Springer Online Journal Archives 1860-2000
Abstract In microbiological studies various methods are employed to estimate fractions from paired counts of organisms. When the fraction (second count divided by first count) is constant among the paired counts, the maximum likelihood estimate is the ratio of the arithmetic means. In many practical applications this fraction may however not be constant, but vary substantially between pairs of counts. We discuss a statistical method that estimates the distribution of the fraction from pairs of counts, to allow for this variation. Four real data sets (concerning viability for growth and infection, recovery of a detection method, and removal in a treatment process) are analyzed by this method. Often, pairs of counts are not determined in the same physical sample, but the first count is made in one sample, and the second count in a second sample. We provide parametric models to deal with such a situation: the desired fraction is still estimated as a binomial probability, but the model includes sampling effects. This approach also allows for analysis of two distinct cases: paired observations, where the counts “before” and “after” are related in some way to each other, and unpaired observations, where they are not. The four models for separate samples: paired or unpaired observations, and binomial probability fixed or variable, are used to analyze the removal data. It is concluded that this approach of statistical analysis of fractions is more appropriate than often used calculations based on the ratio between the (geometric) means “before” and “after”. The implications for risk analysis are briefly discussed.
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