Model-assisted calibration of non-probability sample survey data using adaptive LASSO
Section 2. Calibration

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2.1  Traditional calibration

For an analytical sample $s_A$ (the sample which requires weight calibration) of size $n$ drawn from sample design $\mathcal{A}$ with design weights $\underset{n\times 1}{d},$ and the diagonal matrix of design weights $D,$ calibrated weights $\underset{n\times 1}{w}$ minimize a distance measure

$$E_{\mathcal{A}}\left[{\displaystyle \sum _{i\in s_A}}g\mathrm{<mo\; stretchy="false">(</mo>}{w}_i\mathrm{,}{d}_i\mathrm{<mo\; stretchy="false">)</mo>}/q_i\right](2.1)$$

under the constraint:

$${\displaystyle \sum _{i\in s_A}}{w}_i{x}_i^T\mathrm{<mo>=</mo>}{T}^X(2.2)$$

where $E_{\mathcal{A}}$ is expectation with respect to the analytic (probability) design, $g\left(w_i\mathrm{,}{d}_i\right)$ is a differentiable function with respect to $w_i,$ strictly convex on an interval containing $d_i,$ and $g\left(d_i\mathrm{,}{d}_i\right)\mathrm{<mo>=</mo>\; 0},$ and where $T^X$ is a row vector of known population totals of sample calibration variables $X$ (Deville and Särndal, 1992). The constant $q_i$ is independent of design weight $d_i.$ The commonly used generalized regression (GREG) estimator uses the chi-square distance: $g\left(w_i\mathrm{,}{d}_i\right)\mathrm{<mo>=</mo>}{\left(w_i-d_i\right)}^2/d_i$ with $q_i\mathrm{<mo>=</mo>\; 1.}$ Under this distance measure:

$$w^{\text{GREG}}=d+DX{\left(X^{T}DX\right)}^{-1}{\left(T^X-d^{T}X\right)}^T\text{}\mathrm{.}(2.3)$$

The estimate of population total of outcome $y$ is based on calibrated weights:

$$\begin{array}{ll}{\widehat{T}}_y^{\text{GREG}}\hfill & =w^{\left(\text{GREG}\right)T}y\hfill \\ \hfill & =d^{T}y+\left(T^X-d^{T}X\right){\left(X^{T}DX\right)}^{-1}{X}^{T}Dy\hfill \\ \hfill & ={\widehat{T}}_y^{\text{HT}}+\left(T^X-d^{T}X\right)\widehat{\beta }(2.4)\hfill \end{array}$$

where ${\widehat{T}}_y^{\text{HT}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s_A}}{d}_i{y}_i$ is the standard (weighted) design-based estimator, $\widehat{\beta }\mathrm{<mo>=</mo>}{\left(X^{T}DX\right)}^{-1}{X}^{T}Dy$ is the weighted least squares estimate of the linear regression $E_{\xi }\left[y_i|x_i\mathrm{,}\beta \right]\mathrm{<mo>=</mo>}{x}_i^T\beta ,$ given weights $D.$ (This corresponds to the poststratified estimator when $X$ consists entirely of cell totals for categorical variables.) The calibrated weights defined in equation (2.3) do not rely on any outcome variable. Thus the same set of weights can be applied to all variables in the survey. Note that GREG assumes a working model that is linear. Although ${\widehat{T}}_y^{\text{GREG}}$ is asymptotically design-unbiased for $T_y,$ when the relationship between $y$ and $X$ is non-linear, such as in the case when $y$ is binary, the design variance of ${\widehat{T}}_y^{\text{GREG}}$ can be larger than the design variance ${\widehat{T}}_y^{\text{HT}}\text{}.$

2.2  Model-assisted calibration

Model-assisted calibration estimators can have significant advantage over ${\widehat{T}}_y^{\text{GREG}}$ because model-assisted calibration allows for non-linear models to assist in the construction of calibrated weights.  In model-assisted calibration, we assume a relationship between an outcome $y$ and $X$ through first two moments (Wu and Sitter, 2001):

$$E_{\xi }\left(y_i|x_i\right)\mathrm{<mo>=</mo>}\mu \left(x_i\mathrm{,}\beta \right)\mathrm{,}{V}_{\xi }\left(y_i|x_i\right)\mathrm{<mo>=</mo>}{\nu }_i^2{\sigma }^2(2.5)$$

