Variance estimation under monotone non-response for a panel survey
Section 7. Conclusion

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In this paper, we considered variance estimation accounting for weighting adjustments in panel surveys. We proposed both an approximately unbiased variance estimator and a simplified variance estimator for estimators of totals, complex parameters and measures of change, which covers most cases that may be encountered in practice. Our simulation results indicate that the proposed variance estimator performs well in all cases considered. The simplified variance estimator tends to overestimate the variance of the expansion estimator for totals, and to overestimate the variance for calibrated estimators of totals when the calibration variables lack of explanatory power for the variable of interest. However, the simplified variance estimator performs well for the estimation of ratios and change in totals with calibrated weights, even if the calibration model is not appropriate for the study variable.

The assumption of independent response behaviour is usually not tenable for multi-stage surveys, since units within clusters tend to be correlated with respect to the response behaviour. In this context, estimation of response probabilities based upon conditional logistic regression in the context of correlated responses has been studied by Skinner and D’Arrigo (2011), see also Kim, Kwon and Park (2016). Extending the present work in the context of correlated response behaviour is a challenging problem for further research.

Acknowledgements

We thank the Editors, an Associate Editor and the referees for useful comments and suggestions which led to an improvement of the paper.

Appendix

Estimation of the variance due to non-response for Response Homogeneity Groups

We consider the model of Response Homogeneity Groups introduced in Section 2.5. Recall that this model may be summarized as follows: at each time $\delta \mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}t,$ the sub-sample $s_{\delta -1}$ is partitioned into $C\left(\delta -1\right)$ groups $s_{\delta -1}^c\mathrm{,}c\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}C\left(\delta -1\right).$ The response probabilities are assumed to be constant within the groups.

This model is equivalent to the logistic regression model in (2.18), with

$$z_i^{\delta }={\left[1\left\{i\in s_{\delta -1}^1\right\}\mathrm{,}\dots \mathrm{,}1\left\{i\in s_{\delta -1}^{C\left(\delta -1\right)}\right\}\right]}^{\top }\text{}\mathrm{.}(\text{A}.1)$$

The equation (2.2) leads to the estimated response probabilities

$${\widehat{p}}_i^{\delta }\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\in s_{\delta -1}^c}{k}_i^{\delta }{r}_i^{\delta }}}{{\displaystyle {\sum }_{i\in s_{\delta -1}^c}{k}_i^{\delta }}}\text{for}i\in s_{\delta -1}^c\mathrm{.}(\text{A}.2)$$

We first consider the case when the reweighted estimator is computed at time $t\mathrm{<mo>=</mo>\; 1.}$ In the estimator of the variance due to non-response given in (2.21), the vector ${\widehat{\gamma }}_1^1$ simplifies as

$${\widehat{\gamma }}_1^1={\left(\frac{{\displaystyle {\sum }_{i\in s_1\cap s_0^1}\frac{y_{i1}}{{\pi }_i}}}{{\widehat{p}}_1^1{\displaystyle {\sum }_{i\in s_1\cap s_0^1}{k}_i^1}}\mathrm{,}\dots \mathrm{,}\frac{{\displaystyle {\sum }_{i\in s_1\cap s_0^{C\left(0\right)}}\frac{y_{i1}}{{\pi }_i}}}{{\widehat{p}}_{C\left(0\right)}^1{\displaystyle {\sum }_{i\in s_1\cap s_0^{C\left(0\right)}}{k}_i^1}}\right)}^{\top }\mathrm{.}(\text{A}.3)$$

After some algebra, the variance estimator in (2.21) may be rewritten as

$${\widehat{V}}_1^{\text{nr}}\left({\widehat{Y}}_1\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{c\mathrm{<mo>=</mo>\; 1}}^{C\left(0\right)}}\frac{\left(1-{\widehat{p}}_c^1\right)}{{\left({\widehat{p}}_c^1\right)}^2}{\displaystyle \sum _{i\in s_1\cap s_0^c}}{\left(\frac{y_{i1}}{{\pi }_i}-k_i^1\frac{{\displaystyle {\sum }_{j\in s_1\cap s_0^c}\frac{y_{j1}}{{\pi }_j}}}{{\displaystyle {\sum }_{j\in s_1\cap s_0^c}{k}_j^1}}\right)}^2\mathrm{.}(\text{A}.4)$$

We now consider the case when the reweighted estimator is computed at time $t\mathrm{<mo>=</mo>\; 2.}$ We focus on the simpler case when the same system of RHGs is kept over time. In the estimator of the variance due to non-response given in (2.22), the vectors ${\widehat{\gamma }}_2^1$ and ${\widehat{\gamma }}_2^2$ simplify as

$${\widehat{\gamma }}_2^1\mathrm{<mo>=</mo>}{\left(\frac{{\displaystyle {\sum }_{i\in s_2\cap s_1^1}\frac{y_{i2}}{{\pi }_i}}}{{\widehat{p}}_1^1{\displaystyle {\sum }_{i\in s_2\cap s_1^1}{k}_i^1}}\mathrm{,}\dots \mathrm{,}\frac{{\displaystyle {\sum }_{i\in s_2\cap s_1^{C\left(0\right)}}\frac{y_{i2}}{{\pi }_i}}}{{\widehat{p}}_{C\left(0\right)}^1{\displaystyle {\sum }_{i\in s_2\cap s_1^{C\left(0\right)}}{k}_i^1}}\right)}^{\top }\text{}\mathrm{,}(\text{A}.5)$$

