Search design is searching and estimating for a few nonzero effects in a large set of effects along with estimation of elements in a set of unknown parameters. In presence of noise the probability of discrimination between the true nonzero effect from an alternative one depends on the design and an unknown parameter, say $\rho$ . We develop a new criterion for design com- parison which is independent of $\rho$ and for a family density weight function show that it discriminates and ranks the designs precisely. This criterion is invariance to the variable noise which may be present between designs due to noise factors. This allows us to extend the design comparison to classes of equivalent designs.
