Survey sampling reference guidelines

Transcription

1 ISSN Metodologies and Working papers Survey sampling reference guidelines Introduction to sample design and estimation tecniques 2008 edition EUROPEAN COMMISSION

2 How to obtain EU publications Our priced publications are available from EU Booksop (ttp://booksop.europa.eu), were you can place an order wit te sales agent of your coice. Te Publications Office as a worldwide network of sales agents. You can obtain teir contact details by sending a fax to (352) Europe Direct is a service to elp you find answers to your questions about te European Union Freepone number (*): (*) Certain mobile telepone operators do not allow access to numbers or tese calls may be billed. More information on te European Union is available on te Internet (ttp://europa.eu). Luxembourg: Office for Official Publications of te European Communities, 2008 ISBN ISSN Cat. No. KS-RA EN-N Teme: General and regional statistics Collection: Metodologies and working papers European Communities, 2008 Te publication was prepared by Professor Risto Letonen and Senior Researcer Kari Djerf. Manuscript finalised in August 2007.

3 EUROSTAT L-2920 Luxembourg Tel. (352) website ttp://ec.europa.eu/eurostat Eurostat is te Statistical Office of te European Communities. Its mission is to provide te European Union wit ig-quality statistical information. For tat purpose, it gaters and analyses figures from te national statistical offices across Europe and provides comparable and armonised data for te European Union to use in te definition, implementation and analysis of Community policies. Its statistical products and services are also of great value to Europe s business community, professional organisations, academics, librarians, NGOs, te media and citizens. Eurostat's publications programme consists of several collections: News releases provide recent information on te Euro-Indicators and on social, economic, regional, agricultural or environmental topics. Statistical books are larger A4 publications wit statistical data and analysis. Pocketbooks are free of carge publications aiming to give users a set of basic figures on a specific topic. Statistics in focus provides updated summaries of te main results of surveys, studies and statistical analysis. Data in focus present te most recent statistics wit metodological notes. Metodologies and working papers are tecnical publications for statistical experts working in a particular field. Eurostat publications can be ordered via te EU Booksop at ttp://booksop. europa.eu. All publications are also downloadable free of carge in PDF format from te Eurostat website ttp://ec.europa.eu/eurostat. Furtermore, Eurostat s databases are freely available tere, as are tables wit te most frequently used and demanded sortand long-term indicators. Eurostat as set up wit te members of te European statistical system (ESS) a network of user support centres wic exist in nearly all Member States as well as in some EFTA countries. Teir mission is to provide elp and guidance to Internet users of European statistical data. Contact details for tis support network can be found on Eurostat Internet site.

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5 Contents 1. Introduction 7 2. Survey planning and reporting Basic concepts and definitions Overall survey design Reporting of survey quality Sampling frame issues Tecniques for sample selection and estimation Preliminaries Basic sampling tecniques Simple random sampling Systematic sampling Sampling wit probability proportional to size Stratified sampling and allocation tecniques Cluster sampling Sample size determination Use of auxiliary information in estimation pase Treatment of nonresponse Reweigting to adjust for unit nonresponse Imputation to adjust for item nonresponse Software References Web links 36 Appendix

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7 1. Introduction Te guidelines concern basic principles and metods of survey sampling. Tis includes survey planning, survey quality, sampling and estimation, and nonresponse. Te approac is non-tecnical; only necessary tecnical materials are included. Te metods are illustrated wit practical examples, and references to statistical software are given wen relevant. Because a compreensive treatment of te various aspects of survey sampling is not possible in some brief guidelines, we ave concentrated on selected topics we believe are of importance for readers. We ave aimed at a practical guide intended for experts wose practical experience in survey sampling is limited but wo ave some background knowledge in basic statistics. For furter information on topics covered and extensions, we refer to selected literature. Te guidelines are organized as follows. Capter 2 discusses survey planning and reporting. A number of basic concepts and definitions are given, also including survey quality. Basic sampling tecniques are introduced in Capter 3. We discuss metods suc as simple random sampling, systematic sampling and cluster sampling. Te use of auxiliary information plays a key role in modern survey sampling, and metods are discussed suc as PPS sampling, stratified sampling and model-assisted metods including ratio and regression estimation. Sample size determination is treated and illustrated. Capter 4 covers nonresponse and discusses reweigting and imputation metods. A brief summary of software available for survey sampling and analysis is included in Capter 5. We ave included a compreensive list of references on current survey sampling literature in Capter 6. Capter 7 includes a list of selected links to web materials relevant to te area. 2. Survey planning and reporting 2.1. Basic concepts and definitions Definition of a survey A survey refers to any form of data collection. A sample survey is more restricted in scope: te data collection is based on a sample, a subset of total population - i.e. not total count of target population wic is called a census. However, in sample surveys some sub-populations may be investigated completely wile te most sub-populations are subject to selected samples. In te subsequent capters te term survey is devoted to sample surveys. Descriptive surveys versus analytical surveys Descriptive surveys, including censuses, are typical in statistical offices. Tey tend present information on parameters like totals, averages or proportions at te total population level or some well-defined sub-populations. In surveys were te empasis is on analysis, te interest is focused on connections and interdependences between penomena. Te parameters of interest are connected wit statistical models, suc as linear models, and are represented by correlation or regression coefficients. However, it is important for bot types of surveys to estimate te unknown parameters as reliably as possible. 7

