A design effect measure for calibration weighting in single-stage samples 3. Proposed design-effect measureA design effect measure for calibration weighting in single-stage samples 3. Proposed design-effect measure

• Previous
• Next

We extend Spencer’s (2000) approach in single-stage sampling to produce a new weighting design effect for a calibration estimator. While Spencer’s assumed $y_i=\alpha +\beta p_i+{\epsilon }_i,$ we model $y_i$ as $y_i=\alpha +x_i^T\beta +{\epsilon }_i={\dot{x}}_i^T\dot{\beta }+{\epsilon }_i,$ where ${\dot{x}}_i=\left[\begin{array}{cc}1& x_i\end{array}\right]$ and $\dot{\beta }=\left[\begin{array}{cc}\alpha & \beta \end{array}\right].$ Denote the full finite population estimators of $\alpha$ and $\beta$ by $A=\overline{Y}-\overline{X}B$ and $B={\left(X^{T}X\right)}^{-1}{X}^{T}Y$ where $X$ is the $N\times p$ matrix of auxiliaries for the $N$ units in the finite population and $Y$ is the $N-$ vector of $y$ values. The finite population residuals are defined as $e_i=y_i-\left(A+x_i^{T}B\right)\equiv y_i-{\dot{x}}_i^T\dot{B}$ where $\dot{B}=\left[\begin{array}{cc}A& B\end{array}\right].$

Producing the design effect proposed below involves four steps: (1) constructing a linear approximation to the GREG estimator; (2) obtaining the design-variance of this linear approximation; (3) substituting model-based components into the GREG variance; and (4) taking the ratio of this model-assisted variance to the variance of the $\text{pwr}-$ estimator of the total under $\text{srswr}.$ Since steps $\left(1\right)-\left(4\right)$ produce the theoretical design effect for an estimator, we add the final step: (5) plug-in sample-based estimates for each theoretical design effect component.

Step 1. A linearization of the GREG estimator (Expression 6.6.9 in Särndal et al. 1992) is

$$\begin{array}{ll}{\widehat{T}}_{\text{GREG}}\hfill & \doteq {\widehat{T}}_{\text{HT}y}+{\left(T_x-{\widehat{T}}_{\text{HT}x}\right)}^T\dot{B}\hfill \\ \hfill & =T_x^T\dot{B}+{\displaystyle {\sum }_{i\in s}{e}_i/{\pi }_i}\hfill \end{array}(3.1)$$

where ${\displaystyle {\sum }_{i\in s}{e}_i}/{\pi }_i$ is the HT estimator of the population total of the $e_i,$ $E_U={\displaystyle {\sum }_{i\in U}{e}_i}.$ To obtain a simple variance formula in step 2, we treat the case of with-replacement sampling and replace ${\displaystyle {\sum }_{i\in s}{e}_i}/{\pi }_i$ with the $\text{pwr}-$ estimator $n^{-1}{\displaystyle {\sum }_{i=1}^n{e}_i}/p_i.$ Next, define ${\delta }_i$ to be the number of times that unit $i$ is selected for the sample. Since $E_{\pi }\left({\delta }_i\right)=n{p}_i,$ the second component in (3.1) has design-expectation $E_{\pi }\left(n^{-1}{\displaystyle {\sum }_{i=1}^n{e}_i}/p_i\right)=E_U.$

Step 2. From step 1 with the assumption of with-replacement sampling, ${\widehat{T}}_{\text{GREG}}-T_x^T\dot{B}\doteq n^{-1}{\displaystyle {\sum }_{i=1}^n{e}_i}/p_i,$ with design-variance

$$\begin{array}{ll}{\text{Var}}_{\pi }\left({\widehat{T}}_{\text{GREG}}-T_x^T{\dot{B}}_U\right)\hfill & \doteq {\text{Var}}_{\pi }\left(n^{-1}{\displaystyle {\sum }_{i=1}^n{e}_i}/p_i\right)\hfill \\ \hfill & =n^{-1}{\displaystyle {\sum }_{i\in U}{p}_i{\left(e_i/p_i-E_U\right)}^2}.\hfill \end{array}(3.2)$$

Steps 3 and 4. We follow Spencer’s approach and substitute model values in variance (3.2) to formulate a design-effect measure. However, we substitute in the model-based equivalent to $e_i,$ not $y_i.$ Substituting the GREG residuals $e_i$ into the variance and taking its ratio to the variance of the $\text{pwr}-$ estimator in simple random sampling with replacement, ${\text{Var}}_{\text{srswr}}\left({\widehat{T}}_{\text{srswr}}\right)=N^2{\sigma }_y^2/n,$ where ${\sigma }_y^2=N^{-1}{\displaystyle {\sum }_{i=1}^N{\left(y_i-\overline{Y}\right)}^2},$ will produce our approximate design effect due to unequal calibration weighting. We can simplify things greatly by defining $u_i=A+e_i,$ where $u_i=y_i-x_i^{T}B,$ which implies $\overline{U}=A+{\overline{E}}_U=A.$ The resulting design effect (see Appendix) is

$${\text{Deff}}_H=\frac{n\overline{W}}{N}\left(\frac{{\sigma }_u^2}{{\sigma }_y^2}\right)+\frac{n{\sigma }_w}{N{\sigma }_y^2}\left({\rho }_{u^{2}w}{\sigma }_{u^2}-2A{\rho }_{uw}{\sigma }_u\right)(3.3)$$

where ${\sigma }_u^2=N^{-1}{\displaystyle {\sum }_{i=1}^N{\left(u_i-\overline{U}\right)}^2},{\sigma }_y^2=N^{-1}{\displaystyle {\sum }_{i=1}^N{\left(y_i-\overline{Y}\right)}^2},{\rho }_{u^{2}w}$ is the finite population correlation between $u_i^2$ and $w_i,{\sigma }_{u^2}^2$ is the variance of $u_i^2$ and ${\rho }_{uw}$ is the correlation between $u_i$ and $w_i.$

