ST 762, DATA ANALYSIS PROJECT, FALL 2009N.B.: This is a “closed” take-home project. Thus, you may not collaborate with any other person(whether in the class or not), nor may you consult with anyone except the instructor for help orclarification. You must carry out the analyses on your own, without discussing them with anyoneexcept the instructor, and you must write up your own report in your own words. You mayconsult the class notes, books, or things you find on the web. If you use any written passagesdirectly from any source, you must reference the source appropriately.On the first page of your paper, please write out and sign the following pledge certifying that youhave followed the above restrictions:“I have neither given nor received unauthorized aid on this project.”In the pharm aceutical and biotechnology industries, potency testing of biologic drug pr oducts isan important regulatory requirement mandated by the Food and Drug Admin istration that providescritical information on the consistency of the manufacturing process and that assures delivery of theappropriate dose of the drug. Thus, considerable effort goes into to the development of bioassays,which are procedures to measure the biologic activity of a drug (or other substance), that can beused to determine the relative strength, or potency, of different preparations (e.g., m anufacturedat different sites) of the same product. In a typical bioassay, drug product is formulated at severaldoses, often by serial dilution of a fu ll-strength formulation. At each dose, replicate samples ofbiologic material are subjected to the drug, and the measure of biologic activity is recorded foreach. The resulting dose-response (activity) relationsh ip s that can be deduced are th en used tomake inference formally on the relative potency of various preparations. The measure of biologicactivity may be qualitative (e.g., binary, as in the case where a response such as a change in tumorcells is observed or not) or quantitative, where the measure is some continuous response, such asa extent of bacterial growth or inhibition, tumor cell proliferation, and s o on. Many quantitativebioassays used in industry are based on r adioimmunoassay (RIA) or enzyme linked immunosorbentassay (ELISA) techniques, where a radioactive count or color change related to the extent of biologicactivity is measured.Bioassays are used for different types of potency testing in indu s tr y. One such use is in studiesin which an established bioassay procedure is used to assess the potency of drug product preparedunder different manufacturing conditions relative to a standard preparation, e.g., that on whichregulatory approval was based. Scientists at a small biopharmaceutical company have conducteda such a study and have contacted you for help in addressing a number of questions. Because ofconfidentiality requirements of the company, the scientists will not divulge the name of the drugor th e particular situation that has led them to conduct a bioassay study. They give you only thefollowing information about the study, the data from which are available in the file potency.daton the class web page.A “standard” full-strength formulation of drug X was obtained, along with two full-strength“test” formulations from different sources. Each formulation was diluted to achieve doses of 0.5, 1.0,2.5, 6.4, 16.0, 28.0, 40.0, and 80.0 dose units. For each of the standard and two test formulations,at each of these doses, as well as at 0 dose units, three s amples of biologic material were prepared.Each s ample was placed in a well of a 96-well assay plate and RIA techniques were used to obtainthe response (units not given). Thus, for each of the standard and two test formulations, responsesare available for 27 samples, 3 at each dose (including 0 dose), for a total of n = 81 responses, ascan be seen in the data file, where1Column Description1 formulation indicator, =1 for standard, = 2 or 3 for the two test formulations2 dose (dose un its)3 response (measure of activity)The scientists explain that a standard mo del for dose-response from such a RIA-based bioassayis the four-parameter logistic model. Specifically, the response R(x) at dose x is parameterized asR(x) = β1+β2− β11 + exp{β4(log x − β3)}, (1)where the interpretations of the parameters are given in the class notes. In this parameterization,note that β3has the interpretation as the logarithm of the so-called ED50, the dose at which th eresponse is halfway between β1and β2.An key issue in relative potency testing is whether or not dose-response relationships cor-responding to different formulations (e.g., standard and test here), as characterized by (1), aresimilar. Similarity means that the relationships share the same values of β1, β2, and β4(so havethe same asymptotic behavior at x = 0 and x = ∞ and the s ame shape). Under similarity, the re-lationships, if plotted on the same graph, will appear parallel in the region in which they are linearexcept for horizontal shifts from each other dictated by possibly different values of β3. Similarity ishypothesized to h old when the formulations being compared have the same mechanism of action,which might be reasonable to expect in the present situation where the formulations are standardand test preparations of the s ame dr ug; however, it is not given that similarity must hold.When th e dose-response relationships can reasonably be assumed similar, an approach for as-sessing relative potency of the formulations is easy to appreciate. Because they are parallel overthe linear range, the dose-responses relationships for different formulations differ only by the doseat which the response is the same distance between β1and β2, and the doses at which this occursare separated by the same amount regardless of the distance. Thus, it suffices to base a comparisonamong them on the ED50 values. Accordingly, it is standard to define the relative potency offormulation A to B as the ratio r = ED50B/ED50A. Thus, if the relative potency is r > 1, A ismore potent than B in that one needs a s maller dose of A (in proportion to the factor r) to achievethe same response as a given dose of B, and similarly for r < 1. If r = 1, then the formulationsare of equal potency (i.e, the same d ose of each leads to the same response). If B is a standardpreparation and A is a test preparation, then, relative potency quantifies the extent to which thetest formulation is