EMOS 2015 Spring School Sampling I + II

Transcription

1 EMOS 2015 Spring School Sampling I + II quad Trier, 24 th March 2015 Ralf Münnich 1 (113) EMOS 2015 Spring School Sampling I + II

2 1. Introduction to Survey Sampling Trier, 24 th March 2015 Ralf Münnich 2 (113) EMOS 2015 Spring School Sampling I + II

3 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Business of statistics One major aim (business) of statistics, especially for official statistics, is gathering and provision of data for administrative, political, sociological, economic, and other research purposes. This information is obtained in universes (or samples) of units (persons, households, establishments, machinery etc.). The data are gained in total (complete inventory) or via pre-specified principles partially as samples (randomly, quota). The following shall be considered: Primary or secondary statistics (e.g. register data) Problems of adequacy Errors of different kinds Data protection (anonymisation) Trier, 24 th March 2015 Ralf Münnich 3 (113) EMOS 2015 Spring School Sampling I + II

4 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Surveys and Samples Definition of a survey A survey, in general, is the methodology for gathering information via samples on persons, households, or other units. In a survey planning and development, pretest, survey design, implementation, data gathering, editing and processing, as well as analysis play a vital role. Trier, 24 th March 2015 Ralf Münnich 4 (113) EMOS 2015 Spring School Sampling I + II

5 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Surveys and Samples Definition of a survey A survey, in general, is the methodology for gathering information via samples on persons, households, or other units. In a survey planning and development, pretest, survey design, implementation, data gathering, editing and processing, as well as analysis play a vital role. Trier, 24 th March 2015 Ralf Münnich 4 (113) EMOS 2015 Spring School Sampling I + II

6 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Areas of applications in survey statistics Economics statistics and empirical economic research Social statistics and empirical social research Demography and socio-demographic research Market and opinion research Quality control Medical statistics and biometry Meteorology Environmental statistiscs Forestry and environmental control Traffic control Inventory based on samples Natural science research and measurement Trier, 24 th March 2015 Ralf Münnich 5 (113) EMOS 2015 Spring School Sampling I + II

7 Data gathering in surveys Different kinds of data 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Cross sectional or longitudinal data (point of time, period) Random samples Systematic samples Time series Panel data (cross sectional and longitudinal) Data acquisition (Computer assisted) interviewing Telephone interviewing Use of register data Snowball sampling Quota sampling Trier, 24 th March 2015 Ralf Münnich 6 (113) EMOS 2015 Spring School Sampling I + II

8 Examples for important surveys 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics EU-SILC: Statistics on Income and Living Conditions in Europe. http: //ec.europa.eu/eurostat/web/microdata/european_ union_statistics_on_income_and_living_conditions). ESS: European Social Survey The European Social Survey (the ESS) is an academically-driven social survey designed to chart and explain the interaction between Europe s changing institutions and the attitudes, beliefs and behaviour patterns of its diverse populations. Now in its third round, the survey covers over 20 nations and employs the most rigorous methodologies. It is funded via the European Commission s 5th and 6th Framework Programmes, the European Science Foundation and national funding bodies in each country. Source: Trier, 24 th March 2015 Ralf Münnich 7 (113) EMOS 2015 Spring School Sampling I + II

9 The German Microcensus (MZ) 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Survey of households via interviewer Structure of the population ILO unemployment and labour participation Special aspects (changing) 1% sample since 1957 Reform 1990 Microcensus law from 17. January 1996 Programme of questions, mandatory survey, max. 4 years Since 2005: short-term sampling Microcensus as rotational panel Microcensus law 2005: Trier, 24 th March 2015 Ralf Münnich 8 (113) EMOS 2015 Spring School Sampling I + II

10 Arrangement of statistical units 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics 214 regional classes Federal states, districts Regional classes: min to inh. 5 house size classes Census 1987 and statistics on building activity 1 Small buildings: 1 to 4 dwellings 2 Mid size buildings: 5 to 10 dwellings 3 Large buildings: minimum 11 dwellings 4 Institutions: dw. = 0 or indiv. (dw. + 4) 4 6 New buildings Selection domain sequential arrangement 1 Approx. 12 dwellings ( 10 13); maximal 70 inhabitants 2 One building each dwellings; floor-wise selection 4 Initial letter of last name; approx. 15 inhabitants 6 Selection according 1 4 depending on the type of new building Trier, 24 th March 2015 Ralf Münnich 9 (113) EMOS 2015 Spring School Sampling I + II

11 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Sampling design of the Microcensus 4 -stage design Strata strata strata clusters 1% sample One selection domain from each zone Approx. 1% of individuals and households Semi-systematic random selection 4 zones block (rotation quarters) Replacement of 1 (4) rotation quarters each year Note: The dropped rotation quarter yields the basis for acquiring households for the access panel (EU-SILC and ICT). Trier, 24 th March 2015 Ralf Münnich 10 (113) EMOS 2015 Spring School Sampling I + II

