A generalization of inverse probability weighting
Section 4. The choice of the positive definite matrix $\Sigma$

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Different choices for $\Sigma$  will generally lead to different generalized inverse probability estimators and different generalized calibration estimators. The advantage of the generalization of the inverse probability estimator comes from its use in a generalization of calibration, as seen in Section 3, and the optimality of generalized calibration, as discussed in Section 5. It will be seen that a matrix $\Sigma$  is an appropriate choice to use for ${\widehat{\theta }}_{\text{GIP}}\left(\Sigma \right),$  if a model $\xi$  with $V_{\xi }\left(y\right)=\Sigma$  is an appropriate model for $y.$  Even if the assumption that $V_{\xi }\left(y\right)=\Sigma$  is wrong, the estimator ${\widehat{\theta }}_{\text{GIP}}\left(\Sigma \right)$  remains design unbiased and the estimator ${\widehat{\theta }}_{\text{GCAL}}\left(\Sigma \right)$  remains asymptotically design unbiased. The generalized calibration estimators with $\Sigma =V_{\xi }\left(y\right)$  can be said to be model assisted as opposed to model based or model dependent (see Särndal et al., 1992, Section 6.7). The ordinary calibration estimators, ${\widehat{\theta }}_{\text{CAL}}$  use (3.1) with $\Sigma =I.$  A model that fits the population perfectly is not necessary, but hopefully a better model than one with $V_{\xi }\left(y\right)=I$  can be utilized. In fact, if $\Sigma$  is any positive diagonal matrix, then ${\widehat{\theta }}_{\text{GIP}}\left(\Sigma \right)$  will result in the ordinary inverse probability estimator, and the generalized calibration estimator will result in the ordinary calibration estimator. Often, a more appropriate model for $y$  would have $V_{\xi }\left(y\right)$  non-diagonal. As for the variance of ${\widehat{\theta }}_{\text{GIP}}\left(\Sigma \right),$  it may be higher than that of the ordinary inverse probability estimator, even if $V_{\xi }\left(y\right)=\Sigma .$  It is the calibration of ${\widehat{\theta }}_{\text{GIP}}\left(\Sigma \right)$  that yields, as will be seen in Section 5, an optimal estimator.

The use of a block-diagonal matrix simplifies the computation of inverses needed in (2.3). Blocks may correspond to persons of a household, students of a class, workers of an establishment, dwellings of a block, etc. It is often natural for units belonging to the same block to have a correlated variable of interest. For example, how one worker rates their employer is likely correlated with the rating of another worker of the same employer; the race or religion of a couple is often the same. In such cases, a multistage sampling plan would often be used, but it will be assumed here that a single stage plan is used. This could be because a single stage sampling plan was more suitable for other variables of interest of the same survey, or because some unit level characteristics are so important, that it is desirable to stratify at the population level so that the sample can be targeted at certain strata. For example, it may be important to stratify persons by age, but households can’t be stratified by age.

In the simulation presented in this paper, the vaccination status of individuals in two-person households is made to be correlated. An extreme case presents itself if the blocks are persons of a same household and the variable of interest is household income. In such a case the correlation is perfect, and lines of $\Sigma$ corresponding to persons from a same household should be identical. Such a matrix $\Sigma$ is not positive definite, but it is the limit of a sequence of positive definite matrices, and the limit of the corresponding sequence of generalized inverse probability estimators could be computed. The example (1.2) given in the introduction is based on this idea.

If $\Sigma$ is block-diagonal with blocks ${\Sigma }_1,{\Sigma }_2,\dots ,{\Sigma }_B,$ then because both the Moore-Penrose inverse and the ordinary inverse of a block-diagonal matrix is the block-diagonal matrix of inverses, the estimator ${\widehat{\theta }}_{\text{GIP}}\left(\Sigma \right)$ can be decomposed into

$$\begin{array}{ll}{\widehat{\theta }}_{\text{GIP}}\left(\Sigma \right)\hfill & ={\displaystyle \sum _{b=1}^B{\widehat{\theta }}_{\text{GIP}b}\left({\Sigma }_b\right)}\hfill \\ \hfill & ={\displaystyle \sum _{b=1}^B{y}_b^{\prime }{\left({\Delta }_{s_b}{\Sigma }_b{\Delta }_{s_b}\right)}^{†}{\left(E{\left({\Delta }_{s_b}{\Sigma }_b{\Delta }_{s_b}\right)}^{†}\right)}^{-1}{1}_{N_b\times 1}},(4.1)\hfill \end{array}$$

where $N_b$ is the size of block $b,$ $y_b$ and ${\Delta }_{s_b}$ are the sub-vector and sub-matrix respectively, which correspond to block $b.$

If the population is partitioned into blocks of correlated units, the variable defining the blocks must be on the frame. But that variable need not be perfect. For example, a unit’s household may only be known at the time of the survey, but using an outdated household variable available on the frame will still be useful, while not introducing any bias. It simply means that the strength borrowed by the generalized inverse probability estimator from the correlations will be reduced. On the other hand, the strength borrowed from the correlations by the ordinary inverse probability estimator is nil.

If a positive definite estimator ${\widehat{\Sigma }}_s$ converges to a positive definite $\Sigma$ in probability, then the bias and variance of ${\widehat{\theta }}_{\text{GIP}}\left({\widehat{\Sigma }}_s\right)$ are asymptotically the same as those of ${\widehat{\theta }}_{\text{GIP}}\left(\Sigma \right).$ In practice, even if the general form of $\Sigma$ depends on $N\left(N-1\right)/2$ covariances, the number of parameters in $\Sigma$ should be small compared to the sample size. Using the inverse probability estimator means assuming all covariances are zero. When using the generalized inverse probability estimator, one could assume that those covariances depend on a few parameters, and that those parameters are considered fixed, rather than estimated from the sample. In the examples of Section 6, $\Sigma$ depends on only one parameter, $\rho ,$ and its value is assumed to be 1.

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