Optimal block designs for double cross experiments

ABSTRACT

This paper presents systematic methods of construction of optimal block designs for a double cross experiments for both even and odd values of “p” parental lines. The both even and odd values of designs are derived by using initial block of unreduced balanced incomplete block designs and initial block of row–column designs given by Bose et al. (1953 Bose, R.C., Shrikhande, S.S., Bhattacharya, K.N. (1953). On the construction of group divisible incomplete block design. Ann. Math. Stat. 24:167–195.[Crossref] , [Google Scholar]) and Gupta and Choi (1998 Gupta, S., Choi, K.C. (1998). Optimal row-column designs for complete diallel crosses. Commun. Stat. Theory Methods 27(11):2827–2835.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), respectively. In this attempt we have found some new universally optimal block designs for double cross experiments.     

Some extensions of circular balanced and circular strongly balanced repeated measurements designs

Repeated measurement designs are widely used in medicine, pharmacology, animal sciences, and psychology. In this paper the works of Iqbal and Tahir (2009 Iqbal, I., and M. H. Tahir. 2009. Circular strongly balanced repeated measurements designs. Communications in Statistics—Theory and Methods 38:3686–96.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Iqbal, Tahir, and Ghazali (2010 Iqbal, I., M. H. Tahir, and S. S. A. Ghazali. 2010. Circular first- and second-order balanced repeated measurements designs. Communications in Statistics—Theory and Methods 39:228–40.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) are generalized for the construction of circular-balanced and circular strongly balanced repeated measurements designs through the method of cyclic shifts for three periods.     

Optimum mixture designs in some constrained experimental regions

In a mixture experiment, the response depends on the mixing proportions of the components present in the mixture. Optimum designs are available for the estimation of parameters of the models proposed in such situations. However, these designs are found to include the vertex points of the simplex Ξ defining the experimental region, which are not mixtures in the true sense. Recently, Mandal et al. (2015 Mandal, N.K., Pal, M., Sinha, B.K., and Das, P. (2015). Optimum mixture designs in a restricted region. Stat. Pap. 56(1):105–119.[Crossref], [Web of Science ®] , [Google Scholar]) derived optimum designs when the experiment is confined to an ellipsoidal region within Ξ, which does not include the vertices of Ξ. In this paper, an attempt has been made to find optimum designs when the experimental region is a simplex or is cuboidal inside Ξ and does not contain the extreme points.     

Circular First- and Second-Order Balanced Repeated Measurements Designs

Sharma (1977 Sharma , V. K. ( 1977 ). Change-over designs with complete balance for first and second order residual effect . Canad. J. Statist. 5 : 121 – 132 .[Crossref] , [Google Scholar]) and Aggarwal et al. (2006 Aggarwal , M. L. , Deng , L.-Y. , Jha , M. K. ( 2006 ). Balanced residual treatment effects designs of first and second order . Statist. Probab. Lett. 76 : 597 – 600 .[Crossref], [Web of Science ®] , [Google Scholar]) considered non circular construction of first- and second-order balanced repeated measurements designs. Sharma et al. (2002 Sharma , V. K. , Varghese , C. , Jaggi , S. ( 2002 ). On optimality of change-over designs balanced for first and second order residual effects . Metron 60 : 153 – 162 . [Google Scholar]) constructed circular first- and second-order balanced repeated measurements designs only for a class with parameters (v, p = 3n, n = v ²) and also showed its universal optimality. In this article, we consider circular construction of first- and second-order balanced repeated measurements designs and strongly balanced repeated measurements designs by using the method of cyclic shifts. Some new circular designs with parameters (v, p, n) for cases p = v, p < v and p > v are given.     

