In this work, further relative potency properties and a thorough proof have been attained. The findings are derived using the Taylor’s (Maclaurin’s) series, and the expected value and variance of the relative potency are as a consequence. Alternately, the first and second moment for the relative potency (R)are obtained given the potency.
The common assumption is that the test doses and the standard
doses have the same variance and follow two normal distributions
(Bivariate Normal Distribution) [1]. Suppose we have {XT1, XT2, XT3…
XnT} which are identical and independently distributed N() and {XS1,
XS3, XS3, ... , XnS } are also identical and independently distributed and
N(), where ρ = μS/μT is estimated by
.
where: ns is the number of subjects assigned to the standard
preparation
nT is the number of subjects assigned to the test
preparation.
XSi is the Individual Effect Dose (IED) of the ith subject
receiving the standard preparation.
XTj is the IED of the jth subject receiving the test preparation.
The proof and other properties (first and second moments)
have not been fully explained in the literature. Hence the need to
address these.
Proof: Let {XSi : i = 1, …, ns} be a random sample from a population
(not necessarily normal) with mean and variance while {XTi : i = 1, …,
nT} is a random sample from a population (not necessarily normal)
with mean and variance where the samples are independent. Then
if and ρ = /, we have E(R) ρ and var
Note: R is a ratio of two random variables and so its expected value
is not
Note: R is a ratio of two random variables and so its expected value
is not
Proof: The following results can be derived from the Taylor’s
(Maclaurin’s) series.
If f(x,y) = f(a,b) + (x − a) + (y ̶ b) + ... (1)
where expansion is about x = a and y = b.
Let R = relative potency of the sample
Corollary : Given that the populations in the Theorem above is
normal, then
Remark
a. Homogeneity of Variance: For ease of comparison, the
variance of the two preparations is assumed to be equal. To test
this assumption, we use the F-distribution, where nS – 1 is the
numerator degree of freedom and nT – 1 is the denominator degree
of freedom [2].
where and are the sample variances of the standard and test
preparations respectively.
b. Variance Estimate: Following from Theorem above,
Establishing the Moment Generating Function for the relative
potency, R.
A detailed proof and other properties of relative potency have
been established, hence, certain other expression; the coefficient of
variation, skewness and kurtosis can be obtained [3].