8 Social surveys vs. Business surveys In social surveys te focus is related wit persons and ouseolds: e.g. population statistics, labour force participation, wages and salaries, ouseold consumption, poverty and income distribution, education, cultural activities, ealt and oter interested topics. In business surveys te focus is related wit enterprises, establisments and/or oter business units like te local kind of activity units, including farms. Te interest may vary from production composition and amount to investment plans, employment, use of energy, output waste etc. Social surveys and business surveys differ from eac oter also in oter aspects. In official statistics business surveys are often mandatory wile social surveys tend to be voluntary; te data collection modes are more versatile in social surveys; even te sampling designs can be different Overall survey design In recent years many textbooks ave been publised on survey metodology. Groves et al. (2004) provide a good overview on te wole process from te design to te analysis and interpretation. In addition tere is a number of specific literature on various data collection modes, testing questionnaires and questions, interviewing strategies etc. Operational pases of a survey are described e.g. by Sundgren (1999). It includes various tasks from te definition of te main objectives, data collection strategy, processing of data, production of results, evaluation of quality till arciving. All tasks are important to guarantee te various uses of data and teir quality. Te readers are recommended to obtain more information from appropriate literature like Lyberg et al. (1997), or Biemer & Lyberg (2003). Figure 1. Flow cart of survey process (see e.g. Statistics Finland) 8

9 2.3. Reporting of survey quality Te users sould be reported wit appropriate information on survey quality, preferably from all stages of te survey process. It as been a tradition to report of survey quality by distinguising various sources of error wic may occur during te many stages of survey operations. For example, Biemer & Lyberg (2003) describe following types of errors: Specification, Frame, Nonresponse, Measurement, Processing, and Sampling error. Some may be born randomly but unfortunately various sources tend to introduce systematic errors. Sampling errors Standard errors for te estimable parameters, often point estimates are te oldest quality measures. Tey (and oter estimates derived from tose like coefficients of variation or confidence intervals) were introduced during te rise of survey metodology in 1940s. Measurement errors Besides te sampling errors te oter types of errors were introduced quite early. Te first UN recommendations on reporting survey quality were given already in 1950s and te measurement errors were already included. However, te implementation of systematic reporting took muc longer. Total survey error Te total survey error of a parameter θ is measured by te mean square error (MSE), i.e. sum 2 2 of te variance and squared bias: MSE( ˆ θ ) = E( ˆ θ θ ) = V ( ˆ θ ) + Bias ( ˆ θ ). Sampling variance is derived from te sampling design, te oter components affecting its estimate are sample size, te variability of te parameter of interest and sampling weigts. Sampling error, i.e. square root of sampling variance, is a random error by definition. Bias is te difference between te true value and te expectation of te estimator, and wen nonzero it represents systematic error. Unfortunately te MSE estimation requires repeated sampling and tus cannot easily be carried out wit large-scale surveys. Some subtle metods ave, owever, been suggested to evaluate te total error (see e.g. Lessler & Kalsbeek 1992). Te quality dimensions and standards of te European Statistical System provide a good frame to report on quality. Te quality dimensions are Relevance, Accuracy, Timeliness and Punctuality, Comparability, Coerence, and Accessibility and Clarity. Relevance describes ow te statistical survey meets te user needs and requirements. Accuracy contains te traditional measures on survey quality (like standards errors, confidence intervals and coefficients of variation etc.). Timeliness and punctuality measure te fresness of data and te results. Comparability and coerence are related wit various forms of comparisons: different sources describing te same penomenon, comparability of te same survey over various domains, like geograpical areas, comparability over time etc. Finally, Accessibility and clarity describe te various form data are available and results disseminated, metadata and oter user support etc. Furtermore, a list of quality indicators ave been constructed to make te follow-up easier for tose surveys wic are repeated more or less regularly. Te Eurostat Quality website presents all relevant documents on quality reporting and also some current practices and guidelines on te issue: ttp://ec.europa.eu/eurostat/quality. 9

10 Te International Monetary Fund (IMF) and Organization for Economic Co-operation and Development (OECD) ave created own standards wic are also widely used especially in te field of economic statistics (see International Monetary Fund 2003) Sampling frame issues Population and frame Target population is te population we teoretically are interested in. It is assumed to be fixed (and finite). Frame population is te population we can obtain. Survey population is te intersection of tose above. Tose tree populations do not quite coincide because te frame population tends to contain some erroneous elements called coverage errors. Below we present some typical reasons for coverage errors: time lags between te moment te sample frame was created and it was actually used failure to include new birts in te frame failure to include or exclude elements wic ave moved (pysical removals, enterprises wic ave canged teir industry etc.) failure to remove deats and similar out-of-scope elements Overcoverage means tat our sampling frame contains elements wic do not belong to our target population. Overcoverage can normally be detected during te field-work. Undercoverage is a muc more problematic penomenon since often it cannot be detected and assessed in a reliable manner. Tere may be no realistic way to include all possible differences between te target population and te ultimate sampling frame but tose known sould be included. Kis (1965) advocated a stratum of surprises to include tose cases. Sometimes no good frame exists for te target population and one as to find oter solutions described below. Multiple frames Multiple frames may occur if te target population can be compiled from several independent sources. Use of many frames is not uncommon in developing countries but can also used in developed societies wen new penomena are investigated. Clustered frames It may well appen tat tere is not a good population frame for te ultimate sampling units, or tat te creation of suc would be muc too expensive. Ten te next solution is to seek for an alternative from te combinations of te elements, i.e. seek for clustered frames. 10

