Comments on “Statistical inference with non-probability survey samples”
Section 1. Introduction

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Many thanks to Changbao Wu for his stimulating review and assessment of methods for making inferences from nonprobability samples. I especially appreciate his thoughtful examination of the strong assumptions needed to derive the bias and variance of estimates.

Wu reviews three approaches for estimating the finite population mean ${\mu }_y$ of a variable $y$ that is measured in a nonprobability sample $S_A$ of size $n_A.$ Because this sample is not representative of the population (and hence the sample mean ${\overline{y}}_A$ is likely biased for estimating ${\mu }_y),$ each approach relies on information from a high-quality probability sample $S_B$ of size $n_B\text{:}$ $S_B$  does not measure $y$ but it contains a set of auxiliary variables $x$ that are also observed in $S_A.$

In the model-based predictive approach, a model is developed on $S_A$ to predict $y$ from $x.$ The mass imputation (MI) estimator, for example, uses the model to impute an estimate $y_i^{\text{*}}$ of $y_i$ for every member of the probability sample $S_B.$ Then the population total of $y$ is estimated by ${\displaystyle {\sum }_{i\in S_B}{d}_i^B{y}_i^{\text{*}}},$ where $d_i^B$ is the design weight of unit $i$ in $S_B.$

In the inverse propensity weighting (IPW) approach, a model is developed predicting the probability ${\pi }_i^A$ that population unit $i$ appears in $S_A$ as a function of $x.$ Then unit $i$ in $S_A$ is assigned weight $w_i^A=1/{\widehat{\pi }}_i^A$ and the population total is estimated by ${\displaystyle {\sum }_{i\in S_A}{w}_i^A{y}_i}\text{.}$

Wu also reviews a “doubly robust” estimator of ${\mu }_y$ that, by combining the predictive and IPW estimators, is approximately unbiased under the assumptions if either model is correctly specified. In this discussion, I will concentrate on the predictive and IPW approaches because these methods generalize more easily for multivariate analyses and estimating population characteristics other than means.

In Section 2, I explore assumptions needed for inference from nonprobability samples and diagnostics for assessing them. Then, in Section 3, I look at some questions to ask when deciding which approach (if any) to use for inference.

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