Calculating the standard deviation or standard error of a ratio
 Several scientific assays are based on ratios, but often the raw data is the numerator or the denominator of the ratio.
 Example:
 treated subjects: 50, 60, 70
 control subjects: 15, 25, 20
 To determine (1): What is the ratio of treated / control?
 To determine (2): What is the variability in the ratio of treated / control? In other words, how long should error bars be in a figure?
 The best answer to this "ratio of two means" issue came from Mathis Thoma (former BaRC statistics consultant), who said
 All methods have problems if the mean of y is close to zero.
 With the usual small number of measurements, standard error is a much better measure of variation than standard deviation.
 Several statistical methods can be used to calculate the variability of a ratio.
 the delta method (based on error propagation formula)
 the zero method (ignores the variation in the control group)
 the Fieller method (exact, but more complicated)
 The delta method is the easiest to implement, and appears to be no worse than the others, so we'll use that method.
 Calculations involve the standard error of the mean (SEM) and the coefficient of variation (CV) of each set of measurements
CV(x) = SEM(x) / mean(x) CV(y) = SEM(y) / mean(y) SE = ratio * square_root(CV(x)^2)+ CV(y)^2)
 This can be implemented readily in Excel. For an example, see Delta_method_SE_of_ratio.xls.
 Reference: Beyene and Moineddin, 2005  BMC Med Res Methodol. 2005 Oct 12;5:32. Methods for confidence interval estimation of a ratio parameter with application to location quotients.
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