Integration of data from probability surveys and big found data for finite population inference using mass imputation
Section 3. Methodology

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3.1   Nearest neighbor imputation

For simplicity, we will focus on the Horvitz-Thompson type estimator, although our discussion applies to other type of estimators. If $Y_i$ were observed throughout Sample A, the Horvitz $–$ Thompson estimator ${\widehat{\mu }}_{g\mathrm{,}\text{HT}}\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{i\in A}{\pi }_i^{-1}g\left(Y_i\right)}$ can be used. We consider the imputation estimator of ${\mu }_g,$ given by ${\widehat{\mu }}_{g\mathrm{,}I}\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{i\in A}{\pi }_i^{-1}g\left(Y_i^{*}\right)}\mathrm{,}$ where $Y_i^{*}$ is an imputed value for $Y_i.$ Creating imputed values for the whole data is called mass imputation (Chipperfield et al., 2012; Kim and Rao, 2012).

To find suitable imputed values, we consider nearest neighbor imputation; that is, find the closest matching unit from Sample B based on the $X$ values and use the corresponding $Y$ value from this unit as the imputed value. This approach has been called Sample Matching by Rivers (2007). To investigate the theoretical properties, we first consider matching with replacement with single imputation; the discussion on $k$ nearest neighbor imputation is presented in Section 4.

The nearest neighbor approach to mass imputation can be described in the following steps: 

$${\widehat{\mu }}_{g\mathrm{,}\text{nni}}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}g(Y_{i(1)})}\mathrm{.}(3.1)$$

Remark 1. Our theoretical development applies to a general class of distances ${\Vert X\Vert }_{\Sigma }\mathrm{<mo>=</mo>}{(X^{\text{T}}{\Sigma }^{-1}X)}^{1/2},$  where $\Sigma$  is a positive definite matrix (Abadie and Imbens, 2006). This class includes the standard Mahalanobis distance by taking $\Sigma$  to be the empirical covariance matrix of $X.$  Write $\Sigma \mathrm{<mo>=</mo>}{L}^{\text{T}}L.$  Notice that ${\Vert X\Vert }_{\Sigma }\mathrm{<mo>=</mo>}{\{(LX)LX\}}^{1/2}\mathrm{<mo>=</mo>}\Vert LX\Vert .$  Hence, using ${\Vert \cdot \Vert }_{\Sigma }$  and $X$  is equivalent to using $\Vert \cdot \Vert$  and $LX.$  So, we can carry over the the theoretical result to the case with ${\Vert X\Vert }_{\Sigma }.$

Comparing to model-based imputation, nearest neighbor imputation has several advantages. First, it does not require strong parametric model assumptions and therefore is robust to model misspecification. Second, nearest neighbor imputation is donor-based, where the imputed value is a value that was actually measured and will always be within the bounds of observed values. Third, in contrast to regression imputation approaches, nearest neighbor imputation can retain the complex variance covariance structure of the data. Moreover, for the same imputed dataset, one can estimate different parameters by choosing reasonable $g\left(\text{}\cdot \text{}\right).$ Recall that $p$ is the dimension of $X.$ The asymptotic bias of ${\widehat{\mu }}_{g\mathrm{,}\text{nni}}$ is of order $O_{\text{p}}\left(N_B^{-1/p}\right)$ (Abadie and Imbens, 2006), which is negligible when the number of continuous covariates is fixed at a reasonable number and the size of the matching donor pool is huge as in our big data setup. In the presence of a large dimension of $X,$ variable selection is necessary for the nearest neighbor imputation estimator to have good statistical properties. In this case, we suggest selecting important variables that are associated with the outcome in order to ensure Assumption 1 holds and also to increase estimation precision (Brookhart, Schneeweiss, Rothman, Glynn, Avorn and Stürmer, 2006).

3.2   Asymptotic results

To study the asymptotic properties of ${\widehat{\mu }}_{g\mathrm{,}\text{nni}},$ we impose the following regularity conditions.