$${\widehat{\gamma }}_2^2\mathrm{<mo>=</mo>}{\left(\frac{{\displaystyle {\sum }_{i\in s_2\cap s_1^1}\frac{y_{i2}}{{\pi }_i}}}{{\widehat{p}}_1^1{\widehat{p}}_1^2{\displaystyle {\sum }_{i\in s_2\cap s_1^1}{k}_i^2}}\mathrm{,}\dots \mathrm{,}\frac{{\displaystyle {\sum }_{i\in s_2\cap s_1^{C\left(0\right)}}\frac{y_{i2}}{{\pi }_i}}}{{\widehat{p}}_{C\left(0\right)}^1{\widehat{p}}_{C\left(0\right)}^2{\displaystyle {\sum }_{i\in s_2\cap s_1^{C\left(0\right)}}{k}_i^2}}\right)}^{\top }\mathrm{.}(\text{A}.6)$$

After some algebra, the variance estimator in (2.22) may be rewritten as

$$\begin{array}{ll}{\widehat{V}}_2^{\text{nr}}\left({\widehat{Y}}_2\right)\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{c\mathrm{<mo>=</mo>\; 1}}^{C\left(0\right)}}\frac{\left(1-{\widehat{p}}_c^1\right)}{{\widehat{p}}_c^2}{\displaystyle \sum _{i\in s_2\cap s_1^c}}{\left(\frac{y_{i2}}{{\pi }_i{\widehat{p}}_c^1}-k_i^1\frac{{\displaystyle {\sum }_{j\in s_2\cap s_1^c}\frac{y_{j2}}{{\pi }_j}}}{{\displaystyle {\sum }_{j\in s_2\cap s_1^c}{k}_j^1}}\right)}^2\hfill \\ \hfill & +{\displaystyle \sum _{c\mathrm{<mo>=</mo>\; 1}}^{C\left(0\right)}}\left(1-{\widehat{p}}_c^2\right){\displaystyle \sum _{i\in s_2\cap s_1^c}}{\left(\frac{y_{i2}}{{\pi }_i{\widehat{p}}_c^1{\widehat{p}}_c^2}-k_i^2\frac{{\displaystyle {\sum }_{j\in s_2\cap s_1^c}\frac{y_{j2}}{{\pi }_j}}}{{\displaystyle {\sum }_{j\in s_2\cap s_1^c}{k}_j^2}}\right)}^2\text{}\mathrm{.}(\text{A}.7)\hfill \end{array}$$

If we further assume that $k_i^{\delta }$ is constant over times $\delta \mathrm{<mo>=</mo>\; 1,}2,$ and may thus be rewritten as $k_i,$ the expression in (A.7) simplifies as

$${\widehat{V}}_2^{\text{nr}}\left({\widehat{Y}}_2\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{c\mathrm{<mo>=</mo>\; 1}}^{C\left(0\right)}}\frac{\left(1-{\widehat{p}}_c^{1\to 2}\right)}{{\left({\widehat{p}}_c^{1\to 2}\right)}^2}{\displaystyle \sum _{i\in s_2\cap s_1^c}}{\left(\frac{y_{i2}}{{\pi }_i}-k_i\frac{{\displaystyle {\sum }_{j\in s_2\cap s_1^c}\frac{y_{j2}}{{\pi }_j}}}{{\displaystyle {\sum }_{j\in s_2\cap s_1^c}{k}_j}}\right)}^2\mathrm{.}(\text{A}.8)$$

with ${\widehat{p}}_c^{1\to 2}\mathrm{<mo>=</mo>}{\displaystyle {\prod }_{\delta \mathrm{<mo>=</mo>\; 1}}^2}{\widehat{p}}_c^{\delta }$ for $c\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}C\left(0\right).$ This simplification of the variance estimator can be extended to the reweighted estimator at time $t.$ Assuming that the RHGs are kept over time, and that $k_i^{\delta }\mathrm{<mo>=</mo>}{k}_i$ for any $\delta \mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}t,$ the variance estimator in (2.12) may be written as

$${\widehat{V}}_t^{\text{nr}}\left({\widehat{Y}}_t\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{c\mathrm{<mo>=</mo>\; 1}}^{C\left(0\right)}}\frac{\left(1-{\widehat{p}}_c^{1\to t}\right)}{{\left({\widehat{p}}_c^{1\to t}\right)}^2}{\displaystyle \sum _{i\in s_t\cap s_{t-1}^c}}{\left(\frac{y_{it}}{{\pi }_i}-k_i\frac{{\displaystyle {\sum }_{j\in s_t\cap s_{t-1}^c}\frac{y_{jt}}{{\pi }_j}}}{{\displaystyle {\sum }_{j\in s_t\cap s_{t-1}^c}{k}_j}}\right)}^2(\text{A}.9)$$

with ${\widehat{p}}_c^{1\to t}\mathrm{<mo>=</mo>}{\displaystyle {\prod }_{\delta \mathrm{<mo>=</mo>\; 1}}^t}{\widehat{p}}_c^{\delta }$ for $c\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}C\left(0\right).$

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