Motivation and assumptions
This method relies on framework, where imputed values of the outcome variables are created for whole probability sample (\(S_B\)). In this case we treat big-data sample (\(S_A\)) as a training dataset, which is used to build an imputation model. The mass imputation approach for inference with non-probability samples makes use of model-based prediction methods, but it opens the door for more flexible methods as well. We consider parametric (e.g. regression models) and non-parametric (nearest neighbor algorithm) methods for this problem. General framework consists of building a model based on the \(S_A\) sample and for each \(i \in S_A\) calculate imputed value.
Model-based methods
This method is based on parametric model on sample \(S_A\) in the form of
\[
\begin{equation}
\mathbb{E}\left(y_i \mid \bx_i\right) = m\left(\bx_i, \bbeta_{0}\right)
\end{equation}
\]
for some unknown \(\bbeta_{0}\) and known function \(m(\cdot)\). The specification of m-function typically follows the mean function for generalized linear models. If \(\by\) is continous, we can use linear regression model with \(m\left(\boldsymbol{x}_i, \bbeta_{0}\right) = \bx_i^{T} \bbeta_0\). If \(\by\) is binary, one can use logistic regression model and let \(m\left(\boldsymbol{x}_i, \bbeta_{0}\right) = \frac{\exp\left( \bx_i^{T} \bbeta_0\right)}{\exp\left( \bx_i^{T} \bbeta_0\right) + 1}\). If \(\by\) represents count data, we can use log-linear model, where \(m\left(\boldsymbol{x}_i, \bbeta_{0}\right) = \exp\left(\bx_i^{T} \bbeta_0\right)\).
Basing on this approach we can obtain another approach for population mean estimator. It is worth noting that this time we rely on known design weights and imputation model for units from probability samples. \[
\begin{equation}
\frac{1}{\hat{N}^{\mathrm{B}}} \sum_{i \in \mathcal{S}_{\mathrm{B}}} d_i^{\mathrm{B}} m\left(\boldsymbol{x}_i, \hat{\bbeta}\right)
\end{equation}
\] Theoretical calculations lead to the following variance of the estimator, where the second part is variance of the widely-known Horvits-Thompson population mean estimator, which is implemented in the R survey package.
\[
\text{var}\left(\hat{\mu}_{MI}\right) = \text{var}_A + \text{var}_B
\] The \(\text{var}_A\) can be estimated by \[
\text{var}_A = \frac{1}{n_A} \sum_{i \in S_A} \hat{e}_i^2 \left(\bx_i^{T} \hat{c}\right)^2,
\] where \(\hat{e_i} = y_i - m\left(\boldsymbol{x}_i, \hat{\bbeta}\right)\) and \(\hat{c} = \left\{\frac{1}{n_A} \sum_{i \in S_A} \dot{m}\left(\boldsymbol{x}_i, \hat{\bbeta}\right) \bx_i^T \right\}^{-1} N^{-1} \sum_{i \in S_B} d_i^B \bx_i\). Respectively \(\text{var}_B\) can be estimated by \[
\hat{\text{var}}_B = \frac{1}{N^2} \sum_{i \in \mathcal{S}_{\mathrm{B}}} \sum_{j \in \mathcal{S}_{\mathrm{B}}} \frac{\pi_{i j}^{\mathrm{B}}-\pi_i^{\mathrm{B}} \pi_j^{\mathrm{B}}}{\pi_{i j}^{\mathrm{B}}} d_i^B m\left(\boldsymbol{x}_i, \hat{\bbeta}\right) d_j^B m\left(\boldsymbol{x}_j, \hat{\bbeta}\right)
\]
Nearest neighbor imputation
On the other hand we can applied non-parametric method to this problem, such as nearest neighbor algorithm, that is, find the closest matching unit from sample \(S_B\) based on the \(\bx\) values and use the corresponding \(\by\) value from this unit as the imputed value.
Model
Procedure contains two steps
for each \(i \in S_B\) find the nearest neighbor from sample \(S_A\).
Calculate the nearest neighbor imputation estimator of \(\mu_y\) \[
\begin{equation}
\hat{\mu}_\mathrm{nn}=\frac{1}{N} \sum_{i \in S_B} d_i^B y_{i(1)} .
\end{equation}
\]
This methods can provide robust results, but suffers from so called curse of dimensionanlity. For \(p > 1\) asymptotic bias of the estimator is not negligible (Yang and Kim, 2020). Variance of an estimator
We have \[
V_{\mathrm{nni}}=\lim _{n \rightarrow \infty} \frac{n}{N^2} E\left[\operatorname{var}_p\left\{\sum_{i \in S_B} d_i^B g\left(y_i\right)\right\}\right] .
\]
which can be estimated by \[
\hat{\text{var}}_{\mathrm{nni}}=\frac{n}{N^2} \sum_{i \in S_A} \sum_{j \in S_A} \frac{\pi_{i j}-\pi_i \pi_j}{\pi_i \pi_j} \frac{y_{i(1)}}{\pi_i} \frac{y_{j(1)}}{\pi_j}
\]
K-nearest neighbor imputation
Steps
- For each unit \(i \in S_B\) find k-nearest neighbors from sample \(S_A\). Impute the \(\by\) value for unit \(i\) by \(\hat{\mu}\left(\mathbf{x}_i\right)=k^{-1} \sum_{j=1}^k y_{i(j)}\).
- Calculate k-nearest neighbor imputation estimator of \(\mu_y\) \[
\hat{\mu}_{\mathrm{knn}}=\frac{1}{N} \sum_{i \in S_B} d_i^B \hat{\mu}\left(\mathbf{x}_i\right) .
\]
Variance of an estimator
We have
\[
\text{var}_{\mathrm{knn}}=\lim _{n \rightarrow \infty} \frac{n}{N^2}\left(E\left[\operatorname{var}_p\left\{\sum_{i \in S_B} d_i^B \mu\left(\bx_i\right)\right\}\right]+E\left\{\frac{1-\pi_A(\bx)}{\pi_A(\bx)} \sigma^2(\bx)\right\}\right),
\] where \(\sigma^2(\bx)=\operatorname{var}\{y \mid \bx\}\) and \(\pi_A(\bx) = P\left(R_i=1 \mid \boldsymbol{x}\right)\)