Can we im prove th e two stage procedure?

T he answer to th a t question lies in the percentage of tim e th a t PA R is selected by the two stage procedure. The original scheme, as proposed by Grizzle, selects PA R 10% of th e tim e under the null. It is obvious th a t if TS is modified such th a t th e size of the test for PA R is lowered, then it would be possible to fix the Type

I error ra te at th e nom inal 5% level, b u t it is not clear a t all if th a t alteration will improve the power of th e procedure and make it superior to CROS. One way to ad ju st th e TS procedure is by keeping fixed th e size of th e test for carry-over and th e size of th e test for tre a tm e n t (when CROS is selected) a t their original values, b u t adju stin g the size of PAR. To illu strate the idea, suppose th a t the unconditional size of the te st for PA R [ü p a r)^ is set according to the relation;

(1 — ü s eq) o^cros + f^PAR — 0 05 (3.33)

This relationship is approxim ately valid, since if P(SEQ ,PA R ) defines the prob­ ability th a t PA R and SEQ are jointly significant, th en by requiring Type I error ra te of the procedure to be 5% th e following exact relation holds:

(1 — c l s e q) ^c r o s + P (PAR, SEQ) = 0.05 (3.34) Because P (PAR, SEQ) < üpar we conclude th a t

(1 — asEç) ciCROs + O'PAR < 0.05 (3.35)

where equality holds when PA R and SEQ are perfectly correlated (see Senn [79]). This expression will be close to equality for th e sorts of C orr (PAR, SEQ) com­ m only encountered in practice.

E quation (3.33) implies th a t the original scheme could be corrected in two dif­ ferent ways. According to th e first plan the investigator m ight wish to te st the significance of th e treatm en t diflPerence a t the same pre-specified level (say 5%) irrespectively of which tre a tm e n t estim ato r is chosen by th e procedure a t the second stage (i.e a c R o s = c l p a r)- T his approach requires re-setting the level of

carry-over testing, b u t keeps the sizes of CROS and PA R equal. If we targ et T ype I error a t 5% then th e perm issible range of values for the common size of th e te st for th e tre a tm e n t difference lies in th e interval (2.5%, 5.0%).

An altern ativ e way of am ending the procedure requires fixing the size of th e test for carry-over difference at th e trad itio n al 10% level, while altering sim ultaneously th e sizes of CROS and PA R so th a t equation (3.33) is satisfied. As it is obvious from th a t equation an increase in the size of CROS should be accom panied by a decrease in th e size of PAR, if th a t plan is followed. This approach may have im plem entation difficulties, since th e analyst has to decide different significance

Table 3.2: Perform ance of th e corrected two stage procedure

Type I error Power

P lan 1 P lan 2 Plan 1 Plan 2

O'SEQ 7.6% 14.8% 10.0% 7.6% 14.8% 10.0% O-CROS A 2.6% 2.7% 4.5% 5.0% 5.5% 2.6% 2.7% 4.5% 5.0% 5.5% 0.0 0.047 0.049 0.050 0.050 0.050 0.848 0.821 0.870 0.871 0.852 0.5 0.049 0.051 0.053 0.053 0.053 0.818 0.798 0.849 0.850 0.828 1.0 0.055 0.056 0.061 0.062 0.062 0.782 0.768 0.821 0.821 0.797 1.5 0.064 0.066 0.075 0.076 0.078 0.740 0.731 0.785 0.785 0.758 2.0 0.079 0.079 0.094 0.097 0.099 0.692 0.688 0.742 0.741 0.712 2.5 0.098 0.097 0.119 0.123 0.127 0.640 0.641 0.693 0.691 0.659 3.0 0.122 0.119 0.150 0.155 0.160 0.586 0.591 0.638 0.635 0.601 3.5 0.151 0.145 0.185 0.192 0.199 0.531 0.540 0.581 0.575 0.539 4.0 0.186 0.176 0.225 0.233 0.241 0.477 0.489 0.521 0.513 0.475 4.5 0.224 0.209 0.268 0.277 0.285 0.426 0.440 0.462 0.451 0.410 5.0 0.266 0.245 0.312 0.322 0.331 0.380 0.396 0.405 0.391 0.348

levels for the testing of the tre a tm e n t effect, depending on th e tre a tm e n t estim a­ to r chosen a t the first stage of the procedure. In th a t case the m axim um value th a t acRos can be set a t is 5.5%, while the corresponding range of acceptable values for ür ar is from 0% to 5%.

Applying the first correction scheme leads to an im provem ent of th e power of the procedure, as th e size for testing th e treatm en t difference decreases, con­ trary to the Type I error ra te which looks to deviate from th e desired 5% level. In Table 3.2 b o th power and Type I error rate for P lan 1 are displayed, when

o^CROs = CLpAR = 2.6% (or 2.7%). From equation (3.33) it can easily be derived th a t the size of the test for carry-over should be set at 7.6% and 14.8% respec­ tively. Those values were chosen on the grounds of providing best power values, while keeping the Type I error rate close to 5%.

Moving on now and studying more carefully the perform ance of the second cor­ rection scheme, it can be seen th a t power initially increases as ü c r o s varies from

0% to 5% b u t decreases afterwards. Type I error rate gets closer and closer to 5% as acRos moves from 0% to 5.5%. Once more the values chosen to illu strate the perform ance of the second correction plan give th e highest power values. It is w orth m entioning here th a t th e first correction plan alters th e bias and variance of th e procedure, while the second one leave them unchanged. C om parison of the two plans perform ance show th a t correction P lan 1 is less effective in improving the power of th e procedure if the Type I error ra te is set a t ab o u t th e sam e level for both.

In conclusion all atte m p ts to improve the two stage procedure have failed for the whole range of carry-over values. This indicates th a t this procedure is ra th e r of historical ra th e r th a n actual value and by no m eans should be used in th e future by the analyst of the cross-over experiment.