Thanks to Kit Baum, there is a new package -rori- on the SSC.

rori -- Immediate command for estimation of selection bias through relative odds ratio (ROR) with large sample confidence interval

Response bias is an often neglected consideration when evaluating the results of epidemiologic surveys, usually because of a paucity of information about nonrespondents. The effect of nonparticipation was described by a relative odds ratio (ROR), calculated as the OR (participants)/ OR (source population).
The relative odds ratio (ROR) is computed as the cross product ratio of the participation rates in the four exposure by outcome categories.

-rori- calculates the relative odds ratio (ROR) in a 2x2 table of outcome and exposure. -rori- also calculates the Prevalence ratios of the exposure groups. When exposure isn't binary, it is possible to specify N for the target and response populations.

References
  • Austin MA, Criqui MH, Barrett-Connor E, Holdbrook MJ. The effect of response bias on the odds ratio. Am J Epidemiol. 1981;114:137–43
  • Nohr EA, Frydenberg M, Henriksen TB, Olsen J. Does low participation in cohort studies induce bias? Epidemiology. 2006;17:413–8
  • Nohr EA, Liew Z. How to investigate and adjust for selection bias in cohort studies.