12 Data quality: Eurostat definition 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Relevance of the statistical concept: End-user, user needs, hierarchical structure and contents Accuracy and reliability: Sampling errors: standard error, CI coverage Non-sampling errors: nonresponse, coverage error, measurement errors Timeliness and punctuality: Time and duration from data acquisition until publication Coherence and comparability: Preliminary and final statistics, annual and intermediate statistics (regions, domains, time) Accessibility and clarity: Publication of data, analysis and method reports Completeness (not part of the Code of Practice) Source: european-statistics-code-of-practice Trier, 24 th March 2015 Ralf Münnich 11 (113) EMOS 2015 Spring School Sampling I + II

13 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Errors and types of errors in surveys Frame error Interviewer error Processing error Coding error Nonresponse (NR) Unit-Nonresponse Item-Nonresponse Sampling error Trier, 24 th March 2015 Ralf Münnich 12 (113) EMOS 2015 Spring School Sampling I + II

14 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Evaluation of samples and surveys Main principles in survey sampling Practicability Costs of a survey Accuracy of results Standard errors Confidence interval coverage Disparity of subpopulations Robustness of results In order to adequately evaluate the estimates from samples, appropriate evaluation criteria have to be considered. Trier, 24 th March 2015 Ralf Münnich 13 (113) EMOS 2015 Spring School Sampling I + II

15 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Basic principles in statistics for surveys In addition to the analysis from survey data, we need to investigate the quality of the results. The aim is to generalize results from the sample to the universe. inferential statistics Sampling function and estimator Properties of estimators unbiasedness efficiency normality large sample properties (limit theorems) small sample properties Test procedures based on survey data Details can be drawn from Schaich and Münnich (2001), Chapter 4 or 6, Mittelhammer (2013), or others. Trier, 24 th March 2015 Ralf Münnich 14 (113) EMOS 2015 Spring School Sampling I + II

16 Unemployment in Saarland 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Unemployed Women τ Men τ τ True values in Saarland Estimates from the Microcensus Is the quality of the cell estimates identical? Trier, 24 th March 2015 Ralf Münnich 15 (113) EMOS 2015 Spring School Sampling I + II

17 Unemployment in Saarland 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Unemployed Women τ E τ Men τ E τ τ E τ True values in Saarland Estimates from the Microcensus Is the quality of the cell estimates identical? Trier, 24 th March 2015 Ralf Münnich 15 (113) EMOS 2015 Spring School Sampling I + II

18 Unemployment in Saarland 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Unemployed Women τ E τ Men τ E τ τ E τ True values in Saarland Estimates from the Microcensus Is the quality of the cell estimates identical? Trier, 24 th March 2015 Ralf Münnich 15 (113) EMOS 2015 Spring School Sampling I + II

19 Unemployment in Saarland 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Unemployed Women τ E τ Men τ E τ τ E τ True values in Saarland Estimates from the Microcensus Is the quality of the cell estimates identical? Trier, 24 th March 2015 Ralf Münnich 15 (113) EMOS 2015 Spring School Sampling I + II

20 Unemployed women, Raking estimator 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics variance estimator NR rates: 1: 5%, 2: 10%, 3: 25%, 4: 40% 95% 90% Trier, 24 th March 2015 Ralf Münnich 16 (113) EMOS 2015 Spring School Sampling I + II

21 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Unemployed women, 25 44, distribution of point and variance estimator (25% NR) Dichte 0 e+00 1 e 04 2 e 04 3 e 04 4 e 04 Dichte 0 e+00 1 e 06 2 e 06 3 e 06 4 e Erwerbslose Varianz Trier, 24 th March 2015 Ralf Münnich 17 (113) EMOS 2015 Spring School Sampling I + II

22 Unemployed women, 65 + Raking estimator 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics variance estimator NR rates: 1: 5%, 2: 10%, 3: 25%, 4: 40% 95% 90% Trier, 24 th March 2015 Ralf Münnich 18 (113) EMOS 2015 Spring School Sampling I + II

23 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics Unemployed women, 65 +, distribution of point and variance estimator (25% NR) Dichte Dichte Erwerbslose Varianz Trier, 24 th March 2015 Ralf Münnich 19 (113) EMOS 2015 Spring School Sampling I + II

24 R packages in survey statistics 1.1 Surveys and Samples 1.2 Examples for important surveys 1.3 Quality and error of surveys 1.4 Example: quality of statistical tables 1.5 R packages in survey statistics survey Thomas Lumley R package for survey statistics focusing on estimation and accuracy measurement in complex surveys. Resampling methods and calibration are also considered. ReGenesees ISTAT http: // sampling Yves Tillé The package is focusing on sampling algorithms which help to draw samples from complex designs. Some estimators are also implemented. others Several packages exist on resampling methods (boot). Further emphasis is put on multiple imputation (several packages) and visualizing missing values (vim). Trier, 24 th March 2015 Ralf Münnich 20 (113) EMOS 2015 Spring School Sampling I + II

25 Definitions and notations 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Universe (U) Sample (S) U = {1,..., N} S = {U 1,..., U n } Characteristic of interest Y y i Auxiliary variable X x i (i-th unit) Parameters of the universe Estimates from the sample Total τ τ Proportion θ θ Mean µ µ Ratio ψ = τ 1 τ 2 ψ General parameter π π A distinction between random variables and their outcomes follows from the context. Trier, 24 th March 2015 Ralf Münnich 21 (113) EMOS 2015 Spring School Sampling I + II

26 Population parameters 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Total τ Y = Y = N y i i=1 Mean µ Y = Y = 1 N y i N i=1 Proportion θ Y = P Y = 1 N y i N i=1 Variance σy 2 = 1 N ( yi Y ) 2 N i=1, where y i {0; 1} i Variance (dichotomous variable) σ 2 Y = P Y (1 P Y ) Trier, 24 th March 2015 Ralf Münnich 22 (113) EMOS 2015 Spring School Sampling I + II

27 Sample values 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Total y = n Mean y = 1 n y i i=1 n y i i=1 Proportion p Y = 1 n Variance s 2 y = 1 n 1 n y i i=1 n ( yi y ) 2 i=1, where y i {0; 1} i Variance (dichotomous variable) s 2 y = p Y (1 p Y ) Trier, 24 th March 2015 Ralf Münnich 23 (113) EMOS 2015 Spring School Sampling I + II

28 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Sample selection scheme A sample S consists of elements from the universe U = {1, 2,..., N}: Sampling without replacement: The sample is a subset of the universe. Sampling with replacement: The sample consists of a index set with indices 1,..., N, in which elements may appear several times. Definition 1.1 Let U be a finite population with N elements. A sample S is generated with the help of a sample selection scheme that selects elements from the universe. The set of all possible samples with respect to the sample selection scheme is denoted by S. The probability of drawing a pre-specified sample i is P(S i ). Trier, 24 th March 2015 Ralf Münnich 24 (113) EMOS 2015 Spring School Sampling I + II

29 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Example 2.1 (cf. Lohr, 1999, S. 25) A universe U consists of N = 4 elements. Sampling with replacement with a sample size n = 2 yields: S 1 = {1, 2} S 2 = {1, 3} S 3 = {1, 4} S 4 = {2, 3} S 5 = {2, 4} S 6 = {3, 4} 1. All samples are drawn with equal probability. 2. The probabilities for the samples are: P(S 1 ) = 1/3, P(S 2 ) = 1/6, P(S 6 ) = 1/2, P(S 3 ) = P(S 4 ) = P(S 5 ) = The probabilities for the samples are: P(S 1 ) = 1/3, P(S 2 ) = 1/6, P(S 4 ) = 1/2, P(S 3 ) = P(S 5 ) = P(S 6 ) = 0. What is the sample selection probability of a pre-specified element from the universe? How do the different probabilities affect the inclusion probabilities? Trier, 24 th March 2015 Ralf Münnich 25 (113) EMOS 2015 Spring School Sampling I + II

30 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling First order inclusion probabilities The probability π i for an element i from the universe to be included in a sample is called inclusion probability: S π i = I(i S i ) P(S i ). i=1 The statistic π to estimate the parameter π can be assessed from the set of possible samples S (which may be tedious in practice). We get the (design-based) expected value E( π) from S E( π) = π(s i ) P(S i ) i=1 The same procedure yields the (design-based) variance and MSE. Designs with unequal probabilities. Trier, 24 th March 2015 Ralf Münnich 26 (113) EMOS 2015 Spring School Sampling I + II

31 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Examples for selection schemes SRS Simple random sampling with replacement SRSWOR Simple random sampling without replacement BERN Bernoulli-Sampling (each element of the universe will be selected with probability θ) SYS Systematic sampling One element c from 1,..., a N is drawn. Consequently the elements i a + c will be selected. Note: 1. SRS and SRSWOR yield fixed sample sizes n 2. BERN yields a random sample size with E(n) = θ N. Trier, 24 th March 2015 Ralf Münnich 27 (113) EMOS 2015 Spring School Sampling I + II

32 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Examples for selection schemes SRS Simple random sampling with replacement SRSWOR Simple random sampling without replacement BERN Bernoulli-Sampling (each element of the universe will be selected with probability θ) SYS Systematic sampling One element c from 1,..., a N is drawn. Consequently the elements i a + c will be selected. Note: 1. SRS and SRSWOR yield fixed sample sizes n 2. BERN yields a random sample size with E(n) = θ N. Trier, 24 th March 2015 Ralf Münnich 27 (113) EMOS 2015 Spring School Sampling I + II

33 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Simple random sampling Definition of a random sample A random process in which successively n elements are selected from a finite universe with N elements (n < N) is called random sample selection. The result from a random sample selection is called random sample. Simple random sample Drawing samples from an urn with or without replacement is called simple random sampling. The outcome is a simple random sample. Trier, 24 th March 2015 Ralf Münnich 28 (113) EMOS 2015 Spring School Sampling I + II