where $\beta \mathrm{<mo>=</mo>}{\left({\beta }_1\mathrm{,}\dots \mathrm{,}{\beta }_p\right)}^T$ and $\sigma$ are unknown superpopulation parameters, $\mu \left(x_i\mathrm{,}\beta \right)$ is a known function of $x_i$ and $\beta \text{,}$ and ${\nu }_i$ is a known function of $x_i$ or $\mu \left(x_i\mathrm{,}\beta \right).$ $E_{\xi }$ and $V_{\xi }$ are expectation and variance with respect to the model $\xi .$ Let $B$ be the finite population (or census) estimate of $\beta$ (i.e., the quasilikelihood estimator of $\beta$ based on the entire finite population), and ${\widehat{\mu }}_i\mathrm{<mo>=</mo>}\mu \left(x_i\mathrm{,}\widehat{B}\right),$ where $\widehat{B}$ is the sample estimate of $B.$ The model-assisted calibrated weights $w$ then minimize a distance measure $E_{\mathcal{A}}\left[{\displaystyle {\sum }_{i\in s_A}}g\left(w_i\mathrm{,}{d}_i\right)/q_i\right]$ under the constraints ${\displaystyle {\sum }_{i\in s_A}}{w}_i\mathrm{<mo>=</mo>}N$ and ${\displaystyle {\sum }_{i\in s_A}}{w}_i{\widehat{\mu }}_i\mathrm{<mo>=</mo>}{\displaystyle {\sum }_i^N}{\widehat{\mu }}_i.$ The main conceptual difference between traditional calibration and model-assisted calibration is that in model-assisted calibration, the constraints are based on two quantities: (1) population size, and (2) population total of predicted values ${\widehat{\mu }}_i.$ In traditional calibration, the constraint is a vector of population totals of $X$ (see equation (2.2)). Under chi-square distance measure with $q_i\mathrm{<mo>=</mo>\; 1},$ the model-assisted calibrated weights are:

$$w^{\text{MC}}\mathrm{<mo>=</mo>}d+DM{\left(M^{T}DM\right)}^{-1}{\left(T^M-d^{T}M\right)}^T(2.6)$$

where $T^M\mathrm{<mo>=</mo>}\left[N\mathrm{,}{\displaystyle {\sum }_i^N}{\widehat{\mu }}_i\right]$ and $M\mathrm{<mo>=</mo>}\left[d\mathrm{,}{\left({\widehat{\mu }}_i\right)}_{i\in s_A}\right].$ (In the non-probability setting the vector of design weights $d$ can be replaced with $\mathrm{<mo\; stretchy="false">(</mo>}N/n\mathrm{<mo\; stretchy="false">)</mo>}1.)$ The estimate for the population total based on model-assisted calibrated weights is then:

$$\begin{array}{ll}{\widehat{T}}_y^{\text{MC}}\hfill & \mathrm{<mo>=</mo>}{\left(w^{\text{MC}}\right)}^{T}y\hfill \\ \hfill & \mathrm{<mo>=</mo>}{d}^{T}y+\left(T^M-d^{T}M\right){\left(X^{T}DX\right)}^{-1}{X}^{T}Dy\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\widehat{T}}_y^{\text{HT}}+\left({\displaystyle \sum _i^N}{\widehat{\mu }}_i-{\displaystyle \sum _{i\in s_A}}{d}_i{\widehat{\mu }}_i\right){\widehat{B}}^{\text{MC}}(2.7)\hfill \end{array}$$

where ${\widehat{B}}^{\text{MC}}$ is the calibration slope to satisfy the calibration constraints (different from the model parameter estimates $\widehat{B}):$

$${\widehat{B}}^{\text{MC}}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\in s_A}{d}_i\left({\widehat{\mu }}_i-\widehat{\overline{\mu }}\right)\left(y_i-\overline{y}\right)}}{{\displaystyle {\sum }_{i\in s_A}{d}_i{\left({\widehat{\mu }}_i-\widehat{\overline{\mu }}\right)}^2}}\mathrm{,}\widehat{\overline{\mu }}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in s_A}}{d}_i{\widehat{\mu }}_i/{\displaystyle \sum _{i\in s_A}}{d}_i\text{}\mathrm{,}\overline{y}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in s_A}}{d}_i{y}_i/{\displaystyle \sum _{i\in s_A}}{d}_i\mathrm{.}$$

Unbiasedness and small variances of ${\widehat{T}}_y^{\text{MC}}$ both rely on how well the ${\widehat{\mu }}_i$ approximates the true expected value of $y_i.$

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