Relation of the crude relative risk of a disorder to relative risks in strata of a "susceptible" variable

If tau d is the relative risk of a disorder in the entire population, and tau d,i is the relative risk of the disorder in i = 1, ... m strata, then one may show readily that tau d = sigma ci tau d,i, where ci is the product of two terms tau i, the risk ratio of being in the ith stratum, and pi,unexp d, the proportion of those with the disorder and unexposed who are in the ith stratum. This formulation is of primary interest in epidemiology when relative risks are available on one or only some strata of a variable that itself may be affected by exposure (what one may define as a "susceptible" covariate) such as mortality or hospitalization. Although relative risks within strata of such a variable may be of some intrinsic clinical interest, only the risk ratio unstratified on such a variate may be pertinent to a causal effect (unlike the case for nonsusceptible variables such as sex, age, etc). In some instances, as for birth defects, one may have data from a few strata or only one (for example, livebirths) of a susceptible covariate (for example, conceptus viability). But one may still be able to draw useful inferences about tau d, the risk ratio in the entire population, because if tau d,i > or = 1/ci (or k/ci), one may conclude that tau d is, at least, greater than 1.0 (or k). Similarly, a study of a disorder limited to hospitalized cases and controls may enable investigators to infer, using the same criterion, a positive association in the entire population despite the presence of hospitalization bias of the type described by Berkson.