On combining independent probability samples
Section 5. Discussion
The simulation examples in the previous section are only intended to demonstrate the different approaches and we make no claim of the generality of the result. However, we find it very likely that using pooled variance estimators is better than using separate variance estimators in a linear combination estimator, especially in cases where the separate total estimators are highly correlated to their variance estimators.
We have presented in detail how to combine independent probability samples and derived corresponding design features needed to do unbiased estimation and variance estimation. The danger of using the combined sample approach for very different designs has been illustrated. Moreover, we have shown that there is often a risk for a strong positive correlation between the HT estimator and its variance estimator. Such dependence can be a source for bias if estimated variances are used in a linear combination. Thus, as an alternative approach, we have shown how to use the combined sample to estimate separate variances. This alternative approach can lead to more stable weights in a linear combination of separate estimators, and has potential to reduce both bias and variance.
There are of course limitations to when this methodology can be applied due to our assumption of fully known designs and use of the same frame with identifiable units. Sensitivity for deviations from some of these assumptions, such as having unidentifiable units or using approximate second order inclusion probabilities, needs further investigation.
In particular, knowledge of this methodology is important if an initial sampling effort was proven insufficient. Such situations are common in e.g., environmental monitoring (Christensen and Ringvall, 2013). Then a complementary sample may be designed in such a way that it allows for a combination with improved efficiency.
Acknowledgements
We thank an anonymous reviewer and an associate editor for helpful comments that improved the paper. This work was funded by the Swedish Science Council grant 340-2013-5076.
References
Bankier, M.D. (1986). Estimators based on several stratified samples with applications to multiple frame surveys. Journal of the American Statistical Association, 81(396), 1074-1079.
Christensen, P., and Ringvall, A.H. (2013). Using statistical power analysis as a tool when designing a monitoring program: Experience from a large-scale Swedish landscape monitoring program. Environmental Monitoring and Assessment, 185(9), 7279-7293.
Cochran, W.G. (1954). The combination of estimates from different experiments. Biometrics, 10(1), 101-129.
Cochran, W.G., and Carroll, S.P. (1953). A sampling investigation of the efficiency of weighting inversely as the estimated variance. Biometrics, 9(4), 447-459.
Fridman, J., Holm, S., Nilsson, M., Nilsson, P., Ringvall, A.H. and Ståhl, G. (2014). Adapting National Forest Inventories to changing requirement-the case of the Swedish National Forest Inventory at the turn of the 20th century. Silva Fennica, 48(3), article id 1095.
Grafström, A. (2010). Entropy of unequal probability sampling designs. Statistical Methodology, 7(2), 84-97.
Gregoire, T.G., and Schabenberger, O. (1999). Sampling-skewed biological populations: Behavior of confidence intervals for the population total. Ecology, 80, 1056-1065.
Hansen, M.H., and Hurwitz, W.N. (1943). On the theory of sampling from finite populations. The Annals of Mathematical Statistics, 14(4), 333-362.
Koricheva, J., Gurevitch, J. and Mengersen, K. (Eds.) (2013). Handbook of meta-analysis in ecology and evolution. Princeton University Press.
Lohr, S.L. (2009). Multiple-frame surveys. Handbook of Statistics, 29, 71-88.
Lohr, S.L. (2011). Alternative survey sample designs: Sampling with multiple overlapping frames. Survey Methodology, 37, 2, 197-213. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2011002/article/11608-eng.pdf.
Næsset, E., and Gjevestad, J.G. (2008). Performance of GPS precise point positioning under conifer forest canopies. Photogrammetric Engineering & Remote Sensing, 74(5), 661-668.
O’Muircheartaigh, C., and Pedlow, S. (2002). Combining samples vs. cumulating cases: a comparison of two weighting strategies in NLSY97. In Proceedings of the 2002 Joint Statistical Meetings, American Statistical Association, 2557-2562.
Roberts, G., and Binder, D. (2009). Analyses based on combining similar information from multiple surveys. In Joint Statistical Meetings 2009 Survey Research Methods Section, 2138-2147.
Rubin, D.B., and Weisberg, S. (1974). The variance of a linear combination of independent estimators using estimated weights. ETS Research Bulletin Series, 1974(2), i-5.
Schmidt, F.L., and Hunter, J.E. (2014). Methods of meta-analysis: Correcting error and bias in research findings. Sage publications.
Ståhl, G., Allard, A., Esseen, P.-A., Glimskår, A., Ringvall, A., Svensson, J., Sundquist, S., Christensen, P., Gallegos Torell,å., Högström, M., Lagerqvist, K., Marklund, L., Nilsson, B. and Inghe, O. (2011). National Inventory of Landscapes in Sweden (NILS)-scope, design, and experiences from establishing a multiscale biodiversity monitoring system. Environmental Monitoring and Assessment, 173(1-4), 579-595.
Tomppo, E., Gschwantner, T., Lawrence, M. and McRoberts, R.E. (Eds.) (2009). National Forest Inventories: Pathways for Common Reporting. Springer Science & Business Media.
- Date modified: