# Department of Mathematical Modelling

 Technical University of Denmark Building 321/305 DK-2800 Lyngby Denmark Telephone: +45 4588 1433 Telefax : +45 4588 2673 E-mail : imm@imm.dtu.dk

## Summary

The present thesis describes design and analysis of agricultural experiments utilizing the spatial and temporal correlation between the measurements. The thesis is organized in three parts, spatial experimental design in Part 1, analysis of temporally correlated measurements in Part 2 and a brief introduction to spatio-temporal models in part 3.

Classical statistical analysis normally assumes independent observations. Therefore, knowledge concerning the spatial and temporal relation between plots and between measurements are not included in this kind of analysis. However, agricultural experiments often contain spatial correlations due to a spatial layout and/or temporal correlation due to repeated sampling of measurements at the same experimental unit.

A method for design of field experiments is proposed in Part 1. The residual variance between plots in different layouts is used to compare different layouts. The optimal design and layout from a statistical point of view is the one with the smallest residual variance. The residual variance between plots consists of an error term which depends on the plot size (the dispersion variance) and an error term independent of the plot size (assumed to be the nugget variance). The two error terms are estimated using a semivariogram describing the variation between plots as a function of the distance between them. The method for calculation of the residual variance is based on a uniformity trial. Unfortunately, uniformity trials are seldomly performed.

Therefore, an approach for estimating and removing treatment effects in ordinary field experiments is described. The treatment eliminated residuals obtained in this way can then be used as the base for calculating the residual variance. An example based on a uniformity trial showed a remarkable reduction of the residual variance by choosing among different possible layouts.

In Part 2 different methods are described for the analysis of temporally correlated measurements in field trials. When the assumption of sphericity is satisfied the univariate analysis of variance is a valid, easy and comprehensive method to use. However, the assumption is seldomly satisfied for repeated measurements due to the temporal correlation between the measurements at the same experimental unit. Alternative methods have to be used in this case to obtain a valid analysis. A modified univariate analysis of variance with adjusted F-tests is a simple alternative to the usual univariate analysis of variance. Different multivariate analyses are given both with and without a structured variance-covariance matrix. Ante-dependence and autoregressive variance-covariance structures have been tried. The analysis with a structured variance-covariance matrix is in some sense a compromise between the univariate and multivariate analyses. The latter methods with a structured variance-covariance matrix often bring forth a very informative analysis with few restrictions and reasonable results. An analysis with a structured variance-covariance matrix using an ante-dependence structure is to be preferred to an auto-regressive structure because the ante-dependence structure gives a model which can take the different variations and correlations into account at the cost of only a few extra parameters.

The conclusion of the thesis is that design and analysis of agricultural experiments can be improved by utilization of the spatial and/or temporal correlation between measurements.

## IMM Ph.D Thesis 7, 1994

Last modi fied April 18, 1997