Sankhya: The Indian Journal of Statistics

1998, Volume 60, Series B, Pt. 1, 101-126

MODELLING DIFFERENTIAL NONRESPONSE IN SAMPLE SURVEYS

By

THOMAS c. LITTLE, *Morgan Stanley Dean Witter, New York*

and

ANDREW GELMAN, *Columbia University, New York*

*SUMMARY.* The standard analysis of unit nonresponse in sample surveys is to assume missing at random -- that is, that
the probability a person responds is independent of their response to the question of interest, *y*, conditional on fully-observed
covariates *x* or on sampling weights *w*, In this paper, we discuss weakening these assumptions without the use of
additional covariates in the special case of a binary outcome variable, *y*=0 or 1. We note frequentist
confidence bounds that do not rely on strong assumptions about the response mechanism. From a Bayesian
perspective, we discuss using prior distributions to average over uncertainty in the missing data mechanism.
Surprisingly, a natural -looking "noninformative" prior distribution yields unappealing posterior inferences. We discuss methods of constructing porate unequal sampling
weights into the model using design-based sampling theory. This is important so that the nonresponse
modeling can be an improvement upon rather than merely a replacement for standard weighted analysis of
sample surveys. We illustrate the hierarchical model by applying it to the state-level analysis of a series
of national pre-election opinion polls. The use of a reasonable prior distribution for the relative response probabilities leads to substantial improvements in coverage
of posterior intervals and prediction error of point estimates. We also consider the sensitivity to the prior
distribution and the effect of including sampling weights in the analysis.

*AMS (1991) subject classification. *62C10.

*Key words and phrases. *Bayesian inference, hierarchical model, opinion polls, sampling weights.