Let Zi =1 if unit i is included in the sample and 0 otherwise, with P(Zi = 1)=Ï€i.
Let Ri =1 if unit i responds to the survey and 0 otherwise, with P(Ri = 1)=φi
And

Assume Ri is independent of Zi for each i = 1. . . N. Let Å·R estimate the population mean

using only the respondents:

Where wi = 1/Ï€i. Show that the bias of Å·R is approximately

Where

As a consequence, the non response bias is approximately zero if either (1) φi =φU for all i, that is, the response propensity is the same for all units, or (2) the propensity to respond is uncorrelated with the response yi (Tremblay, 1986).

YU VR i= I