Estimation within the new integrated system of household surveys in Germany

Abstract

In 2015, the European Commission has drafted a framework regulation for integrated European social statistics. This integration covers the Labour Force Survey, the Statistics on Income and Living conditions, and others. In order to avoid an inappropriate response burden, administrative and other sources shall be considered to achieve accurate survey estimates. Combining information from different data sources has become a field of growing research interest among statistical offices and other institutions. In the statistical literature this problem is known as data fusion or statistical matching, and is widely considered as a particular missing-data pattern. Assuming that budgets are limited, and that only some additional information can be obtained to improve the quality of the data fusion, we investigate different scenarios of using these limited resources within an integrated system of household surveys. Our main objective is to develop a framework that fosters on the one hand the estimation of statistical models using several surveys, and on the other hand classical totals for different sub-classes and areas which are of special interest for official statistics.

Appendices

Appendix A: simulation study steps

Here, we briefly describe all steps used in our simulation study.

• 1. A fixed population (size N), variables of interest and auxiliary information are defined.

• 2. The parameters of interest including the model parameters (regression coefficients of a specified model) and the small area means (as local parameters) are defined .

• 3. The proposed designs, D0, D1, D2, D3 and D4 are defined.

• 4. A sample, called MC sample, of size n is selected from the fixed population (SRSWOR).

• 5. For D0, all parameters of interest are estimated based on the complete information from the MC sample.

• 6. For each design, D1, D2, D3 or D4, the following steps are performed.

• 7. The subsample sizes and the overlap size (if needed) are determined. For D1 and D2, the subsample sizes are n1 and n2, where $n1+n2=n$. For D3 and D4, the subsample sizes are n1 and n2, and the horizontal overlap sample size is n3, where $n1+ n2-n3=n$.

• 8. The MC sample (S) is randomly split into two disjoint (for D1 and D2) or overlap subsamples (for D3 and D4), called $S_1$ and $S_2$, where $S_1 \cup S_2 = S$. The $S_1 \cap S_2$ is denoted as $S_3$, where $S_3=\emptyset$, for D1 and D2, and $S_3 \ne \emptyset$ for D3 and D4.

• 9. The design is applied on the complete information (available from the MC sample). According to definition of the design, NA values are inserted into dataset for those variables which are not asked from the corresponding sample units.

• 10. We use the function mice in the mice package of the statistical software R to impute the NA values of dataset. Here, the number of imputations (M), number of iterations, and method of imputation (e.g. predictive mean matching) are determined. Then, M completed datasets are constructed by mice function.

• 11. In order to estimate the model parameters, the combined point estimates and the corresponding confidence intervals and fractions of missing information are obtained based on M completed dataset, using the function pool of the mice package.

• 12. In order to estimate the small area means based on M completed datasets, we obtain the estimator for each completed dataset, using different methods (HT, GREG, SAE under unit level model, SAE under area-level model). Then, for each method, the resulting estimators have been combined (using Rubin’s combination formula defined in Sect. 3.1.3 ) to obtain the overall small area estimates, $\hat{\mu }_{d, \mathrm{HT}}$, $\hat{\mu }_{d, \mathrm{GREG}}$, $\hat{\mu }_{d, \mathrm{BHF}}$ and $\hat{\mu }_{d, \mathrm{FH}}$.

• 13. As a design-based Monte-Carlo simulation study, we repeat steps 4–12, R times.

• 14. Finally, all measures of interest are derived.

Appendix B: convergence diagnostics

Boxplots for groups of 10 subsequent iterations (for the version with 100 iteration) and groups of 100 subsequent iterations (for the version with 1000 iterations) help to assess if convergence in distribution can be assumed (Figs. 9 and 10).

Fig. 9

Convergence diagnostics (100 and 1000 iterations) for $\beta _1$, based on D3 and D4

Fig. 10

Convergence diagnostics (100 and 1000 iterations) for standard deviations of $\beta _1$, based on D3 and D4

Appendix C: coverage probabilities for small area estimates

The area-specific sample sizes vary mainly around 200–300 with outliers of 16, 68, and 97 for small areas (shown as red crosses) as well as 406 and 800 for large areas (shown as blue triangles). The separation between small, medium-size, and large areas was done by the first and third quartile of area-specific sample sizes (Fig. 11).

Fig. 11

Coverage probabilities (CP) versus log mean length of confidence intervals by design and method for $\mu _{d}$ of Y (dashed lines denote nominal coverage of 95%). (Color figure online)