= 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 [attachment:Delta_method_SE_of_ratio.xls].

   * Reference: [[http://www.ncbi.nlm.nih.gov/pubmed/16221306|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.