Purpose: for resolvable row-column or lattice rectangle designs, a variety of analysis options are given below. These are for a randomized complete block design with rows (columns) as blocks, standard textbook analysis, differential gradients within rows (columns), and trend analysis using orthogonal polynomial regression functions of the rows and columns and their interactions. The example used maybe found in Table 12.5 ofW.G. Cochran and G.M. Cox's 1957 book Experimental Designs as well as in other reports in this volume. There are 156 insecticide treatments arranged in four rows and four columns within each of five complete blocks (replicates) to for a balanced lattice square. The data are averages of three counts of plants infected with boll weevil. The trend analysis is the most appropriate analysis for these data. The code can also be used for incomplete block design by either deleting the row or the column category.

Publisher: School of Statistics, Renmin University of China, Journal: Journal of Data Science, Title: Exploratory Model Selection for Spatially Designed Experiments – Some Examples, Authors: Walter T. Federer

AbstractThe subject of statistics deals with variability and how to deal with it. In the planning and conduct of an environmental or ecological investigation, the items used to control variability are (a) refinement of experimental technique, (b) selection of homogeneous material and/or environments, (c) grouping (blocking, stratifying) material into homogeneous subgroups (blocks, strata), and (d) measurement of related variables and use of covariance. There are many ways of blocking (arranging) the experimental units (EUs) in a comparative experiment withvtreatments. If the sample of EUs is from a homogeneous population, then no blocking is required and a completely randomized experiment design (ED) of thevtreatments randomly allotted to thervEUs is used. The replicate number (sample size) for each treatment isrunless unequal replication is desired. If homogeneous blocks of sizevare available to accommodate allvtreatments, a randomized complete block ED (allvtreatments in each block, not necessarily an equal number of times) is used. In many situations, the block size,k, is less thanvand an incomplete block ED (not all treatments in each of the blocks) is used.

Abstract Intercropping is defined as the simultaneous or sequential growing of more than one crop on the same area of land. It may involve intermixing the plants of the crops, growing them in alternate rows, or growing them sequentially through time. When dealing with a cropping system, it is necessary to study the response for each component of the mixture (when available) as well as for the combined data for the mixture. This will involve statistical procedures not found in standard statistics textbooks. A popular statistical procedure for analyzing results from experiments involving several intercropping treatments is a relative yield ratio or land equivalent ratio (LER). The yields of the individual crops in a mixture need to be available in order to compute a LER. An LER is defined as the sum of the ratios of the yield of the i th crop in a mixture to its yield as a sole crop.

Crop breeding programs using conventional approaches, as well as new biotechnological tools, rely heavily on data resulting from the evaluation of genotypes in different environmental conditions (agronomic practices, locations, and years). Statistical methods used for designing field and laboratory trials and for analyzing the data originating from those trials need to be accurate and efficient. The statistical analysis of multi-environment trails (MET) is useful for assessing genotype × environment interaction (GEI), mapping quantitative trait loci (QTLs), and studying QTL × environment interaction (QEI). Large populations are required for scientific study of QEI, and for determining the association between molecular markers and quantitative trait variability. Therefore, appropriate control of local variability through efficient experimental design is of key importance. In this chapter we present and explain several classes of augmented designs useful for achieving control of variability and assessing genotype effects in a practical and efficient manner. A popular procedure for unreplicated designs is the one known as “systematically spaced checks.” Augmented designs contain “c” check or standard treatments replicated “r” times, and “n” new treatments or genotypes included once (usually) in the experiment.

Traditional methods for covariate adjustment of treatment means in designed experiments are inherently conditional on the observed covariate values. In order to develop a coherent general methodology for analysis of covariance, we propose a multivariate variance components model for the joint distribution of the response and covariates. It is shown that, if the design is orthogonal with respect to (random) blocking factors, then appropriate adjustments to treatment means can be made using the univariate variance components model obtained by conditioning on the observed covariate values. However, it is revealed that some widely used models are incorrectly specified, leading to biased estimates and incorrect standard errors. The approach clarifies some issues that have been the source of ongoing confusion in the statistics literature.

