In statistical books for the analysis of designed experiments one can finds sometimes also the computation of the number of replications for balanced one-factor and two-factors designs.Later there were papers published concerning the computation of the number of replications of at most three-factors crossed or nested balanced designs.In 2011 the book "Optimal experimental design with R" was published; further a special R-program OPDOE was made to do the computation for these designs and the OPDOE program was used in this book.In this paper an extension of the determination of the minimum number of replications for balanced designs is given for four-factor crossed designs.The balanced cross classification of the four-way analysis of variance of the following models are investigated: Model 1 The factors A, B, C and D are all fixed; Model 2 D is random A, B and C are fixed; Model 3 C and D are random, A and B are fixed; Model 4 B, C and D are random, A is fixed.For these models small R-programs are given to compute the minimal number of the replications for testing the fixed effects using the non-centrality parameter λ of the non-central F-distribution F(df 1 , df 2 , λ).Further balanced Split-Plot design with one or two fixed factors in the main-plots are considered.The Blocks are denoted with B. The F statistics for testing the significance of the fixed factors are described and small R-programs for the determination of the minimal number of replications are given using the non-centrality parameter λ of the non-central F-distribution F(df 1 , df 2 , λ).

In the paper for all classifications, the models of analysis of variance (ANOVA) have up to three factors and all have the fixed factor A which has to be tested. R-programs of the package OPDOE for the determination of the experimental size are given.

In selecting a candidate with the largest expectation $$\mu_{a}$$ (the best one) from a huge number a of candidates $$P_{a}$$ with expectations $$\mu_{a}$$, we can first select a smaller subset using Gupta´s procedure and then from this subset the best one by Bechhofer’s approach. A simulation experiment (Rasch and Yanagida in J Stat Theory Pract 13:3, 2019) for normal distributions indicated that such a two-stage selection is size-optimal and preferred over using each procedure alone. In this paper, we investigate the robustness of the procedure against several kinds of non-normality and for selecting more than the largest expectation. It is shown that in all cases, the two-stage procedure is better than Bechhofer´s approach.

In Section 1 the approach of improving crop yields by the development of agriculture and addition of various mineral or organic substances in the last 200–300 years is investigated. In Section 2 the principle of randomized experiments is treated. Section 3 describes the variety trials of field crops. The elimination of effects in two dimensions is shown in Section 4 on Row–Column designs. Fertilizer trials are treated in Section 5 (qualitative factors) and in Section 6 (quantitative factors). Field trials with spatial analysis are discussed in Section 7. In Section 8 remarks on analysis and optimal experimental design are made. Supplementary materials accompanying this paper appear on-line

This chapter considers the general structure of analysis of variance (ANOVA) models for the case that all factor levels have randomly been drawn from a universe of factor levels. It characterises methods of variance component estimation for the simplest case, the one-way ANOVA, and demonstrates most of them by some data set. R. L. Anderson and T. A. Bancroft introduced a restricted maximum likelihood (REML) method. There are extensions by Thompson and a generalisation by H. D. Patterson and R. Thompson. This method uses a translation invariant restricted likelihood function depending on the variance components to be estimated only and not on the fixed effect. This restricted likelihood function is a function of the sufficient statistics for the variance components. The latter is then derived with respect to the variance components under the restriction that the solutions are non-negative. The chapter also considers mainly the estimators of the variance components with the ANOVA method and the REML method.

In this chapter, the authors discuss problems of variance component estimation and of estimating and testing fixed effects. They also discuss the cross-classification where they consider the factor A to be fixed without loss of generality. If the factor B is fixed, they rename both factors. In the nested classification, two-mixed models occur when the super-ordinate factor or the nested factor is random. The authors write random factors with their elements in bold. In mixed models, it may happen that variance components of the random effects as well as the fixed effects have to be estimated. The authors differentiate the likelihood function with respect to the fixed effects and the variance components. They also focus on models with at least one fixed factor and for determining the minimum size of the experiment. For three-way analysis of variance, there are three types of classification: cross-classification, nested classification, and mixed classification.

The conjecture: “If the case (v,k) = (8, 3) is the only one where the trivial balanced incomplete block design (BIBD) is elementary” is formulated. It is supported by the theorem: Theorem The conjecture is correct if at least one of the following conditions holds: v < 26, k < 6, for v > 8 if v is prime or a prime power.