Optimum Mixture Design Using Deficiency Criterion

In a mixture experiment the measured response is assumed to depend only on the relative proportion of ingredients or components present in the mixture. Scheffe (1958 Scheffe , H. ( 1958 ). Experiments with mixtures . J. R. Statist. Soc. B 20 : 344 – 360 . [Google Scholar], 1963 Scheffe , H. ( 1963 ). Simplex – centroid design for experiments with mixtures . J. R. Statist. Soc. B 25 : 235 – 263 . [Google Scholar]) first systematically considered this problem and introduced different models and designs suitable in such situations. Optimum designs for the estimation of parameters of different mixture models are available in the literature. The problem of estimating the optimum proportion of mixture components is of great practical importance. Pal and Mandal (2006 Pal , M. , Mandal , N. K. ( 2006 ). Optimum designs for optimum mixtures . Statist. Probab. Lett. 76 ( 13 ): 1369 – 1379 .[Crossref], [Web of Science ®] , [Google Scholar], 2007 Pal , M. , Mandal , N. K. (2007). Optimum mixture design via equivalence theory. communicated. JCISS 32:107–126. [Google Scholar]) attempted to find a solution to this problem by adopting a pseudo-Bayesian approach and using the trace criterion. Subsequently, Pal and Mandal (2008 Pal , M. , Mandal , N. K. ( 2008 ). Minimax designs for optimum mixtures . To appear in Statist. Probab. Lett.  [Google Scholar]) solved the problem using minimax criterion. In this article, the deficiency criterion due to Chatterjee and Mandal (1981 Chatterjee , S. K. , Mandal , N. K. ( 1981 ). Response surface designs for estimating the optimal point . Calcutta Statist. Assoc. Bull. . 30 : 145 – 169 .[Crossref] , [Google Scholar]) has been used as a measure for comparing the performance of competing designs.     

Construction and Nonparametric Testing of Orthogonal Arrays through Algebraic Strata and Inequivalent Permutation Matrices

In this article we use a new methodology, based on algebraic strata, to generate the class of all the orthogonal arrays of given size and strength. From this class we extract all the non isomorphic orthogonal arrays. Then, using all these non isomorphic orthogonal arrays, we suggest a method based on the inequivalent matrices permutations testing procedures Basso et al. (2004 Basso , D. , Evangelaras , H. , Koukouvinos , C. , Salmaso , L. ( 2004 ). Nonparametric testing for main effects on inequivalent designs. Proc. 7th Int. Workshop Model-Oriented Design Anal. Heeze, Netherlands, June 14–18 . [Google Scholar]) in order to obtain separate permutation tests for the effects in unreplicated mixed level fractional factorial designs. In order to validate the proposed method we perform a Monte Carlo simulation study and find out that the permutation tests appear to be a valid solution for testing effects, in particular when the usual normality assumptions cannot be justified.     

Efficient Block Designs for Dependent Observations—A Computer-Aided Search

In this article, the exchange and interchange algorithm of Zergaw (1989 Zergaw , G. ( 1989 ). A sequential method of constructing optimal block designs . Austral. J. Statist. 31 : 333 – 342 .[Crossref] , [Google Scholar]) and Martin and Eccleston (1992 Martin , R. J. , Eccleston , J. A. ( 1992 ). Recursive formulae for constructing block designs with dependent errors . Biometrika 79 : 426 – 430 .[Crossref], [Web of Science ®] , [Google Scholar]) have been modified and used for searching efficient block designs for making all possible pairwise treatment comparisons when observations are dependent. The lower bounds to the A- and D-efficiencies of the designs in a given class of the designs have been obtained for correlated observation structure and the procedure of computing lower bounds to A- and D-efficiencies has been incorporated in the algorithm. The algorithm has been translated into a computer program using Microsoft Visual C++. Using this program, a search for efficient designs for making all possible pairwise treatment comparisons has been made for v ≤ 10, b ≤ 33, k ≤ 10 such that bk ≤ 100 and v > k. The block designs considered are usual block designs (rectangular block designs) and circular block designs. Nearest neighbor (NN), autoregressive of order 1 (AR(1)) correlation structures are studied. The ranges of correlation coefficients for different correlation structures investigated are |ρ|≤0.50 for NN correlation structure in rectangular blocks, |ρ|≤0.45 for NN correlation structure in circular blocks, and |ρ|≤0.95 for AR(1) correlation structure. For these ranges, the matrix of correlation coefficients among observations within a block is positive definite. Robustness aspects of designs that are efficient for a given value of correlation have been investigated against other values of correlation coefficients. Robustness aspects of designs that are efficient for independent observations have also been studied for experimental situations with dependent observations.     

