Of interest is the analysis of results of a series of experiments repeated at several environments with the same set of plant varieties. Suppose that the experiments, multi-environment variety trials, are all conducted in resolvable incomplete block (IB) designs. Following the randomization approach adopted in Caliński and Kageyama (2000, Lecture Notes in Statistics, 150), two models for analyzing such trial data can be considered. One is derived under a complete additivity assumption, the other takes into account possible different responses of the varieties to variable environmental conditions. The analysis under the first, the standard model, does not provide answers to questions related to the performance of the individual varieties at different environments. These can be considered when using the more general second model. The purpose of this article is to devise interesting parameter estimation and hypothesis testing procedures under that more realistic model. Its application is illustrated by a thorough analysis of a set of data from a winter wheat series of trials.

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This article has been published with an erroneous version of Eq. 15. Please find the correct Eq. 15 in this document. Derivation of (11) Let T = [T 1 T 2 ] be an (n 9 n) non-singular transformation matrix such that T 1 and T 2 , of dimension (n 9 t) and (n 9 (n-t)), satisfy T T 1 X ¼ I t T T 2 X ¼ 0 , RðT 2 Þ ? RðXÞ: Likewise, let Q = [Q 1 Q 2 ] be an ((n-t) 9 (n-t)) non-singular transformation matrix such that Q 1 and Q 2 , of dimension ((n-t) 9 d) and ((n-t) 9 (n-t-d)), satisfy Q T 1 T T 2 X g ¼ I d Q T 2 T T 2 X g ¼ 0 , RðQ 2 Þ ? RðT T 2 X g Þ: ð15Þ Should read: Derivation of (11) Let T = [T 1 T 2 ] be an (n 9 n) non-singular transformation matrix such that T 1 and T 2 , of dimension (n 9 t) and (n 9 (n-t)), satisfy T T 1 X ¼ I t T T 2 X ¼ 0 , RðT 2 Þ ? RðXÞ: Likewise, let Q = [Q 1 Q 2 ] be an ((n-t) 9 (n-t)) non-singular transformation matrix such that Q 1 and Q 2 , of dimension ((n-t) 9 d) and ((n-t) 9 (n-t-d)), satisfy Q T 1 T T 2 X g ¼ I d Q T 2 T T 2 X g ¼ 0 , RðQ 2 Þ ? RðT T 2 X g Þ: ð15Þ The original article can be found online at https://

Independence of observations is one of the basic assumptions of the analysis of variance. Performed randomizations prevent results from being biased in cases when independence is violated. The objective of the present paper is to find out the predominant shape of spatial relationships in Polish wheat variety testing trials. One of the possibilities is to apply some geo-statistical method (Cressie Noel 1993, Grondona andCressie 1991). Such an approach is used in this paper. Using the results of nearly 200 trials on wheat varieties, conducted either in generalized lattice (GL) designs (Patterson and Hunter 1983) or in incomplete split-block designs, the empirical values of semivariance have been calculated. Residuals were computed from a model with fixed effects for varieties and replicates, ignoring incomplete blocks, then 10 different geo-statistical models have been fitted to empirical semivariances. With spatial models convergence problems occurred in some cases. The linear and bounded-linear models were the ones most often successfully fitted. Inclusion of revealed relationships into classical models for incomplete blocks did not improve substantially the effectiveness of analyses.

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