Walter T. Federer

Augmented Designs with One-Way Elimination of Heterogeneity

One of the principal problems in plant breeding and in biochemical research of new pesticides, herbicides, soil fumigants, drugs, etc., is the evaluation of the new strain or chemical. Efficient experimental designs and efficient screening procedures are necessary in order to make the most efficient use of available resources. In some instances sufficient material of a new strain or a new chemical is available for only one or two observations (plots). Hence, the experimenter should use an experimental design and a screening procedure suitable for these conditions. In other cases, the experimenter may wish to limit his observations to a single observation on the new material. In still other cases (e.g., in physics), a single observation on new material may be desirable because of relatively low variability in the experimental material. Furthermore, it may be desired to combine screening experiments on new material and preliminary testing experiments on promising material. The experimental design should be selected to meet the requirements of such experiments rather than selecting the material and experiments to meet the requirements of the experimental design. The experimnental designs described in the present paper were developed to satisfy requirements such as those described above. The class of experimental designs known as augmented designs was Introduced by the author in 1955 to fill a need arising in screening new strains of sugar cane and soil fumigants used in growing pineapples2 (Federer [1956a, 1956b, 1956c, 1958]). An augmented experimental design is any standard design augmented with additional treatments in the complete block, the incomplete block, the row, the column, etc.

On Augmented Designs

When some treatments (checks) are replicated r times and other treatments (new treatments) are replicated less than r times, an augmented desigii may be used. These designs may be minimum variance designs for estimating contrasts of check effects, of new variety effects, of new variety versus checks, or of all check and new varieties simultaneously. In this paper optimal augmented block and optimal augmented row-column designs for estimating certain contrasts of new treatments are presented.

Two-Treatment Crossover Designs for Estimating a Variety of Effects

Abstract The variances of contrasts among direct, residual, and cumulative treatment effects are compared for a variety of two-treatment crossover designs. Estimation of contrasts among second-order residual effects and among direct-by-period and direct-by-first-order-residual interaction effects is also considered.

Recovery of Interblock, Intergradient, and Intervariety Information in Incomplete Block and Lattice Rectangle Designed Experiments

Spatial analysis and blocking analysis of experimental results are treated separately in the literature. Here we combine these analyses into a single analysis. The information arising from the random nature of different gradients within incomplete blocks is used to adjust treatment means. We extend Coxs (1958, Journal of the Royal Statistical Society, Series B 20, 193-204) idea of differential gradients within columns of a Latin square to within blocks for incomplete block and row-column designed experiments and, in addition, treat them as random effects. With this analysis, the restrictions on randomization due to blocking are taken into consideration whereas they are often ignored in spatial analysis literature. Some comments on designing experiments and analyzing experimental results to control heterogeneity are presented. A numerical example illustrates the computational procedure and indicates effect of alternative analyses. The class of augmented experiment designs has been found useful for experiments involving comparisons of standard check treatments with a set of new and untried treatments, usually with one replicate. Interreplicate, interblock, interrow, and/or intercolumn information is available to use in obtaining solutions for new treatment effects. Since the new treatment effects are often considered to be random effects, their distributional properties may be used to increase the efficiency of the experiment. We demonstrate the statistical procedures for recovering this information in block and row-column designs using mixed model procedures.

Statistical Design and Analysis for Intercropping Experiments

Statistical design and analysis for intercropping experiments , Statistical design and analysis for intercropping experiments , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

Procedures and Designs Useful for Screening Material in Selection and Allocation, with a Bibliography

This paper represents a brief review of results on screening. The references listed in the paper were selected to illustrate diversity of topics related to screening procedures with particular attention being paid to the results for plant and animal breeding screening programs and from truncation, fragmentation, and censoring of samples. Also, the large number of journals involved should be noted. In this form it is hoped that researchers involved in screening various drugs, pesticides, fumigants, herbicides, etc. (and also with students, enlistees, missiles, etc.) will become acquainted with the topics and some of the literature on screening in breeding fields, and vice versa. With this in mind, the similarities in biochemical and breeding programs were stressed. In particular, the developmental phase, the evaluation phase, and the production or maintenance phase are common to both fields. Many of the statistical results obtained in one field are applicable to the other. No discussion is given concerning unsolved statistical problems related to screening. This subject is discussed in papers listed at the end of the paper. Some comments are given on experimental designs useful for multistage screening experiments. Particular reference is made to the class of designs known as augmented designs.

