A generalization of inverse probability weighting
Section 7. Simulation results

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In this simulation, estimators from the preceding section will be compared to the ordinary inverse probability estimator and the ordinary calibrated estimator. A population of 2,000 individuals grouped into 1,000 two-person households was generated. Persons $2i$  and $2i-1$  for $i=1,2,\dots ,\text{1,000}$  belong to the same household. A variable of interest $y$  takes the value 1 to represent a vaccinated person, and it takes the value 0 to represent an unvaccinated person. To simulate how vaccination status can be correlated within household, the method of Lunn and Davies (1998) was used to generate pairs of correlated Bernoulli variables with a probability of 0.7 of a value of 1 and a correlation of 0.8. The actual population generated has 254 households where neither person is vaccinated, 660 households where both are vaccinated, 44 households where only the person with an odd label is vaccinated, and 42 households where only the person with an even label is vaccinated. The total number of persons vaccinated is $660\times 2+44+42=\text{1,406}$  for a vaccination rate of 0.703. The correlation between persons of the same household is $\left(\text{0}\text{.66}-\text{0}\text{.704}\times \text{0}\text{.702}\right)/\left(\sqrt{\text{0}\text{.704}\times \text{0}\text{.296}}\times \sqrt{\text{0}\text{.702}\times \text{0}\text{.298}}\right)=\text{0}\text{.7941}\text{.}$

The population was sampled 10,000 times. Each household $i\left(i=1,2,\dots ,N_p\right)$ was sampled independently; the probability of selecting both units was 0.05, the probability of selecting only unit $2i-1$ was 0.10, and the probability of selecting only unit $2i$ was 0.05. Thus, for each sample, the probabilities of inclusion were ${\pi }_{2i-1}=\text{0}\text{.15,}$ ${\pi }_{2i}=\text{0}\text{.1}$ and ${\pi }_{2i-12i}=\text{0}\text{.05}\text{.}$ This means ${\pi }_{\text{diff}i}=\left({\pi }_{2i}-{\pi }_{2i-1}\right)/{\pi }_{2i-12i}$ in (1.2) was chosen to not be zero. This is because when ${\pi }_{\text{diff}i}$ is zero, ${\widehat{\theta }}_{\text{LIM}}$ is a somewhat obvious choice: it is an inverse probability estimator based on pairs where the pair is given a value of $2y_{2i-1}$ if only unit $2i-1$ is sampled, a value of $2y_{2i}$ if only unit $2i$ is sampled, and a value of $y_{2i-1}+y_{2i}$ if both units are sampled. Combined with calibration, it is an obvious competitor to the ordinary calibration estimator. Why not base the estimator on pairs in this way, rather than units, if there is a strong correlation between units of a pair? When ${\pi }_{\text{diff}i}$ is zero, it is also true that ${\widehat{\theta }}_{\text{LIM}}={\widehat{\theta }}_{\text{ALIM}}.$ It is interesting to find out how ${\widehat{\theta }}_{\text{LIM}}$ compares when ${\pi }_{\text{diff}i}$ is not zero. For each sample, eight estimators of the total were calculated: the inverse probability estimator, the ordinary calibrated estimator, the generalization of the inverse probability estimator and its calibrated version, the modified generalized inverse probability estimator and its calibrated version, and finally the alternative estimator and its calibrated version. For the generalized and modified generalized estimators, including their calibrated versions, $\Sigma \left(\rho \right)$ with $\rho \to 1$ was used, as explained in the examples of the preceding section. The simple closed-form formulae of that section can thus be used. For the calibration, $X=1_{N\times 1}$ with $T=1$ and $U=I_{N\times N}$ yields $\widehat{y}=X\widehat{\beta }=\frac{{\displaystyle {\sum }_{k\in s}{y}_k}}{{\displaystyle {\sum }_{k\in s}{x}_k}}{1}_{N\times 1}.$ The average total and the variance over the 10,000 repetitions are given in Table 7.1.

All eight estimators are either unbiased or asymptotically unbiased, so as expected, the observed bias of each estimator is negligible, since the real population total is 1,406.

The observed variances show that only the four calibrated estimators have reasonable variances. With the sampling plan used for this simulation, only the calibrated estimators can estimate the known population total with zero variance.

The calibrated generalized inverse probability estimator, with a variance of 3,419, performs best. This despite being calculated assuming that the correlation between the units of a pair is one. It should be remembered that the calibrated inverse probability estimator, with a variance of 3,856, is a special case of the calibrated generalized inverse probability estimator, but it is computed assuming that the correlation between the units of a pair is zero. The calibrated alternative estimator, which contrary to the other estimators, has been defined only for a household size of 2, has a variance somewhere in between that of the calibrated versions of the inverse probability and generalized inverse probability estimators. Finally, the calibrated modified generalized estimator had the highest variance of the four calibrated estimators.

The generalized Godambe-Joshi lower bound with the variance matrix, $V_{\xi }\left(y\right),$ of the model $\xi$ used to generate $y$ is 3,408. This is the asymptotic variance that could be expected of the calibrated generalized estimator, if it had been calculated with a matrix $\Sigma =V_{\xi }\left(y\right)$ based on the correct model $\xi ,$ where the correlation between units of a pair is 0.8. If $\Sigma \left(\rho \right)$ is defined as in the preceding section, and $\text{GJ}\left(\Sigma \left(\rho \right)\right)$ is the generalized Godambe-Joshi lower bound for the positive definite variance matrix $V_{\xi }\left(y\right)=\Sigma \left(\rho \right),$ then the limit as $\rho \to 1$ of $\text{GJ}\left(\Sigma \left(\rho \right)\right)$ is 3,360. This is the variance that could be expected of the generalized calibration estimator, if the correlation between units of a same pair was one.

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