This module is part of the. Memobust Handbook. on Methodology of Modern Business Statistics

Transcription

1 This moule is part of the Memobust Hanbook on Methoology of Moern Business Statistics 26 March 2014

2 Metho: Balance Sampling for Multi-Way Stratification Contents General section Summary General escription of the metho Preparatory phase Examples not tool specific Example: the multi-way stratification esign for controlling the sample size Example: the multi-way stratification esign to retain the sample allocation Example: the multi-way stratification esign for reucing the response buren Examples tool specific Glossary References... 7 Specific section... 8 Interconnections with other moules Aministrative section... 11

3 General section 1. Summary Balance sampling is a class of techniques using auxiliary information at the sampling esign stage. Many types of sampling esigns can be interprete as balance sampling, such as simple ranom sampling with fixe size, stratifie simple ranom sampling an unequal probability sampling. Furthermore, the balance sampling can be applie to efine a multi-way stratification esign also known as incomplete stratification or marginal stratification. Multi-way stratification allows to plan the sample sizes of the omains of interest belonging to two or more non-neste partitions of the population in question without using the stanar solution base on a stratifie sample in which strata are ientifie by cross-classifying the variables efining the ifferent partitions (one-way stratifie esign). The stanar solution in many Structural Business Surveys (SBSs) may have rawbacks from the view-point of cost-effectiveness. In fact, SBSs prouce typically estimates for a great number of very etaile omains forming several non-neste partitions of the population an creating really small cross-classifie strata. 2. General escription of the metho Balance sampling can be applie accoring to two ifferent inferential approaches: the moel base approach (Royall an Herson, 1973, Valliant et al. 2000) an the esign base or ranomisation assiste approach (Deville an Tillé, 2004). The first approach bases the inference on a statistical superpopulation moel an it may be performe by probability or non-probability sample. In this framework a sample is balance when the sample means of a set of auxiliary variables (balancing variables) are equal to the known population means (Valliant et all, 2000). Balance samples are use to follow a robust sampling strategy. The esign base approach nees a sampling frame an uses a probability sample to make inferences. In this secon context a sample is balance when the Horvitz- Thompson (H-T) sample estimates for the auxiliary variables are equal to their known population totals. The selection of a balance sample generally improves the efficiency of the sampling estimates (Cochran, 1977). This section focuses on this secon inferential approach. A wiely use application of the metho is stratifie simple ranom sampling. As known, it has been introuce in the sampling methoology to enhance the efficiency of the estimates. Nevertheless, stratifie sampling can be use as an operative tool in the surveys as well. An instrumental use of stratifie sampling is when the objective of the survey is to prouce estimates for some subpopulations (or omains) forming two or more non-neste partitions of the population an a fixe or planne sample size for each omain is require. A stanar sampling esign solution efines strata by crossclassifying the variables efining the ifferent partitions. In this case stratification is not strictly use to improve estimation quality. It is use to implement a ranom selection metho guaranteeing the selecte sample sizes corresponing to the planne ones. This stanar solution, hereinafter enote as one-way stratifie esign, may have some rawbacks, especially in the SBSs. When the number of cross-classifie strata is too large, there are some immeiate consequences, escribe as follows: (i) the overall sample size coul easily be too large for the survey economic constrains; 3

4 (ii) when the population size in many strata is small, the stratification scheme becomes inefficient; in other wors the sample allocation may be far from the theoretically esire allocation; (iii) when there are strata containing only few units in the population, a not equally istribute response buren may arise in surveys repeate over time. Many methos have been propose in the literature to keep that the sample size uner control in all the omains without using one-way stratifie esigns. This means that sample size of each cross-classifie stratum is a ranom variable. These approaches may be roughly ivie into two main categories. The first category contains methos commonly known as controlle selection. Seminal papers have been propose by Bryant et al. (1960) an Jessen (1970). Other methos base on controlle rouning problems via linear programming have been propose by Causey et al. (1985), Rao an Nigam (1990; 1992), Sitter an Skinner (1994) an Winkler (2001). In the secon category there are methos base on sample coorination. A separate sample is selecte for each partition in orer to guarantee the maximum overlap among the ifferent samples (Ohlsson, 1995; Ernst an Paben, 2002). We efine all these methos as multi-way stratifie esigns. Literature shows that these methos pose theoretical an operative problems especially for large scale surveys as in the SBSs. A recently propose metho, the Cube algorithm (Deville an Tillé, 2004) overcomes these rawbacks. The metho, inclue in the first category, has been originally efine for rawing balance samples with a large number of balancing variables for large population size. Multiway stratification is a special case of balance sampling. Given the population U of size N, let π k be the inclusion probability of k-th population unit (k=1,, N) an let δ k be the value of the inicator variable of the omain U, being δ k =1 if the unit k belongs to omain U an equal to zero otherwise. Then, by efinition the sample size of U is N k =1 δ k π = n. The Cube metho assumes that the inclusion probabilities are known, an it selects a ranom sample achieving the consistency among the known totals an the H-T estimates. We efine the following auxiliary variables zk = δ π for each omain. When the sample is balance on the z variables the H-T estimate k N = k = k ˆ sk z π has to be equal to the known population total 1 k k n Z k Z = with s k being a ranom variable equal to 1 if the k-th unit belongs to the sample an equal to 0 otherwise. For satisfying the balancing equations, Z ˆ = Z, the Cube algorithm has to select n units from U. When the expecte sample sizes are integer numbers the Cube algorithm applie to obtain a multi-way stratification always fins the solution. Some illustrative examples of multi-way sampling esigns are given in Falorsi an Righi (2008). 3. Preparatory phase 4

