govern the relationship between the split GWP of a blocked factorial design and that of its blocked consulting design. These identities work for regular and nonregular designs. Furthermore, the authors establish general rules for identifying generalized minimum aberration (GMA) blocked designs through their blocked consulting designs. Finally they tabulate and compare some GMA blocked designs from Hall's orthogonal array OA(16,2¹5,2) of type III.     

The robust economic statistical design of the Hotelling's T2 chart

ABSTRACT

Economic statistical designs aim at minimizing the cost of process monitoring when a specific scenario or a set of estimated process and cost parameters is given. But, in practice the process may be affected by more than one scenario which may lead to severe cost penalties if the wrong design is used. Here, we investigate the robust economic statistical design (RESD) of the T² chart in an attempt to reduce these cost penalties when there are multiple scenarios. Our method is to employ the genetic algorithm (GA) optimization method to minimize the total expected monitoring cost across all distinct scenarios. We illustrate the effectiveness of the method using two numerical examples. Simulation studies indicate that robust economic statistical designs should be encouraged in practice.     

Isomorphism check in fractional factorial designs via letter interaction pattern matrix

Two fractional factorial designs are considered isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and/or switching the levels of factors. To identify the isomorphism of two designs is known as an NP hard problem. In this paper, we propose a three-dimensional matrix named the letter interaction pattern matrix (LIPM) to characterize the information contained in the defining contrast subgroup of a regular two-level design. We first show that an LIPM could uniquely determine a design under isomorphism and then propose a set of principles to rearrange an LIPM to a standard form. In this way, we can significantly reduce the computational complexity in isomorphism check, which could only take O(2^p)+O(3k³)+O(2^k) operations to check two 2^k−p designs in the worst case. We also find a sufficient condition for two designs being isomorphic to each other, which is very simple and easy to use. In the end, we list some designs with the maximum numbers of clear or strongly clear two-factor interactions which were not found before.     

Double-objective economic statistical design of the VP T2 control chart: Wald's identity approach

Research has shown that applying the T² control chart by using a variable parameters (VP) scheme yields rapid detection of out-of-control states. In this paper, the problem of economic statistical design of the VP T²control chart is considered as a double-objective minimization problem with the statistical objective being the adjusted average time to signal and the economic objective being expected cost per hour. We then find the Pareto-optimal designs in which the two objectives are met simultaneously by using a multi-objective genetic algorithm. Through an illustrative example, we show that relatively large benefits can be achieved by applying the VP scheme when compared with usual schemes, and in addition, the multi-objective approach provides the user with designs that are flexible and adaptive.     

On the problem of augmented fractional factorial designs

Let D be a saturated fractional factorial design of the general K₁ x K₂ ...x K_t factorial such that it consists of m distinct treatment combinations and it is capable of providing an unbiased estimator of a subvector of m factorial parameters under the assumption that the remaining k-m,t (k = H it ) factorial parameters are negligible. Such a design will not provide an unbiased estimator of the varianceσ² Suppose that D is an optimal design with respect to some optimality criterion (e.g. d-optimality, a-optimality or e-optimality) and it is desirable to augment D with c treatmentcombinations with the aim to estimate ² Suppose that D is an optimal design with respect to some optimality criterion (e.g. d-optimality, a-optimality or e-optimality) and it is desirable to augment D with c treatment combinations with the aim to estimate σ² unbiasedly. The problem then is how to select the c treatment combinations such that the augmented design D retains its optimality property. This problem, in all its generality is extremely complex. The objective of this paper is to provide some insight in the problem by providing a partial answer in the case of the 2^tfactorial, using the d-optimality criterion.     

Analytic connections between a double design and its original design in terms of different optimality criteria

In recent years, there has been increasing interest in the study of double designs. Various popular optimality criteria have been proposed from different principles for design construction and comparison, such as E(s²), generalized minimum aberration (GMA), minimum moment aberration (MMA), and minimum projection uniformity (MPU). In this article, these criteria are reviewed, and analytic connections between a double design and its original design in terms of these criteria are investigated. These connections are suitable for general original two-level factorial design, whether regular or non regular. In addition, these results provide strong insight into the relationship between double design and original design from different viewpoints.     

