Statistical inference with non-probability survey samples
Section 3. Model-based prediction approach

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Model-based prediction methods for finite population parameters require two critical ingredients: the amount of auxiliary information that is available at the estimation stage and the reliability of the assumed model for inference. In the absence of any auxiliary information, the common mean model $E_{\xi }\left(y_i\right)\mathrm{<mo>=</mo>}{\mu }_0,$ $V_{\xi }\left(y_i\right)\mathrm{<mo>=</mo>}{\sigma }^2,$ $i\mathrm{<mo>=</mo>}1,\dots \mathrm{,}N$ may be viewed as reasonable but the model-based prediction estimator ${\widehat{\mu }}_y\mathrm{<mo>=</mo>}{\overline{y}}_A\mathrm{<mo>=</mo>}{n}_A^{-1}{\displaystyle {\sum }_{i\in S_A}{y}_i},$ although unbiased under the model since $E_{\xi }\left({\overline{y}}_A-{\mu }_y\right)\mathrm{<mo>=</mo>}0,$ is generally not an acceptable estimator of ${\mu }_y.$ The variance ${\sigma }^2$ for the common mean model is typically large and it renders the estimator ${\widehat{\mu }}_y\mathrm{<mo>=</mo>}{\overline{y}}_A$ with a prediction variance that is too large to be practically useful.

3.1  Semiparametric outcome regression models

Without loss of generality, we assume that $x$ contains 1 as its first component corresponding to the intercept of a regression model. Under the setting described in Section 2, we consider the following semiparametric model for the finite population, denoted as $\xi :$

$$E_{\xi }\left(y_i|x_i\right)\mathrm{<mo>=</mo>}m\left(x_i\mathrm{,}\beta \right),\text{and}{V}_{\xi }\left(y_i|x_i\right)\mathrm{<mo>=</mo>}v\left(x_i\right){\sigma }^2\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots ,N,(3.1)$$

where the mean function $m(\cdot \mathrm{,}\cdot )$ and the variance function $v(\cdot )$ have known forms, and the $y_i’\text{s}$ are also assumed to be conditionally independent given the $x_i’\text{s}\text{.}$ Let ${\beta }_0$ and ${\sigma }_0^2$ be the true values of the model parameters $\beta$ and ${\sigma }^2$ under the assumed model. The first major implication of assumption A1 is that $E_{\xi }\left(y_i|x_i,R_i\mathrm{<mo>=</mo>}1\right)\mathrm{<mo>=</mo>}{E}_{\xi }\left(y_i|x_i\right)$ and $V_{\xi }\left(y_i|x_i,R_i\mathrm{<mo>=</mo>}1\right)\mathrm{<mo>=</mo>}{V}_{\xi }\left(y_i|x_i\right).$ The model (3.1) which is assumed for the finite population also holds for the units in the non-probability survey sample $S_A.$ The quasi maximum likelihood estimator $\widehat{\beta }$ of ${\beta }_0$ is obtained using the dataset $\left\{(y_i\mathrm{,}{x}_i),i\in S_A\right\}$ from the non-probability survey sample as the solution to the quasi score equations (McCullagh and Nelder, 1989) given by

$$\text{S}\left(\beta \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in S_A}\frac{\partial m\left(x_i\mathrm{,}\beta \right)}{\partial \beta }}{\left\{v\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}\right\}}^{-1}\left\{y_i-m(x_i\mathrm{,}\beta )\right\}\mathrm{<mo>=</mo>}0.(3.2)$$

The semiparametric model (3.1) can be extended to replace $v(x_i)$ by a general variance function $v({\mu }_i)$ where ${\mu }_i\mathrm{<mo>=</mo>}m(x_i\mathrm{,}\beta ).$ The quasi maximum likelihood estimation theory covers linear or nonlinear regression models with the weighted least square estimators, the logistic regression model and other generalized linear models. Let $m_i\mathrm{<mo>=</mo>}m(x_i\mathrm{,}{\beta }_0)$ and ${\widehat{m}}_i\mathrm{<mo>=</mo>}m(x_i\mathrm{,}\widehat{\beta }),$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}N.$

3.2  Two general forms of prediction estimators

There are two commonly used model-based prediction estimators for ${\mu }_y$ in the presence of complete auxiliary information $\left\{x_1,\dots \mathrm{,}{x}_N\right\};$ see Chapter 5 of Wu and Thompson (2020). Note that $E_{\xi }({\mu }_y)\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{i=1}^N{m}_i}.$ The two prediction estimators are constructed as

$${\widehat{\mu }}_{y1}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i=1}^N{\widehat{m}}_i}\text{and}{\widehat{\mu }}_{y2}\mathrm{<mo>=</mo>}\frac{1}{N}\left\{{\displaystyle \sum _{i\in S_A}{y}_i}-{\displaystyle \sum _{i\in S_A}{\widehat{m}}_i}+{\displaystyle \sum _{i=1}^N{\widehat{m}}_i}\right\}.(3.3)$$

