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The Confidence Interval and Limits of Potency
A bioassay provides an estimate of the true potency of an Unknown. This estimate falls within a confidence interval, which is computed so that the odds are not more than 1 in 20 (P = 0.05) that the true potency either exceeds the upper limit of the confidence interval or is less than its lower limit. Since this interval is determined by a number of factors that may influence the estimate of potency, the required precision for most bioassays is given in the monograph in terms of the confidence interval, related either to the potency directly or to its logarithm.
General Calculation— Despite their many forms, bioassays fall into two general categories: (1) those where the log-potency is computed directly from a mean or a mean difference, and (2) those where it is computed from the ratio of two statistics.
(1) When the log-potency of an assay is computed as the mean of several estimated log-potencies that are approximately equal in precision, the log-confidence interval is
where s is the standard deviation of a single estimated log-potency, t is read from Table 9 with the n degrees of freedom in s, and k is the number of estimates that have been averaged. The same equation holds where the log-potency is computed as the mean bar(x) of k differences x, with s the standard deviation of a single x. In either case, the estimated log-potency M is in the center of its confidence interval, so that its confidence limits are
The upper and lower limits are converted to their antilogarithms to obtain the limits as explicit potencies.
(2) More often, the log-potency or potency is computed from a ratio, and in these cases the length of the confidence interval is typified by the log-interval in the equation
where M¢ is the log-relative potency as defined (see Calculation of Potency from a Single Assay), i is the log-interval between successive doses, and c¢ is a constant characteristic of the assay procedure. The remaining term C depends upon the precision with which the slope of the dosage-response curve has been determined. (This is sometimes expressed in terms of g = (C – 1)/C.) In factorial assays, it is computed as
where s2 is the error variance of a single observation, t2 is read from Table 9 with the degrees of freedom in s2, f is the number of responses in each Tt used in calculating Tb, and Tb and eb are computed with the factorial coefficients for row b in Tables 6 to 8. The s2 in Equation 26 depends upon the design of the assay, as indicated for each drug in the next section. In a valid assay, C is a positive number.
In an assay of two or more Unknowns against a common Standard, all with dosage-response curves that are parallel within the experimental error, C may be computed with the error variance s2 for the assay and with the assay slope as
The slope factor Tb¢ = S(x1Tt) or S(x1y) for each of the h¢ preparations, including the Standard, is computed with the factorial coefficients x1 for the Standard in the appropriate row b of Table 6 or 8. If a treatment total Tt includes one or more replacements for a missing response, replace eb f in Equation 27, or eb fh¢/2 in Equation 28, by f2S(x12/f¢), where each x1 is a factorial coefficient in row b of Tables 6 to 8, in this chapter, and f¢ is the number of responses in the corresponding Tt before adding the replacement. With this C, compute the confidence interval as
In assays computed from a ratio, the most probable log-potency M is not in the exact center of the confidence interval. The upper and lower confidence limits in logarithms are
C is often very little larger than unity, and the more precise the assay, the more nearly C approaches 1 exactly. R = zS / zU is the ratio of corresponding doses of the Standard and of the Unknown or the assumed potency of the Unknown. The upper and lower confidence limits in log-potencies are converted separately to their antilogarithms to obtain the corresponding potencies.
Confidence Intervals for Individual Assays— Since the confidence interval may vary in detail from the above general patterns, compute it for each assay by the special directions given under the name of the substance in the paragraphs following.
Antibiotic Assays— The confidence interval may be computed by Equations 24 and 25.
Calcium Pantothenate— For log-potencies obtained by interpolation from the Standard curve, the confidence interval may be computed with Equations 19 and 24. For log-potencies calculated with Equation 8 or 10, s2 may be computed with Equation 15, C with Equation 27 or 28, and the confidence interval L with Equation 26 or 29.
Corticotropin Injection— Compute the log confidence interval by Equations 26 and 27, with the coefficients and constants in Table 6 for a 3-dose assay, and s2 as determined by Equation 13 or 14.
Digitalis— Compute the confidence interval as where fU and fS are the number of observations on the Unknown and on the Standard, and is determined with s2 from Equation 11. The confidence limits for the potency in USP Units are then in which R is as defined in the Glossary of Symbols.
Glucagon for Injection— Compute the error variance s2 by Equation 15a, C by Equation 27 with eb f = 16n¢, and the log confidence interval L by Equation 26 with c¢i2 = 0.09062.
Chorionic Gonadotropin— Proceed as directed under Corticotropin Injection.
Heparin Sodium— If two independent determinations of the log-potency M differ by more than 0.05, carry out additional assays and compute the error variance among the N values of M as with n = N – 1 degrees of freedom. Given this value, determine the confidence interval in logarithms (L) by Equation 24.
Insulin Injection— Compute the error variance (s2) of y by Equation 16 and C as where t 2 from Table 9 depends upon n = 4( f – 1) degrees of freedom in s2 and N = 4f is the total number of differences in the four groups. By Equation 26, compute the confidence interval L in logarithms, where c¢i 2 = 0.09062. The upper and lower confidence limits in USP Units of insulin are given by the antilogarithms of XM from Equation 30.
Oxytocin Injection— Compute the approximate log confidence interval by Equation 26, in which where s2 is defined by Equation 18, and
Tubocurarine Chloride Injection— Compute the error variance by Equation 12, and the confidence interval by Equation 24.
Vasopressin Injection— Compute the error variance s2 by Equation 16, C by Equation 35, and the log confidence interval by Equation 26, where c¢ = 1 and i is the log-interval separating the two dosage levels.
Vitamin B12 Activity— Proceed as directed under Calcium Pantothenate.