Coordination of spatially balanced samples
Section 7. Conclusions

• Previous
• Next

New methods are proposed to coordinate spatially balanced samples based on PRNs. The objective is two-fold: first, to achieve a good coordination degree between samples, and second to preserve a good spatial balance degree. With the coordination of LPM and SCPS a good degree of spatial balance is ensured. SCPS with PRNs is less memory consuming since only a PRN vector of size $N$ is used, while for LPM one uses a matrix of dimension $N\times N\mathrm{.}$ Our examples concern moderate size populations, and a large $N$ quickly introduces limits in the calculations. In practice, a large $N$ leads to an oversized matrix to be employed in the coordination of LPM samples. In these cases, the method can be implemented using dynamic allocation of the computer memory. Despite this solution, limits of the proposed method are possible in practice.

In our simulations, SCPS tends to perform better than LPM in terms of overlap expectation and variance, for both positive and negative coordination. A good coordination of LPM samples is more difficult to achieve than of SCPS samples, because the same pairs of units should be considered in the sample selection process, instead of single units. If births or deaths appear in the population, the pairs used for the selection of $s_1$ may not be available any more for the selection of $s_2\mathrm{.}$ Thus, the sample coordination becomes poor. SCPS does not have this weakness, but instead the coordination level may drop as well if the weights are distributed very differently the second time compared to the first time. This is the reason why SCPS with PRNs performs worse than Poisson sampling with PRNs. LPM with PRNs may have better behavior in terms of overlap than SCPS with PRNs if changes in the population are not detected. This situation is exemplified in Table 5.1 when the static MU284 population is used.

As shown in our examples in Section 5.1 both methods show a weaker performance in terms of expected overlap than Poisson sampling. This is a normal feature of these methods since one imposes the fixed sample size constraint for LPM and SCPS. In order to overcome this weakness, we introduced a new family of designs, based on a transformation of SCPS and a choice of scalar $\alpha ,$ $0\le \alpha \le 1.$ Each value of $\alpha$ leads to a member of this family. For $\alpha \mathrm{<mo>=</mo>\; 1}$ one obtains SCPS, while for $\alpha \mathrm{<mo>=</mo>\; 0}$ Poisson sampling. This family of designs reminds us another family depending upon a scalar, the Pomix design (Kröger et al., 1999). The Pomix design is a mixture between Bernoulli and Poisson sampling, also used for coordination with PRNs.

For the transformed version of SCPS, the degree of coordination and spatial balance depend on the choice of $\alpha \mathrm{.}$ Being a mixture of Poisson sampling and SCPS, it achieves a better coordination degree than SCPS. However, the improved degree of coordination comes at the cost of increased variance of sample size and reduced spatial balance as our examples in Section 5.1 and Section 4 showed. Based on our results, for the transformed SCPS, our recommendation is to use $\alpha \mathrm{<mo>=</mo>}$0.5 that represents a compromise between a good spatial balance degree and a good coordination degree. On the other hand, $\alpha \mathrm{<mo>=</mo>}$0.5 seems a good all-purpose suggestion since the results for variance estimation of differences and averages shown in Section 5.3 also indicate this value as a reasonable choice.

In our results shown in Section 5.3 LPM with PRNs, SCPS with PRNs and the TSCPS family reduce the Monte-Carlo variance of the differences when positively coordinated samples are used compared to independent samples’ selection. In both used settings, it seems, however, that in the case of LPM with PRNs and SCPS with PRNs, variance reduction comes mainly from the combined effect of spatial balance and fixed sample size rather than from the effect of positive coordination. The Monte-Carlo variance of the averages is not always reduced in our examples when negatively coordinated samples are selected compared to independent samples; LPM with PRNs and SCPS with PRNs show in this case negligible improvement when negative coordination is used.

All the proposed methods can also be applied in the case where the spatial distance is replaced by a distance between auxiliary variables like the Mahalanobis distance. Thus, the sample coordination can be performed in the space spanned by these variables. The proposed methods allow thus not only a spatial sample coordination, but also the coordination of representative samples, in the terminology used by Grafström and Schelin (2014).

Aknowledgements

The authors wish to thank the Associate Editor and two referees for their valuable comments and suggestions that helped in improving the quality of the paper significantly.

References

Benedetti, R., Piersimoni, F. and Postiglione, P. (2017). Spatially balanced sampling: A review and a reappraisal. International Statistical Review, 85, 439-454.

Bondesson, L., and Thorburn, D. (2008). A list sequential sampling method suitable for real-time sampling. Scandinavian Journal of Statistics, 35, 466-483.

Brewer, K., Early, L. and Joyce, S. (1972). Selecting several samples from a single population. Australian Journal of Statistics, 3, 231-239.

Cotton, F., and Hesse, C. (1992). Tirages coordonnés d’échantillons. Technical Report E9206, Direction des Statistiques Économiques, INSEE, Paris, France.

Deville, J.-C., and Tillé, Y. (1998). Unequal probability sampling without replacement through a splitting method. Biometrika, 85, 89-101.

Dickson, M.M., Benedetti, R., Giuliani, D. and Espa, G. (2014). The use of spatial sampling designs in business surveys. Open Journal of Statistics, 04, 345-354.

Dubin, R.A. (1992). Spatial autocorrelation and neighborhood quality. Regional Science and Urban Economics, 22, 3, 433-452.

GeoDa Center for Geospatial Analysis and Computation (2017). Sample data. http://spatial.uchicago.edu/ sample-data. Accessed: 6-April-2017.

Grafström, A. (2012). Spatially correlated Poisson sampling. Journal of Statistical Planning and Inference, 142, 139-147.

Grafström, A., and Matei, A. (2015). Coordination of Conditional Poisson samples. Journal of Official Statistics, 31, 4, 649-672.

Grafström, A., and Schelin, L. (2014). How to select representative samples. Scandinavian Journal of Statistics, 41, 2, 277-290.

Grafström, A., and Tillé, Y. (2013). Doubly balanced spatial sampling with spreading and restitution of auxiliary totals. Environmetrics, 14, 2, 120-131.

Grafström, A., Lundström, N.L.P. and Schelin, L. (2012). Spatially balanced sampling through the pivotal method. Biometrics, 68, 2, 514-520.

Haziza, D. (2013). Sampling and estimation procedures in business surveys: A discussion of some specific features. Seminar of the Royal Statistical Society, London, England.

Kröger, H., Särndal, C.-E. and Teikari, I. (1999). Poisson mixture sampling: A family of designs for coordinated selection using permanent random numbers. Survey Methodology, 25, 1, 3-11. Paper available at https://www150.statcan.gc.ca/n1/pub/12-001-x/1999001/article/4707-eng.pdf.

Kröger, H., Särndal, C.-E. and Teikari, I. (2003). Poisson mixture sampling combined with order sampling. Journal of Official Statistics, 19, 59-70.

Mach, L., Reiss, P.T. and Şchiopu-Kratina, I. (2006). Optimizing the expected overlap of survey samples via the northwest corner rule. Journal of the American Statistical Association, 101, 476, 1671-1679.

Matei, A., and Tillé, Y. (2005). Maximal and minimal sample co-ordination. Sankhyā, 67, part 3, 590-612.

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model Assisted Survey Sampling. Springer Verlag, New York.

Stevens, D.L.J., and Olsen, A.R. (2004). Spatially balanced sampling of natural resources. Journal of the American Statistical Association, 99, 262-278.

• Previous
• Article main page
• Next