34 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling With or without replacement sampling N n many different simple random samples with replacement can be drawn. Each of them is drawn with identical probability. Each element can be drawn 0 to n times. The draws are stochastically independent. N!/(N n)! many different simple random samples without replacement can be drawn. Each of them is drawn with identical probability. Each element can be drawn 0 or 1 times. The draws are stochastically dependent. The first order inclusion probabilities are π i = n N for with and without replacement. Trier, 24 th March 2015 Ralf Münnich 29 (113) EMOS 2015 Spring School Sampling I + II

35 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling The sample mean µ for simple random sampling 1. E( µ) = µ (WR/WOR) 2. V( µ) = σ2 n σ2 (WR) and V( µ) = n N n N 1 3. If U N(µ; σ 2 ) (WR): µ N(µ; σ2 n ) 4. If U is normal (WR): µ µ S t n 1 distributed. n mit S 2 = 1 n 1 5. Without loss of generality for large n n; µ µ µ µ σ σ N n N 1 n (N n large); µ µ S are approximately standard normal. (WOR) resp. n (Y i Y ) 2 is i=1 n; µ µ S N n N 1 n Trier, 24 th March 2015 Ralf Münnich 30 (113) EMOS 2015 Spring School Sampling I + II

36 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling The estimator µ is unbiased for µ (WR/WOR) consistent for µ (only WR is relevant) is BLUE for µ is ML estimator for µ if U is normally distributed (WR) Similar results are derived for totals τ = N µ via the corresponding estimator τ = N µ. For SRSWOR, the random variables are slightly negatively correlated: σ2 CovX i ; X j = N 1 Trier, 24 th March 2015 Ralf Münnich 31 (113) EMOS 2015 Spring School Sampling I + II

37 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Confidence intervals for the mean (WR) CI for µ; U normally distributed; σ 2 known: [ µ z(1 α/2) CI for µ; U normally distributed: [ µ t(1 α/2; n 1) σ n ; µ + z(1 α/2) ] σ n s n ; µ + t(1 α/2; n 1) CI for µ; σ 2 known; n > 30 (CLT): [ ] σ 2 σ 2 µ z(1 α/2) ; µ + z(1 α/2) n n CI for µ; n > 30 (CLT): [ µ z(1 α/2) s ; µ + z(1 α/2) s ] n n ] s n Trier, 24 th March 2015 Ralf Münnich 32 (113) EMOS 2015 Spring School Sampling I + II

38 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Confidence interval for the mean (WOR) CI for µ; σ 2 known; n > 50; sample fraction not close to 1 (CLT): [ σ 2 N n µ z(1 α/2) n N 1 ; µ + z(1 α/2) σ 2 n ] N n N 1 CI for µ; n > 50; sample fraction not close to 1 (CLT): [ µ z(1 α/2) ] s N n n N ; µ + z(1 α/2) s N n n N Confidence intervals for variances are rarely considered in practice and, hence, are omitted here. Trier, 24 th March 2015 Ralf Münnich 33 (113) EMOS 2015 Spring School Sampling I + II

39 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Necessary sample size I CI estimation for µ with confidence level (1 α); e is given; WR; no distributional assumptions are made. e is the half CI length for central CIs; this leads to e = z(1 α/2) σ n. With given prior information ˆσ for σ we get and finally n = z(1 α/2) ˆσ e, n (z(1 α/2)) 2 ˆσ2 e 2. Trier, 24 th March 2015 Ralf Münnich 34 (113) EMOS 2015 Spring School Sampling I + II

40 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Necessary sample size II As before, but now considering WOR sampling: Analogously we have e = z(1 α/2) With (N 1 N) we get σ N n n N 1. and finally n = (z(1 α/2)) 2 ˆσ2 e 2 N n N n(1 + 1 N (z(1 α/2))2 ˆσ2 e 2 ) (z(1 α/2))2 ˆσ2 e 2, n (z(1 α/2))2 N ˆσ 2 (z(1 α/2)) 2 ˆσ 2 + Ne 2. Trier, 24 th March 2015 Ralf Münnich 35 (113) EMOS 2015 Spring School Sampling I + II

41 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Necessary sample size III Again, with prior information θ (WR) we get θ(1 e z(1 α/2) θ) n, and hence n z(1 α/2) θ(1 θ) n (z(1 α/2)) 2 θ(1 θ) e 2. e WOR analogously. Trier, 24 th March 2015 Ralf Münnich 36 (113) EMOS 2015 Spring School Sampling I + II

42 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Stratified random sampling The universe U is decomposed into L pairwise disjoint non-empty subpopulations (primary sampling units) G 1,..., G L : G q1 G q2 = for all q 1, q 2 = 1,..., L; q 1 q 2 and L G q = G q=1 In stratified random sampling the subpopulations are called strata. For the qth stratum we get: µ q = 1 Nq N q i=1 Nq y qi mean of stratum q σq 2 = 1 (y qi µ q ) 2 variance of stratum q N q i=1 Sampling from stratified populations is called stratified random sampling (StrRS). Trier, 24 th March 2015 Ralf Münnich 37 (113) EMOS 2015 Spring School Sampling I + II

