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Experimental designs with orthogonal block structures are commonly used in many areas of science in order to control the external sources of variability. The aim of this study is to compare several analysis of variance (ANOVA) methods applicable to such structures. Comparing these approaches is of practical importance, as the choice of the analytical method may influence inference about treatment effects and the estimation of variance components in complex experimental structures. The experiments were conducted using a nested block design, where the treatments are distributed over blocks. Each block consists of a certain number of experimental units (plots), which are further grouped into superblocks. This framework allows the total variation to be decomposed into orthogonal components corresponding to successive strata, which naturally leads to the mixed model representation of treatment and block effects.
Three analytical methods were compared in this study. The first method is based on decomposing the model into several submodels, in accordance with the stratification of the experimental units. Then, information from the individual strata was then taken into account in the analysis of variance. This method was thoroughly described by Caliński and Kageyama (2000). The second method is based on the residual maximum likelihood (REML) approach. The Kenward–Roger method for estimating degrees of freedom was applied. Finally, taking advantage of the orthogonal block structure, the analysis of variance can be performed directly, without combining results from intra-block and inter-block analyses, as described by Caliński and Siatkowski (2018).
The main goal of this research was to assess the effectiveness of the aforementioned methods. They were applied to datasets gathered from several experiments involving different numbers of plots, different block sizes, different numbers of treatments, and various sizes of superblocks. The results of analyses of variance were compared and discrepancies were investigated by identifying issues arising from the estimation of variance components, especially cases where the REML method omitted certain effects. Moreover, the run-times of the software and the numbers of iterations required to obtain variance component estimates were compared.
References:
1. Caliński, T., Kageyama, S. (2000) Block Designs: A Randomization Approach: Volume I: Analysis. Springer New York
2. Caliński, T., Siatkowski, I. (2018). On a new approach to the analysis of variance for experiments with orthogonal block structure. II. Experiments in nested block designs. Biometrical Letters, 55(2), 147-178.
3. Searle, S. R., Casella, G., & McCulloch, C. E. (2009). Variance components. John Wiley & Sons.
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