11 Consider, for example, a study of scool cildren: even if a population frame would be available covering all cildren attending te scools, te field work will become muc ceaper if te scools and/or classes are selected instead of te pupils randomly over te wole population. In large population and ouseold surveys we most often deal wit clusters wic comprise to some natural combination of elements, e.g. people living in enumeration districts or administrative regions. Oter issues Double listings of te same elements sould always be removed from te frame if found. Small sub-populations may sometimes be quite impossible to reac altoug tey are known, e.g. people living in remote mountainous villages. For cost and oter reasons tey may be removed from te sampling frame. Ten a difficult question arises: do te estimates from te reduced population reflect te properties of tose from removed sub-populations? Cut-off samples are anoter example related wit te same problem. Normally cut-off samples are applied in business surveys were te smallest units do not contribute too muc to te parameter of interest. However, since one part of te target population is deliberately excluded tere is a cance to obtain bias in estimation. Auxiliary information Information obtained from background variables to be used eiter at te sampling stage (e.g. to create strata or clusters, calculate measure of size etc.) or after data collection to calculate weigts etc. Sometimes auxiliary data cannot be obtained from te sampling frame but can be available after te survey from oter sources, suc as official statistics. 3. Tecniques for sample selection and estimation 3.1. Preliminaries In a sample survey, a probability sample is drawn from te frame population by using a specified sampling design. Typically, te sampling design consists of a combination of various sample selection tecniques. A complex sampling design can involve clustering and stratification and several stages of sampling. In simple cases, sampling of elements is carried out directly from te sampling frame. In all cases, some of te well-documented sample selection tecniques are used in te sampling procedure. Good examples of relevant literature on sampling tecniques are Kis (1965), Cocran (1977); Lor (1999), and Letonen & Pakinen (2004), wic is te primary source for tis section. Helpful supplemental materials on survey sampling and estimation, including computational examples using real survey data, can be found in VLISS-virtual laboratory in survey sampling, representing a web extension of te Letonen and Pakinen textbook. Te application can be accessed freely at ttp:// Many of te common sample selection tecniques can be readily implemented by statistical software products, suc as te SAS procedure SURVEYSELECT. Te properties of sampling tecniques vary wit respect to statistical efficiency and certain practical aspects, suc as suitability to a given sampling task, requirements for application and 11

12 user friendliness. Often te study design and time and budget constraints affect te coice of te sampling design in a given survey setting. An important additional aspect is te role of auxiliary information in a given sampling procedure. Let us first discuss te standard sample selection scemes from tis point of view. Use of auxiliary information in sampling and estimation It is often useful to incorporate auxiliary information on te population in a sampling procedure. In practice, tere are different ways to obtain auxiliary information. For example, in te so-called register countries (e.g. Scandinavian countries), sampling frames used in official statistics production often include auxiliary information on te population elements, or tese data are extracted from administrative registers and are merged wit te sampling frame elements at te micro level. In oter cases, aggregate-level auxiliary information can be obtained from different sources suc as publised official statistics. Use of auxiliary information in sampling and estimation is an expanding feature in official statistics production. Auxiliary information can be useful in te construction of an efficient sampling design and furter, at te estimation stage for improved efficiency for te actual sample. To be useful, auxiliary information sould be related to te variation of te study variables. In simple random sampling (SRS), te sample is drawn witout using auxiliary information on te population. Terefore, SRS provides a reference sceme wen assessing te gain from te use of auxiliary information in more complex designs or in improving te efficiency of estimation for a given sample. Auxiliary information does not play a role in standard application of systematic sampling (SYS). Tus, te efficiency of SYS tends to be similar tan tat of SRS. Tis also olds if population elements in te sampling frame are in random sort order wit respect to te study variable. In a metod called implicit stratification, auxiliary information can be used in te form of te list order of elements in te frame. Now, SYS can be more efficient tan SRS if tere is a certain relationsip between te ordering of elements in te sampling frame and te values of te study variable. Sampling wit probability proportional to size (PPS) is a metod were auxiliary information as a key role. An auxiliary variable is assumed to be available as a measure of te size of a population element. Varying inclusion probabilities for population elements can be assigned using te size variable. Efficiency improves relative to SRS if te relationsip between te study variable and te size variable is strong. PPS is often used in business surveys and in general, for situations were te sampling units vary wit a size measure. Stratified sampling (STR) relies strongly on te use of auxiliary information. In STR, te frame population is first divided into non-overlapping subpopulations called strata, and sampling is executed independently witin eac stratum. If te strata are internally omogeneous wit respect to te study variable, i.e. if te witin-stratum variation of te study variable is small and a large sare of te total variation is captured by te variation between te strata, ten STR can be more efficient tan SRS. In cluster sampling (CLU), te population is assumed to be readily divided into naturally formed subgroups called clusters. A sample of clusters is first drawn from te population of clusters. In te next stage, all elements of te sampled clusters are taken in te element sample (one-stage cluster sampling), or a sample of elements is drawn from eac sample cluster (twostage cluster sampling). If te clusters are internally omogeneous, wic is usually te case, 12