Reverse Foldovers for Blocked 2 m−k Fractional Factorial Designs

Two-level regular fractional factorial designs are often used in industry as screening designs to help identify early on in an experimental process those experimental or system variables which have significant effects on the process being studied. When the experimental material to be used in the experiment is heterogenous or the experiment must be performed over several well-defined time periods, blocking is often used as a means to improve experimental efficiency by removing the possible effects of heterogenous experimental material or possible time period effects. In a recent article, Li and Jacroux (2007 Li , F. , Jacroux , M. (2007). Optimal foldover plans for blocked 2 ^m?k fractional factorial designs. J. Statsist. Plann. Infer 137:2434–2452. [Google Scholar]) suggested a strategy for constructing optimal follow-up designs for blocked fractional factorial designs using the well-known foldover technique in conjunction with several optimality criteria. In this article, we consider the reverse foldover problem for blocked fractional factorial designs. In particular, given a 2^(m+p)?(p+k) blocked fractional factorial design D, we derive simple sufficient conditions which can be used to determine if there exists a 2^{(m+p?1)?(p?1+k+1)} initial fractional factorial design d which yields D as a foldover combined design as well how to generate all such d. Such information is useful in developing an overall experimental strategy in situations where an experimenter wants an overall blocked fractional factorial design with “desirable” properties but also wants the option of analyzing the observed data at the halfway mark to determine if the significant experimental variables are obvious (and the experiment can be terminated) or if a different path of experimentation should be taken from that initially planned.     

Minimal Markov Basis for Tests of Main Effect Models for 2p-1 Fractional Factorial Designs of Resolution p

We consider conditional exact tests of factor effects in design of experiments for discrete response variables. Similarly to the analysis of contingency tables, Markov chain Monte Carlo methods can be used to perform exact tests, especially when large-sample approximations of the null distributions are poor and the enumeration of the conditional sample space is infeasible. In order to construct a connected Markov chain over the appropriate sample space, one approach is to compute a Markov basis. Theoretically, a Markov basis can be characterized as a generator of a well-specified toric ideal in a polynomial ring and is computed by computational algebraic software. However, the computation of a Markov basis sometimes becomes infeasible, even for problems of moderate sizes. In the present article, we obtain the closed-form expression of minimal Markov bases for the main effect models of 2^{p ? 1} fractional factorial designs of resolution p.     

A lower bound for the centred L 2-discrepancy on combined designs under the asymmetric factorials

The foldover is a useful technique in the construction of two-level factorial designs for follow-up experiments. To search an optimal foldover plans is an important issue. In this paper, for a set of asymmetric fractional factorials such as the original designs, a lower bound for centred L ₂-discrepancy of combined designs under a general foldover plan is obtained, which can be used as a benchmark for searching optimal foldover plans. All of our results are the extended ones of Ou et al. [Lower bounds of various discrepancies on combined designs, Metrika 74 (2011), pp. 109–119] for symmetric designs to asymmetric designs. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.     

Orthogonal three-level parallel flats designs for user-specified resolution

Orthogonal factorial and fractional factorial designs are very popular in many experimental studies, particularly the two-level and three-level designs used in screening experiments. When an experimenter is able to specify the set of possibly nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design belonging to the class of parallel flats designs, that has a smaller run-size than a suitable design from the class of classical fractional factorial designs belonging to the class of single flat designs. Sri-vastava and Li (1996) proved a fundamental theorem of orthogonal s-level, s being a prime, designs of parallel flats type for the user-specified resolution. They also tabulated a series of orthogonal designs for the two-level case. No orthogonal designs for three-level case are available in their paper. In this paper, we present a simple proof for the theorem given in Srivastava and Li (1996) for the three-level case. We also give a dual form of the theorem, which is more useful for developing an algorithm for construction of orthogonal designs. Some classes of three-level orthogonal designs with practical run-size are given in the paper.