460 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS The normit model may be extended to the two classes I and II separ- ately so that in the mixed population 0 = II1 01 + 172 02 III = rl• (P (a• + • z) + r12 (p (a2 + •2 z) IV where (P (al) (P (a2). The quantities of especial interest to be determined by a clinical trial are i. the proportion 1-] 1, of placebo reactors in the population, ii. the dose, Z0, such that, say, 99.9 per cent of nonreactors to the placebo will be successfully treated by the drug, where Z,, = ( 3.09 -- a•) / [•2. V If these quantities can be estimated satisfactorily then the potency of a drug may be assessed without the complications that a drug may be rejected if the proportion H1, of placebo reactors is high and that the effective dose of an accepted drug may be underestimated for the same reason. Lasagna et al (10) have stressed the importance of these complica- tions and have also suggested that the presence of placebo reactors may alter the dose response relationship and so alter the sensitivity of the clinical trial. If [•1 is non zero then the placebo reactors are also subject to the pharmacological action of the drug whereas if [•1=0 then the placebo reaction is independent of the dose and there is no advantage in admin- istering the drug to this group. MODIFICATIONS TO THE MODEL If the population contains any placebo reactors then l-I1 (I)(a l) will not be zero and therefore Ill • (a 1-+-[•l Z) is not negligibly small for all negative values of Z. Although the model may fit the data from a clinical trial satisfactorily it is difficult to interpret the model when Z is negative. This objection may be overcome by putting 121)= 1-[1{,- •- f (21-I)-' exp (--t2/2)dt } + 1-I2 (I)((1 2 -¾ "2Z), VI o Z•0. The dose response relationship for many drugs is given by the probit transformation in terms of the logarithm of the dose as metameter (12, p 23). If this is the case then either Z may be replaced by log ()• + g) for some )• 0 or • 1 may be deleted from the model.

THE EVALUATION OF PLACEBOS IN CLINICAL TRIALS 461 The two classes of patient taken separately may not have a linear relationship between the probit and the dose and it may be necessary to consider polynomial forms. In this case it is preferable to use the logit transformation. y = log p -- log ( 1 -- p) VII where p is the observed proportion of successes and then to put y -- a + [I Z + 7 Z2 VIII for example. The logit transform is equivalent to taking 0 = ey/(1 + ey) IX where y may be given any one of a number of functional forms, possibly different in the two classes. THE ESTIMATION OF THE PARAMETERS IN THE MODEL If the population contains only one homogeneous class the parameters for the probit or normit and the logit models may be estimated by the method of maximum likelihood, by the method of minimum chi-square (13), and by weighted least squares (14). In the case of a population consisting of two classes the maximum likelihood method is the most appropriate since the other methods lose any computational advantage they may have in the generalization to two groups. The method of maximum likelihood will be applied to the model given in equation (IV) the various modifications may be treated similarly. Let the observed frequencies of successes be r i when n i individuals are treated and the dose is • i, for i--1,2, . . . , k. The likelihood is the joint probability of the observed results taken as a function of the unknown parameters. The maximum likelihood is found by solving the following equations for A A A A A the estimates k • ri -- ni Oi 0 • L 1-I• Z. X = •a--• = 0i ( 1 -- 0i) i•-I k Z ri -- ni 0i • L II2 0i ( 1 -- 03 Z2i XI 0 i--1