The estimator ${\widehat{\mu }}_{y2}$ is built based on ${\mu }_y=N^{-1}\left\{{\displaystyle {\sum }_{i\in S_A}{y}_i}+{\displaystyle {\sum }_{i\notin S_A}{y}_i}\right\}$ and uses ${\displaystyle {\sum }_{i\notin S_A}{\widehat{m}}_i}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i=1}^N{\widehat{m}}_i}-{\displaystyle {\sum }_{i\in S_A}{\widehat{m}}_i}$ to predict the unobserved term ${\displaystyle {\sum }_{i\notin S_A}{y}_i}.$ Under a linear regression model where $m\left(x,\beta \right)=x^{\prime }\beta ,$ the two estimators given in (3.3) reduce to

where ${\mu }_x\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{i=1}^N{x}_i}$ is the vector of the population means of the $x$ variables and ${\overline{x}}_A=n_A^{-1}{\displaystyle {\sum }_{i\in S_A}{x}_i}$ is the vector of the simple sample means of $x$ from the non-probability sample $S_A.$ If the linear regression model contains an intercept and $\widehat{\beta }$ is the ordinary least square estimator, we have ${\widehat{\mu }}_{y2}\mathrm{<mo>=</mo>}{\widehat{\mu }}_{y1}\mathrm{<mo>=</mo>}{\mu }_x^{\prime }\widehat{\beta }$ since ${\overline{y}}_A-{\overline{x}}_A^{\prime }\widehat{\beta }\mathrm{<mo>=</mo>}0$ due to the zero sum of fitted residuals. The prediction estimators in (3.4) under a linear model only require the population means ${\mu }_x$ in addition to the non-probability sample $S_A.$ Under the setting described in Section 2 with auxiliary information on $x$ provided through a reference probability sample $S_B,$ we simply replace ${\displaystyle {\sum }_{i=1}^N{\widehat{m}}_i}$ by ${\displaystyle {\sum }_{i\in S_B}{d}_i^B{\widehat{m}}_i}$ for the estimators in (3.3) and substitute ${\mu }_x$ by ${\widehat{\mu }}_x\mathrm{<mo>=</mo>}{\widehat{N}}_B^{-1}{\displaystyle {\sum }_{i\in S_B}{d}_i^B{x}_i}$ for the estimators in (3.4), where ${\widehat{N}}_B\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in S_B}{d}_i^B}.$ The population size $N$ appearing in (3.3) or (3.4) should also be replaced by ${\widehat{N}}_B$ even if it is known.

3.3  Mass imputation

Model-based prediction estimators of ${\mu }_y$ using a non-probability survey sample on $(y\mathrm{,}x)$ and a reference probability survey sample on $x$ have traditionally been presented as the mass imputation estimator. The study variable $y$ is not observed for any units in the reference survey sample $S_B$ and hence can be viewed as missing for all $i\in S_B.$ Let $y_i^{*}$ be an imputed value for $y_i,$ $i\in S_B.$ The mass imputation estimator of ${\mu }_y$ is then constructed as

$${\widehat{\mu }}_{y\text{MI}}\mathrm{<mo>=</mo>}\frac{1}{{\widehat{N}}_B}{\displaystyle \sum _{i\in S_B}{d}_i^B{y}_i^{*}},(3.5)$$

where ${\widehat{N}}_B$ is defined as before and the subscript “MI” indicates “Mass Imputation” (not “Multiple Imputation”). Under the deterministic regression imputation where $y_i^{*}\mathrm{<mo>=</mo>}{x}_i^{\prime }\widehat{\beta },$ the estimator ${\widehat{\mu }}_{y\text{MI}}$ reduces to the model-based prediction estimator ${\widehat{\mu }}_x^{\prime }\widehat{\beta }$ as discussed in Section 3.2.

The mass imputation approach to analyzing non-probability survey samples has the same spirit as model-based prediction methods but it opens the door for using more flexible models and imputation techniques that have been developed in the existing literature on missing data problems. The approach was first examined by Rivers (2007) through the so-called sample matching method. For each $i\in S_B,$ the “missing” $y_i$ is imputed as $y_i^{*}=y_j$ for some $j\in S_A,$ where $j$ is a matching donor from $S_A$ selected through the nearest neighbor method as measured by the distance between $x_i$ and $x_j.$ The underlying model $\xi$ for the nearest neighbor imputation method is nonparametric, i.e., $E_{\xi }\left(y_i|x_i\right)=m\left(x_i\right)$ for some unknown function $m(\cdot ).$ The matching value $y_j$ can be viewed as the predicted value of the missing $y_i$ under the model. Theoretical properties of estimators based on nearest neighbor imputation were discussed by Chen and Shao (2000, 2001) for missing survey data problems.