Minimum cost linear trend-free 12-run fractional factorial designs

Time trend resistant fractional factorial experiments have often been based on regular fractionated designs where several algorithms exist for sequencing their runs in minimum number of factor-level changes (i.e. minimum cost) such that main effects and/or two-factor interactions are orthogonal to and free from aliasing with the time trend, which may be present in the sequentially generated responses. On the other hand, only one algorithm exists for sequencing runs of the more economical non-regular fractional factorial experiments, namely Angelopoulos et al. [1 P. Angelopoulos, H. Evangelaras, and C. Koukouvinos, Run orders for efficient two-level experimental plans with minimum factor level changes robust to time trends, J. Statist. Plann. Inference 139 (2009), pp. 3718–3724. doi: 10.1016/j.jspi.2009.05.002[Crossref], [Web of Science ®] , [Google Scholar]]. This research studies sequential factorial experimentation under non-regular fractionated designs and constructs a catalog of 8 minimum cost linear trend-free 12-run designs (of resolution III) in 4 up to 11 two-level factors by applying the interactions-main effects assignment technique of Cheng and Jacroux [3 C.S. Cheng and M. Jacroux, The construction of trend-free run orders of two-level factorial designs, J. Amer. Statist. Assoc. 83 (1988), pp. 1152–1158. doi: 10.1080/01621459.1988.10478713[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] on the standard 12-run Plackett–Burman design, where factor-level changes between runs are minimal and where main effects are orthogonal to the linear time trend. These eight 12-run designs are non-orthogonal but are more economical than the linear trend-free designs of Angelopoulos et al. [1 P. Angelopoulos, H. Evangelaras, and C. Koukouvinos, Run orders for efficient two-level experimental plans with minimum factor level changes robust to time trends, J. Statist. Plann. Inference 139 (2009), pp. 3718–3724. doi: 10.1016/j.jspi.2009.05.002[Crossref], [Web of Science ®] , [Google Scholar]], where they can accommodate larger number of two-level factors in smaller number of experimental runs. These non-regular designs are also more economical than many regular trend-free designs. The following will be provided for each proposed systematic design:

• (1) The run order in minimum number of factor-level changes.

• (2) The total number of factor-level changes between the 12 runs (i.e. the cost).

• (3) The closed-form least-squares contrast estimates for all main effects as well as their closed-form variance–covariance structure.

In addition, combined designs of each of these 8 designs that can be generated by either complete or partial foldover allow for the estimation of two-factor interactions involving one of the factors (i.e. the most influential).     

MULTI-LEVEL FACTORIAL DESIGNS WITH MINIMUM NUMBERS OF LEVEL CHANGES

The order of experimental runs in a fractional factorial experiment is essential when the cost of level changes in factors is considered. The generalized foldover scheme given by [1] Coster, D. C. and Cheng, C. S. 1988. Minimum cost trend free run orders of fractional factorial designs. The Annals of Statistics, 16: 1188–1205. [Crossref], [Web of Science ®] , [Google Scholar]gives an optimal order to experimental runs in an experiment with specified defining contrasts. An experiment can be specified by a design requirement such as resolution or estimation of some interactions. To meet such a requirement, we can find several sets of defining contrasts. Applying the generalized foldover scheme to these sets of defining contrasts, we obtain designs with different numbers of level changes and then the design with minimum number of level changes. The difficulty is to find all the sets of defining contrasts. An alternative approach is investigated by [2] Cheng, C. S., Martin, R. J. and Tang, B. 1998. Two-level factorial designs with extreme numbers of level changes. The Annals of Statistics, 26: 1522–1539. [Crossref], [Web of Science ®] , [Google Scholar]for two-level fractional factorial experiments. In this paper, we investigate experiments with all factors in slevels.     

Construction of regular 2n41 designs with general minimum lower-order confounding

Mixed-level designs, especially two- and four-level designs, are very useful in practice. In the last two decades, there are quite a few literatures investigating the selection of this kind of optimal designs. Recently, the general minimum lower-order confounding (GMC) criterion (Zhang et al., 2008 Zhang, R.C., Li, P.F., Zhao, S.L., Ai, M.Y. (2008). A general minimum lower order confounding criterion for two-level regular design. Stat. Sin. 18:1689–1705.[Web of Science ®] , [Google Scholar]) gave a new approach for choosing optimal factorials. It is proved that the GMC designs are more powerful than other criteria in the widely practical situations. In this paper, we extend the GMC theory to the mixed-level designs. Under the theory we establish a new criterion for choosing optimal regular two- and four-level designs. Further, a construction method is proposed to obtain all the 2ⁿ4¹ GMC designs with N/4 + 1 ? n + 2 ? 5N/16, where N is the number of runs and n is the number of two-level factors.     

