Donald A. Anderson

Bio: Donald A. Anderson is an academic researcher. The author has contributed to research in topics: Fractional factorial design & Box–Behnken design. The author has an hindex of 2, co-authored 2 publications receiving 34 citations.

Some Basic Properties of Multidimensional Partially Balanced Designs

Abstract: An experimental design is called a multidimensional (MD) design, if it involves more than one factor; see e.g. Potthoff (1964a, b). A number of two-, three-, and four-dimensional designs are now in common use. For example, the ordinary balanced and partially balanced incomplete block designs are two-dimensional. The Latin squares, Youden squares, and the designs of Shrikhande (1951) are three-dimensional. Finally, the Graeco-Latin square designs are four-dimensional. The construction of multidimensional designs involving three or more factors has been discussed by several authors both when additivity is assumed and when interactions are present. To mention a few, we cite Plackett and Burman (1946), Plackett (1946), Potthoff (1964a, b), Agarwal (1966), Anderson (1968), and Causey (1968). A general class of multidimensional designs, which are partially balanced (PB), has been introduced in Srivastava (1961) and Bose and Srivastava (1964). These designs are called multidimensional partially balanced (MDPB) designs. The (MDPB) designs are useful for economizing on the number of observations to be taken, retaining at the same time a relative ease in analysis. MDPB designs for models containing interaction terms have been considered by Anderson (1968). The purpose of this paper is to obtain a class of necessary combinatorial conditions satisfied by the parameters of MDPB designs, and to provide a relatively easy method of determining whether a given design is "completely connected". This latter concept, also of a combinatorial nature, is a generalization of the concept of "connected" block-treatment designs. It signifies that for every factor included under the design, the "true" difference between any two factor levels possesses a best linear unbiased estimate. In a succeeding communication, Srivastava and Anderson (1968), general methods of construction of MDPB designs are considered.

Designs with Partial Factorial Balance

Abstract: In this paper a class of multidimensional experimental designs said to have partial factorial balance is introduced. These designs are shown to belong to the more general class of multidimensional partially balanced designs. The analysis of designs with partial factorial balance is given in detail and several series of three, four and five dimensional designs are presented.

