462 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS k 0 --/) L _ • ri -- ni 0i •[5• I-l• Oi(1 --oi) Z•izi i=l XII k 0- 3 L • ri_•_ niOi • [h -- Fie • Oi ( 1 -- Oi) Z2i •i i--1 XIII k 0 • L • _ri_--_niOi • l-I1 •.-•Oi ( l • Oi) (01i -- 02i) XIV xvhere 0i, 01i, 02i are the values of 0, 0•, 02 when Z = Zi, respectively, and Z. = (211)-* exp (--(el + [• Zi)z/2 ) XV Z2i -- ( 211 )- • exp (-- ( a2 + [•2 Zi)2 / 2 ) XVI for L • log likelihood = k k constant + • ri log 0i + •(ni-- ri) log (1 -- i=l i=l Oi) XVII There are no explicit solutions for equations X through XIV and it is necessary to use a method of successive approximations. The following equations have been obtained by a generalization of the A A A A A Bliss and Fisher solution of the probit equations (12). If el, [•, a2, [t2, iI• are approximate solutions then better solutions are obtained by taking A A A A R + A•I,•I + A•l,•t2+A•2,•2 + A•2, 1 +Alii, where A a•, A a2, A •, A •2, A Ill are obtained as the solutions of the equations: A -- •-- Zli • A (•1 lil •'• 0)i Zli -•- A (•2 li2 •'• o)i Zli 2i -•- tOi ( 1 --Oi) A A A2 A A A A •1 lil Z 0)i Zli Zi 21- A [•2 I-I2 Y• COl Zli Z2i •,i & A A + A li• Z (oi Zli ( 0ii -- •) XVIII

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