5 4. Examples not tool specific 4.1 Example: the multi-way stratification esign for controlling the sample size In orer to explain the problem, we consier the population of 165 schools reporte in Table 1 (Cochran, 1977, p. 124). We assume that the parameters of interest are the totals of a variable, relate to school, separately for the Size of city (5 categories: I,II,III,IV,V) an for the Expeniture per pupil (4 categories: A,B,C,D). Two istinct partitions of the population are efine: the size of city (first partition) efining 5 non-overlapping omains, an the expeniture per pupil efining 4 omains. We have 9 omains of interest. Table 1. Size of city Expeniture per pupil A B C D Totals I II III IV V Totals The stanar one-way stratifie esign (or cross-classification esign) efines 20=5 4 strata by crossing the categories of the omains of the two partitions. Due to bugetary constraints, we suppose that the sample size coul be up to 10 units. Nevertheless, in each stratum at least one school shoul be selecte (or two schools for estimating the sampling variance without any bias) an, consequently, accoring to this esign the sample size shoul amount to 20 (or 40) schools at least. Hence, the crossclassification esign becomes unfeasible. 4.2 Example: the multi-way stratification esign to retain the sample allocation We consier the above population of schools. We plan a sample of 20 schools an we want to allocate the sample proportionally to the omain size. Table 2 shows the planne size an the integer roune sample allocation. We note that the sample sizes in the cross-classifie strata are not constraine to be integers. Table 2. Size of city Expeniture per pupil A B C D Domain Roune planne weight sample size I II III IV V Domain weight Roune planne sample size

6 Accoring to the one-way stratifie esign of size 20 (Table 3), we obtain a sample allocation far from the planne one. Table 3. Size of city Expeniture per pupil A B C D Domain Roune planne weight sample size I II III IV V Domain weight Roune planne sample size Example: the multi-way stratification esign for reucing the response buren We consier the population of schools escribe in section 4.1 an we suppose that the population istribution to be fixe over time. We have to select a sample of size 40 on several survey occasions. Moreover, we want to compute unbiase variance estimates. Accoring to the one-way stratifie esign we have to select 2 schools per stratum (Table 4). Table 4. Size of city Expeniture per pupil A B C D Domain Roune planne weight sample size I II III IV V Domain weight Roune planne sample size Then, we can see that the schools in stratum V-B are rawn with certainty on each survey occasion an the schools in strata IV-B an V-A have a high probability to be inclue in the samples. That happens because in stratum V-B there are only two schools in the population, while in the strata IV-B an V-A there are three schools in the population an the inclusion probability is Hence, the response buren is not equally istribute in the population of schools (is high for the schools belonging to small population size strata) an this buren oes not epen on efficiency issues. 5. Examples tool specific 6

7 6. Glossary For efinitions of terms use in this moule, please refer to the separate Glossary provie as part of the hanbook. 7. References Bryant, E. C., Hartley, H. O., an Jessen, R. J. (1960), Design an Estimation in Two-Way Stratification. Journal of the American Statistical Association 55, Causey, B. D., Cox, L. H., an Ernst, L. R. (1985), Applications Transportation Theory to Statistical Problem. Journal of the American Statistical Association 80, Cochran, W. G. (1977), Sampling Techniques. Wiley, New York. Deville J.-C. an Tillé, Y. (2004), Efficient Balance Sampling: the Cube Metho. Biometrika 91, Ernst, L. R. an Paben, S. P. (2002), Maximizing an Minimizing Overlap When Selecting Any Number of Units per Stratum Simultaneously for Two Designs with Different Stratifications. Journal of Official Statistics 18, Falorsi, P. D. an Righi, P. (2008), A Balance Sampling Approach for Multi-Way Stratification Designs for Small Area Estimation. Survey Methoology 34, Jessen, R. J. (1970), Probability Sampling with Marginal Constraints. Journal of the American Statistical Association 65, Lu, W. an Sitter, R. R. (2002), Multi-Way Stratification by Linear Programming Mae Practical. Survey Methoology 28, Ohlsson, E. (1995), Coorination of Samples using Permanent Ranom Numbers. In: Business Survey Methos (es. Cox, B. G., Biner, D. A., Chinnappa, B. N., Christianson, A., College, M. J., an Kott, P. S.), Wiley, New York, Chapter 9. Rao, J. N. K. an Nigam, A. K. (1990), Optimal Controlle Sampling Design. Biometrika 77, Rao, J. N. K. an Nigam, A. K. (1992), Optimal Controlle Sampling: a Unifying Approach. International Statistical Review 60, Royall, R. an Herson, J. (1973), Robust Estimation in Finite Population. Journal of the American Statistical Association 68, Valliant, R., Dorfman, A. H., an Royall, R. M. (2000), Finite Population Sampling an Inference: A Preiction Approach. Wiley, New York. Winkler, W. E. (2001), Multi-Way Survey Stratification an Sampling. RESEARCH REPORT SERIES, Statistics # , Statistical Research Division U.S. Bureau of the Census Washington D.C

