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.