A Comparative Study of Statistical Methods to Assess Dilutional Similarity

The four parameter logistic model is a better control method than the dilution effect statistic
Sep 30, 2005

Eloi P. Kpamegan, Ph.D., MSF
Bioassays are an important component of an effective quality control program in biopharmaceutical development and manufacture. Proper statistical analysis can optimize interpretation of the information they provide. Two statistics are currently being used to assess dilutional similarity between the test and reference samples: the dilution effect (DE) and F-statistics. These two tests do not always agree. In this paper, we compare the two statistics in parallel logistic assay and assess the results (F-statistics are superior) and suggest implications for the pharmaceutical industry. Biological assays used to determine biopharmaceutical potency have come under increasing regulatory scrutiny. Potency is the ability of a material to exert its intended activity. During assay development, scientists rely on potency as the single most important parameter to confirm lot-to-lot consistency. Proper statistical assessment of the dilutional similarity between the test and reference samples can provide reliable estimates of bioassay potency.


Often, the dilution effect measure confirms dilutional similarity of a test and reference sample even though the four parameter logistic curves are not parallel (intersect each other or don't have the same asymptotes). Recently, a full curve analysis was introduced to test dilutional similarity between the test sample and the reference standard using F-statistics.1

Emmens proposed that a logistic (semi-logarithmic) function might provide a good fit to the regression of quantitative response when the range of response is too great for a simple linear regression.2 Finney showed that under many but not all circumstances a four-or five parameter logistic will fit data well over a wide range of doses.3 The four parameter logistic equation is extensively used in bioassays due its similarity to equations used in various types of bioassays; it facilitates a uniform approach to problems of similar logical content. In addition to quality control of biopharmaceuticals, the use of non-linear logistics models such as a four-parameter logistic model is more common in clinical serology and pre-clinical evaluation of immunogenicity (including animal potency tests). According to Finney, the logistic equation has no theoretical pretensions but is simple to estimate.4

The four parameter logistic model of response Y values (e.g., counts/min, delta, optical density) versus the assay concentration (conc) is a standard model utilized in many immunological and biological assays. This model has several equivalent forms. The form used in this article is Equation (1):

where E(Y) is the expected response, a and d are the asymptotes, that is to say: E(Y)d when conc → ∞ and E(Y) a when conc → 0; b is the "slope" or the shape parameter. The variable xmid (also called EC50 or IC50 in the literature) is the dose at which 50% of the maximal response is observed. All symbols are also listed in a separate box .