How to decompose the non-response variance: A total survey error approach
Section 2. Inference framework

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Assume a sample $s$ of size $n$ is drawn from a population $U$ of size $N.$ Define the population total by

$$t_d={\displaystyle {\sum }_{k\in U}{d}_k{y}_k}(2.1)$$

for a variable, $y,$ and a domain indicator, $d_k,$ which takes the value $d_k=1$ if unit $k$ belongs to the domain $d,$ and $d_k=0$ otherwise. In the context of full response, $t_d$ is estimated by ${\widehat{t}}_d^0={\displaystyle {\sum }_{k\in s}{d}_k{w}_k{y}_k}$ where $w_k$ could be the sampling weight or a calibrated weight if calibration is performed. Because surveys are generally subject to non-response, both unit or item, a sample unit is classified into either a responding or a nonresponding unit with regard to the variable $y$ at any given point during data collection. The subset $s_r$ contains item-responding units whereas $s_m$ contains item-nonresponding units. Note that $s_r$ and $s_m,$ respectively of size $n_r$ and $n_m,$ form a partition of the sample $s,$ $P_s=\left\{s_r,s_m\right\},$ with $s_r\cup s_m=s$ and $s_r\cap s_m=\varnothing .$

The approach proposed in this paper assumes that imputation is used in case of non-response, which is the common approach in business surveys. Moreover, this approach can be considered for both item and unit non-response as long as imputation is used. However, since only one variable of interest $y$ is considered here for simplicity, then no distinction is made if the $y$ variable is imputed because of item or unit non-response. Also, the set $s_r$ and $s_m$ are not indexed by an item number for simplicity without loss of generality. However, the action following the calculation of a unit score might be different depending on whether the unit is responding or not.

2.1 Estimation under imputation

The framework requires linear imputation methods. In other words, the imputed value, $y_k^{*},$ can be written as a linear combination of the values reported by the other units. This linear combination is given by $y_k^{*}={\phi }_{0k}+{\displaystyle {\sum }_{l\in s_r}{\phi }_{lk}{y}_l}.$ The quantities, ${\phi }_{0k}$ and ${\phi }_{lk}$ do not depend on the values of variable of interest, $y,$ but they may depend on $s,$ $s_r$ and auxiliary data from the nonrespondents available on the frame, registers or elsewhere. Linear imputation methods cover most methods used in practice like auxiliary value imputation (Beaumont, Haziza and Bocci, 2011) and linear regression imputation, as well as donor imputation, which is often used to impute categorical variables. 

It is common practice to use several imputation methods, referred to as composite imputation, applied sequentially to the same variable. More than one linear imputation method can be used to impute nonresponding units. Section 2 of Beaumont and Bissonnette (2011) defines composite imputation in detail. Briefly, suppose that the set of nonrespondents is broken down into two or more groups and that a different imputation method is used within each group. For example, let $x_k$ be the complete vector of auxiliary variables for unit $k,$ and suppose regression imputation is used to impute the variable of interest. However, if, for some cases, $x_k$ were incomplete, another imputation method, based on the available subset of $x_k,$ would be used. The approach presented in our paper can be generalized to include composite imputation as long as linear imputation methods are used. For simplicity of notation, the case of a single linear imputation method is presented.

The estimator of the domain total after imputation is given by

$${\widehat{t}}_d={\displaystyle {\sum }_{l\in s_r}{w}_l{d}_l{y}_l}+{\displaystyle {\sum }_{k\in s_m}{w}_k{d}_k{y}_k^{*}}(2.2)$$

where $w_k$ is the sampling weight or a calibrated weight. The estimator presented in equation (2.2) can be rewritten as

$$\begin{array}{ll}{\widehat{t}}_d\hfill & ={\displaystyle {\sum }_{l\in s_r}{w}_l{d}_l{y}_l}+{\displaystyle {\sum }_{k\in s_m}{w}_k{d}_k{y}_k^{*}}\hfill \\ \hfill & ={\displaystyle {\sum }_{l\in s_r}{w}_l{d}_l{y}_l}+{\displaystyle {\sum }_{k\in s_m}{w}_k{d}_k\left({\phi }_{0k}+{\displaystyle \sum _{l\in s_r}{\phi }_{lk}{y}_l}\right)}\hfill \\ \hfill & ={\displaystyle {\sum }_{l\in s_r}{w}_l{d}_l{y}_l}+{\displaystyle {\sum }_{k\in s_m}{w}_k{d}_k{\phi }_{0k}+{\displaystyle {\sum }_{l\in s_r}{y}_l{\displaystyle {\sum }_{k\in s_m}{w}_k{d}_k{\phi }_{lk}}}}\hfill \\ \hfill & =W_{0d}+{\displaystyle {\sum }_{l\in s_r}{w}_l{d}_l{y}_l}+{\displaystyle {\sum }_{l\in s_r}{y}_l}{W}_{dl}\hfill \\ \hfill & =W_{0d}+{\displaystyle {\sum }_{l\in s_r}{y}_l\left(w_l{d}_l+W_{dl}\right)}.\hfill \end{array}$$

