Lila F. Knudsen

The use of the angular transformation in biological assays.

Abstract A comparison is made of the use of the angular transformation and the use of the probit transformation in evaluating results from biological assays having percentage responses. It is found that the angular transformation results in making the weights dependent on the number of animals used on each dose and if these are equal the weights are eliminated. The logit transformation is also cited. A two-dose assay design used in conjunction with the angular transformation results in a simplified calculation which can be put in the form of a graph and nomograph for use in the laboratory to estimate both potency and error of the assay as a per cent of the standard. A comparison of the calculation time involved shows that the probit method requires about twelve times as long as the angular transformation method involving graph and nomograph. A comparison of results obtained shows very little difference.

STATISTICS IN MICROBIOLOGICAL ASSAY

It is the purpose here to give a brief explanation of the logic and common sense that serve as a basis for the statistical methods applicable to results obtained from microbiological assays. In an attempt to present a simplified explanation, only a few formulas and illustrations are cited, and these to illustrate important classes of technics that are now in general use. Bacteriologists and biochemists are prone to avoid the use of computations involving complicated mathematical formulas. Fortunately, these are not necessary. Simpler formulas are often a satisfactory compromise and can be used to obtain both an estimate of potency from a set of assay data and some measure of the precision of that estimate. There are, of course, several different ways they may be expressed. The use of “short cut” statistical procedures often helps to reduce either the series of determinations required to give a desired precision or the number of assays necessary to give assurance as to acceptability of a given lot of material. Saving time by this means makes for greater’efficiency and better utilization of an analyst’s time in any analytical laboratory. Also, the use of shortened calculation procedures means that more time can be devoted to additional assays. The applied statistician is continually beset with requests to simplify these calculations so that they may be easily applied by laboratory technicians. The original procedures rest on certain assumptions, and still others are required in their simplification. The latter assumptions are usually easier to test, and the validity of those underlying the simplifications presented here has been the subject of appropriate tests. Statistics can also be invaluable in giving an objective measure of the validity of an assay by means of testing the linearity of the dosage-response curve, where such linearity is assumed, or testing whether the slope of the dosage-response curve is significantly different from zero. However, no attempt will be made to discuss this phase of statistical methods. Further detailed procedures can be obtained from some of the references given a t the end of this discussion. In general, there are a t least three statistical approaches commonly applied to the resuIts of microbiological assay for calculating (1) an estimate of the potency, and (2) some measure of how much variation may be expected in a number of estimates of the potency of a given substance assayed in the same laboratory. These three approaches depend on the type of response in a particular assay: (1) assays having an undefined dosage-response relationship, such as those involving a daily standard curve relating dose and response; (2) assays, such as that for nicotinic acid, involving a linear relationship between dose and response and the straight lines for standard and unknown intersect a t zero dose; and (3) assays, such as the penicillin plate assay, wherein there is a linear relationship between the response and the logarithm of the dose but with parallel straight lines for standard and