Dalzell's theorem and the analysis of proportions: A methodological note

The golden section is a well-known proportion that occurs when something (e .g. a line! is divided into two unequal parts such that the smaller (m) is to the larger (M) as the larger is to the sum of the two (i.e. m/M = M/(M+m ) = .618). Dalzell's theorem holds that the absolute value of the differenc e between M/(M + m) and .618 will tend to be smaller than the corresponding difference between m/M and .618. This means that the use of M/(M+m) ratios leads to results that are more supportive of the golden section hypothesis than does the use of m/M ratios. Notice that M/(M+m) corresponds to the pr oportion of ns that will occur; while m/M corresponds to the odds that m wi ll occur. While these are mathematically equivalent, in practice they may l ead to different interpretations of the same data. Although originally envi saged as applying to the golden section, Dalzell's theorem map have implica tions for any study that uses either a proportion or the odds as a dependen t measure. The use of proportions may produce results that are closer to a predicted value than will the use of the odds as a dependent measure.