43 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling For the universe U holds (γ q := N q /N): 1. Introduction to Survey Sampling µ = θ = σ 2 = σ 2 e = 1 L = 1 L L γ q µ q ; τ = N µ = q=1 L γ q θ q q=1 L γ q σq 2 + q=1 L N q µ q q=1 L γ q (µ q µ) 2 = σw 2 + σb 2 q=1 L (N q µ q Nµ L )2 = 1 L q=1 L Nqµ 2 2 q ( 1 L q=1 L Nqµ 2 2 q N2 µ 2 q=1 L N q µ q ) 2. q=1 Trier, 24 th March 2015 Ralf Münnich 38 (113) EMOS 2015 Spring School Sampling I + II L 2

44 Parameters in the sample Sample allocation 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Given a decomposition of the universe U into L primary sampling units, the breakdown of the total sample size n into L stratum-specific sample sizes n 1,..., n q,..., n L with q n q = n is called allocation. For the qth primary sampling unit we get ȳ q = 1 n q n q i=1 s 2 q = 1 n q 1 y qi n q stratum mean (y qi ȳ q ) 2 stratum variance i=1 Trier, 24 th March 2015 Ralf Münnich 39 (113) EMOS 2015 Spring School Sampling I + II

45 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Estimating means, totals, and proportions in StrRS Sample mean: µ StrRS = L γ q µ q = q=1 L q=1 γ q 1 n q n q i=1 y iq! = µ StrRSWOR The estimates do not differ from WR and WOR in StrRS. Lemma 2.1 The estimator µ StrRS is unbiased for µ (WR and WOR). The variance of the estimator µ is: V( µ StrRS ) = V( µ StrRSWOR ) = L q=1 L q=1 γ 2 q σ2 q n q γq 2 σ2 q Nq n q n q N q 1 (WR) (WOR) Trier, 24 th March 2015 Ralf Münnich 40 (113) EMOS 2015 Spring School Sampling I + II

46 Since E(S 2 q) = σ 2 q, we get V( µ StrRS ) = s 2 µ = 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling L q=1 γ 2 q s2 q n q as an unbiased estimate for V( µ StrRS ). Since E(Sq) 2 = N N 1 σ2 q, V( µ StrRSWOR ) = s 2 µ = L q=1 γ 2 q s2 q n q Nq n q N q (WR) (WOR) is an unbiased estimate for V( µ StrRSWOR ). In case of large stratum-specific sample sizes n q, the estimator µ StrRS is approximately normal (CLT). This yields the (1 α) 100% - CI for µ: [ µ StrRS z(1 α/2) V( µ StrRS ); µ StrRS + z(1 α/2) ] V( µ StrRS ) Trier, 24 th March 2015 Ralf Münnich 41 (113) EMOS 2015 Spring School Sampling I + II

47 Example Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling A universe is split into 4 strata with stratum sizes 10000, 40000, and Drawing a stratified random sample yields the following table: Stratum n q ȳ q s q a) An unbiased estimate for µ is derived as µ StrRS = γ q ȳ q = Trier, 24 th March 2015 Ralf Münnich 42 (113) EMOS 2015 Spring School Sampling I + II

48 Example 2.2 (continued) b) In sampling WR we get 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling V( µ StrRS ) = and thus L q=1 γ 2 q s2 q n q = V( µ StrRS ) 0,3791. The 95%-CI for µ is then: [ ; ] = [55.09; 56.59] In the case of small sample fractions n h /N h (h = 1,..., L), sampling WOR yields (approximately) the same figures. Trier, 24 th March 2015 Ralf Münnich 43 (113) EMOS 2015 Spring School Sampling I + II

49 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Example 2.3 (cf. Example 2.2) The sample results remain now out of consideration. We assume known true stratum-specific variances σ 2 1 = 100, σ2 2 = 55, σ2 3 = 40 and σ 2 4 = 150. Calculate V(µ StrRS) under WR a) using the allocation given in example 2.2 and b) with stratum-specific sample sizes n 1 = 60, n 2 = 180, n 3 = 110, and n 4 = 150. Further, comment on the results. In a) we get V( µ StrRS ) = and for b) V( µ StrRS ) = Trier, 24 th March 2015 Ralf Münnich 44 (113) EMOS 2015 Spring School Sampling I + II

50 2.4.1 Equal allocation 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling n q = n L q = 1,..., L µ StrRS,eq = 1 L L q=1 ȳ q V( µ StrRS, eq ) = L n V( µ StrRS, eq ) = 1 n L γq 2 σq 2 q=1 L q=1 γ 2 q σ 2 q NqL n N q 1 (WR) (WOR) Trier, 24 th March 2015 Ralf Münnich 45 (113) EMOS 2015 Spring School Sampling I + II

51 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Proportional allocation (Bowley) n q = γ q n µ StrRS, prop = 1 L n q ȳ q n V( µ StrRS, prop ) = 1 n V( µ StrRSWOR, prop ) = 1 n = q=1 L γ q σq 2 q=1 L q=1 ( γ q σq 2 1 n ) N q N N q 1 ( 1 n ) 1 N n q = 1,..., L (WR) (WOR) L γ q σq 2 (N q N q 1) q=1 Trier, 24 th March 2015 Ralf Münnich 46 (113) EMOS 2015 Spring School Sampling I + II

