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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