The first component in (3.3) is $O\left(1\right);$ the factor $n\overline{W}/N$ is related to the Kish $\text{deff}$ as described below. The factor ${\sigma }_u^2/{\sigma }_y^2$ is an adjustment based on the effectiveness of the covariates in predicting $y.$ The second component in (3.3) is $O\left(n/N\right)$ and incorporates terms related to the strength of the relationship between the calibration covariates and the weights.

Note that the derivation of (3.3) assumes with-replacement (WR) sampling was used. Although without replacement (WOR) sampling is more common in practice, the WOR variance of an estimated total is complicated since it involves joint selection probabilities. The WR variance formula is simple enough to provide insights into the effect of calibration on a $\text{deff}\text{.}$ In cases where there are gains in precision from using WOR sampling, an ad hoc finite population correction factor can be incorporated in (3.3), i.e., $\left(1-n/N\right){\text{Deff}}_H.$

Step 5. To estimate (3.3), we use

$${\text{deff}}_H\approx {\text{deff}}_K\left(w\right)\frac{{\widehat{\sigma }}_u^2}{{\widehat{\sigma }}_y^2}+\frac{n{\widehat{\sigma }}_w}{N{\widehat{\sigma }}_y^2}\left({\widehat{\rho }}_{u^{2}w}{\widehat{\sigma }}_{u^2}-2\widehat{\alpha }{\widehat{\rho }}_{uw}{\widehat{\sigma }}_u\right),(3.4)$$

where the model parameter estimate $\widehat{\alpha }$ is obtained using survey-weighted least squares, ${\widehat{\sigma }}_y^2$ was defined in Section 2.3, ${\widehat{\sigma }}_u^2={\displaystyle {\sum }_{i\in s}{w}_i{\left({\widehat{u}}_i-{\overline{u}}_w\right)}^2}/{\displaystyle {\sum }_{i\in s}{w}_i},$ ${\widehat{\overline{u}}}_w={\displaystyle {\sum }_{i\in s}{w}_i{\widehat{u}}_i}/{\displaystyle {\sum }_{i\in s}{w}_i},{\widehat{u}}_i=y_i-x_i^T\widehat{\beta },$ and $\widehat{\beta }={\left(X_s^{T}W{X}_s\right)}^{-1}{X}_s^{T}W{y}_s$ is the survey-weighted least-squares estimate of $\beta ,$ with $W=\text{diag}\left(w_1,\dots ,w_n\right),$ and other terms defined in Section 2.1.

If the correlations in (3.3) are negligible or the sampling fraction $n/N$ is small, the first term dominates and we obtain

$${\text{Deff}}_H\approx \frac{n\overline{W}}{N}\left(\frac{{\sigma }_u^2}{{\sigma }_y^2}\right),$$

which can be estimated with

$${\text{deff}}_H\approx {\text{deff}}_K\left(w\right){\widehat{\sigma }}_u^2/{\widehat{\sigma }}_y^2.(3.5)$$

Note that in samples without calibration weight adjustments, we have ${\widehat{u}}_i=y_i-x_i^T\widehat{\beta }\approx y_i$ and ${\sigma }_u^2\approx {\sigma }_y^2.$ In this case expression (3.5) becomes ${\text{Deff}}_H\approx n\overline{W}/N,$ which we estimate with Kish’s measure ${\text{deff}}_K=1+{\left[\text{CV}\left(w\right)\right]}^2.$ However, when the relationship between the calibration covariates $x$ and $y$ is stronger, the variance ${\sigma }_u^2$ should be smaller than ${\sigma }_y^2.$ In this case, measure (3.5) is smaller than Kish’s estimate. Variable weights produced from calibration adjustments are thus not as “penalized” (shown by overly high design effects) as they would be using the Kish and Spencer measures. However, if we have “ineffective” calibration, or a weak relationship between $x$ and $y,$ then ${\sigma }_u^2$ can be greater than ${\sigma }_y^2,$ producing a design effect greater than one. The Spencer measure only accounts for an indirect relationship between $x$ and $y$ if there was only one $x$ and it was used to produce $p_i.$ This is illustrated in Section 4. We also examine the extent to which the correlation components in our proposed design effect (3.3) are large enough to influence the exact measure. Calculation of (3.3) requires only the sample $y-$ values, covariates, and calibration weights. This measure can, thus, be produced more quickly than measure (2.3), whose components are often available later in data processing after a variance estimation system is set up.

• Previous
• Next