52 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Optimal allocation (Neyman-Tschuprov) n q = n n q = n γ q σ q = n L γ h σ h h=1 N q σ q L N h σ h h=1 γ q θq (1 θ q ) L γ h θh (1 θ h ) h=1 V( µ StrRS, opt ) = 1 ( L ) 2 γ q σ q n V( µ StrRS, opt ) = 1 n q=1 ( L q=1 γ q σ q ) 2 1 N L γ q σq 2 q=1 q = 1,..., L (U is dichotomous) (WR) (WOR) Trier, 24 th March 2015 Ralf Münnich 47 (113) EMOS 2015 Spring School Sampling I + II

53 Example Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling N q σq Given a total sample size of n = 500, we calculate the Neyman-Tschuprov allocation via L γ q σ q = q=1 as n 1 = 60, n 2 = 179, n 3 = 114 and n 4 = 147. The variance estimate µ results in (WR) V( µ StrRS, opt ) = This variance is smaller than the variance in Example 2.3! Trier, 24 th March 2015 Ralf Münnich 48 (113) EMOS 2015 Spring School Sampling I + II

54 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Problems in deriving the optimal allocation In sampling WOR one should consider the WOR correction: n q = n N q σ q L h=1 Nq N q 1 N h σ h Nh N h 1 In sampling WOR, an overallocation may result. In practice, the corresponding stratum is sampled completely and the remaining sample size will be reallocated optimally. Theory: box-constrained optimal allocation (later). Theoretically, the optimal allocation is an integer-optimization problem under constraints. Rounding may result in small errors since the sum of the rounded sample sizes is not equal to the total sample size. Trier, 24 th March 2015 Ralf Münnich 49 (113) EMOS 2015 Spring School Sampling I + II

55 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Cost-optimal allocation (Yates-Zacopanay) n q = γ q σ q / cq (C C 0 ) L γ h σ h ch h=1 q = 1,..., L n q = γ q θq (1 θ q )/ cq L γ h θh (1 θ h ) c h h=1 (U is dichotomous) 1 ( L ) 2 V( µ StrRS, YZ ) = γ q σ q cq C C 0 q=1 1 ( L ) 2 1 V( µ StrRS, YZ ) = γ q σ q cq C C 0 N q=1 L γ q σq 2 q=1 (WR) Trier, 24 th March 2015 Ralf Münnich 50 (113) EMOS 2015 Spring School Sampling I + II (WOR)

56 Example 2.5 q N q σq 2 C q We calculate 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling A universe is split into L = 4 strata according to the table alongside. Calculate the cost-optimal allocation given C 0 = 2000 and C = n 1 = = n 2 = n 3 = n 4 = The total sample size results in n = Trier, 24 th March 2015 Ralf Münnich 51 (113) EMOS 2015 Spring School Sampling I + II

57 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Problems in deriving the cost-optimal allocation Again, an overallocation may result in case of sampling WOR. As in the case of the optimal allocation, we have a integer-valued optimization under constraints. Rounding may lead to inadequate sample sizes that are (slightly) too expensive. Trier, 24 th March 2015 Ralf Münnich 52 (113) EMOS 2015 Spring School Sampling I + II

58 Accuracy comparisons I 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling V( µ SRS ) = σ2 n = 1 n (σ w 2 + σb 2 ) 1 = n = V( µ StrRS, prop ) + 1 n σ2 b ( L ) γ q σq 2 + σb 2 q=1 = V( µ SRS ) V( µ StrRS, prop ) After multiplying out we get L q=1 ( γ q σ q L ) 2 γ ι σ ι = ι=1 L γ q σq 2 q=1 }{{} n V( µ StrRS, prop ) and finally V( µ StrRS, prop ) V( µ StrRS, opt ) ( L ) 2 γ q σ q q=1 }{{} n V( µ StrRS, opt ) Trier, 24 th March 2015 Ralf Münnich 53 (113) EMOS 2015 Spring School Sampling I + II

59 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Accuracy comparisons II Given the sample size n the following holds: Note: V( µ StrRS, opt ) V( µ StrRS, prop ) V( µ SRS ) 1. The more homogeneous strata are, the higher is the gain in efficiency by using StrRS instead of SRS. This results from σ 2 w (variance within) being considerably small in contrast to σ 2 b (variance between). This is called the effect of stratification. 2. The more heterogeneous the stratum-specific variances are, the higher is the gain in efficiency while applying the optimal rather than the proportional allocation. Trier, 24 th March 2015 Ralf Münnich 54 (113) EMOS 2015 Spring School Sampling I + II

60 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Box-constrained optimal allocation Aim: Minimise 2-norm of the RRMSE-vector: RRMSE < > ( τ) 2 = G 2 RRMSE(τ <g> ) 2 subject to: g=1 Lower and upper sampling fractions are given in each stratum Maximum number of sampling units Solution via a specialised box-constraints optimization algorithm Exact: Gabler, Ganninger and Münnich (2012), Metrika Numerical: Münnich, Sachs and Wagner (2012), AStA Remark: Overallocation of optimal allocation may be handled easily! Trier, 24 th March 2015 Ralf Münnich 55 (113) EMOS 2015 Spring School Sampling I + II

