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ISSN: 2644-1381

Current Trends on Biostatistics & Biometrics

Review Article(ISSN: 2644-1381)

A Note on Relative Potency Theorem Volume 3 - Issue 5

Ogoke Uchenna Petronilla* and Nduka Ethelbert Chinaka

  • University of Port Harcourt, Nigeria

Received: September 21, 2022   Published: October 07, 2022

*Corresponding author: Ogoke Uchenna Petronilla, University of Port Harcourt, Nigeria

DOI: 10.32474/CTBB.2022.03.000171

Abstract PDF

Abstract

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.

Introduction

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 = μST 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.

Given

where

Conclusion

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].

References

  1. Ogoke UP, Nduka EC (2021) Statistical Theory and Analysis in Bioassay IPS Intelligentsia publishing Services.
  2. Taylor series (2001) Encyclopedia of Mathematics, EMS Press.
  3. Weisstein Eric W Taylor Series. MathWorld.

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