Assumption 3. (i) $f\left(\text{}X\text{}\right)$  and ${\mu }_g\left(\text{}X\text{}\right)\mathrm{<mo>=</mo>}E\left\{g\left(\text{}Y\text{}\right)|X\right\}$  are continuously differentiable for any continuous and bounded $g\left(\text{}Y\text{}\right),$  and (ii) $E\left\{g{(Y)}^{\beta }|X\right\}$  is bounded for $\beta \mathrm{<mo>=</mo>}\mathrm{1,}2.$

Assumption 4. (i) There exist positive constants $C_1$  and $C_2$  such that $C_1\le N{n}^{-1}{\pi }_i\le C_2\mathrm{,}$  for $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}N;$  (ii) the sampling fraction for Sample A is negligible, $n{N}^{-1}\mathrm{<mo>=</mo>}o\left(\text{}1\text{}\right);$  and (iii) the sequence of the Horvitz-Thompson estimators ${\widehat{\mu }}_{g\mathrm{,}\text{HT}}$  satisfies ${\text{var}}_p\left({\widehat{\mu }}_{g\mathrm{,}\text{HT}}\right)\mathrm{<mo>=</mo>}O\left(\text{}{n}^{-1}\text{}\right)$  and ${\left\{{\text{var}}_p\left({\widehat{\mu }}_{g\mathrm{,}\text{HT}}\right)\right\}}^{-1/2}\left({\widehat{\mu }}_{g\mathrm{,}\text{HT}}-{\mu }_g\right)|F_N\to N\left(\mathrm{0,}1\right)$  in distribution, as $n\to \infty ,$  where ${\text{var}}_p\left(\text{}\cdot \text{}\right)\mathrm{<mo>=</mo>}\text{var}\left(\cdot |F_N\right)$  is the variance under the sampling design for Sample A.

For clarification, the probability distribution underpinning the notation $E\left(\text{}\cdot \text{}\right),$ $\text{var}\left(\text{}\cdot \text{}\right),$ $o_{\text{p}}\left(\text{}\cdot \text{}\right)$ and $O_{\text{p}}\left(\text{}\cdot \text{}\right)$ is the joint distribution of the superpopulation model and the sampling processes for Samples A and B. Assumption 3 is a technical condition imposed on the functional continuity and finite moments, which holds for common models; see, e.g., Mack (1981). Assumption 4 holds for standard sampling designs in survey practice (Fuller, 2009; Chapter 1). It requires the sampling weights to behave well in the sense that there do not exist extremely large weights that dominate other weights. This occurs when subjects when certain characteristics are largely underrepresented in the sample. Sufficient conditions for Assumption 4 (iii) can be found in Chapter 3 of Fuller (2009).

We derive the asymptotic theory for ${\widehat{\mu }}_{g\mathrm{,}\text{nni}}$ in the following theorem and defer its proof to the Supplementary Material.

Theorem 1. Under Assumptions 1 $–$ 3 and $N{N}_B^{-1}\mathrm{<mo>=</mo>}O\left(\text{}1\text{}\right),$   ${\widehat{\mu }}_{g\mathrm{,}\text{nni}}$  has the same distribution as ${\widehat{\mu }}_{g\mathrm{,}\text{HT}}$  as $N_B\to \infty .$  Furthermore, under Assumption 4, ${\widehat{\mu }}_{g\mathrm{,}\text{nni}}$  is consistent for ${\mu }_g,$  and

$$n^{1/2}\left({\widehat{\mu }}_{g\mathrm{,}\text{nni}}-{\mu }_g\right)\to N\left(\mathrm{0,}{V}_{\text{nni}}\right)\mathrm{,}(3.2)$$

where

$$V_{\text{nni}}\mathrm{<mo>=</mo>}\underset{n\to \infty }{{\displaystyle \lim }}\frac{n}{N^2}E\left[{\text{var}}_p\left\{{\displaystyle \sum _{i\in A}{\pi }_i^{-1}g\left(Y_i\right)}\right\}\right]\mathrm{.}$$

Theorem 1 implies that the standard point estimator can be applied to the imputed data $\{(X_i\mathrm{,}{Y}_{i(1)})\text{:}i\in A\}$ as if the $Y_{i(1)}’s$ were observed values. Let ${\pi }_{ij}$ be the joint inclusion probability for units $i$ and $j.$ We show in the Supplementary Material that the direct variable estimator based on the imputed data

$${\widehat{V}}_{\text{nni}}\mathrm{<mo>=</mo>}\frac{n}{N^2}{\displaystyle \sum _{i\in A}{\displaystyle \sum _{j\in A}\frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_i{\pi }_j}\frac{g(Y_{i(1)})}{{\pi }_i}\frac{g(Y_{j(1)})}{{\pi }_j}}}$$

is consistent for $V_{\text{nni}}.$

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