The quantities $W_{dl}$ and $W_{0d}$ denote the compensatory weights (or adjustment weights) defined as

$$\begin{array}{ll}{W}_{dl}\hfill & ={\displaystyle \sum _{k\in s_m}{w}_k{d}_k{\phi }_{lk}}\hfill \\ W_{0d}\hfill & ={\displaystyle \sum _{k\in s_m}{w}_k{d}_k{\phi }_{0k}.}\hfill \end{array}$$

They represent the effect of the non-response in the domain, $d,$ carried by the respondent unit, $l\in s_r,$ with a reported value, $y_l.$

2.2  Variance estimation

Consider an imputation model, $\eta ,$ describing the relationship between variable $y$ and the vector of observed auxiliary variables $x^{\text{obs}}.$ Let $E_{\eta }(.),$ ${\mathrm{Var}}_{\eta }(.)$ and ${\mathrm{cov}}_{\eta }(.)$ denote respectively the expectation, the variance, and the covariance with respect to the imputation model $\eta .$ The imputation model is

$$\begin{array}{ll}{E}_{\eta }\left(y_k|X^{\text{obs}}\right)\hfill & ={\mu }_k\hfill \\ V_{\eta }\left(y_k|X^{\text{obs}}\right)\hfill & ={\sigma }_k^2\hfill \\ {\text{cov}}_{\eta }\left(y_k,y_{k^{\prime }}|X^{\text{obs}}\right)\hfill & =0\hfill \end{array}$$

where $k,k^{\prime }\in U$ and $k\ne k^{\prime }.$ The matrix $X^{\text{obs}}$ contains all observed vectors $x^{\text{obs}}\text{}.$ The quantities ${\mu }_k$ and ${\sigma }_k^2$ can be estimated by ${\widehat{\mu }}_k$ and ${\widehat{\sigma }}_k^2$ respectively. We assume that these estimators are unbiased with respect to the imputation model $\eta .$ These estimators will be useful later for estimating the total variance components and the unit decompositions of those components.

The total error of the estimator (2.2) can be expressed as

$${\widehat{t}}_d-t_d=\left({\widehat{t}}_d^0-t_d\right)+\left({\widehat{t}}_d-{\widehat{t}}_d^0\right),(2.3)$$

where ${\widehat{t}}_d^0$ is the estimator under complete response given by (2.1). The first term on the right-hand side of (2.3) is usually referred to as the sampling error and the second term is called the non-response error. As proposed in Särndal (1992) and in Beaumont and Bissonnette (2011), the mean square error of ${\widehat{t}}_d$ using (2.3) can be decomposed in three components and is given by

$$\begin{array}{ll}{E}_{\eta pq}{\left({\widehat{t}}_d-t_d\right)}^2\hfill & =E_{\eta }{V}_p\left({\widehat{t}}_d\right)+E_{pq}{E}_{\eta }\left[{\left({\widehat{t}}_d-{\widehat{t}}_d^0\right)}^2|s,s_r\right]\hfill \\ \hfill & +2E_{pq}{E}_{\eta }\left[\left({\widehat{t}}_d-{\widehat{t}}_d^0\right)\left({\widehat{t}}_d^0-t_d\right)|s,s_r\right],(2.4)\hfill \end{array}$$

under imputation model, $\eta ,$ sampling design, $p,$ and response mechanism, $q.$ $E_{\eta pq}{\left({\widehat{t}}_d-t_d\right)}^2$ is approximately equivalent to the variance $V_{\eta pq}\left({\widehat{t}}_d-t_d\right)$ assuming that the overall bias is negligible. Thus, the equation (2.4) is equivalent to $V_{\eta pq}\left({\widehat{t}}_d-t_d\right)\equiv V_{\text{TOT}}\left({\widehat{t}}_d\right)=V_{\text{SAM}}\left({\widehat{t}}_d\right)+V_{\text{NR}}\left({\widehat{t}}_d\right)+V_{\text{MIX}}\left({\widehat{t}}_d\right),$ where:

• $V_{\text{SAM}}\left({\widehat{t}}_d\right)\equiv E_{\eta }{V}_p\left({\widehat{t}}_d\right)$ is the sampling variance;
• $V_{\text{NR}}\left({\widehat{t}}_d\right)\equiv E_{pq}{E}_{\eta }\left[{\left({\widehat{t}}_d-{\widehat{t}}_d^0\right)}^2|s,s_r\right]$ is the non-response variance;
• $V_{\text{MIX}}\left({\widehat{t}}_d\right)\equiv 2E_{pq}{E}_{\eta }\left[\left({\widehat{t}}_d-{\widehat{t}}_d^0\right)\left({\widehat{t}}_d^0-t_d\right)|s,s_r\right]$ is the covariance between sampling and non-response error terms, also called the mixed variance component.

