Register-based sampling for household panels 8. DiscussionRegister-based sampling for household panels 8. Discussion

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Households, due to their instability over time, are inappropriate as sampling units in panels conducted to collect information at the level of households or persons. In this paper, a sample design is proposed where persons are drawn through a self-weighted sample design. At each point in time, the household members of these so-called core persons are included in the sample. This results in a sample where households can be drawn more than once but with a maximum that is equal to the household size. Households are included with expectations proportional to the household size. First and second order inclusion expectations for households are derived under an equal probability sample design for selecting core persons. These inclusion expectations can be used in a similar way to the more common inclusion probabilities in design-based and model-assisted inference.

The sample design in this paper is a special case of indirect sampling (Lavallée 1995, 2007). In the case of a self-weighted sample design it is shown that first and second order inclusion expectations for this sample design can be derived in a relatively straightforward manner from the household composition of the core persons at each point in time. In the case of more complex sample designs, the Generalized Weight Share method (Lavallée 1995, 2007), is required to construct inclusion weights at each point in time.

The advantage of the proposed sample design is that the estimation procedure is simpler than the Generalized Weight Share method. The design is particularly useful if core persons are selected with a self-weighted sampling design. If, due to, e.g., minimum precision and maximum cost requirements, an unequal probability design for the selection of core persons is required, then the Generalized Weight Share method is required. Since core persons remain in the panel indefinitely, this sample design is particularly appropriate for register-based household panels where all the required information is derived from administrative data. For interview-based household panels some kind of rotating design is required to cope with problems like panel attrition.

In the paper the so called average standard error measure, defined as the square root of the mean over the variances of the estimated income classes of an income distribution, is proposed as a precision measure for minimum sample size determination. It is shown that the maximum value of this precision measure corresponds with a distribution where the proportions in the categories are equal. It is also shown that this result can be seen as generalization of the variance of a fraction taking its maximum value at 0.5. An expression for the minimum required sample size to meet a pre-specified precision for estimated distributions is derived. Since households can be included more than once in the sample, an expression for the expected number of unique households in a sample is also derived.

A topic for further research is to combine this mean standard error measure with a Neyman allocation or power allocations to have expressions for the minimum sample size based on precision requirements for estimated distributions at aggregates of strata. This results in an unequal inclusion probability design for the core persons and requires the Generalized Weight Share method for deriving appropriate weights.

In the context of household surveys and panels, weighting procedures that enforce equal regression weights for persons within the same household are relevant in order to enforce consistency between person based and household based estimates. In this paper an integrated weighting approach based on Lemaître and Dufour (1987) is applied to the RIS. In this application standard errors obtained with Lemaître and Dufour (1987) are smaller than a non-integrated weighting procedure for household based estimates. For person based estimates, standard errors can be slightly larger. These results are in line with Steel and Clark (2007), who show that the large-sample design variance of integrated weighting at the household level is smaller than or equal to the design variance obtained with non-integrated weighting at the person level. In their simulation they also report small increases of the design-variances due to integrated weighting in the case of small sample sizes.

Integrated weighting of Lemaître and Dufour (1987) at the household level is obtained by assuming a variance structure for the residuals that is proportional to the household size (Nieuwenbroek 1993). If household characteristics are proportional to household size, then it can be anticipated that such a variance structure better explains the variation of the household variables in the population compared to a variance structure that assumes equal residual variance for the households. For person based variables such a variance structure might be less efficient but the additional advantage of integrated weighting is that totals for household and person based income, which can be derived directly from their means, are consistent.

Acknowledgements

The views expressed in this paper are those of the author and do not reflect the policies of Statistics Netherlands. The author is grateful to the Associate Editor and the unknown referees for giving constructive comments on two former drafts of the paper. The author also thanks Drs. M. van den Brakel-Hofmans for making the RIS data available.

