Comments on “Statistical inference with non-probability survey samples” – Miniaturizing data defect correlation: A versatile strategy for handling non-probability samples
Section 4. Quasi-randomization or super-population implementations

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In a nutshell, the quasi-randomization approach focuses on making $W_I{\pi }_I$ a constant variable (induced by FPI $I).$ When our sample is genuinely selected by a probabilistic scheme by design, then ${\pi }_i=\mathrm{Pr}\mathrm{<mo\; stretchy="false">(</mo>}{R}_i=1|x_i\mathrm{<mo\; stretchy="false">)</mo>},$ for $i\in N,$ is a design probability, free of $y_i,$ but it can depend on $x_i$ for example when $x_i$ includes a stratifying variable. When the design probability is unavailable, we first need to invoke a divine probability. This could be a natural one given by the finite population, such as the propensity ${\pi }_i={\mathrm{Pr}}_I\mathrm{<mo\; stretchy="false">(</mo>}{R}_I=1|A_I=A_i\mathrm{<mo\; stretchy="false">)</mo>}$ induced by FPI, where $A_i=\mathrm{<mo>\{</mo>}{y}_i\mathrm{,}{x}_i\mathrm{<mo>\}</mo>},$ or an imagined super-population one such as the $R_i’\text{s}$ being generated independently from $\text{Ber}\mathrm{<mo\; stretchy="false">(</mo>}{\pi }_i\mathrm{<mo\; stretchy="false">)</mo>},$ where ${\pi }_i=\mathrm{Pr}\mathrm{<mo\; stretchy="false">(</mo>}{R}_i=1|A_i\mathrm{<mo\; stretchy="false">)</mo>}>0.$ This positivity assumption is necessary if the finite population is pre-specified, or its imposition defines the finite population that can be studied. (This is a practically rather relevant consideration, such as in election polling, where the finite population may not be always pre-specified even theoretically.) Since these divine probabilities are unknown and serve as our estimand, we need to assume some device probabilities, such as via a generalized linear model ${\pi }_i=g\mathrm{<mo\; stretchy="false">(</mo>}{y}_i\mathrm{,}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}$ to proceed, even though we don’t really believe in any particular choice of $g.$

For our current discussion, suppose our divine probability is given by the super-population Bernoulli model. Let $n_R={\displaystyle {\sum }_{i\mathrm{<mo>=</mo>}1}^N}{R}_i,$ and $\tilde{p}\mathrm{<mo\; stretchy="false">(</mo>}A\mathrm{<mo\; stretchy="false">)</mo>}=\mathrm{Pr}\mathrm{<mo\; stretchy="false">(</mo>}{n}_R>0|A\mathrm{<mo\; stretchy="false">)</mo>}=1-{\Pi }_{i\in N}\mathrm{<mo\; stretchy="false">(</mo>1}-{\pi }_i\mathrm{<mo\; stretchy="false">)</mo>},$ where $A=\mathrm{<mo>\{</mo>}{A}_i\mathrm{,}i\in N\mathrm{<mo>\}</mo>.}$ Because the $R_i$ here is controlled by a divine probability, the sample size $n_R$ is no longer a design variable to be conditioned upon in our replication scheme; it is generally no longer an ancillary statistic. Nevertheless, we should condition on $n_R>0,$ a universal requirement for constructing data-driven estimates for $\overline{G}.$ Fortunately this conditioning does not create mathematical complications to the simplicity granted by the independence among ${\pi }_i\mathrm{,}i\in N$ as functions of $A_i.$ This is because ${\tilde{\pi }}_i\mathrm{<mo\; stretchy="false">(</mo>}A\mathrm{<mo\; stretchy="false">)</mo>}\equiv \mathrm{Pr}\mathrm{<mo\; stretchy="false">(</mo>}{R}_i=1|A\mathrm{,}{n}_R>\mathrm{0<mo\; stretchy="false">)</mo>}={\pi }_i/\tilde{p}\mathrm{<mo\; stretchy="false">(</mo>}A\mathrm{<mo\; stretchy="false">)</mo>},$ but the normalizing constant $\tilde{p}\mathrm{<mo\; stretchy="false">(</mo>}A\mathrm{<mo\; stretchy="false">)</mo>}$ ‒ which depends on the entire $A$‒ is not relevant for the developments in this article, such as assigning weights that are proportional to ${\tilde{\pi }}_i^{-1}\mathrm{<mo\; stretchy="false">(</mo>}A\mathrm{<mo\; stretchy="false">)</mo>.}$