61 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Box-constraints optimization algorithm Idea: The set of strata is split into three groups. The first group L l uses the lower bounds, the second group L u uses the upper bounds, and the third group L R is allocated optimally. Given d h = N h S h, a necessary and sufficient condition of the box-constraints optimization problem are given by d h d h h L > R M h n m h M h h L l L u and d h m h < d h h L R n for all h L u h L l m h L u M h for all h L l. The optimal solution can be drawn while finding the right breakdown constant (r.h.s.) which splits the strata into the three sets Trier, 24with Marchlinear 2015 effort. Ralf Münnich 56 (113) EMOS 2015 Spring School Sampling I + II

62 Stratification 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Stratification is the unique division of the universe into strata. Two problems arise: 1. The number of strata is given; 2. The stratum boundaries are given. Aim: Gain in efficiency of the estimator Problem: Prior information on universe level Official Statistics Sampling inventory Note: In many cases the number of strata is given in practice. Trier, 24 th March 2015 Ralf Münnich 57 (113) EMOS 2015 Spring School Sampling I + II

63 Stratification 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Stratification is the unique division of the universe into strata. Two problems arise: 1. The number of strata is given; 2. The stratum boundaries are given. Aim: Gain in efficiency of the estimator Problem: Prior information on universe level Official Statistics Sampling inventory Note: In many cases the number of strata is given in practice. Trier, 24 th March 2015 Ralf Münnich 57 (113) EMOS 2015 Spring School Sampling I + II

64 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Dalenius stratification Continuous variable of interest Y with density f (y); Number of strata L is given; Variable of interest is used for stratification; Strata are non-overlapping (division of D Y ) Stratification problem: The L 1 inner stratum boundaries ξ q (q = 1,..., L 1) are of interest, such that V ( µ StrRS ; ξ 1,..., ξ L 1 ) min. Under proportional allocation we get: V ( µ ) L ( ) ( ) StrRS ; ξ 1,..., ξ L 1 = γ q ξ1,..., ξ L 1 σ 2 q ξ1,..., ξ L 1 q=1 min ξ 1,...,ξ L 1 (cf. Münnich, 1997, S. 78 ff.). Trier, 24 th March 2015 Ralf Münnich 58 (113) EMOS 2015 Spring School Sampling I + II

65 Approximate solution 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling cum f rule (Dalenius and Hodges, 1957) The stratum boundaries will be set such that the cumulative values f (ξq ξ q 1 ) are approximately equal while applying the optimal allocation. equal aggregate σ rule (Wright, 1983) The stratum boundaries will be set such that i I q σ i = 1 L N σ i i=1 approximately holds. I q is the set of all indices in stratum q and σ i the individual standard deviation which are expected to be monotonous. This model-based approach uses the equal allocation (which is optimal for the variable of interest assuming the existence of an auxiliary variable). Trier, 24 th March 2015 Ralf Münnich 59 (113) EMOS 2015 Spring School Sampling I + II

66 Example 2.5 Class rel. freq. rel. freq. cum. over until Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Empirical frequencies of the variable of interest (20 classes) with L = 5 strata according to the Dalenius-Hodges approximation. Here, we have / 5 = 873. The stratum boundaries can be allocated at 8,73; 17,64; 26,19; and 34,92. In case the class widthes differ from each other, the computation has to be corrected. Trier, 24 th March 2015 Ralf Münnich 60 (113) EMOS 2015 Spring School Sampling I + II

67 Example Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling 20 units were observed in a survey (Y is the variable of interest): The aim is to build 4 strata. The assumptions σ I y and σ II y 2 are to be considered as true. We get and as aggregate σ. This yields the stratum boundaries 11.43, and as well as 42.27, and respectively. i σ I,i σi,i σ II,i σii,i Note: the information was obtained from extremely few observations (teaching example!). The equal allocation will be applied. Trier, 24 th March 2015 Ralf Münnich 61 (113) EMOS 2015 Spring School Sampling I + II

68 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Poststratification The stratification is used after drawing a simple random sample (Lohr, 1999) SRS yields approximately equal proportions with respect to a stratification Correction of over- and under-representation of the categories (strata) Variance formulae of the proportional allocation are applied Poststratification is applied when the necessary information for stratification is not available in the sampling frame but from other sources (e.g. register). Example: Nationality (German, non-german) and gender Note: Stressing poststratification with many variables may lead to erroneous results Trier, 24 th March 2015 Ralf Münnich 62 (113) EMOS 2015 Spring School Sampling I + II

69 Example 2.7 A simple random sample yields: 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling n y s 2 W M The proportion of women in the universe is 60%. Hence: SRS µ SRS = ( ) = V ( µ SRS ) = ( ) = PostStr µ PostStr = = V ( µ PostStr ) = L q=1 γ 2 q s2 q n q = The variability of the sample size in the denominator of V ( µ PostStr ) was not considered (may be ignored in practice if not too low)! Trier, 24 th March 2015 Ralf Münnich 63 (113) EMOS 2015 Spring School Sampling I + II

