458 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS placebo reaction as well as of pharmacological action the observed response is likely to be quantal. The distinction between graded and quantal responses to the action of drugs is not absolute since any quantitative response may be made quantal. Conversely, quantal data may be made numerical by the use of scores, particularly if there are more than two possible outcomes. Hewlett and Plackett (9) have studied the relation between the quantal and graded responses to a drug in terms of a bivariate Normal distribution of graded response and critical graded response. Since the placebo reaction is usually observed quantally it is appropriate that the corresponding drug action is also discussed in the same terms, although it might be possible in a further investigation to consider a trivariate distribution of graded response and critical graded response to drug action jointly with quantal response to placebo action. THE EFFECT OF PLACEBO REACTION ON DRUG ACTION In clinical trials as reported in the literature, where placebo reaction is considered, the usual design is equivalent, in the simplest case of one treat- ment, to allocating individuals to a group of placebo reactors or to a group of nonreactors according to the results of tests with placebos. Each of the groups is then separately tested for responses to a given dose of a drug. In a suitably designed trial the observed proportion of placebo reactors is an unbiased estimate of the proportion in the populatkm. However placebo reactors are not always consistent, for example Lasagna (10) found that 55 per cent of patients receiving the placebo were inconsistent, that is 79 per cent of reactors to the placebo did not react on every occasion they received the placebo. Similarly the response to drug action is not consistent in given individuals Berkson (11) has compared this variability with the variation in the weights of human beings from time to time and Hewlett and Plackett (9) have stated that this variation does not invalidate the interpretation of drug action on a particular occasion. It follows that if the groups are allocated by means of single observations on the placebo then each group may be contaminated by the other and so the difference in drug action on the two groups is liable to be underestimated. Moreover even if individuals in a trial are correctly allocated as reactors or not, the difference in response to an active drug will depend on the dose given in general. This is illustrated in Fig. I for a high dose A with a small difference between the groups and for a low dose B with a large difference between them.

THE EVALUATION OF PLACEBOS IN CLINICAL TRIALS 459 It is clear that in order to evaluate drug action in the presence of placebo action it is necessary to study a range of doses, preferably spanning from zero level to a dose corresponding to about 95 per cent response. In view of the difficulty in separating the placebo reactors from the non- reactors it is proposed to analyse data from mixed groups. O o B A Figure I Comparison between placebo reactors (2) and non reactors (2) A MODEL OF PLACEBO REACTION The population of persons to be treated by drugs for a particular disease or condition may be regarded as composed of a proportion rI 1 of placebo reactors, I, and a proportion II 2 = 1-- II • of nonreactors to placebo, II. A common model for the quantal response of living organisms to given doses, Z, of a drug is the probit or normit transformation (12) defined by 0=•(% It) :•(a+•iZ) I i• 2 r- •Z = (I2}-1 (0) = yp -- 5 = y II where 0 is the probability of a response to a dose Z, ß denotes the standard normal probability integral and it, • are two parameters which are often transformed to a, •. The empirical probit of an observed proportion p of successes is given by y• and the normit by y when 0 is replaced by p. The parameters it and , are sometimes interpreted as the mean and standard deviation, respectively, of an underlying normal distribution of tolerances to the drug in the population but this interpretation is not essential to the use of the probit method.