A note on propensity score weighting method using paradata in survey sampling
Section 2. Basic setup

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Consider a finite population of size $N,$ where $N$ is known. The finite population ${\mathcal{F}}_N=\left\{u_1\mathrm{,}{u}_2\mathrm{,}\dots \mathrm{,}{u}_N\right\}\mathrm{,}$ $u_i\mathrm{<mo>=</mo>}\left(x_i\mathrm{,}{y}_i\right)$ is assumed to be a random sample from a superpopulation distribution $F\left(x\mathrm{,}y\right).$ In addition, we assume that $x$ is always observed and $y$ is subject to missingness. Let $\delta$ be the response indicator function that takes the value one if $y$ is observed and takes the value zero otherwise. Note that $x,$ $y$ and $\delta$ are all considered as random.

Suppose a sample of size $n$ is drawn from the finite population using a probability sampling design, where inclusion in the sample is represented by the indicator variables $I_i\mathrm{,}$ with $I_i\mathrm{<mo>=</mo>\; 1}$ if unit $i$ is included in the sample and $I_i\mathrm{<mo>=</mo>\; 0}$ otherwise. Let $A$ be the index set of the sample and $w_i\mathrm{<mo>=</mo>}{\pi }_i^{-1}$ be the design weight, where ${\pi }_i$ is the first-order inclusion probability.

We are interested in estimating parameter $\theta$ that is implicitly defined through an estimating equation $E\left\{U\left(\theta \mathrm{<mo>;</mo>}X\mathrm{,}Y\right)\right\}\mathrm{<mo>=</mo>\; 0.}$ Under complete response, an estimator of $\theta$ is obtained by solving

$${\displaystyle \sum _{i\in A}}{w}_{i}U\left(\theta \mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_i\right)\mathrm{<mo>=</mo>\; 0.}$$

In the presence of missing data, assuming that the response probabilities are known, the propensity-score adjusted estimator is obtained by solving

$${\displaystyle \sum _{i\in A}}{w}_i\frac{{\delta }_i}{p_i}U\left(\theta \mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_i\right)\mathrm{<mo>=</mo>\; 0,}(2.1)$$

where $p_i$ is the response probability of unit $i.$ Unfortunately, (2.1) is not applicable in practice because $p_i$ are generally unknown.

Now suppose that there exists additional variable $z$ obtained from paradata, which is always observed and satisfies

$$P\left({\delta }_i\mathrm{<mo>=</mo>\; 1}|x_i\mathrm{,}{y}_i\mathrm{,}{z}_i\right)\mathrm{<mo>=</mo>}P\left({\delta }_i\mathrm{<mo>=</mo>\; 1}|x_i\mathrm{,}{z}_i\right)\mathrm{.}(2.2)$$

As $x,$ $y,$ $\delta$ and $z$ are considered as random, we can use $z$ to make inference about $\theta$ under nonresponse. Such variable $z$ is sometimes called surrogate variable (Chen, Leung and Qin, 2008). By including a suitable surrogate variable, we can make the response mechanism missing at random (MAR) in the sense of Rubin (1976). We call assumption (2.2) as the Augmented MAR (AMAR) since MAR holds only under the augmented model that includes surrogate variable $z.$

Under (2.2), we can build a parametric model for the response mechanism and construct a propensity score weighted (PSW) estimator that is obtained from

$${\displaystyle \sum _{i\in A}}{w}_i\frac{{\delta }_i}{\widehat{\pi }\left(x_i\mathrm{,}{z}_i\right)}U\left(\theta \mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_i\right)\mathrm{<mo>=</mo>\; 0,}$$

where $\widehat{\pi }\left(x_i\mathrm{,}{z}_i\right)$ is a consistent estimator of $\pi \left(x_i\mathrm{,}{z}_i\right)\mathrm{<mo>=</mo>}P\left({\delta }_i\mathrm{<mo>=</mo>\; 1}|x_i\mathrm{,}{z}_i\right).$ Such PSW approach incorporating $z$ variable has been discussed in Peress (2010) and Kreuter and Olson (2013).

