Is there a ‘safe area’ where the nonresponserate has only a modest effect on nonresponsebias despite non-ignorable nonresponse?

Rising nonresponse rates in social surveys makes the issue of nonresponse bias highly contentious. Nonresponse may induce bias and increase data collection costs. We study the relationship between response rate and bias, assuming non-ignorable nonresponse and focusing on estimates of totals or means. We show that there is a ‘safe area’ enclosed by the response rate on the one hand and the correlation between the response propensity and the study variable on the other hand; in this area, 1) the response rate does not greatly affect the nonresponse bias and 2) the nonresponse bias is small.