Robust and Efficient Bayesian Inference for Non-Probability Samples

Abstract

The declining response rates in probability surveys along with the widespread availability of unstructured data has led to growing research into non-probability samples. Existing robust approaches are not well-developed for non-Gaussian outcomes and may perform poorly in presence of influential pseudo-weights. Furthermore, their variance estimator lacks a unified framework and rely often on asymptotic theory. To address these gaps, we propose an alternative Bayesian approach using a partially linear Gaussian process regression that utilizes a prediction model with a flexible function of the pseudo-inclusion probabilities to impute the outcome variable for the reference survey. By efficiency, we mean not only computational scalability but also superiority with respect to variance. We also show that Gaussian process regression behaves as a kernel matching technique based on the estimated propensity scores, which yields double robustness and lowers sensitivity to influential pseudo-weights. Using the simulated posterior predictive distribution, one can directly quantify the uncertainty of the proposed estimator and derive associated 95% credible intervals. We assess the repeated sampling properties of our method in two simulation studies. The application of this study deals with modeling count data with varying exposures under a non-probability sample setting.

Robust Model-based Inference for Non-Probability Samples

“…Given the observed outcome for the entire population, one can directly quantify the unknown population parameter of interest. This idea clearly eliminates the need for the remaining design-based term in Rafei et al (2022)'s adjusted estimator, and therefore not only fills the above-mentioned gaps in bias adjustment when using more flexible models but fully satisfies the likelihood principle (Berger and Wolpert, 1988;Little, 2004). As a direct advantage, one can easily expand such a method to a Bayesian setting.…”

“…The prior literature proposed various robust approaches for finite population inference based on a non-probability sample under situations where a parallel "reference survey" is available as the external benchmark (Chen et al, 2019;Rafei et al, 2022Rafei et al, , 2021. One common assumption among these approaches is that units of the reference survey have been selected independently with unequal probabilities of selection.…”

“…Furthermore, to avoid imposing a pseudo-likelihood structure under a Bayesian framework while adjusting for unequal sampling weights of the reference survey in estimating the PS, Rafei et al (2022) employed a two-step pseudo-weighting approach proposed by Elliott and Valliant (2017). A noteworthy complication arose from the first step where the sampling weights of the reference survey had to be modeled conditional on the observed auxiliary information explicating the selection mechanism in the non-probability sample.…”

“…More importantly, the ultimate form of the adjusted estimators in previous literature mostly involve design-based terms, and therefore, are subject to the general drawbacks of design-based methods. For instance, both the QR and PM terms in the proposed estimator of Rafei et al (2021) appeared as (pseudo-)weighted sums, and that proposed in Rafei et al (2022) still contained a similar structure in the PM term. Despite being design-consistent, it is well-understood that any presence of influential (pseudo-)weights may cause inflated variance in such estimators, especially when the sample size is small (Chen et al, 2017;Zangeneh, 2012;Zhang and Little, 2011).…”

“…To this end, a partially linear Gaussian process regression is employed to non-parametrically link the estimated propensity scores (PS) to the response surface (Kammann and Wand, 2003). This approach was termed the Gaussian Process of Propensity Prediction (GPPP) by Rafei et al (2022). However, because the auxiliary variables are unmeasured for the non-sampled units, we generate synthetic populations beforehand by undoing the sampling mechanism of the reference survey through a Pólya urn scheme (Dong et al, 2014).…”