Reply to “Comments on ‘Evaluation of Statistical Estimation Methods for Lognormally Distributed Variables’”

Distributions of many chemical, physical, and microbiological properties of soils appear to be lognormal. Several conflicting recommendations exist in the soil science and statistical literature on how to best estimate the population mean, variance, and coefficient of variation of lognormally distributed data. We chose to determine with statistical certainty which of the following three methods is best: (i) the method of moments (method 1); (ii) maximum likelihood (method 2); and (iii) Finney's method (method 3). We assessed the efficacy of these three methods for estimating the mean, variance, and coefficient of variation of lognormal data in the range of sample sizes from n = 4 to 100. Three test lognormal populations were used in our evaluation with coefficients of variation that span the range seen for many soil variables (CVs of 50%, 100%, and 200%). We found Finney's method was best for estimating the mean and variance of lognormal data when the coefficient of variation of the underlying lognormal frequency distribution exceeds 100%, below this value the extra computational effort required to implement Finney's technique buys little, relative to the method of moments. Finney's method has not been previously applied by soil scientists, but its superiority over maximum likelihood suggests that the latter should not be generally recommended for estimating the mean, variance and coefficient of variation of lognormal data. Additional Index Words: Lognormal, Mean square error, Bias, Efficiency, Soil variables, Monte Carlo simulation. T VARIABILITY of soil properties has received increased interest (Nielsen & Bouma, 1985). The combination of low cost computers and automated analysis systems has enabled scientists to generate large databases for particular soil variables. These large daT.B. Parkin, J.J. Meisinger, and J.L. Starr, USDA-ARS, BARC. Soil Nitrogen and Environmental Chemistry Lab., Beltsville, MD 20705; S.T. Chester and J.A. Robinson, The Upjohn Company, Kalamazoo, MI 49001. Contribution of the USDA-ARS and The Upjohn Company. Received 29 May 1987. "Corresponding author. Published in Soil Sci. Soc. Am. J. 52:323-329 (1988). tabases have, in turn, allowed the characterization of the variability and frequency distributions of soil variables. Such analyses indicate that the frequency distributions of many physical, chemical, and microbiological soil properties are skewed to the right and are better approximated by the lognormal frequency distribution than by the normal (Gaussian) probability density function (Table 1). Confusion exists on how to best estimate the mean, variance, and coefficient of variation of lognormally distributed data. This stems from the fact that several statistical procedures for estimating these population parameters have appeared in soil science and statistical literature. (Warrick & Nielsen, 1980; Koch & Link, 1970). A statistically complete evaluation of the most commonly applied methods has not been published in a source accessible to the majority of soil scientists. We undertook the present study to determine the statistical efficacy of three methods (method of moments, maximum likelihood, and Finney's approximation) for estimating the population mean, variance, and coefficient of variation of lognormal data. Of these three, method 2 (maximum likelihood) has been recommended for use by some soil scientists (Warrick & Nielsen, 1980;Folurenso&Rolston, 1984; Parkin et al. 1985), although, as we will show, it is generally inferior to Finney's method (method 3) as well as the more commonly applied method of moments (method 1). Finney's method for estimating the mean, variance, and coefficient of variation of lognormal data has only rarely been applied to soils data. (White et al., 1987; Parkin, 1987; Parkin et al. 1987). The present study is a theoretical one in the sense that we investigated the properties of the three estimation methods in a "world" where the answers were known. Description of any new statistical estimation method often incorporates an evaluation of the method for a family of known probability density functions that cover the range of distributions seen or expected 324 SOIL SCI. SOC. AM. J., VOL. 52, 1988 Table 1. Abbreviated survey of soil variables which have been reported to be approximately lognormally distributed.