In Bioequivalence, for ln-transformed data with the 2x2 crossover default model, the intrasubject CV is computed as: CV = sqrt( exp(Var(Residual)) – 1) A common mistake when using log10-transformed data is to substitute 10^(Var(Residual) for exp(Var(Residual) in the above formula. That is, the formula, CV = sqrt(10^(Var(Residual)) – 1), is incorrect. To derive the correct formula for log10-transform, first consider the above formula for CV for the ln-transform case and how it is derived from the first and second moments of the log-normal distribution. The log-normal distribution is defined as continuous probability distribution of a random variable X whose natural logarithm is normally distributed. Its probability density function is derived by applying the change-of-variables rule on the density function of a normal distribution (i.e., ln(x) is changed for x in the normal distribution). Its second moment or variance is derived to be: Var = ( exp(sigma^2) – 1 ) * E^2 where sigma^2 is the variance of the variable’s natural logarithm. CV is defined as sqrt(Var(X)) / E(X), so with E(X) cancelling out, this yields: CV = sqrt( exp(sigma^2) – 1 ). Since sigma^2 is the variance of the variable’s natural logarithm, in Bioequivalence, Var(Residual) can be used as an estimate of sigma^2 when ln-transform is used on the data. To instead use a log10-transform of the values from a log-normal distribution, use the fact that ln(a) = ln(10) * log10(a) to get, for X from a log-normal distribution: E(ln(X)) = ln(10) * E(log10(X)) Var(ln(X)) = (ln(10))^2 * Var(log10(X)) So instead of using Var(Residual) from ln-transformed data (i.e., Var(ln(X)) as the estimate of sigma^2, if log10-transform was used when computing Var(Residual), it is necessary to use (ln(10))^2 * Var(Residual) as the estimate for sigma^2 in the formula CV = sqrt( exp(sigma^2) – 1), i.e., CV = sqrt( exp[ (ln(10))^2 * Var(Residual) ] – 1) By cancelling exp with one ln, this is equivalent to: CV = sqrt( 10[1] – 1) When this formula is used with the Data 2x2.csv file shipped with Phoenix, with the default Bioequivalence model and with log10-transform, this yields intrasubject CV = 0.233928 (since Var(Residual) = 0.010048748). This matches the intrasubject CV that is obtained if ln-transform were used instead.
ln(10)* Var(Residual) ↩︎