70 Example 2.7 A simple random sample yields: 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling n y s 2 W M The proportion of women in the universe is 60%. Hence: SRS µ SRS = ( ) = V ( µ SRS ) = ( ) = PostStr µ PostStr = = V ( µ PostStr ) = L q=1 γ 2 q s2 q n q = The variability of the sample size in the denominator of V ( µ PostStr ) was not considered (may be ignored in practice if not too low)! Trier, 24 th March 2015 Ralf Münnich 63 (113) EMOS 2015 Spring School Sampling I + II

71 Example 2.7 A simple random sample yields: 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling n y s 2 W M The proportion of women in the universe is 60%. Hence: SRS µ SRS = ( ) = V ( µ SRS ) = ( ) = PostStr µ PostStr = = V ( µ PostStr ) = L q=1 γ 2 q s2 q n q = The variability of the sample size in the denominator of V ( µ PostStr ) was not considered (may be ignored in practice if not too low)! Trier, 24 th March 2015 Ralf Münnich 63 (113) EMOS 2015 Spring School Sampling I + II

72 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Further remarks on stratified random sampling In general, stratification is set up as regards content (cf. microcensus) Stratification and estimation variable have to be different (cf. Dalenius) Multidimensional allocation (cf. Schaich and Münnich, 1993): Possible conflict of targets while applying the optimal allocation decision problem Multidimensional stratification: rarely of interest Number of strata: Theory: L high ( variance reduction) integer valued solution! Practice: 5-8 strata High sensitivity to data (and estimators) Trier, 24 th March 2015 Ralf Münnich 64 (113) EMOS 2015 Spring School Sampling I + II

73 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Single stage cluster sampling A universe is split into L primary sampling units (PSU; clusters). The selection of PSUs will follow SRS at the first stage; all secondary sampling units (SSU) in the selected PSUs (indicated by a superscript s) are sampled totally at the second stage. At the first-stage, l PSUs are selected by SRSWOR (single stage cluster sampling: SIC). The total sample size of sampling units is random. µ SIC = L l l q=1 γ sel q µ sel q = L l l q=1 N sel q L N r r=1 N 1 q sel Nq sel i=1 Y iq. is an unbiased estimator for the mean µ in the universe (here: µ sel q = µ sel q ) Trier, 24 th March 2015 Ralf Münnich 65 (113) EMOS 2015 Spring School Sampling I + II

74 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Variance of SIC Single stage cluster sampling equals random sampling without replacement at the first stage; hence, the observed values can easily be summed up. The resulting survey is equivalent to stratified random sampling. It follows: V( µ SIC ) = L2 N 2 σ2 e l L l L 1 which can be (asymptotically unbiasedly) estimated via where V ( µ ) L 2 SIC = N 2 s2 e l s 2 e = 1 l 1 L l L l ( r=1 N sel r, µ sel r, N µ ) SIC 2. L Trier, 24 th March 2015 Ralf Münnich 66 (113) EMOS 2015 Spring School Sampling I + II

75 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Estimation of totals under SIC We have τ SIC = N µ SIC and hence V( τ SIC ) = L 2 σ2 e l L l L 1 or V( τsic ) = L 2 s2 e l L l L respectively. Estimation of proportions under SIC Here, we have µ r q = p r q. Then, the relation of interest follows from σ 2 e;θ N 2 = 1 L (γ q θ q θ L L )2. q=1 Trier, 24 th March 2015 Ralf Münnich 67 (113) EMOS 2015 Spring School Sampling I + II

76 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Accuracy comparisons I In general, we assume (l L): This yields V( µ SRS ) V( µ SIC ) V( µ SIC ) 1 l σ 2 b = 1 l (σ2 σ 2 w) 1. If σ 2 b 0 (and hence σ2 w = σ 2 ), the variance V( µ SIC ) becomes considerably small. SIC is then approximately equivalent to SRSWOR. 2. In contrast, if σ 2 w 0, the variance V( µ SIC ) may become very large. This effect is called cluster effect. Trier, 24 th March 2015 Ralf Münnich 68 (113) EMOS 2015 Spring School Sampling I + II

77 2.1 Notational background 2.2 Simple random sampling 2.3 Stratified random sampling 2.4 Single stage cluster sampling 2.5 Two-stage cluster sampling Accuracy comparisons II Whenever the PSUs are of approximately the same size (N q N), one can show that ( V( µ SIC ) = 1 l L N L q SSTOT = q=1 i=1 ) N SSB L 1 l (Y qi Y q ) 2 +, where L N q (Y q Y ) 2 = SSW + SSB. q=1 This leads to the intraclass correlation coefficient (ICC) and finally to ICC = 1 N N 1 SSW SSTOT V( µ SIC ) = LN 1 N(L 1) (1 + (N 1) ICC ) V( µ SRSWOR ) Trier, 24 th March 2015 Ralf Münnich 69 (113) EMOS 2015 Spring School Sampling I + II