THE EVALUATION OF PLACEBOS IN CLINICAL TRIALS 463 A i -- nii Z2i = A ct• 1-I• • roi Eli Z2i -¾ A {12 rI2 • {9i Z2i•, - •0i ( 1 A A A A A A A2 A [•1 l'II Z •Oi Zli Z2i •i 21- A [•2 1-I 2 Z (-0i Z2i Xi A A A A + A H• I toi Z2i (0. -- 02i) XIX A A A A + A 11i Z toi Z. Zi (0. -- 02i) XX A i . Z2iZ i =• A 0tl HI Z (.0 i Zli Z2i Zi -• A {12 II 2 •0i ( 1 -- 0i) n n a a 2 a a a2 2 A [1• II• Z •oi Z•i Z2i Zi + A [32 112 Z A A A A + A H• Z •oi Z:i •i (0. -- 02i) XXI A a ni where toi - n A 0i (1 -- 0i) A A A A A A 02i ) ---• A al 1-If Z mi (01i- 02i) Zli -•- A A A A A A •t 2 112 Z •Oi (01i -- 02i) Z2i -• A A A A A A [•1 111 Z (0i (01i- 02i) Zli Zi -¾ fi A A A A A [•2 2 Z fOi (01i -- 02i) Z2i Zi -• n n n 2 A 111 Z (0i (01i -- 02i) XXII The calculations are iterated until the estimates do not change and a goodness of fit test may be used as a criterion of convergence as in probit analysis (12). The complexity of the calculations makes it necessary to use a digital computer when solving these equations. INITIAL APPROXIMATIONS FOR THE SOLUTIONS In order to apply the method of successive approximations given above it is necessary to choose suitable starting values. These may be

464 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS obtained graphically from an arithmetic probability plot of the observed responses. The case of fifty per cent reactors with the two groups differing only in the value of a which is sufficient to clearly separate the groups is given in Fig. 2. It can be seen that the curve for the mixed groups tends to the first line for low dosage and to the second for high dosage. Fig. 3 illustrates the behaviour when the two groups are not so clearly separated ,•o ' ' ' i. ' • Figu•'e 2 50% placebo reactors. and where the values of [I are different• In this case also the mixture curve tends to the separate groups at the extreme values of the dosage. The initial values for a•, a 2, [31, •2 may be obtained by drawing lines through the points on the A.P. plot at the ends of the range of dosage and these values used with an intermediate point to estimate II•. It is advisable to take a• • a2 and [l• • 132 since otherwise equations XVIIl through XXII become identical with the equations for fitting a single straight line to the A.P. plot and the above method is liable to converge to the wrong solutions. When the observed responses are plotted the points will be scattered about the population curve and straight lines should be drawn so that the points lie close and slightly below line II and above line I.