8 Specific section 8. Purpose of the metho Balance sampling is use for selecting a multi-way stratifie esign, which is a sampling esign planning the sample sizes for omains of interest belonging to ifferent partitions of the population without using a one-way stratifie or cross-classifie stratification esign. 9. Recommene use of the metho 1. The metho can be applie when the one-way stratifie esigns (those where strata are obtaine by combining the omains of ifferent partition of the population) can be inefficient or can prouce statistical buren for surveys repeate over time. 2. The metho may be applie in large scale surveys, with large population an a lot of omains. 3. The metho may be useful in the small area estimation problem when the membership inicator variables for small areas are known at population level. Planning the sample size for each omain allows to estimate specific small area effects improving the efficiency of inirect moel base small area estimators. 10. Possible isavantages of the metho 1. The metho nees to know the inclusion probabilities. The efinition of the optimal inclusion probabilities is less intuitive than in case of one-way stratification esign. 2. Analytic expression of the variance of the estimates is unknown. Approximations (shown in the literature) are neee. 3. Some ifficulties when a complex estimator is use for the computation of sampling errors. 11. Variants of the metho 1. Balance Sampling for multi-way stratification is efine to select a planne sample size for each omain. In aition, it may be worthwhile incluing other balancing variables to enhance the estimation efficiency accoring to the calibration estimation theory. 12. Input ata 1. Data incluing the omain membership inicator variable an the inclusion probability for each population units are neee. 13. Logical preconitions 1. Missing values 1. Not allowe. 2. Erroneous values 1. Not allowe. 3. Other quality relate preconitions 8

9 1. 4. Other types of preconitions 1. The sum of the inclusion probabilities over each omain must be an integer. 2. The sum over population omains of the inclusion probabilities must be equal for each partitions. 14. Tuning parameters 1. Depening on the origin of the inclusion probabilities a calibration step coul be neee. The calibration step moifies the probabilities for satisfying sample size consistency among the partitions an for achieving an integer expecte sample size in each omain (see ). 15. Recommene use of the iniviual variants of the metho 1. n/a 16. Output ata 1. Sample membership inicator variable is ae in the input ata set. 17. Properties of the output ata 1. The sum of the sample membership inicator variable over each omain is equal to the expecte sample size. 18. Unit of input ata suitable for the metho Processing full ata set. 19. User interaction - not tool specific 1. Definition of the set of inclusion probabilities. 2. Before execution of the metho, verify that the planne sample sizes for each omain are integer numbers an consistent. 3. When performing a multi-way stratification esign consiering also other balancing variables in the sample selection process, the inicators of the quality of balancing have to be analyse. 20. Logging inicators 1. No specific inicators. 21. Quality inicators of the output ata 1. When use only for multi-way stratification, the theory shows that the metho selects exactly a sample satisfying the planne sample size. When other balance variables are ae, the ratio among the H-T estimates an the known totals are use as quality inicators. 2. No other quality inicators are use to strictly evaluate the performances of the methos. 9

10 22. Actual use of the metho 1. Balance sampling is wiely use in the Insee not specifically for implementing multi-way stratification. 2. Istat has use balance sampling for a population survey. 3. An Istat research project is stuying the optimal allocation for multi-way stratifie esign. Interconnections with other moules 23. Themes that refer explicitly to this moule 1. Sample Selection Main Moule 2. Sample Selection Sample Co-orination 24. Relate methos escribe in other moules Mathematical techniques use by the metho escribe in this moule 1. The metho mainly implements a balancing martingale theory, with the aim to roun off each inclusion probabilities ranomly to 0 or 1. From the mathematical point of view that correspons to the maximisation of the entropy measure (maximisation of the ranomness) uner linear constraints (balancing equations). 26. GSBPM phases where the metho escribe in this moule is use Design Frame & Sample methoology 2. Partially 4.1 Select sample 27. Tools that implement the metho escribe in this moule 1. Sampling R package 2. SAS Macro ownloaable Insee site 28. Process step performe by the metho Sample planning an selection 10

11 Aministrative section 29. Moule coe Sample Selection-M-Balance Sampling 30. Version history Version Date Description of changes Author Institute first version Paolo Righi ISTAT secon version Paolo Righi ISTAT thir version Paolo Righi ISTAT preliminary release page numbering ajuste final version within the Memobust project 31. Template version an print ate Template version use 1.0 p Print ate :42 11