Expected number of active effects

Box and Meyer (1986) examined several published analyses of experiments and concluded that, in these, between 10% and 30% of main effects were identified as active, with an average of 20% active effects. Similar arguments may be made for screening experiments, with some modification for interactions and higher-order effects. The suggestions made here utilize calculations for the expected number of active effects. Such expectations should be easier to specify than prior probabilities. 

The value of the hyperparameter jt may be chosen by considering the prior expected number of active effects. Illustrative calculations are now given for a full second-order model with / factors, and for subsets of active effects that include linear and quadratic main effects and linear x linear interactions. Thus, the full model contains / linear effects, / quadratic effects, and ( ) linear x linear interaction effects. Prior probabilities on the subsets being active have the form of (22) and (23) above. A straightforward extension of the calculations of Bingham and Chipman (2002) yields an expected number of active effects as... 

This simple expression is invaluable for elicitation of a prior distribution instead of specifying the probability of activity for a variety of different effects, an expected number of effects can be specified, along with values of do, , a4 (for example, the defaults suggested below (23)). The expression for the expected number of active effects (24) is then solved for n. A data analyst could also experiment with alternative values of do,..., a4 and see the impact of these choices in terms of the expected number of linear effects and interactions. 

Table 7. Expected numbers of active effects, corresponding hyperparameter n, and breakdown by linear main effects and linear x linear interaction effects, for the simulated experiment with 11 factors...

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See also in sourсe #XX -- [ ]

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