I/EC
Statistical Design
Group Screening Designs by W. S. Connor, The Research Triangle Institute
In the l/EC Statistical Design Column for February of this year, a method of group-screening designs was discussed, for example where experimental error is negligible. All factors were correctly classified according to whether or not they were effective; thus, efficiency of the design could be judged solely by comparing the number of runs it required with the number of runs which a comparable single-stage experiment required. In this article the example is discussed where experimental error is appreciable, and thus the concept of efficiency is more complex.
IOR
T H E EXAMPLE
where
experi-
mental error is appreciable, the concept of efficiency is more complex because of wrong decisions about whether or not a factor is effective. A design should be j u d g e d by the dual criteria of the expected n u m b e r of correct decisions a n d the expected n u m b e r of runs. In this article references a r e m a d e to the Statistical Design column in the February 1961 issue of I / E C . In that column four assumptions were m a d e about the factors which must be realized, at least approximately, if the group screening d e signs a r e to be used. Here, three more assumptions are added—i.e., Nos. 5, 6, a n d 7. A d d i t i o n a l Assumptions
For the case when t h e experimental error is appreciable, 0. This assumption implies that the effect of a group factor is one of the values 0, A, 2A, . . . , kA a n d that if the effect is .fA, (s = 0, . . . , k), then the group factor contains s effective factors and (k — s) ineffective factors. But, of course, in the real problem, the effect of a group factor m a y be some value other t h a n A, a n d effects sA m a y be achieved by adding
effects from s' p^ s factors. Although this assumption is somewhat arbitrary a n d unrealistic, it results perhaps in shedding some light on the characteristics of group screening designs. I t is akin to the real problem, in that levels of the factors may be chosen in such a way that there is a c o m m o n least change in response, say A, which is worth detecting. Another assumption is that 6. Errors of all observations are independently normal with a constant known variance a2. T h e procedure is further specified by assuming that 7. Estimated m a i n effects of group factors are tested at significance level a, a n d if one or more of them is significantly different from zero, a second-stage experiment is carried out. Tests of whether the m a i n effects of the factors are zero are m a d e a t significance level /3.
B u r m a n give orthogonal designs having At, (t — 1, 2, . . . ) treatment combinations, which will accommodate At — \ factors. For the example under discussion, suppose that the first stage design is the one described in the February issue. If only one group factor is significant, . t h e n three factors are varied in the second experiment, and the same design c a n be used. If two group factors are significant, then eight treatment combinations are needed at the second stage. T a b l e I gives such a design. A seventh factor could be included at the levels shown in the last column, or the three remaining factors all could be held constant throughout. If all three group factors turn out to be significant, then 12 treatment combinations are required at the second stage. A suitable design is given in T a b l e I I . This design would accommodate two other factors by assigning levels as indicated in the last two columns. Expected Performance
As for the case when cr = 0, the expected n u m b e r of runs, R, is of interest. Also of interest are the expected n u m b e r of effective factors detected, E, a n d the expected n u m ber of ineffective factors, E, wrongly declared to be effective. These quantities have been computed for A/ < Q-
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2 a n d for a = 0.005 or 0.025. T h e r e is a substantial saving in average n u m b e r of runs from the 12 runs which would be needed in a single stage design.
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Expected Values
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Plackett and Bur man Table III.
CONTEST:
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E E E E E E E E E E E
A fair comparison between effectiveness of the one-stage and the two-stage designs would be afforded by choosing the sizes of the critical regions to maximize the expected n u m b e r of correct decisions. However, to the knowledge of the writer, it is not yet known which design would have the larger expected number. Further, it should not be overlooked that the criterion of efficiency must also take into account the expected n u m b e r of runs a n d that the better design with respect to correct decisions might not be better with respect to runs. Background Reference Plackett, R. L., Burman,' J. P., metrika 33, 305-25 (1946).
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