Abstract A new class of augmented experiment designs is introduced. This is a follow-up on the 2005 paper by the author on augmented split block experiment designs. The designs are presented to expand the possibilities for experimenters for use in screening untested or partially screened material. These designs allow testing of new treatments over other factors, such as tillage, weed control, fertilizer, density, etc. This is desirable in the later stages of screening new genotypes or other treatments.

The subject of statistics deals with variability and how to deal with it. In the planning and conduct of an environmental or ecological investigation, the items used to control variability are (a) refinement of experimental technique, (b) selection of homogeneous material and/or environments, (c) grouping (blocking, stratifying) material into homogeneous subgroups (blocks, strata), and (d) measurement of related variables and use of covariance. There are many ways of blocking (arranging) the experimental units (EUs) in a comparative experiment with v treatments. If the sample of EUs is from a homogeneous population, then no blocking is required and a completely randomized experiment design (ED) of the v treatments randomly allotted to the rv EUs is used. The replicate number (sample size) for each treatment is r unless unequal replication is desired. If homogeneous blocks of size v are available to accommodate all v treatments, a randomized complete block ED (all v treatments in each block, not necessarily an equal number of times) is used. In many situations, the block size, k, is less than v and an incomplete block ED (not all treatments in each of the blocks) is used.

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.

This chapter contains sections titled: Introduction Covariance analysis for a standard split plot design Covariance analysis for a split block experiment design Covariance analysis for a split split plot experiment design Covariance analysis for variations of designs Discussion Problems 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

The need for a new analytical approach was encountered in the course of characterizing newly developed tomato lines resistant to late blight. Late blight resistant tomato lines were created in independent breeding programs using the accession Solanum pimpinellifolium L. (formerly Lycopersicon pimpinellifolium (L.) Miller) L3708 as the source of the resistance. However, initial field observation suggested that the late blight resistance in the lines produced by two independent breeding programs differed. Possible causes included a partial transfer of the late blight resistance derived from S. pimpinellifolium L3708 or the possibility of race specificity of this resistance. A crucial issue was determining the most appropriate and robust analytical method to use with data from laboratory analyses of the responses of nine tomato lines against five P. infestans isolates. Prior analysis by standard ANOVA revealed significant differences across tomato lines but could not determine whether the disease responses in the CLN-R lines were different from those of the heterozygous F-1 hybrids, created by crossing susceptible tomatoes with the fixed CU-R lines. A different analytical method was needed. Therefore, sporangia numbers/leaflet and diseased area data were analyzed using a half-normal probability plot and regression analysis. The results of this analysis show its utility for genetic or pathology studies. Considering only populations of the uniform tomato lines, this method confirms the results obtained by using a standard ANOVA, but provides a clearer demonstration of the distributions of the individuals within the populations and how this distribution impacts variance and the difference among the populations. This method also allows a joint analysis of the uniform lines with an additional population that is less uniform, because it is segregating. Such an analysis would be invalid using a standard ANOVA. The results of this joint analysis determined that the additional population was divergent from the fixed CU-R lines, and, against some isolates, against the CLN-R lines as well. Half-normal probability plot analysis method would be applicable more broadly beyond analysis of disease resistance data. It could be useful for data from populations that are not normally distributed, for traits which are affected by epistatic gene action, and could be useful for selection of extremes.

Some of the topics covered are statistical design axioms, plot technique, experiment design selection, block experiment designs, row-column experiment designs, unreplicated or screening experiment designs, exploratory model selection, multi-site/year trials, and parsimonious experiment designs. In line with the axiom "design for the experiment, do not experiment for the design," two simple methods for constructing block experiment designs are described. In addition, a software toolkit for constructing optimal or near-optimal randomized plans of experiment designs for many situations is discussed. The effect of response model selection is illustrated with examples. Suggestions for increasing the efficiency of plant-breeding programs are given. Augmented and parsimonious experiment designs can be utilized to increase the efficiency of plant-breeding programs. Some remarks on the construction of these designs are given. (C) 2005 by The Haworth Press, Inc. All rights reserved.