Designs Robust against Violation of Normality Assumption in the Standard F-test

In this article, designs are found for which the F-test of analysis of variance is insensitive to violation of normality assumption. Atiqullah (1962 Atiquallah, M. (1962). The estimation of residual variance in quadratically balanced least-squares problems and the robustness of the F-test. Biometrika 49: 83–91.[Crossref], [Web of Science ®] , [Google Scholar]) proved that the F-test for treatments adjusting for blocks in the intra-block analysis of a balanced incomplete block design is robust against non-normality in the observations. Here an attempt has been made to identify other designs robust in this sense. In particular, it is observed that for testing relevant hypothesis, a partially balanced incomplete block design in block design setup, under certain conditions, is robust. Robustness of a balanced treatment incomplete block design and a partially balanced treatment incomplete block design (Biswas, 2012 Biswas, A. (2012). Block designs robust against the presence of an aberration in a treatment-control setup. Commun Statist.Theor Methods 41: 920–933.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), in treatment-control design setup, is also studied. Moreover, a new measure of robustness is introduced for further study. The performance of the F-test in presence of non-normality in the observations for a quadratically balanced design is also examined.     

On generalized exponential chain ratio estimators under stratified two-phase random sampling

Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]) have suggested generalized exponential chain ratio estimators under stratified two-phase sampling scheme for estimating the finite population mean. However, the bias and mean square error (MSE) expressions presented in that work need some corrections, and consequently the study based on efficiency comparison also requires corrections. In this article, we revisit Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]) estimator and provide the correct bias and MSE expressions of their estimator. We also propose an estimator which is more efficient than several competing estimators including the classes of estimators in Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]). Three real datasets are used for efficiency comparisons.     

Multivariate Bayesian U-type asymmetric designs for non parametric response surface prediction under correlated errors

The objective of this paper is to study U-type designs for Bayesian non parametric response surface prediction under correlated errors. The asymptotic Bayes criterion is developed in terms of the asymptotic approach of Mitchell et al. (1994 Mitchell, T., Sacks, J., Ylvisaker, D. (1994). Asymptotic Bayes criteria for nonparametric response surface design. Ann. Stat. 22:634–651.[Crossref], [Web of Science ®] , [Google Scholar]) for a more general covariance kernel proposed by Chatterjee and Qin (2011 Chatterjee, K., Qin, H. (2011). Generalized discrete discrepancy and its applications in experimental designs. J. Stat. Plann. Inference 141:951–960.[Crossref], [Web of Science ®] , [Google Scholar]). A relationship between the asymptotic Bayes criterion and other criteria, such as orthogonality and aberration, is then developed. A lower bound for the criterion is also obtained, and numerical results show that this lower bound is tight. The established results generalize those of Yue et al. (2011 Yue, R.X., Qin, H., Chatterjee, K. (2011). Optimal U-type design for Bayesian nonparametric multiresponse prediction. J. Stat. Plann. Inference 141:2472–2479.[Crossref], [Web of Science ®] , [Google Scholar]) from symmetrical case to asymmetrical U-type designs.     

Conditions of Slope-Rotatability for Third-Order Polynomial Regression Models

In experimental design for response surface analysis, it is sometimes of interest to estimate the difference of responses at two points. If differences at points close together are involved, the design that reliably estimates the slope of the response surface is important. In particular, Hader and Park (1978 Hader , R. J. , Park , S. H. ( 1978 ). Slope-rotatable central composite designs . Technometrics 20 : 413 – 417 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) suggested the concept of slope-rotatability and studied slope rotatable central composite designs. Until now, many response surface designs including central composite designs have been suggested for fitting second order response surface models. However, we often need to fit third-order polynomial regression models. In this article, we suggest extended central composite designs (ECCDs) to fit third-order models and find the necessary and sufficient conditions for slope-rotatability over all directions in the third-order polynomial models.