13 ten CLU is less efficient tan SRS. Tis clustering effect can be reduced by stratifying te population of clusters, tending to improve efficiency. Te sampling tecniques introduced above can be used to construct a manageable sampling design for a sample survey, eiter using a particular metod or more usually a combination of metods. In all metods excluding SRS, auxiliary information in te form of auxiliary variables can be incorporated in te sampling procedure. Note tat te use of auxiliary information in SRS, SYS and stratified sampling requires tat te values of auxiliary variables must be available for every population element. Auxiliary information in cluster sampling concerns at least te grouping of te population elements into clusters. If additional auxiliary data are available on te population of clusters, tese data can be used for example for stratification or PPS sampling purposes. Use of auxiliary information in te sampling pase is typical in descriptive surveys were te number of study variables is small. Efficiency gains can be obtained if te association between te study variable(s) and te auxiliary variables is strong. Auxiliary information can be used for te selected sample in te estimation pase. Use of auxiliary information in te estimation pase involves flexibility: te sample design can be kept simple and in te estimation pase, te use of auxiliary information can be tailored for diverse study variables. In addition, requirements for auxiliary data in standard metods are weaker tan in te previous case, because unit-level auxiliary data only are needed for te sampled elements, and te auxiliary data can be incorporated at an aggregate level in te estimation procedure. Some of te standard metods are ratio estimation, regression estimation and post-stratification. All tese metods use statistical models as assisting or working models wen incorporating te auxiliary data in te estimation procedure. Te metods tus are called model-assisted. In ratio and regression estimation, te population total of a continuous auxiliary variable is assumed known. Te assisting model is of regression-type linear model. In ratio estimation, te model is witout an intercept term, i.e. te intercept is assumed zero. Efficiency can improve if te study variable and te auxiliary variable are correlated. But te metod can be ineffective if tere is a nonzero intercept term in te true model. In regression estimation, te assisting model is again of regression-type, but now wit an intercept term. Efficiency can improve if te study variable and te auxiliary variable are correlated. Post-stratification resembles stratified sampling, but te stratification is carried out after te sample selection. Te selected sample is divided into non-overlapping subgroups called poststrata according to a categorical or classified auxiliary variable (or several suc variables), and te estimation follows tat of stratified sampling. Similarly as in stratified sampling, efficiency can improve if te post-strata are internally omogeneous wit respect to te study variable. Post-stratification is often used for adjusting for unit nonresponse (see Section 4.1). Tus, auxiliary information on te population can be used in te construction of te sampling design and, for a given sample, to improve te efficiency in te estimation pase. As a rule, efficiency of estimation can improve by te proper use of auxiliary information. Parameters, estimators and quality measures Let our parameter of interest be a fundamental parameter in survey sampling, te population N total T = y k = 1 k of study variable y. In te formula for te total, y k are te (unknown) values of te study variable and N is te number of elements in te population. Many parameters 13

14 routinely used in survey sampling, suc as means, proportions, ratios and regression coefficients, can be expressed as functions of totals. To ave an estimate for te unknown population total T, a sample is drawn from te population and te sample values of te study variable are measured. An estimator of te population total T is denoted by ˆt. Te concept estimator refers to a calculation formula or algoritm tat is used for te sample to obtain a numerical value for te estimate. A simple example is te sample mean 14 n = k = 1 k, y y / n wic is calculated using te n sample measurements. Using te sample mean, an estimate for te population total is calculated as ˆt = N y. Tese derivations old for simple sampling designs; more complex derivations are needed for complex sampling designs. In survey sampling, estimators are preferred tat fulfil certain teoretical properties. Tese are unbiasedness, meaning tat te expectation of an estimator coincides wit te target parameter, i.e. E( tˆ ) = T, and te bias is defined as Bias( tˆ ) = E( tˆ ) T. Consistency is a somewat weaker property, referring to te beaviour of an estimator to better matc wit te value of te target parameter wen sample size n increases, and to reproduce te target parameter wen te sample size coincides N, te population size. Precision of an estimator refers to its variability and is measured by te design variance Var( t ˆ). Te smaller is te design variance, te better is te precision. A precise estimator is called efficient. And accuracy of an estimator refers to combined bias and precision properties of an estimator and 2 is measured by te mean square error: MSE( tˆ ) = Var( tˆ ) + Bias ( tˆ ). In survey sampling practice, estimators are used tat are unbiased or at least consistent. A callenge for survey statistician is for a given sampling task to obtain efficient estimators wose design variances are as small as possible. Tis is for ig reliability of te results calculated by using te collected sample survey data. Te standard error (s.e), coefficient of variation (c.v) and design effect (deff) of an estimator are commonly used quality measures of estimators. Te quality measures are derived from te teoretical properties introduced above. For an estimator ˆt of population total, te measures are defined as follows. Estimated standard error: s.e( tˆ ) = vˆ ( tˆ ), were vˆ( t ˆ) is te estimated design variance or sampling variance of te total estimate ˆt. Estimated coefficient of variation or relative standard error: c.v( tˆ ) = s. e( tˆ ) / tˆ, i.e. te estimated standard error divided by te estimate itself. Coefficient of variation is often expressed in percentages, 100 c.v%. Coefficient of variation is routinely reported in official statistics. C.v is often used as a quality standard in te context of te ESS (see Section 3.3). Design effect (deff) (Kis 1965) measures te statistical efficiency of te sampling design wit respect to simple random sampling (SRS) and is given by ˆ( ˆ ˆ v t ) deff ( t ) = vˆ ( tˆ ), were te numerator is te sampling variance of te total estimator under te actual (possibly complex) sampling design and te denominator represents te sampling variance under an assumption of simple random sampling of a sample of similar size. Using te design effect, SRS

