1. Optimal Allocation in the Multi-way Stratification Design for Business Surveys (*) Paolo Righi, Piero Demetrio Falorsi  parighi@istat.it; falorsi@istat.it  Italian National Statistical Institute (*) Research of National Interest n.2007RHFBB3 (PRIN) “Efficient use of auxiliary information at the design and at the estimation stage of complex surveys: methodological aspects and applications for producing official statistics””

2. Outline Statement of the problem Multi-way Sampling Design Multi-way optimal allocation algorithm Monte Carlo simulation

3. Statement of the problem  Large scale surveys in Official Statistics usually produce estimates for a set of parameters by a huge number of highly detailed estimation domains  These domains generally define not nested partitions of the target population  When the domain indicator variables are available at framework level, we may plan a sample covering each domain  Fixing the sample sizes: Help to control the sampling errors of the main estimates; When direct estimators are not reliable (small area problem), having the units in the domains allows to:  bound the bias of small area indirect estimators; use models with specific small area effects.

4. Statement of the problem  Standard solution for fixing the sample sizes stratifies the sample with strata given by cross-classification of variables defining the different partitions (cross-classified or one- way stratified design)  Main drawback: Too detailed stratification:  Risk of sample size explosion;  Inefficient sample allocation (2 units per stratum constraint);  Risk of statistical burden (e.g. repeated business surveys).

5. Statement of the problem Domain of Interest Parameter of interest and estimator: Multivariate (r=1,…,R) and multidomain (d =1, …, D) context

6. Statement of the problem

9. Multi-way Sampling Design  Main problem of MWD: define a procedure for random selection  We propose to use the Cube method (Deville and Tillé, 2004): Select random sample of multi-way stratified design; For a large population and a lot of domains.

10. Multi-way Sampling Design ITACOSM 2011 - 27-29 June 2011, Pisa, Italy - 6

11. Optimal allocation algorithm ITACOSM 2011 - 27-29 June 2011, Pisa, Italy - 6

12. Optimal allocation algorithm

18. Monte Carlo simulation  Objectives of simulation: Test the convergence of the optimization algorithm (optimization step) Comparison between the expect AV and the Monte Carlo empirical AV Comparison with standard cross-classified stratified design ITACOSM 2011 - 27-29 June 2011, Pisa, Italy - 12

19. Monte Carlo simulation  Data: Subpopulation of the Istat Italian Graduates’ Career Survey (3,427 units); Driving allocation variables:  employed status (yes/no) ;  actively seeking work (yes/no). We generate the values of the two variables by means a logistic additive model (Prediction model); Explicative variables: degree mark, sex, age class and aggregation of subject area degree The parameters are estimated by the data from the previous survey ITACOSM 2011 - 27-29 June 2011, Pisa, Italy - 13

20. Monte Carlo simulation  Survey target estimates: Two partitions define the most disaggregate domains:  First partition: university by subject area degree (9 classes);  Second partition: degree by sex; Domains in real survey:448+94; Strata 2,981 (university, degree, sex); In the simulation: domains 20+15;strata 91.  Errors thresholds fixed in terms of CV(%) ITACOSM 2011 - 27-29 June 2011, Pisa, Italy - 14

21. Monte Carlo simulation  Results: Assuming as known values  Iterations (outer process): 6;  Optimal sample size 171 (after calibration 182). Assuming predicted values:  Iterations (outer process): 3;  Optimal sample size 699 (after calibration 707). ITACOSM 2011 - 27-29 June 2011, Pisa, Italy - 15

22. Monte Carlo simulation  Analysis of the allocation with the predicted values: The sample allocation procedure uses an approximation of the AV  The simulation confirms the input AV is an upward approximation of the real AV ITACOSM 2011 - 27-29 June 2011, Pisa, Italy - 16

23. Monte Carlo simulation  Comparison with the standard approach: The implicit model (one-way stratification model) is similar to the model used in our approach; The allocation differences depend on the unit minimum number constraint (2) in each stratum; The sample size is 751 units (+7.4%); Taking into account the domains with small population strata (<10 units in average per stratum) standard approach produces +14.4% sample size. ITACOSM 2011 - 27-29 June 2011, Pisa, Italy - 17

24. References  Bethel J. (1989) Sample Allocation in Multivariate Surveys, Survey Methodology, 15, 47-57.  Chromy J. (1987). Design Optimization with Multiple Objectives, Proceedings of the Survey Research Methods Sec-tion. American Statistical Association, 194-199.  Deville J.-C., Tillé Y. (2004) Efficient Balanced Sampling: the Cube Method, Biometrika, 91, 893-912.  Deville J.-C., Tillé Y. (2005) Variance approximation under balanced sampling, Journal of Statistical Planning and Inference, 128, 569-591  Falorsi P. D., Righi P. (2008) A Balanced Sampling Approach for Multi-way Stratification Designs for Small Area Estimation, Survey Methodology, 34, 223- 234  Falorsi P. D., Orsini D., Righi P., (2006) Balanced and Coordinated Sampling Designs for Small Domain Estimation, Statistics in Transition, 7, 1173-1198  Isaki C.T., Fuller W.A. (1982) Survey design under a regression superpopulation model, Journal of the American Statistical Association, 77, 89-96 ITACOSM 2011 - 27-29 June 2011, Pisa, Italy - 18