Consequently, under this divine probability, which corresponds to (the true model for) the $q$ -model setting in Wu (2022), we have for any chosen $W_I,$ by (3.1)

$$\begin{array}{ll}\text{E}\mathrm{<mo\; stretchy="false">(</mo>}{c}_{\tilde{R},z}|A\mathrm{,}{n}_R>\mathrm{0<mo\; stretchy="false">)</mo>}\hfill & \mathrm{<mo>=</mo>}{\text{Cov}}_I\mathrm{<mo\; stretchy="false">(</mo>}{W}_I\text{E}\mathrm{<mo\; stretchy="false">[</mo>}{R}_I|A\mathrm{,}{n}_R>\mathrm{0<mo\; stretchy="false">]</mo>,}{y}_I-m\mathrm{<mo\; stretchy="false">(</mo>}{x}_I\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\hfill \\ \hfill & ={\tilde{p}}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}A\mathrm{<mo\; stretchy="false">)</mo>}{\text{Cov}}_I\mathrm{<mo\; stretchy="false">(</mo>}{W}_I{\pi }_I\mathrm{,}{y}_I-m\mathrm{<mo\; stretchy="false">(</mo>}{x}_I\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}(4.1)\hfill \end{array}$$

where $\text{E}$ is with respect to the (unknown) divine probability over $R_I$ (for fixed $I).$ It follows then that, regardless of whether we want to ensure zero expectation in (3.2) or in (4.1), we will impose $W_I{\pi }_I\propto 1,$ that is, $W_I\propto {\pi }_I^{-1},$ the well-known inverse probability weighting. Therefore, if our postulated model $q$ permits us to reliably capture ${\pi }_i$ in reality, then $c_{\tilde{R},z}=O_p\mathrm{<mo\; stretchy="false">(</mo>}{N}^{-1/2}\mathrm{<mo\; stretchy="false">)</mo>}$ because it has mean zero (with respect to the divine probability), and it is a weighted average of $N$ essentially independent Bernoulli variables, as seen in (3.1).

This is a randomization oriented approach because it treats the entire finite population attribute values $A$ as fixed, and the hypothetical replications are generated only by repeated realizations of the recording indicator $R_I.$ Of course, in general, the values of $\mathrm{<mo>\{</mo>}{\pi }_i\mathrm{,}i\in N\mathrm{<mo>\}</mo>}$ are unknown, and worse they are inestimable from a non-probability sample without further assumptions. To proceed, we pose assumptions such as missing at random, i.e., $\mathrm{Pr}\mathrm{<mo\; stretchy="false">(</mo>}{R}_i=1|A_i\mathrm{<mo\; stretchy="false">)</mo>}=\mathrm{Pr}\mathrm{<mo\; stretchy="false">(</mo>}{R}_i=1|x_i\mathrm{<mo\; stretchy="false">)</mo>},$ and the requirement of an auxiliary sample so that we have some values of $x_i$ with $R_i=0.$ We also have choices on how to estimate the inclusion propensity ${\pi }_i=\mathrm{Pr}\mathrm{<mo\; stretchy="false">(</mo>}{R}_i\mathrm{<mo>=</mo>}1|x_i\mathrm{<mo\; stretchy="false">)</mo>},$ parametrically or non-parametrically. These assumptions, requirements, and estimation methods are all essential for practical implementation, as carefully reviewed and discussed by Wu (2022); also see Tan (2010) for a detailed comparison of various estimation strategies. Nevertheless, the overarching idea of quasi-randomization methods is to choose $W_I$ to free ${\tilde{R}}_I=W_I{R}_I$ from $I$ in expectation over the posited hypothetical replications, to regain the freedom guaranteed by probability sampling.

