Criteria for choosing between calibration weighting and survey weighting
Section 3. Proposed criterion for measuring the impact of using calibration weights

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Calibration weights are used to improve the precision of estimates for survey parameters of interest. This improvement depends largely on how strongly the variable of interest is linked to the calibration variables. To assess the impact of using calibration weights, we can compare the AMSE for estimators ${\widehat{t}}_{yC}$ and ${\widehat{t}}_{y\pi }$ given respectively by (2.5) and (2.10). The impact of using calibration weights can then be measured through the following criterion:

$$\text{Weff}=\frac{{\displaystyle {\sum }_{k\in U}{\sigma }_k^2\left[\frac{V_k}{d_k}+R_k^2\left(d_k-1\right)+{\left(R_k-1\right)}^2\right]}}{V_{\text{Approx}}+{\displaystyle {\sum }_{k\in U}{\sigma }_k^2{d}_k\left(1-{\pi }_k\right)}}(3.1)$$

where calibration weights are chosen in cases where the Weff value is less than 1. Note that the Weff expression (3.1) depends on the population and must be estimated. Furthermore, for any $k\in U,$ $V_k$ represents the variance of calibration weight $w_{kS\text{},C},$ considering the $s$ set of samples containing unit $k.$ Variance $V_k$ is generally not zero since the $w_{kS\text{},C}$ weights depend on the calibration variables and the $s$ sample selected. In order to take variance $V_k$ into account in measuring the impact of using calibration weights $w_{kS\text{},C}\text{},$ we propose estimating the quantity

$$V_w={\displaystyle \sum _{k\in U}{\sigma }_k^2\frac{V_k}{d_k}}(3.2)$$

by

$${\widehat{V}}_w={\displaystyle \sum _{k\in S}{\widehat{\sigma }}_k^2{\left(w_{kS\text{},C}-d_k\right)}^2}(3.3)$$

where ${\widehat{\sigma }}_k^2$ is the White estimator for ${\sigma }_k^2$ defined by $n{\widehat{\epsilon }}_k^2/\left(n-p\right)$ with ${\widehat{\epsilon }}_k=Y_k-x_k^{\prime }\widehat{\beta }.$ The estimator (3.3) is obtained by replacing $V_k$ by ${\left(w_{kS\text{},C}-d_k\right)}^2,$ which can be viewed as a first-order approximation of $V_k.$ For any unit $k\in U,$ the use of calibration produces weight $w_{kS\text{},C},$ which varies from one sample to another, but for which the design-based expectation can be approximated by sampling weight $d_k.$ The simulations discussed in Section 4 show that ${\widehat{V}}_w$ is a good $V_w$ estimator since it helps to deduct an effective estimator of the Weff criterion. The Weff criterion that we propose for choosing between calibration weights $w_{kS\text{},C}$ and sampling weights $d_k$ can be estimated by

$${\widehat{\text{Weff}}}_S=\frac{{\displaystyle {\sum }_{k\in S}{d}_k{\widehat{\sigma }}_k^2\left[\frac{{\left(w_{kS\text{},C}-d_k\right)}^2}{d_k}+{\widehat{R}}_{kS}^2\left(d_k-1\right)+{\left({\widehat{R}}_{kS}-1\right)}^2\right]}}{{\widehat{V}}_{\text{Approx}\text{},S}+{\displaystyle {\sum }_{k\in S}{d}_k{\widehat{\sigma }}_k^2\left(d_k-1\right)}}(3.4)$$

where ${\widehat{R}}_{kS}=w_{kS}/d_k$ and ${\widehat{V}}_{\text{Approx}\text{},S}$ is an estimator for ${\mathrm{var}}_p\left({\displaystyle {\sum }_{k\in S}{d}_k{x}_k^{\prime }\beta }\right)$ resulting from the approximation (2.8). It is produced by:

$${\widehat{V}}_{\text{Approx}\text{},S}={\displaystyle \sum _{k\in S}{\tilde{c}}_k{\left(d_k{x}_k^{\prime }\widehat{\beta }\right)}^2}-\frac{1}{\widehat{h}}{\left({\displaystyle \sum _{k\in S}{\tilde{c}}_k{d}_k{x}_k^{\prime }\widehat{\beta }}\right)}^2(3.5)$$

with ${\tilde{c}}_k=n\left(1-{\pi }_k\right)/\left(n-1\right)$ and $\widehat{h}={\displaystyle {\sum }_{k\in S}{\tilde{c}}_k}.$ The proposed ${\widehat{\text{Weff}}}_S$ criterion has the benefit of considering bias due to the use of calibration weights, through ${\widehat{R}}_{kS},$ as well as the quality of the linear regression model representing the link between the variable of interest and the calibration variables, through variance ${\widehat{\sigma }}_k^2.$ For some survey designs, the weighting traditionally used for estimates effectively leads to an unbiased estimator for the design, but it is not necessarily the HT estimator. This is the case, for example, with a two-stage design where the second stage design depends on the sample from the first stage and the weighting used is the product of the sampling weights for each stage. It is important to note that the ${\widehat{\text{Weff}}}_S$ criterion proposed in this paper is not linked to the HT estimator, since it enables us to compare the calibration estimator with any other estimator using the sampling weights once it is unbiased.

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