In survey sampling, the surrogate variable $z$ can be obtained from paradata which is not of direct interest. The information on $z,$ however, can be helpful in making model assumptions for the response mechanism. In some cases, the surrogate variable $z$ can satisfy

$$f\left(y|x\mathrm{,}z\right)\mathrm{<mo>=</mo>}f\left(y|x\right)\mathrm{.}(2.3)$$

Condition (2.3) means that the surrogate variable $z$ is not related to the study variable $y$ that is subject to missingness. The model satisfying (2.3) can be called the reduced outcome model. If condition (2.3) does not hold, we call $f\left(y|x\mathrm{,}z\right)$ the full outcome model.

If condition (2.3) holds in addition to condition (2.2), we can use this information to obtain a more efficient PSW estimator. Note that, by (2.2) and (2.3), we can establish

$$\begin{array}{ll}P\left(\delta \mathrm{<mo>=</mo>\; 1}|x\mathrm{,}y\right)\hfill & \mathrm{<mo>=</mo>}{\displaystyle \int }P\left(\delta =<mtext>\hspace{0.17em}</mtext>\; 1|x\mathrm{,}y,z\right)f\left(z|x\mathrm{,}y\right)dz\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \int }P\left(\delta \mathrm{<mo>=</mo>\; 1}|x\mathrm{,}z\right)f\left(z|x\mathrm{,}y\right)dz\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{{\displaystyle \int }P\left(\delta \mathrm{<mo><mtext>\hspace{0.17em}</mtext>=<mtext>\hspace{0.17em}</mtext></mo>\; 1}|x\mathrm{,}z\right)f\left(y|x,z\right)f\left(z|x\right)dz}{{\displaystyle \int }f\left(y|x,z\right)f\left(z|x\right)dz}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{{\displaystyle \int }P\left(\delta \mathrm{<mo><mtext>\hspace{0.17em}</mtext>=<mtext>\hspace{0.17em}</mtext></mo>\; 1}|x\mathrm{,}z\right)f\left(y|x\right)f\left(z|x\right)dz}{{\displaystyle \int }f\left(y|x\right)f\left(z|x\right)dz}\hfill \\ \hfill & \mathrm{<mo>=</mo>}P\left(\delta \mathrm{<mo>=</mo>\; 1}|x\right)\mathrm{,}\hfill \end{array}$$

where the second equality follows from assumption (2.2) and the fourth equality follows from assumption (2.3). Thus, assumption (2.2) and (2.3) imply

$$f\left(y|x,\delta \mathrm{<mo>=</mo>\; 1}\right)\mathrm{<mo>=</mo>}f\left(y|x\right)\mathrm{.}(2.4)$$

Under the reduced model assumption (2.3), then we can use another type of PSW estimator of the form

$${\displaystyle \sum _{i\in A}}{w}_i\frac{{\delta }_i}{{\widehat{\pi }}_1\left(x_i\right)}U\left(\theta \mathrm{<mo>;</mo>}{x}_i\mathrm{,}{y}_i\right)\mathrm{<mo>=</mo>\; 0,}(2.5)$$

where ${\widehat{\pi }}_1\left(x_i\right)\mathrm{<mo>=</mo>}{\displaystyle \int }\widehat{\pi }\left(x_i\mathrm{,}{z}_i\right)\widehat{f}\left(z_i|x_i\right)d{z}_i$ and $\widehat{f}\left(z|x\right)$ is an estimated conditional density of $z$ given $x.$ The estimator obtained from (2.5) can be called the smoothed PSW estimator (Beaumont, 2008). Note that ${\widehat{\pi }}_1\left(x\right)$ is the smoothed version of $\widehat{\pi }\left(x\mathrm{,}z\right)$ averaged over the conditional distribution $f\left(z|x\right).$

The smoothed PSW estimator obtained by solving the equation (2.5) is justified under MAR condition in (2.4). In this case, use of paradata for nonresponse adjustment is not necessarily useful, which will be justified in Section 3.

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