Complementarily, the super-population approaches aim to miniaturize $c_{\tilde{R},z}$ via making the other variable in $c_{\tilde{R},z},$ that is, $z_I$ free of $I$ in expectation, but over a different hypothetical replication scheme. Here the idea is to choose an $m\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}$ that is a good approximation to $y_i$ such that the residual $z_i=y_i-m\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}$ will be zero in expectation conditioning on $x.$ Typically, this is done by considering a joint model for $\mathrm{<mo>\{</mo>}{R}_i\mathrm{,}{y}_i\mathrm{<mo>\}</mo>}$ given $x_i,$ and with a specific regression model $\xi \mathrm{<mo\; stretchy="false">(</mo>}y|x\mathrm{<mo\; stretchy="false">)</mo>},$ using the notation in Wu (2022). It is important to recognize that, although we only specify the regression model $y_i$ given $x_i,$ we must include $R_i$ in the replications in order to capture the possible dependence of $R_i$ on the entire $A_i=\mathrm{<mo>\{</mo>}{y}_i\mathrm{,}{x}_i\mathrm{<mo>\}</mo>},$ which is the key concern for non-probability samples. Indeed, it is this joint specification that permits the adoption of the missing at random assumption to reduce $P\mathrm{<mo\; stretchy="false">(</mo>}{y}_i|x_i\mathrm{,}{R}_i\mathrm{<mo\; stretchy="false">)</mo>}=P\mathrm{<mo\; stretchy="false">(</mo>}{y}_i|x_i\mathrm{<mo\; stretchy="false">)</mo>},$ which in turn permits us to focus on specifying a single regression model $\xi \mathrm{<mo\; stretchy="false">(</mo>}{y}_i|x_i\mathrm{<mo\; stretchy="false">)</mo>}$ for both observed and unobserved individuals. Therefore, when we write ${\text{E}}_{\xi },$ we mean the expectation with respect to 

$$P\mathrm{<mo\; stretchy="false">(</mo>}{R}_i\mathrm{,}{y}_i|x_i\mathrm{<mo\; stretchy="false">)</mo>}=P\mathrm{<mo\; stretchy="false">(</mo>}{R}_i|x_i\mathrm{<mo\; stretchy="false">)</mo>}P\mathrm{<mo\; stretchy="false">(</mo>}{y}_i|R_i\mathrm{,}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}={\pi }_i^{R_i}{\mathrm{<mo\; stretchy="false">(</mo>1}-{\pi }_i\mathrm{<mo\; stretchy="false">)</mo>}}^{1-R_i}\xi \mathrm{<mo\; stretchy="false">(</mo>}{y}_i|x_i\mathrm{<mo\; stretchy="false">)</mo>,}(4.2)$$

where ${\pi }_i=\mathrm{Pr}\mathrm{<mo\; stretchy="false">(</mo>}{R}_i=1|x_i\mathrm{<mo\; stretchy="false">)</mo>}$ is left unspecified, unlike with the quasi-randomization approach.

It follows then that, conditioning on $X=\mathrm{<mo>\{</mo>}{x}_i\mathrm{,}i\in N\mathrm{<mo>\}</mo>}$ and $n_R>0,$ which does not alter $P\mathrm{<mo\; stretchy="false">(</mo>}y|X\mathrm{<mo\; stretchy="false">)</mo>}$ because $y$ and $R$ are independent given $X,$ we have

$$\text{E}\mathrm{<mo\; stretchy="false">(</mo>}{c}_{\tilde{R},z}|X\mathrm{,}{n}_R>\mathrm{0<mo\; stretchy="false">)</mo>}={\mathrm{<mo\; stretchy="false">[</mo>}\tilde{p}\mathrm{<mo\; stretchy="false">(</mo>}X\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">]</mo>}}^{-1}{\text{Cov}}_I\mathrm{<mo\; stretchy="false">(</mo>}{W}_I{\pi }_I\mathrm{,}\text{E}\mathrm{<mo\; stretchy="false">[</mo>}{y}_I|x_I\mathrm{<mo\; stretchy="false">]</mo>}-m\mathrm{<mo\; stretchy="false">(</mo>}{x}_I\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>.}(4.3)$$

Clearly, (4.3) becomes zero when we choose $m\mathrm{<mo\; stretchy="false">(</mo>}{x}_I\mathrm{<mo\; stretchy="false">)</mo>}={\text{E}}_{\xi }\mathrm{<mo\; stretchy="false">[</mo>}{y}_I|x_I\mathrm{<mo\; stretchy="false">]</mo>}$ and that the $\xi$ model is (first-order) correctly specified, that is, ${\text{E}}_{\xi }\mathrm{<mo\; stretchy="false">[</mo>}{y}_I|x_I\mathrm{<mo\; stretchy="false">]</mo>}=\text{E}\mathrm{<mo\; stretchy="false">[</mo>}{y}_I|x_I\mathrm{<mo\; stretchy="false">]</mo>.}$ This summarizes the super-population approach, and it renders $c_{\tilde{R},z}=O_p\mathrm{<mo\; stretchy="false">(</mo>}{N}^{-1/2}\mathrm{<mo\; stretchy="false">)</mo>}$ for similar reasons as given for the quasi-randomization framework.

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