Ratio Estimator

Devising an appropriate estimator for the response ratio was especially tricky. Each value of GPx activity usually consisted of five imperfectly measured, unpaired values: three measurements of activity and two measurements of protein. Thus we estimated $SpecificActivity\, =\, Activity/Protein$, where $Activity$ and $Protein$ are the true values. However, if $Activity$ and $Protein$ are each normally distributed with mean 0, $SpecificActivity$ has a Cauchy distribution [6], which has no finite moments of order one or more, that is to say, no mean value. In this case the expectation of $SpecificActivity$ would not exist, making its estimation problematical. Furthermore, Jensen's inequality [6,1] applied to the reciprocal function and independence of the numerator and denominator together imply that the naive estimator of the ratio of the sample averages of $Activity$ and $Protein$ is not an unbiased estimator of $SpecificActivity$. That is, $E\left[ \frac{X}{Y}\right]\,=\,E\left[X\frac{1}{Y}\right]\,=\,E[X]E[\frac{1}{Y}]\, \geq \,E[X]\frac{1}{E[Y]}\,=\, \frac{E[X]}{E[Y]}$. Moreover, using as an estimator the average of ratios obtained directly from the data is not possible, since the repeated measurements from the numerator and denominator have no natural pairing. Using an average of ratios from all possible combinations of numerator and denominator circumvents the problem of pairing, but then observations from a particular experiment are no longer independent. Therefore, values were resampled from the empirical distribution of specific activity from each experiment. For an experiment with $m$ activity measurements and $n$ protein measurements, we resampled with replacement $mn$ times from the $mn$ possible specific activity measurements, thus obtaining independent replications that reflected the measurement error inherent in the original values. The resulting estimate can be had analytically, but it is easier to rely upon computing methods.