Variations on Split Plot and Split Block Experiment Designs: Federer/Variations on Split Plot and Split Block Experiment Designs

Preface. Chapter 1. The standard split plot experiment design. 1.1. Introduction. 1.2. Statistical design. 1.3. Examples of split-plot-designed experiments. 1.4. Analysis of variance. 1.5. F-tests. 1.6. Standard errors for means and differences between means. 1.7. Numerical examples. 1.8. Multiple comparisons of means. 1.9. One replicate of a split plot experiment design and missing observations. 1.10. Nature of experimental variation. 1.11. Repeated measures experiments. 1.12. Precision of contrasts. 1.13. Problems. 1.14. References. Appendix 1.1. Example 1.1 code. Appendix 1.2. Example 1.2 code. Chapter 2. Standard split block experiment design. 2.1. Introduction. 2.2. Examples. 2.3. Analysis of variance. 2.4. F-tests. 2.5. Standard errors for contrasts of effects. 2.6. Numerical examples. 2.7. Multiple comparisons. 2.8. One replicate of a split block design. 2.9. Precision. 2.10. Comments. 2.11. Problems. 2.12. References. Appendix 2.1. Example 2.1 code. Appendix 2.2. Example 2.2 code. Appendix 2.3. Problems 2.1 and 2.2 data. Chapter 3. Variations of the split plot experiment design. 3.1. Introduction. 3.2. Split split plot experiment design. 3.3. Split split split plot experiment design. 3.4. Whole plots not in a factorial arrangement. 3.5. Split plot treatments in an incomplete block experiment design within each whole plot. 3.6. Split plot treatments in a row-column arrangement within each whole plot treatment and in different whole plot treatments. 3.7. Whole plots in a systematic arrangement. 3.8. Split plots in a systematic arrangement. 3.9. Characters or responses as split plot treatments. 3.10. Observational or experimental error? 3.11. Time as a discrete factor rather than as a continuous factor. 3.12. Inappropriate model? 3.13. Complete confounding of some effects and split plot experiment designs. 3.14. Comments. 3.15. Problems. 3.16. References. Appendix 3.1. Table 3.1 code and data. Chapter 4. Variations of the split block experiment design. 4.1. Introduction. 4.2. One set of treatments in a randomized complete block and the other in a Latin square experiment design. 4.3. Both sets of treatments in split block arrangements. 4.4. Split block split block or strip strip block experiment design. 4.5. One set of treatments in an incomplete block design and the second set in a randomized complete block design. 4.6. An experiment design split blocked across the entire experiment. 4.7. Confounding in a factorial treatment design and in a split block experiment design. 4.8. Split block experiment design with a control. 4.9. Comments. 4.10. Problems. 4.11. References. Appendix 4.1. Example 4.1 code. Chapter 5. Combinations of SPEDs and SBEDs. 5.1. Introduction. 5.2. Factors A and B in a split block experiment design and factor C in a split plot arrangement to factors A and B. 5.3. Factor A treatments are the whole plot treatments and factors B and C treatments are in a split block arrangement within each whole plot. 5.4. Factors A and B in a standard split plot experiment design and factor C in a split block arrangement over both factors A and B. 5.5. A complexly designed experiment. 5.6. Some rules to follow for finding an analysis for complexly designed experiments. 5.7. Comments. 5.8. Problems. 5.9. References. Appendix 5.1. Example 5.1 code. Appendix 5.2. Example 5.2 data set, code, and output. Chapter 6. World records for the largest analysis of variance table (259 lines) and for the most error terms (62) in one analysis of variance. 6.1. Introduction. 6.2. Description of the experiment. 6.3. Preliminary analyses for the experiment. 6.4. A combined analysis of variance partitioning of the degrees of freedom. 6.5. Some comments. 6.6. Problems. 6.7. References. Appendix 6.1. Figure 6.1 to Figure 6.6. Chapter 7. Augmented split plot experiment design. 7.1. Introduction. 7.2. Augmented genotypes as the whole plots. 7.3. Augmented genotypes as the split plots. 7.4. Augmented split split plot experiment design. 7.5. Discussion. 7.6. Problems. 7.7. References. Appendix 7.1. SAS code for ASPED, genotypes as whole plots, Example 7.1. Appendix 7.2. SAS code for ASPEDT, genotypes as split plots, Example 7.2. Appendix 7.3. SAS code for ASSPED, Example 7.3. Chapter 8. Augmented split block experiment design. 8.1. Introduction. 8.2. Augmented split block experiment designs. 8.3. Augmented split blocks for intercropping experiments. 8.4. Numerical example 8.1. 8.5. Comments. 8.6. Problems. 8.7. References. Appendix 8.1. Codes for numerical Example 8.1. Chapter 9. Missing observations in split plot and split block experiment designs. 9.1. Introduction. 9.2. Missing observations in a split plot experiment design. 9.3. Missing observations in a split block experiment design. 9.4. Comments. 9.5 Problems. 9.6. References. Appendix 9.1. SAS code for numerical example in Section 9.2. Appendix 9.2. SAS code for numerical example in Section 9.3. Chapter 10. Combining split plot or split block designed experiments over sites. 10.1. Introduction. 10.2. Combining split plot designed experiments over sites. 10.3. Combining split block designed experiments over sites. 10.4. Discussion. 10.5. Problems. 10.6. References. Appendix 10.1. Example 10.1. Appendix 10.2. Example 10.2. Chapter 11. Covariance analyses for split plot and split block experiment designs. 11.1. Introduction. 11.2. Covariance analysis for a standard split plot design. 11.3. Covariance analysis for a split block experiment design. 11.4. Covariance analysis for a split split plot experiment design. 11.5. Covariance analysis for variations of designs. 11.6. Discussion. 11.7. Problems. 11.8. References. Appendix 11.1. SAS code for Example 11.1. Appendix 11.2. SAS code for Example 11.2. Appendix 11.3. SAS code for Example 11.3. Index.