Advanced experimental design

Abstract: Preface. 1. General Incomplete Block Design. 1.1 Introduction and Examples. 1.2 General Remarks on the Analysis of Incomplete Block Designs. 1.3 The Intrablock Analysis. 1.4 Incomplete Designs with Variable Block Size. 1.5 Disconnected Incomplete Block Designs. 1.6 Randomization Analysis. 1.7 Interblock Information in an Incomplete Block Design. 1.8 Combined Intra- and Interblock Analysis. 1.9 Relationships Among Intrablock, Interblock, and Combined Estimation. 1.10 Estimation of Weights for the Combined Analysis. 1.11 Maximum-Likelihood Type Estimation. 1.12 Efficiency Factor of an Incomplete Block Design. 1.13 Optimal Designs. 1.14 Computational Procedures. 2. Balanced Incomplete Block Designs. 2.1 Introduction. 2.2 Definition of the BIB Design. 2.3 Properties of BIB Designs. 2.4 Analysis of BIB Designs. 2.5 Estimation of rho. 2.6 Significance Tests. 2.7 Some Special Arrangements. 2.8 Resistant and Susceptible BIB Designs. 3. Construction of Balanced Incomplete Block Designs. 3.1 Introduction. 3.2 Difference Methods. 3.3 Other Methods. 3.4 Listing of Existing BIB Designs. 4. Partially Balanced Incomplete Block Designs. 4.1 Introduction. 4.2 Preliminaries. 4.3 Definition and Properties of PBIB Designs. 4.4 Association Schemes and Linear Associative Algebras. 4.5 Analysis of PBIB Designs. 4.6 Classification of PBIB Designs. 4.7 Estimation of rho for PBIB(2) Designs. 5. Construction of Partially Balanced Incomplete Block Designs. 5.1 Group-Divisible PBIB(2) Designs. 5.2 Construction of Other PBIB(2) Designs. 5.3 Cyclic PBIB Designs. 5.4 Kronecker Product Designs. 5.5 Extended Group-Divisible PBIB Designs. 5.6 Hypercubic PBIB Designs. 6. More Block Designs and Blocking Structures. 6.1 Introduction. 6.2 Alpha Designs. 6.3 Generalized Cyclic Incomplete Block Designs. 6.4 Designs Based on the Successive Diagonalizing Method. 6.5 Comparing Treatments with a Control. 6.6 Row-Column Designs. 7. Two-Level Factorial Designs. 7.1 Introduction. 7.2 Case of Two Factors. 7.3 Case of Three Factors. 7.4 General Case. 7.5 Interpretation of Effects and Interactions. 7.6 Analysis of Factorial Experiments. 8. Confounding in 2n Factorial Designs. 8.1 Introduction. 8.2 Systems of Confounding. 8.3 Composition of Blocks for a Particular System of Confounding. 8.4 Detecting a System of Confounding. 8.5 Using SAS for Constructing Systems of Confounding. 8.6 Analysis of Experiments with Confounding. 8.7 Interblock Information in Confounded Experiments. 8.8 Numerical Example Using SAS. 9. Partial Confounding in 2n Factorial Designs. 9.1 Introduction. 9.2 Simple Case of Partial Confounding. 9.3 Partial Confounding as an Incomplete Block Design. 9.4 Efficiency of Partial Confounding. 9.5 Partial Confounding in a 23 Experiment. 9.6 Partial Confounding in a 24 Experiment. 9.7 General Case. 9.7.1 Intrablock Information. 9.8 Double Confounding. 9.9 Confounding in Squares. 9.10 Numerical Examples Using SAS. 10. Designs with Factors at Three Levels. 10.1 Introduction. 10.2 Definition of Main Effects and Interactions. 10.3 Parameterization in Terms of Main Effects and Interactions. 10.4 Analysis of 3n Experiments. 10.5 Confounding in a 3n Factorial. 10.6 Useful Systems of Confounding. 10.7 Analysis of Confounded 3n Factorials. 10.8 Numerical Example. 11. General Symmetrical Factorial Design. 11.1 Introduction. 11.2 Representation of Effects and Interactions. 11.3 Generalized Interactions. 11.4 Systems of Confounding. 11.5 Intrablock Subgroup. 11.6 Enumerating Systems of Confounding. 11.7 Fisher Plans. 11.8 Symmetrical Factorials and Finite Geometries. 11.9 Parameterization of Treatment Responses. 11.10 Analysis of pn Factorial Experiments. 11.11 Interblock Analysis. 11.12 Combined Intra- and Interblock Information. 11.13 The sn Factorial. 11.14 General Method of Confounding for the Symmetrical Factorial Experiment. 11.15 Choice of Initial Block. 12. Confounding in Asymmetrical Factorial Designs. 12.1 Introduction. 12.2 Combining Symmetrical Systems of Confounding. 12.3 The GC/n Method. 12.4 Method of Finite Rings. 12.5 Balanced Factorial Designs (BFD). 13. Fractional Factorial Designs. 13.1 Introduction. 13.2 Simple Example of Fractional Replication. 13.3 Fractional Replicates for 2n Factorial Designs. 13.4 Fractional Replicates for 3n Factorial Designs. 13.5 General Case of Fractional Replication. 13.6 Characterization of Fractional Factorial Designs of Resolution III, IV, and V. 13.7 Fractional Factorials and Combinatorial Arrays. 13.8 Blocking in Fractional Factorials. 13.9 Analysis of Unreplicated Factorials. 14. Main Effect Plans. 14.1 Introduction. 14.2 Orthogonal Resolution III Designs for Symmetrical Factorials. 14.3 Orthogonal Resolution III Designs for Asymmetrical Factorials. 14.4 Nonorthogonal Resolution III Designs. 15. Supersaturated Designs. 15.1 Introduction and Rationale. 15.2 Random Balance Designs. 15.3 Definition and Properties of Supersaturated Designs. 15.4 Construction of Two-Level Supersaturated Designs. 15.5 Three-Level Supersaturated Designs. 15.6 Analysis of Supersaturated Experiments. 16. Search Designs. 16.1 Introduction and Rationale. 16.2 Definition of Search Design. 16.3 Properties of Search Designs. 16.4 Listing of Search Designs. 16.5 Analysis of Search Experiments. 16.6 Search Probabilities. 17. Robust-Design Experiments. 17.1 Off-Line Quality Control. 17.2 Design and Noise Factors. 17.3 Measuring Loss. 17.4 Robust-Design Experiments. 17.5 Modeling of Data. 18. Lattice Designs. 18.1 Definition of Quasi-Factorial Designs. 18.2 Types of Lattice Designs. 18.3 Construction of One-Restrictional Lattice Designs. 18.4 General Method of Analysis for One-Restrictional Lattice Designs. 18.5 Effects of Inaccuracies in the Weights. 18.6 Analysis of Lattice Designs as Randomized Complete Block Designs. 18.7 Lattice Designs as Partially Balanced Incomplete Block Designs. 18.8 Lattice Designs with Blocks of Size Kl. 18.9 Two-Restrictional Lattices. 18.10 Lattice Rectangles. 18.11 Rectangular Lattices. 18.12 Efficiency Factors. 19. Crossover Designs. 19.1 Introduction. 19.2 Residual Effects. 19.3 The Model. 19.4 Properties of Crossover Designs. 19.5 Construction of Crossover Designs. 19.6 Optimal Designs. 19.7 Analysis of Crossover Designs. 19.8 Comments on Other Models. Appendix A: Fields and Galois Fields. Appendix B: Finite Geometries. Appendix C: Orthogonal and Balanced Arrays. Appendix D: Selected Asymmetrical Balanced Factorial Designs. Appendix E: Exercises. References. Author Index. Subject Index.

Factorial Association Schemes with Applications to the Construction of Multidimensional Partially Balanced Designs

Abstract: In this paper, some new multidimensional partially balanced (MDPB) association schemes are defined, and the various parameters of the scheme are obtained. Using such schemes, we discuss procedures for the construction of multidimensional partially balanced designs. The theory so developed is illustrated with actual examples of construction of MDPB designs.

Non-orthogonal graeco-latin designs

Abstract: Statisticians are interested in designs for two non-interacting sets of treatments. These designs present many interesting combinatorial problems. The subject is reviewed from a combinatorial viewpoint, and unsolved problems are indicated. An extensive bibliography is appended.