15 effective sample size is determined as n ˆ eff = n / deff ( t ), tat is, te actual sample size n divided by te design effect of te total estimate. Te formula for deff gives rise to te following remarks: (a) deff < 1 (b) deff = 1 (c) deff > 1 Te actual sampling design is more effective tan SRS. Correspondingly, effective sample size is larger tan te actual sample size. Te efficiency of te actual sampling design is similar to tat of SRS. Te actual sampling design is less effective tan SRS. In tis case, effective sample size is smaller tan te actual sample size. In survey sampling practice, a natural goal is te case (a). In tis effort, te use of te available auxiliary information in te sampling design is beneficial. Stratified sampling and PPS sampling are often used for tis purpose. In addition, efficiency can be improved in te estimation pase by incorporating auxiliary data in te estimation procedure via modelassisted tecniques. In cluster sampling, te case (c) is often encountered because of te internal omogeneity of te clusters wit respect to te variables of interest Basic sampling tecniques Basic sampling tecniques include simple random sampling, systematic sampling and sampling wit probabilities proportional to size (PPS). Tese metods are used in sampling designs as te final metods for selecting te elementary or primary sampling units (PSU:s) and for working out randomization. A manageable sampling design for a survey often involves stratification, clustering and multiple stages of sampling. Stratification of te population into non-overlapping subpopulations is a popular tecnique were auxiliary information can be used to improve efficiency. In cluster sampling, te practical aspects of sampling and data collection are te main motivation for te use of auxiliary information in te sampling design Simple random sampling Simple random sampling (SRS) is often regarded as te basic form of probability sampling. SRS is applicable to situations were tere is no previous information available on te population structure. Simple random sampling directly from te frame population ensures tat eac population element as an equal probability of selection. Tus, SRS is an equal-probability sampling design. As a basic sampling tecnique, simple random sampling can be included as an inerent part of a sampling design. In addition, simple random sampling sets a baseline for comparing te relative efficiency of a sampling design by using te design effect statistic introduced above. In simple random sampling of n elements, every element k in te population frame of N elements as exactly te same inclusion probability, tat is, π k = π = n / N. Recall tat inclusion probability is te probability of a population element to be included in a n element sample. An inclusion probability is assigned for every population element before carrying out te sampling procedures. Inclusion probabilities depend on te sampling design and are by definition greater tan zero for all population elements. 15

16 In practice, SRS can be performed eiter witout replacement (SRS-WOR) or wit replacement (SRS-WR). WOR type sampling refers to te case were a sampled element is not replaced in te population; tis also means tat a population element can be sampled only once. In a WR sceme, a sampled element is replaced in te population. In bot cases, te inclusion probability π = n / N remains, and te only difference is in te variance formula of te statistic of interest. As a general rule, WOR-type SRS is more efficient tat WR-type SRS, tat is, te variance in SRS-WOR tends to be smaller tan tat in a SRS-WR counterpart. Tis property also olds for te oter sampling designs and explains te frequent use of witout replacement type designs in survey sampling practice. Under SRS, an estimator of te target parameter T can be written simply as ˆ n t = N y / k n = Ny, k = 1 were n y = y / n is te sample mean. Alternatively, by using te SRS inclusion k = 1 k probabilities π, te estimator can be expressed in te form, n n n tˆ = y / π = y /( n / N) = w y k = 1 k k = 1 k k = 1 k k were wk = N / n is te sampling weigt, i.e. te inverse inclusion probability. Note tat in SRS, te sampling weigts are equal for all sample elements. In more complex designs to be addressed, te sampling weigts can vary between elements (as in PPS sampling) or groups of elements (as in stratified sampling). Using te estimated total, te population average or mean 16 N Y = y / N can be estimated by y = tˆ / N. Note tat we assumed ere a known population size N, wic is a realistic n assumption in practice. But if N is unknown at te estimation stage, an estimator Nˆ w can be used for te population size. For an estimator ˆt of population total under SRS-WOR, te sampling variance of ˆt is given by 2 2 vˆ ( tˆ ) = N (1 n / N)(1/ n) sˆ, 2 2 were sˆ = n ( y ) /( 1) k 1 k y n is te sample variance of te study variable y. Te quantity = (1 n / N) in te sampling variance formula is called te finite population correction (fpc). Note tat if te sampling fraction n / N is small, as is te case in typical sampling designs for persons or ouseolds, practical importance of te fpc is minor, because fpc is close to one. But tis is not necessarily so in sampling designs for business surveys were sampling fractions can be muc larger. For SRS-WR, te only difference in te sampling variance vˆ( t ˆ) is tat te fpc is given by (1 1/ N). Tis difference also indicates better efficiency for te SRS-WOR design: te design effect of ˆt under SRS-WR is deff ( tˆ ) = (1 1/ N) /(1 n / N) > 1, assuming tat sample size n is larger tan one and smaller tan population size N. Note tat we used SRS-WOR as te reference SRS design in te deff formula; tis is a natural coice but sometimes, SRS-WR is put in tis role in certain statistical software. k = 1 k = k = 1 k

