Statistical inference with non-probability survey samples
Section 2. Assumptions and inferential frameworks

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Suppose that the target population $U\mathrm{<mo>=</mo>}\left\{\mathrm{1,}\mathrm{2,}\dots \mathrm{,}N\right\}$ consists of $N$ labelled units. Associated with unit $i$ are values $x_i$ and $y_i$ for the auxiliary variables $x$ and the study variable $y.$ The discussions focus on a single $y$ but the dataset most likely contains multiple study variables. Let ${\mu }_y=N^{-1}{\displaystyle {\sum }_{i=1}^N{y}_i}$ be the population mean which is the parameter of interest. Let $\left\{(y_i\mathrm{,}{x}_i)\mathrm{,}i\in S_A\right\}$ be the dataset for the non-probability survey sample $S_A$ with $n_A$ participating units. For most practical scenarios, the simple sample mean ${\overline{y}}_A\mathrm{<mo>=</mo>}{n}_A^{-1}{\displaystyle {\sum }_{i\in S_A}{y}_i}$ is a biased estimator of ${\mu }_y$ and hence is invalid.

2.1  Assumptions

Let $R_i\mathrm{<mo>=</mo>}I(i\in S_A)$ be the indicator variable for unit $i$ being included in the non-probability sample $S_A$ . Note that the variable $R_i$ is defined for all $i$ in the target population. Let

                                     ${\pi }_i^A\mathrm{<mo>=</mo>}P\left(i\in S_A|x_i\mathrm{,}{y}_i\right)\mathrm{<mo>=</mo>}P\left(R_i\mathrm{<mo>=</mo>}1|x_i\mathrm{,}{y}_i\right)\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}N\mathrm{.}$

We call the ${\pi }_i^A$ the propensity scores, a term borrowed from the missing data literature (Rosenbaum and Rubin, 1983). Some authors use the term participation probabilities; see, for instance, Beaumont (2020) and Rao (2021), among others. The propensity scores ${\pi }_i^A$ characterize the sample inclusion and participation mechanisms. They are unknown and require suitable model assumptions for the development of valid estimation methods. The following three basic assumptions were used by Chen, Li, and Wu (2020), which were adapted from the missing data literature.

A1  The sample inclusion and participation indicator $R_i$ and the study variable $y_i$ are independent given the set of covariates $x_i,$ i.e., $\left(R_i\perp y_i\right)|x_i.$

A2  All the units in the target population have non-zero propensity scores, i.e., ${\pi }_i^A>0,$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}N.$

A3  The indicator variables $R_1\mathrm{,}{R}_2,\dots \mathrm{,}{R}_N$ are independent given the set of auxiliary variables $\left(x_1\mathrm{,}{x}_2\mathrm{,}\dots \mathrm{,}{x}_N\right).$

Assumption A1 is similar to the missing at random (MAR) assumption for missing data analysis. Under A1, we have ${\pi }_i^A\mathrm{<mo>=</mo>}P\left(R_i\mathrm{<mo>=</mo>}1|x_i\mathrm{,}{y}_i\right)\mathrm{<mo>=</mo>}P\left(R_i\mathrm{<mo>=</mo>}1|x_i\right)\mathrm{<mo>=</mo>}\pi \left(x_i\right).$ Assumption A2 can be problematic in practice; see Section 7 for further discussions. Assumption A3 typical holds when participants are approached one at a time but can be questionable when clustered selections are used. It is shown in Section 4 that estimation of ${\pi }_i^A\mathrm{<mo>=</mo>}\pi \left(x_i\right)$ under assumption A1 requires auxiliary information from the target population. The ideal scenario is that the complete auxiliary information $\left(x_1\mathrm{,}{x}_2,\dots ,x_N\right)$ is available. The more practical scenario is that auxiliary information can be obtained from an existing probability survey.

A4  There exists a probability survey sample $S_B$ of size $n_B$ with information on the auxiliary variables $x$ (but not on $y)$ available in the dataset $\left\{(x_i\mathrm{,}{d}_i^B)\mathrm{,}i\in S_B\right\},$ where $d_i^B$ are the design weights for the probability sample $S_B.$

The $S_B$ is called the reference probability survey sample. The most crucial part of assumption A4 is that the set of auxiliary variables $x$ is observed in both the non-probability sample $S_A$ and the probability sample $S_B.$ A reference probability survey sample is often available in practice but the common set of auxiliary variables may not contain all the components to satisfy assumption A1.

2.2  Inferential frameworks

There are three possible sources of variation under the general setting of two samples $S_A$ and $S_B:$ (i) The model $q$ for the propensity scores on the sample inclusion and participation in the non-probability survey sample $S_A;$ (ii) The model $\xi$ for the outcome regression $\left(y|x\right)$ or imputation; and (iii) The probability sampling design $p$ for the reference probability survey sample $S_B.$ For the three approaches to inference to be discussed in Sections 3 and 4, the reference probability sample $S_B$ is always involved. Each of the three approaches requires a joint randomization framework involving $p$ and one of $\left(q\mathrm{,}\xi \right).$

1. Model-based prediction approach: The $\xi p$ framework under the joint randomization of the outcome regression model $\xi$ and the probability sampling design $p.$
2. Inverse probability weighting using estimated propensity scores: The $qp$ framework under the joint randomization of the propensity score model $q$ and the probability sampling design $p.$
3. Doubly robust inference: The $qp$ framework or the $\xi p$ framework, with no specification of which one.

The inferential framework is the foundation for theoretical development. Consistency of point estimators needs to be established under the suitable joint randomization. Theoretical variances typically involve two components, one from each source of variation, and correct derivations of the two components are the key to the construction of consistent variance estimators under the designated inferential framework.

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