Generalised Structure Preserving (GSPREE) Models in Small Area Estimation.

1. Generalised Structure Preserving (GSPREE) Models in Small Area Estimation. An application in estimation of ethnic group size for local authorities. Philip Clarke, Alison Whitworth and Kirsten Piller, ONS in conjunction with Angela Luna-Hernandez, Southampton University

2. Motivation and outline • Need for contemporary good local ethnic population estimates • Standard Small area estimation methods unsuited • Overview of GSPREE method • Application to ethnic group estimation • Validation of method • Conclusions

3. Ethnicity - Typical census output for local authorities Hartlepool 89899 550 1075 229 170 105 2028 Middlesbrough 122055 2362 9864 904 1731 1496 8412 Redcar and Cleveland 133203 853 714 155 122 130 135177 Stockton-on- Tees 181299 1997 5766 866 1133 549 19161 Darlington 101595 1146 1856 349 357 261 105564 ….. ……….. ……. ……. …… …… …… …….. • Detailed cross tabulation • But can become dated

4. What other sources of ethnicity data are available? • English School Census Annual census so contemporary timing. Detailed cross tabulation but for age 5-15 only. Fully comprehensive though omits independent schools. • Annual Population Survey (APS) Annual survey so contemporary timing Detailed cross tabulation but small sample sizes. So imprecise estimates by local authority. Some cells may be zero. Good estimates of national ethnic group totals

5. SAE: Two Approaches 1) Regression model • Combines sample survey data with auxiliary correlating variables available for all areas. • Fit model to define relationship. • Apply model to obtain estimates. • Suitable for continuous data and for categorical variables where one category is of prime interest e.g. households in poverty.

6. SAE: Two Approaches • Suitable for multi-category data and where correlating auxiliary data for the separate categories not readily identifiable. • Sample survey data is combined instead with auxiliary/proxy data of same structure as target variable is taken. • Model is fitted to the cross classified data to define the structure. 2) Structural model Proxy data, X Categories Areas Sample survey data, Y

7. Summary of data for ethnicity Census/school census/APS Ethnic groups Current time LA Population total estimates from good source, e.g. Mid year population estimates Local Authorities Current time ethnic group totals from APS

8. Objective • To enhance the current time APS cross classified estimates of ethnic group populations • Add precision • Eliminate bias How? • Combine the APS data with the good quality census, school census and margins data to borrow strength. • Take special note of the “structure” of the cross classification – known as association structure

9. (Generalised) Structure preserving estimation – (G)SPREE Original SPREE • The structure of proxy data (X data) is held constant for current estimates. GSPREE (Zhang, L.C. and Chambers, R., 2004) • The structure is allowed to be partially modified to reflect tendencies in the APS (Y data). • Modification is determined by a fitted model. • This allows for updating ethnic group proportions

10. Table Association Structure Take the natural log of the table cells and margin means. Centre them so that the cells, rows and columns sum to 0. For a proxy table (Xaj). Areas a=1 to A, Ethnic groups j=1 to J Define the association structure as follows -: For grand total For LA area margins For ethnic group margins For table cells Called interaction terms

11. GSPREE model Model fitted is log linear using APS cell table ,Y -: Parameter of interest for estimation is . are other nuisance parameters. Fit model and estimate parameters using maximum likelihood Then apply parameter to estimate This gives updated association structure. (For SPREE is 1)

12. Next steps • Exponentiate the updated interaction cell terms to obtain a table of modelled estimates. • Benchmark the table of modelled estimates to known row and column totals using iterative proportional fitting. • Precision estimates are obtained by bootstrap.

13. Using two proxy tables • Estimating ethnic populations can use two proxy tables – Census and English Schools Census. • To implement this combine them using a weight • So now : • Value of chosen to minimise deviance in model fit.

14. Estimating for 2014 - Data sources

15. Estimated Association Structure 2014 Several modelling strategies considered with similar results

16. Validation exercise Estimate for 2011 using 2001 Census and compare results with 2011 census

17. Estimated Association Structure 2011

18. 2011 GSPREE and 2011 Census Estimates White Mixed Asian Chinese Black Other

19. Uncertainty in 2011 estimates - proportions

20. Uncertainty in 2011 estimates - counts

21. Coefficients of Variation

22. Conclusions… • Aggregate estimates of the population by ethnic group and LA + quality measures • Statistical framework for combining sources for population attributes and a flexible mechanism for introducing new administrative and survey data • GSPREE shows good performance (proof of concept) • More precise estimates compared to direct estimates • GSPREE estimates produced where no sample for producing direct estimates • Some further investigation on bias and precision measures needed • Best use of information available across all sources • Potential to consider for other variables of same nature.

23. References ONS, (2016). Assessing the Generalised Structure Preserving Estimator (GSPREE) for Local Authority Population Estimates by Ethnic Group in England. GSS Methodology Series No 42. Luna-Hernandez, A., Zhang, L., Whitworth, A. and Piller, K. (2015) Small Area Estimates of the Population Distribution by Ethnic Group in England: A Proposal Using Structure Preserving Estimators. Statistics in Transition New Series and Survey Methodology Joint Issue: Small Area Estimation 2014. Vol. 14, No.4, pp. 585-602. Luna-Hernandez, A. (2014). On Small Area Estimation for Compositions Using Structure Preserving Models. Unpublished PhD upgrade document, Department of Social Statistics and Demography, University of Southampton. Zhang, L.C. and Chambers, R. (2004). Small area estimates for cross-classifications. Journal of the Royal Statistical Society, B, 66, 479–496.