## Ratio estimators in agricultural research : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Statistics at Massey University, Palmerston North, New Zealand

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##### Date

2002

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Massey University

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The Author

##### Abstract

This thesis addresses the problem of estimating the ratio of quantitative variables from several independent samples in agricultural research. The first part is concerned with estimating a binomial proportion, the ratio of discrete counts, from several independent samples under the assumption that there is a single underlying binomial proportion p in the population of interest. The distributions and properties of two linear estimators, a weighted average and an arithmetic average, are derived and merits of the approaches discussed. They are both unbiased estimators of the population proportion, with the weighted average having lower variability than the arithmetic average. These findings are obtained through a first principles analysis, with a geometrical interpretation presented. This variability result is also a consequence of the Rao-Blackwell theorem, a well-known result in the theory of statistical inference. Both estimators are used in the literature but we conclude that the weighted average estimate should always be used when the sample sizes are unequal. These results are illustrated by a simulation experiment and are validated using survey data in the study of lodging percentage of sunflower cultivar, Improved Peredovic, in Jilin Province, China in 1994. The second part of the research addresses the problem of estimating the ratio μͯ / μ, of the means of continuous variables in agricultural research. The distributional properties of the ratio X/Y of independent normal variables are examined, both theoretically and using simulation. The results show that the moments of the ratio do not exist in general. The moments exist, however, for a punctured normal distribution of the denominator variable if we only sample points for which | Y |>ε, ε being a small positive quantity. We draw out the practical rule-of-thumb that the ratio of two independent normal variables can be used to estimate μͯ / μ, when the coefficient of variation of the denominator variable is sufficiently small (less than or equal to 0.2). Lastly the thesis evaluates the relative merits of two common estimators of the ratio of the means of continuous variables in agricultural research, an arithmetic average and a weighted average, via simulation experiments using normal distributions. In the first simulation, the ratio and common coefficient of variation are changed while the sample size is kept moderately large. In the second simulation, the ratio and sample size are changed while the coefficient of variation is held constant. Results show that the weighted average always provides a better estimate of the true ratio and has lower variability than the arithmetic average. It is recommended that the weighted average be used for estimating the ratio from several pairs of observations. These results are tested using research data from rice breeding multi-environment trials in Jilin Province, China in 1995 and 1996. These data are used to demonstrate the diagnostic approach developed for assessing the 'safety' use of the arithmetic and the weighted average methods for estimating the ratio of the means of independent normal variables.

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##### Keywords

Agriculture, Statistical methods, Estimation theory