The semiparametric model (3.1) can be used for deterministic regression mass imputation. Under assumption A1, a consistent estimator $\widehat{\beta }$ of $\beta$ is first obtained from the non-probability sample dataset $\left\{(y_i\mathrm{,}{x}_i)\mathrm{,}i\in S_A\right\},$ and the estimator $\widehat{\beta }$ is then used to compute the imputed values $y_i^{*}=m\left(x_i\mathrm{,}\widehat{\beta }\right)$ for $i\in S_B.$ In other words, the assumption A1 implies the so-called model transportability by Kim, Park, Chen and Wu (2021): the model which is built for the non-probability sample can be used for prediction with the reference probability sample. The resulting mass imputation estimator ${\widehat{\mu }}_{y\text{MI}}$ is identical to one of the model-based prediction estimators presented in Section 3.2. Asymptotic properties and variance estimation for the estimator ${\widehat{\mu }}_{y\text{MI}}$ using the semiparametric model (3.1) were discussed by Kim et al. (2021).

Under the mass imputation approach, the only role played by the observed $y_i$ for $i\in S_A$ is to estimate the model parameters $\beta .$ The estimator ${\widehat{\mu }}_{y\text{MI}}$ is constructed using the fitted model and auxiliary information from the reference probability sample $S_B.$ It seems that we did not fully use the information on the observed $y_i$ given that ${\mu }_y$ is the main parameter of interest. This led to the research question described in Chapter 17 of Wu and Thompson (2020) on “reverse sample matching”. The proposed estimator is constructed as ${\widehat{\mu }}_{y\text{A}}={({\widehat{N}}^{*})}^{-1}{\displaystyle {\sum }_{i\in S_A}{d}_i^{*}{y}_i}$ using all the observed $y_i$ in the non-probability sample, where ${\widehat{N}}^{*}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in S_A}}{d}_i^{*}.$ The $d_i^{*}$ is a matched survey weight from $S_B$ such that $d_i^{*}=d_j^B$ with $j\in S_B$ being the nearest neighbor of $i\in S_A$ as measured by $\Vert x_i-x_j\Vert .$ Theoretical properties of the reverse matched estimator ${\widehat{\mu }}_{yA}$ using the nearest neighbor $j\in S_B$ to match $d_i^{*}$ with $d_j^B$ have not been formally investigated in the existing literature.

Wang, Graubard, Katki and Li (2020) proposed a kernel weighting approach to reverse sample matching using $d_i^{*}\propto {\displaystyle {\sum }_{j\in S_B}{K}_{ij}{d}_j},$ where $K_{ij}$ is a kernel distance between ${\widehat{p}}_i$ and ${\widehat{p}}_j;$ see the adjusted logistic propensity (ALP) weighting method discussed at the end of Section 4.1.1 on the calculation of ${\widehat{p}}_i.$ They showed that the estimator ${\widehat{\mu }}_{y\text{A}}$ is consistent under certain regularity conditions. In a recent working paper posted on arXiv by Liu and Valliant (2021), the authors discussed issues with the bias and the variance of the reverse matched estimator under different randomization frameworks involving one, two or all three of the sources $\left(p\mathrm{,}q\mathrm{,}\xi \right).$ The authors also proposed a calibration step over the matched weights, which seems to be a promising idea. Further research on this topic is needed.

The mass imputation approach to analyzing non-probability survey samples leads to an interesting research question that is currently under investigation by a doctoral student at University of Waterloo: Is it theoretically feasible and practically useful to create a mass-imputed dataset $\left\{(y_i^{*}\mathrm{,}{x}_i\mathrm{,}{d}_i^B)\mathrm{,}i\in S_B\right\}$ based on the reference probability survey sample that can be used for general statistical inferences? The answer clearly depends on the types of inferential problems to be conducted over the imputed dataset. A minimum requirement is that the conditional distribution of the study variable $y$ given the covariates $x$ is preserved for the mass-imputed dataset. The nearest neighbor imputation method and the random regression imputation method can be useful for this purpose. Fractional imputation is another possibility, especially for binary or ordinal study variables. Multiple imputation is also potentially useful in this direction to create multiple mass-imputed datasets. The subscript “MI” in this case might need to be changed to “MI²”, meaning “Mass Imputation with Multiple Imputation”.

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