Technical appendix

Proof of equation (4.4)

An expression for the variance of the estimated fraction of households in income class $l$ can be derived from the general expression for the variance of the HT estimator (Särndal et al. 1992, Section 2.8):

$$V\left({\widehat{P}}_{lh}\right)=\frac{1}{M_h^2}{\displaystyle \sum _{k=1}^{M_h}{\displaystyle \sum _{k^{\prime }=1}^{M_h}\left({\pi }_{k{k}^{\prime }h}-{\pi }_{kh}{\pi }_{k^{\prime }h}\right)}\frac{y_{khl}}{{\pi }_{kh}}}\frac{y_{k^{\prime }hl}}{{\pi }_{k^{\prime }h}}.(\text{A}\text{.1})$$

Inserting first and second order inclusion expectations specified in (3.3) through (3.6), and taking advantage of the property that $y_{khl}=y_{khl}^2$ since the values of the target variable are restricted to zero or one, it follows after some algebra that (A.1) can be simplified to

$$V\left({\widehat{P}}_{lh}\right)=\frac{N_h-n_h}{n_h}\frac{1}{N_h-1}\left(\frac{N_h}{M_h^2}{\displaystyle \sum _{k=1}^{M_h}\frac{y_{khl}}{g_{kh}}-{\left(\frac{M_{lh}}{M_h}\right)}^2}\right).(\text{A}\text{.2})$$

Result (4.4) is obtained by inserting (A.2) into (4.3).

Proof of equation (4.5)

The population of households in stratum $h$ can be divided into $T$ subpopulations of equally sized households. Let $M_{th}$ denote the number of households of size $t$ in stratum $h.$ Now it follows for the double summation between brackets for the expression of $s$ in (4.4) that

$${\displaystyle \sum _{l=1}^L{\displaystyle \sum _{k=1}^{M_h}\frac{y_{khl}}{g_{kh}}={\displaystyle \sum _{l=1}^L{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{k=1}^{M_{th}}\frac{y_{khl}}{t}}}}}}={\displaystyle \sum _{t=1}^T\frac{M_{th}}{t}}.(\text{A}\text{.3})$$

According to the Cauchy-Schwartz inequality (Cochran 1977, Section 5.5) it follows for the single summation between brackets for the expression of $s_h$ in (4.4) that

$${{\displaystyle \sum _{l=1}^L\left(\frac{M_{lh}}{M_h}\right)}}^2={\displaystyle \sum _{l=1}^L{P}_{lh}^2}\ge \frac{1}{L}.(\text{A}\text{.4})$$

Result (4.5) is obtained by inserting (A.3) and (A.4) in the expression for $s$ in (4.4).

Proof of equation (4.7)

Let ${\tilde{\pi }}_{tkh}$ denote the inclusion probability for household $k$ from stratum $h$ of size $t.$ Since equally sized households share the same first order probabilities, it follows that ${\tilde{\pi }}_{tkh}={\tilde{\pi }}_{t{k}^{\prime }h}\equiv {\tilde{\pi }}_{th}.$ Let $I_{tkh}$ denote an indicator variable, taking value 1 if household $k$ from stratum $h$ of size $t$ is included in the sample and zero otherwise. The expected number of unique households can be derived as

$$\begin{array}{ll}{D}_h\hfill & =\text{E}\left({\displaystyle \sum _{t=1}^T{\displaystyle \sum _{k=1}^{M_{th}}{I}_{tkh}}}\right)={\displaystyle \sum _{t=1}^T{M}_{th}{\tilde{\pi }}_{th}}\hfill \\ \hfill & ={\displaystyle \sum _{t=1}^T{M}_{th}\left(1-\frac{\left(\begin{array}{c}{N}_h-t\\ n_h\end{array}\right)}{\left(\begin{array}{c}{N}_h\\ n_h\end{array}\right)}\right)}={\displaystyle \sum _{t=1}^T{M}_{th}\left(1-\frac{\left(N_h-n_h\right)\left(N_h-n_h-1\right)\mathrm{....}\left(N_h-n_h-t+1\right)}{N_h\left(N_h-1\right)\mathrm{...}\left(N_h-t+1\right)}\right).}\hfill \end{array}$$

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