17 To summarize, if te sampling fraction (n/n) is small te fpc for SRS-WOR will be close to 1. And vice versa: if te sample size n approaces te population size N, te variance estimate vˆ( t ˆ) will reduce. Tus, in a census te sampling variance is zero. In practice SRS is executed wit an appropriate piece of software. For example, te SAS procedure SURVEYSELECT can be used for bot SRS-WR and SRS-WOR. In real life sampling wit SRS we mostly deal wit te witout-replacement type SRS design. Example. Bernoulli sampling provides an example of an SRS-WOR type sampling sceme. In tis metod, te sample size is not fixed in advance but is a random variate wose expectation is n, te desired sample size. Tis property leads to a variation in te sample size wit te expected value Nπ and variance N(1 π)π, were π stands for te inclusion probability. Te randomness in te sample size is relatively unimportant in large samples. Let us briefly introduce te tecnique. To carry out Bernoulli sampling, we need to carry out te following steps: Step 1. Fix te value of te inclusion probability π, were 0 < π < 1, so tat te expected sample size will be Nπ, te product of te population size and te inclusion probability. If te desired sample size is n, ten π = n/n. Step 2. Append tree variables, let say PROB, IND and UNI, to te sampling frame data set. PROB is set equal to te cosen value of π, and IND is set to zero, for all N population elements. For UNI, a value from a uniform distribution over te range (0, 1) is drawn independently for eac population element, starting from te first element. A pseudo random number generator can be used in generating te random numbers. Step 3. Te decision rule for inclusion of a population element in te sample is te following. Te kt population element is included in te sample if UNI < π, and correspondingly, we set IND = 1 for te selected element (oterwise, te value of IND remains zero). Step 4. Treat all population elements sequentially by using Step 3. Wen Steps 1 to 4 are completed, te sum of IND over te sampling frame appears to be close (or, equal) to te desired sample size n. Te elements aving IND = 1 constitute te Bernoulli sample. Te procedure can be easily programmed for example wit Excel, SAS or SPSS. Appendix 1. contains a sort example of Bernoulli sampling Systematic sampling Systematic sampling (SYS) is a widely used sampling tecnique in situations were te sampling frame is an ordinary electronic (or manual) data base, suc as a population register, a register of business firms or farms, or a list of scools. SYS also is an equal probability sampling design because te inclusion probability of a population element in an n element sample is π = n / N. Steps in te selection of a systematic sample of n elements from a population of N elements are te following: 1. Define te sampling interval q = N/n, were an integer q is assumed. 17

18 2. Select a random integer a wit an equal probability of 1/q between 1 and q (a pseudo random number generator for uniform distribution over te range (1, q) of e.g. Excel, SAS, SPSS can be used). 3. Select elements numbered a, a + q, a +2q, a +3q,..., a + (n 1)q in te sample. Tus, wit an integer q, SYS results in an n element sample. If q is not an integer, all sampling intervals can be defined as of equal lengt except one. In practice, tere are several ways of selecting a systematic sample. Te one we introduced above represents an example of SYS sampling wit one random start. Alternatively, two, or more generally m, independent systematic samples can be taken using te procedure above. Te size of eac SYS sample is ten n/m elements and te lengt of te sampling interval is m q. Tis tecnique is suitable if variance estimation is to be carried out using so-called replication tecniques (see Wolter 2007). Furter, a systematic sample can be drawn by treating te elements in te sampling frame as a closed loop. Beginning from te randomly selected integer A from [1, N], te selection proceeds successively by drawing elements A + q, A + 2q,, till te end of te frame, and ten te selection continues from te beginning of te frame. Te loop will be closed wen n elements ave been drawn. Tese random start metods lead to te selection of a SYS sample of n elements, and te tecniques are equivalent wit respect to te estimation. In statistical software products, suc as te SAS procedure SURVEYSELECT, tere are advanced sampling algoritms for SYS tat use fractional intervals to provide exactly te specified sample size n. For SYS, tere is no known analytical variance estimator for te design variance, even for suc a simple estimator as te total. Terefore, approximate variance estimators are used in practice (see e.g. Wolter 2007; Letonen and Pakinen 2004, Section 2.4). Estimation under systematic sampling depends on te knowledge on te sorting order of te sampling frame: 1. If te sorting order of te sampling frame can be assumed random wit respect to te study variables and all auxiliary variables, estimation wit SYS will correspond to tat of SRS- WOR. Tus, formulas derived for SRS can be used. 2. If te sampling frame is sorted by an auxiliary variable (or, several suc variables), SYS sampling will produce a sample wic tends to mirror correctly te structure of population wit respect to te variables used in sorting. Sorting te frame before SYS sampling is called implicit stratification. For example, in some cases it is a good idea to sort te frame according to te regional population structure. Ten a systematic sample will retain te appropriate population distribution across regions. Additional cases are tose were te population is already stratified or a trend exists tat follows te population ordering, or tere is a periodic trend (all tese situations can also be reaced by appropriate sorting procedures). Periodicity may be armful in some cases, especially if armonic variation coincides wit te sampling interval. Te estimation under implicit stratification corresponds to te estimation under stratified sampling. Systematic sampling, including implicit stratification, can be carried out for example wit te SAS procedure SURVEYSELECT. 18