Beaumont and Bissonnette (2011) proposed the following estimators for $V_{\text{SAM}}\left({\widehat{t}}_d\right),$ $V_{\text{NR}}\left({\widehat{t}}_d\right)$ and  $V_{\text{MIX}}\left({\widehat{t}}_d\right).$

1. ${\widehat{V}}_{\text{SAM}}\left({\widehat{t}}_d\right)={\widehat{V}}_{\text{ORD}}\left({\widehat{t}}_d\right)+{\widehat{V}}_{\text{DIF}}\left({\widehat{t}}_d\right)$ where:
   • ${\widehat{V}}_{\text{ORD}}\left({\widehat{t}}_d\right)$ is the naive sampling variance estimator using the imputed values as though they were reported values.
   • ${\widehat{V}}_{\text{DIF}}\left({\widehat{t}}_d\right)={\displaystyle {\sum }_{k\in s_m}\left(1-{\pi }_k\right)w_k^2{d}_k{\widehat{\sigma }}_k^2}$ is a correction to ${\widehat{V}}_{\text{ORD}}\left({\widehat{t}}_d\right)$ in order to reduce the bias of ${\widehat{V}}_{\text{ORD}}\left({\widehat{t}}_d\right),$ as proposed by Beaumont and Bocci (2009), since the variance component ${\widehat{V}}_{\text{ORD}}\left({\widehat{t}}_d\right)$ relies on the use of imputed values, usually more homogeneous than the reported values.
2. ${\widehat{V}}_{\text{NR}}\left({\widehat{t}}_d\right)={\displaystyle {\sum }_{l\in s_r}{W}_{dl}^2{\widehat{\sigma }}_l^2}+{\displaystyle {\sum }_{k\in s_m}{w}_k^2{d}_k{\widehat{\sigma }}_k^2}$ is the estimator of the non-response component of variance.
3. ${\widehat{V}}_{\text{MIX}}\left({\widehat{t}}_d\right)=2{\displaystyle {\sum }_{l\in s_r}{W}_{dl}\left(w_l-1\right)d_l{\widehat{\sigma }}_l^2}-2{\displaystyle {\sum }_{k\in s_m}{w}_k\left(w_k-1\right)d_k{\widehat{\sigma }}_k^2}$ is the estimator of the mixed variance component.

Under complete response, $s_m=\varnothing ,$ the compensation weights are $W_{dl}=0,$ and the variance components, ${\widehat{V}}_{\text{DIF}}\left({\widehat{t}}_d\right),$ ${\widehat{V}}_{\text{NR}}\left({\widehat{t}}_d\right),$ and ${\widehat{V}}_{\text{MIX}}\left({\widehat{t}}_d\right),$ are also equal to 0, leaving the total variance as ${\widehat{V}}_{\text{TOT}}\left({\widehat{t}}_d\right)={\widehat{V}}_{\text{ORD}}\left({\widehat{t}}_d\right).$ Under a census, $s=U,$ the variance components, ${\widehat{V}}_{\text{DIF}}\left({\widehat{t}}_d\right),$ ${\widehat{V}}_{\text{ORD}}\left({\widehat{t}}_d\right),$ and ${\widehat{V}}_{\text{MIX}}\left({\widehat{t}}_d\right),$ are equal to 0, leaving the total variance as ${\widehat{V}}_{\text{TOT}}\left({\widehat{t}}_d\right)={\widehat{V}}_{\text{NR}}\left({\widehat{t}}_d\right).$

2.3  Non-response bias

The reduction of non-response bias is always a desirable goal. It can be achieved through an adaptive design and/or through an appropriate method of dealing with missing values. Our framework assumes that the non-response bias is removed through imputation methods that use relevant auxiliary information. In practice, it is likely that imputation will only reduce non-response bias, not eliminate it. We may then wonder whether adaptive designs could be used to reduce further the bias. In the context of non-response weighting, Beaumont, Bocci and Haziza (2014) argued that auxiliary information used in an adaptive design to reduce non-response bias can also be used in non-response weighting to reduce the same amount of bias. Their argument can also be made in the context of imputation. This justifies our focus on variance reduction rather than bias reduction. We acknowledge that some bias may remain after imputation but ignore this bias because it may not be possible to reduce it further through an adaptive design without the availability of additional auxiliary information. However, it is possible to reduce the variance through an adaptive design. 

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