Comparison of Means of Spatial Data in Unreplicated Field Trials

The measurement technologies of site-specific agriculture provide precision instruments at the landscape scale, permitting investigators and farmers to obtain data from experiments in commercial fields at previously unheard of spatial resolution. However, the statistical properties of data arising from these technologies are different from those of traditional small-plot trials data. These differences provide the opportunity to analyze data in different ways. The objective of this paper is to develop a method for estimating the effective sample size and P value of a significance test of the difference between two means in an unreplicated large-scale trial in a commercial field. The method is based on a modification of a well-known correction for spatial autocorrelation of the data when testing the significance of the correlation coefficient between two random variables. An example, the comparison of readings taken from a yield monitor on two different days, is given to demonstrate the properties of the method. P RECISION AGRICULTURE has been defined as the use of modern tools of information technology in crop management (Lowenberg-DeBoer and Erickson, 2000). While the technologies of precision agriculture, such as the yield monitor, remote sensing, and soil electrical conductivity measurement, together with the use of the global positioning system, have greatly facilitated the practice of site-specific crop management, these technologies have another use as well. They provide what might be called precision measuring instruments at the landscape scale, permitting investigators (and farmers) to obtain data from experiments in commercial fields at previously unheard of spatial resolution. These data, however, differ in their statistical properties from the data collected in traditional small-plot agronomic trials in that the data often consist of hundreds or thousands of correlated values instead of a small number of independent values. Moreover, the investigators or the commercial grower cooperator may, for logistical or economic reasons, be unable or unwilling to lay out an experiment in a commercial field using a traditional replicated plot design. It is therefore useful to explore new statistical methods to analyze precision agriculture data (Nielsen and Wendroth, 2003). Arguably the simplest form of an experiment is a comparison of the mean responses to two levels of a single factor. For example, Lee et al. (2006) provide a report of an experiment to compare the response of C sequestration rate to tillage method (conventional vs. minimum till) in a commercial corn field. The experiment involves the measurement of C sequestration rate using the eddy covariance method. The fetch of the measurement instruments is sufficiently large that no more than two can be placed in a single field. The cost of the experiment precludes more than one trial at this initial stage of the program, but it is nevertheless desirable to obtain some statistical measure of the difference between various quantities, such as yield and normalized difference vegetation index (NDVI), in the conventional and minimum till plots of the field. The characteristic of an experiment such as this is that a single large field is divided into two halves. Half the field receives Factor Level 1 and the other half Factor Level 2. The 2N samples are then extracted from a set of locations, N from one side and N from the other. The data are autocorrelated, but not perfectly so. Depending on the level of autocorrelation, it may not be appropriate to model the experiment either as a nested design or a model in which the samples independent. In calculating degrees of freedom in a comparison of factor level means, the N pairs of measurements provide an effective sample size of Ne pairs, where 1 # Ne # N. The objective of this paper is to develop a method for estimating this value Ne, and to use this to estimate the P value in a test of the null hypothesis of no difference between the population means. The method is based on an extension of a method of Clifford and Richardson (1985) and Clifford et al. (1989) for estimating the effective sample size in a significance test of the correlation coefficient of autocorrelated data. The Clifford et al. (1989) approximation has achieved the status of the standard approach for use with autocorrelated data (Sokal, 2004), and has been used in the analysis of data from agricultural field experiments (Plant et al., 1999). The extension of this method to comparison of treatment means is first developed using Monte Carlo simulation. We then give an example using data from a field experiment and compare the results with a mixed model analysis.