19 Example. Let us consider SYS sampling of n = 200 elements from a population of N = 2000 elements. Te sampling interval is q = N/n = 2000/200 = 10. We next draw a random integer a between 1 and 10, let a = 7. Te SYS sample of n = 200 elements consists of population elements numbered 7, 17, 27,,1997. Te inclusion probability for every population element is π = π = n / N = 200 / 2000 = 0.1 and te constant sampling weigt for te sampled elements k is w = w = 10. k Sampling wit probability proportional to size In sampling wit probability proportional to size (PPS), te inclusion probability depends on te size of te population element. Reduction in variance can ten be expected if te size measure and te study variable are closely related. It is assumed tat te value Z k of te auxiliary size variable z is known for every population element k. Typical size measures are variables tat pysically measure te size of a population element. In business surveys, for example, te number of employees in a business firm can be used as a measure of size, and in a scool survey te total number of pupils in a scool is also a good size measure. PPS sampling can be very efficient, especially for te estimation of te total, if a good size measure is available. In PPS sampling, te inclusion probability of an element in a n element sample is π = np = nz / T, were T k k k z = Z is te sum of size measures over te N element population and k z N k = 1 k p is called te single-draw selection probability. In PPS, te inclusion probabilities π k vary between elements and tus, PPS is an unequal probability sampling design. A PPS sample can be drawn eiter witout or wit replacement. Calculation of te inclusion probabilities is easier to manage under WR type sampling, because te population remains uncanged after eac draw. In PPS-WOR, te population canges after eac draw and te inclusion probabilities must be re-calculated for te remaining elements. Te basic principles of estimation under PPS sampling are introduced ere only briefly. Under PPS-WOR, an unbiased estimator of te population total T is given by, t n n ˆ = w y = y / π k = 1 k k k = 1 k k were wk = 1/ π k is te sampling weigt. Te estimator is called te Horvitz-Tompson (HT) estimator or expansion estimator. Te HT estimator is design unbiased and is very popular in practice. An estimator of te variance of te estimated total is n n ( ˆ) = 1 1 k l kl k l, v ˆ t ( w w w ) y y k = l= kl kl were w = 1/ π. Te variance estimator of te HT estimator contains te second-order inclusion probabilities π kl (i.e. probabilities to include bot elements k and l in te sample), wose computation is often impractical, especially for large samples. Terefore, approximations are often used in practice. One alternative is 2 n 2 ( ˆ) k k vˆ t = N (1/ n) ( y /( Np ) y) /( n 1), k = 1 19

20 wic corresponds to a wit-replacement PPS sceme were te second-order inclusion probabilities are zero, because te draws are mutually independent. Tere are different versions of PPS sampling scemes available for practical purposes. Examples are te cumulative total metod wit replacement or witout replacement, systematic PPS sampling wit unequal probabilities and Poisson sampling. For example, Poisson sampling as a witout-replacement type design resembles Bernoulli sampling were te sample size is a random quantity; te difference is in te calculation of te inclusion probabilities. Despite of te property of a random sample size, Poisson sampling is sometimes considered attractive because te second-order inclusion probabilities reduce to π kl = π kπ l wic simplifies te calculation of te sampling variance. Te book by Brewer & Hanif (1983) provides a good source for te various PPS metods. Te most commonly used PPS tecniques are implemented in te SAS procedure SURVEYSELECT Stratified sampling and allocation tecniques In stratified sampling (STR) te target population is divided into non-overlapping subpopulations called strata. Tese are regarded as separate populations in wic sampling of elements can be performed independently. Witin te strata, some of te basic sampling tecniques, SRS, SYS or PPS, are used for drawing te sample of elements. Stratification involves flexibility because it enables te application of different sampling tecniques for eac stratum. In general, tere are several reasons for te popularity of stratified sampling: 1. For administrative reasons, many frame populations are readily divided into natural subpopulations tat can be used in stratification. For example, strata are identified if a country is divided into regional administrative areas tat are non-overlapping. 2. Stratification allows for flexible stratum-wise use of auxiliary information for bot sampling and estimation. For example, PPS tecnique can be used in sampling witin te stratum, and ratio or regression estimation can be used for te selected sample, depending on te availability of additional auxiliary information in te stratum. 3. Stratification can involve improved efficiency if eac stratum is omogeneous wit respect to te variation of te study variables. Hence, te witin-stratum variation will be small, wic is beneficial for efficiency. 4. Stratification can guarantee representation of small subpopulations or domains in te sample if desired. Tis means tat inclusion probabilities can vary between strata. Te variation is controlled by te so-called allocation tecniques. In stratified sampling, te population is divided into H non-overlapping subpopulations of size N 1, N 2,, N,..., N H elements suc tat teir sum is equal to N. For stratification, auxiliary information is required in te sampling frame. Regional, demograpic and socioeconomic variables are typical stratifying variables. A sample is selected independently from eac stratum, were te stratum sample sizes are n 1, n 2,, n,..., n H elements, and teir sum is equal to n, te overall sample size. Tere are alternative strategies to determine stratum sample sizes for a given survey. In some cases, te overall sample size n is first fixed and ten allocated to te strata. Tis is typical in cases were te strata temselves are not of interest (i.e. producing statistics for te separate 20