Construction and Analysis of an Augmented Lattice Square Design

Augmented designs are useful for screening experiments involving large numbers of new and untried treatments. Since resolvable row-column designs are useful for controlling extraneous variation, it is desirable to use such designs for the check or standard treatments to construct augmented lattice square experiment designs. A simple procedure is described for constructing such designs using c = 2k and c = 3k check treatments and n = rk(k -— 2) and n = rk(k — 3) new treatments, respectively, r being the number of complete blocks. A trend analysis for these designs, which allows for solutions of fixed effects, is presented. The random effects case is also discussed. A SAS computer code and the output from this code illustrated with a small numerical example are available from the author.

New Combinatorial Designs and their Applications to Group Testing

Abstract A class of designs with property C(t) are introduced for the first time, and their applications in group testing of samples are studied.

A parsimonious statistical design and breeding procedure for evaluating and selecting desirable characteristics over environments

The concept of stability as described in the literature does not meet all of the desirable criteria required by growers of cultivars. Various types of possible responses are discussed, and these are divided into those desirable from a growers viewpoint and those not. Measures of stability appearing in the literature are based on variances, linear regression slopes, and/or deviations from regression. The most desirable response type would be denoted as unstable by current concepts of stability. It is shown how to simulate environments that exceed the ranges found in practice. A statistical design is described which is the height of parsimony and has the advantage that the conditions varied are known. The experimental results can then be interpreted in light of the known conditions. The design is optimally cost effective in terms of funds, material, and personnel. A breeding procedure is presented for such characteristics as desired response, stability under current definitions, tolerance (to pests, cold, drought, etc.), protein, quality, fiber, etc.