New and Traditional Methods for the Analysis of Unreplicated Experiments

This paper reviews some traditional and more recent methods for analyzing unreplicated experiments. Such experiments have presented a challenge to statisticians throughout their involvement in agricultural research. At Rothamsted this began in 1919, when R.A. Fisher was appointed to analyze the accumulated data from the classical field experiments. Fisher’s experiences with the classicals, which had virtually no replication, must have contributed to his inclusion of replication as one of the key features of a well-designed experiment. Nevertheless, Fisher made good use of Rothamsted’s data, for example in his study of the influence of rainfall on yields from the Broadbalk. He also devised the randomization test, which can be used to analyze unreplicated data. More recently, Broadbalk has also been used to study climate change and sustainability. Newer developments have been concerned to find alternatives to use, instead of blocking, to take account of the spatial variation within an experiment. The resulting methods for modeling spatial correlations have allowed experimenters to obtain more precise estimates of treatment effects—or to decrease numbers of replicates— and they can also provide reliable analyses of unreplicated treatments. THE DEVELOPMENT OF METHODS for the analysis of experiments began in 1919, when R.A. Fisher was appointed as the original statistician at Rothamsted. His remit was to study the accumulated results of the Rothamsted classical field experiments, which began in 1843 with the Broadbalk experiment on winter wheat (Triticum aestivum L.). This illustrates another (more unfortunate!) tradition, of the statistician being called in to do the analysis long after the design of an experiment. However, Fisher rose to the challenge and, as the 1971 Guide to Rothamsted Experimental Station notes, “he soon realized the need for improved statistical techniques over the whole range of agricultural and biological research, and the groundwork for modern statistics was laid by him during the 1920s and 1930s.” Fisher recognized the importance of replication, as a way of allowing the underlying random variation to be estimated. However, he also made good use of the Rothamsted classical experiments, which essentially were unreplicated. RESULTS AND DISCUSSION Relationship of Yield to Climate Fisher (1924) used data from Broadbalk to study the relationship of yield to climatic variation. Broadbalk was set up by Sir John Lawes and Sir Henry Gilbert in autumn 1843 to study the effects of inorganic fertilizers on winter wheat (see Leigh and Johnston, 1994). It still continues, with minor modifications generally to reflect changing practices (e.g., of varieties), and provides a resource for many research activities not anticipated by Lawes and Gilbert. The main treatments on the plots, shown in Table 1, have remained largely unchanged since 1852. In 1926, subsequent to the period studied by Fisher, each plot was split into sections to allow fallowing as a means of weed control. More recently, some of the sections have been used to study crop rotations. Nevertheless, there are still sections that have grown wheat continuously since 1843. From a statistical point of view it is intriguing to notice that the design contains some factorial structure (e.g., plots 03, 05, 10, and 09). Thus, Lawes and Gilbert were already thinking about how one fertilizer will respond to the presence and absence of other fertilizers, although this was long before the theory of factorial experiments was devised by Fisher (1926). However, there is no replication. The long-term experiments provided a uniquely useful resource for this purpose, giving a long series of data in controlled conditions and with the same treatments year on year. It should therefore be only climate that was affecting yields, other than perhaps a few other aspects that Fisher called “progressive changes.” Specifically, Fisher was aware that there would be long-term trends in yield as well as long-term trends in climate, and that the yield trends could be caused by progressive changes in aspects unrelated to climate (e.g., improvements in husbandry or varieties). Fisher therefore fitted 5th-order Legendre polynomials of years to the annual yields to remove the long-term trends from the annual yields. The fluctuations about the Legendre model might reasonably be assumed to arise mainly from annual climatic variations, and he sought to model these by using the rainfall trends within each year. These rainfall trends were characterized by fitting further Legendre polynomials to the rainfall summarized during 6-d intervals. Fitting a multiple regression, using the Legendre rainfall coefficients of the rainfall trends, accounted for between 11 and 40% of the variance of the yield fluctuations. Furthermore, Fisher found consistent responses to rainfall according to the plots’ fertilizer regimes. The work is impressive not only in terms of the computations that were required using just an electric calculator, but also in terms of current methodology. These days we would probably use smoothing splines to model the trends rather than Legendre polynomials, but the underlying ideas remain the same. Further analyses of the Broadbalk yields were done by Chmielewski and Potts (1995), who retained the longterm trends so that they could study long-term climate variations in order to assess the implications of climate VSN International, Hemel Hempstead, Hertfordshire HP1 1ES, U.K., and Biomathematics and Bioinformatics Division, Rothamsted Research, Harpenden, Herts, U.K. Received 27 Apr. 2006. Corresponding author ([email protected]). Published in Crop Sci. 46:2476–2481 (2006). Analysis of Unreplicated Experiments (Symposium) doi:10.2135/cropsci2006.04.0273 a Crop Science Society of America 677 S. Segoe Rd., Madison, WI 53711 USA R e p ro d u c e d fr o m C ro p S c ie n c e . P u b lis h e d b y C ro p S c ie n c e S o c ie ty o f A m e ri c a . A ll c o p y ri g h ts re s e rv e d . 2476 Published online November 21, 2006