21 strata is not te primary aim). If te survey involves statistics production for eac stratum (e.g. a regional area or industrial group), ten it is important to ascertain large enoug stratum sample sizes. In tis case, te stratum sample sizes n are first determined (see Section 3.3). Te most common allocation tecniques for defining te stratum sample sizes are proportional allocation, equal allocation, optimal or Neyman allocation and power or Bankier allocation. To give an idea of allocation, let us introduce briefly te tree first mentioned metods (Bankier allocation requires more detailed additional information on te population distribution witin strata, see for example Letonen & Pakinen 2004, Section 3.1). Proportional allocation is te simplest allocation sceme and is widely used in practice. It presupposes knowledge of te stratum sizes, since te sampling fraction n / N is constant for eac stratum. Te number of sample elements n in stratum is given by n n W =, were W = N / N is te stratum weigt, and n is te specified overall sample size. Proportional allocation guarantees an equal sare of te sample in all te strata and involves an equal probability sampling design were te inclusion probability π k = π = n / N of population element k in stratum is constant. Tus, te sampling weigt also is a constant w = w = N / n, and te design is called self-weigted. k Equal allocation provides an equal sample size n = n / H for eac stratum, were H is te number of strata. If te stratum sizes N vary, inclusion probabilities also vary and are given by π = n / N = n /( H N ) for element k in stratum. Tus, sampling weigts are k wk = H N / n. If all stratum sizes N are equal, ten π k = π = n / N and an equalprobability design is obtained. Optimal or Neyman allocation is usable if te population standard deviations S for individual strata of te study variable y are known or a reliable figure is available. In practice, close approximations to te true standard deviations may be made from experience gained in past surveys. Tus, Neyman allocation is often used in continuous business surveys. Te stratum sample sizes are first calculated. Te number of sample units n in stratum under optimal allocation is calculated as n N S = n. H = 1 N S Te overall sample size n is ten te sum of stratum sample sizes. In optimal allocation, a stratum wic is large or as a large witin-stratum variance as more sampling units tan a smaller or more internally omogeneous stratum. Te tree allocation scemes are illustrated in an example below. Allocation under STR sampling is furter illustrated, wit additional computational examples, in te VLISS application, te web extension of Letonen & Pakinen (2004). In stratified sampling, an estimator ˆt of population total T y is te sum of stratum total estimators, given by tˆ = tˆ, were t n n ˆ = y / π = w H y = 1 is te Horvitz-Tompson k = 1 k k k = 1 k k estimator of te stratum total T. Because te samples are drawn independently from eac 21

22 stratum, te sampling variance of ˆt is te sum of witin-stratum variances vˆ( t ˆ ), tat is, H vˆ ( tˆ ) = vˆ ( tˆ ). Because in STR sampling, te sampling variance only depends on te = 1 witin-stratum variances, it is a good idea to try to construct internally omogeneous strata wit respect to te study variable y. =, were / = 1 k = 1 k For example, assuming SRS-WOR in eac stratum, te total estimator is given by t H n ˆ N / n y N n is te stratum-specific sampling weigt. Wit proportional allocation tis simplifies as, because te tˆ = N / n y = N / n y H n H n = 1 k = 1 k = 1 k = 1 k weigts N / n are equal to constant N / n. Tis reflects te self-weigting property of proportional allocation. Estimation under stratified sampling is discussed in more detail in standard sampling textbooks; good sources are Kis (1965) and Lor (1999). Stratified sampling can be carried out for example wit te SAS procedure SURVEYSELECT, wic allows for several discrete variables as stratification variables. Example. As a simple example, consider STR sampling wit proportional, equal and Neyman allocation scemes. A stratified SRS-WOR sample of n = 200 elements is drawn from a population of N = 2000 elements (Table 1). Tere are H = 5 strata in te population. In proportional allocation, a 10% sample is drawn from eac stratum, involving a constant sampling weigt w = w = 2000 / 200 = 10 for every k k sample element. In equal allocation, a sample of n = 200 / 5 = 40 elements is drawn from eac stratum, involving varying sampling weigts w = N / n for eac stratum. For Neyman allocation, we assume tat reliable knowledge on S, te population standard deviation (Std. Dev.) of y, is available, and tat figure is equal to all strata except Stratum 3, wose standard deviation is larger indicating larger variation for te study variable. Stratum-wise sample sizes are calculated as n = NS / 64000, = 1,,5. Tis allocation sceme provides larger relative sample size for Stratum 3 and correspondingly, smaller sampling weigt, wen compared to te oter strata. In tose strata, te weigts are nearly equal resembling proportional allocation. Table 1. Proportional, equal and Neyman allocation scemes for STR sampling of n = 200 elements from an N = 2000 element population. Stratum Stratum size N N / N Proportional allocation Sample size n Sampling weigt w k Sample size n Equal allocation Sampling weigt w k Std. Dev. S NS Neyman allocation Sample size n Sampling weigt All w k 22