2000 Mathematics Subject Classi cation: 62D05.This article presents an application of representative method in public opinion polls

Dabala, Wieslawa

Lednicki, Bronislaw

Bulgarian Digital Mathematics Library at IMI-BAS

Pliska Stud. Math. Bulgar. 18 (2007), 91–102STUDIA MATHEMATICABULGARICAESTIMATION OF FRACTION OF DISTINGUISHEDELEMENTS IN POPULATION BASED ON PARTIALLYREALIZED RANDOM SAMPLEWies lawa Daba la Bronis law LednickiThis article presents an application of representative method in public opin-ion polls.1. IntroductionThe finite population comprised persons aged 18 or above, residing in the territoryof Poland. For organizational reasons, individual sampling could not be used.Therefore, the sample was selected using a three-stage sampling scheme, where:– first-stage sampling units were census units (little areas with no more than300 housing units);– second-stage sampling units were housing units (dwellings, flats or housesin rural areas);– third-stage sampling units were adults from the housing units.2000 Mathematics Subject Classification: 62D05Key words: applications of mathematical statistics, applications of representative methodin public opinion polls92 W. Daba la, B. Lednicki2. Stratification of the populationFirst-stage sampling units, i.e. census units, were divided into strata h (h =1, 2, 3, . . . , 64). One stratum comprised the census units belonging to the sameclass with regard to the population of a province (voivodship). In each of the16 administrative provinces of Poland, the localities have been divided into thefollowing categories:– provincial capitals,– other cities/towns,– villages.The number of strata was bigger in the provinces with a bigger adult population.The predetermined number of census units to be sampled was stratified pro ratato the number of second-stage sampling units in a stratum, i.e. proportional tothe number of housing units.Let:n∗ – total number of census units to be sampled,n∗h – number of census units to be sampled in the hth stratum,n∗ =64∑h=1n∗h,h – number of a stratum, h = 1, 2, 3, . . . , 64then:n∗h =MhMn∗(1)where:Mh – number of secondary sampling units (dwellings) in the hth stratum,Mh =Nh∑j=1MhjMhj – number of dwellings in the jth census unit belonging to the hth stratum;M – total number of dwellings in the population.M =64∑h=1Nh∑j=1Mhj =64∑h=1MhNh – number of census units in the hth stratum;N – total number of census units, N =64∑h=1NhEstimation of Fraction of Distinguished Elements in Population . . . 933. Sampling of first-stage units, i.e. sampling of census unitsFirst-stage units were selected using a procedure proposed by W. G. Madow in1949 and subsequently modified by H. O. Hartley and J. N. K. Rao in 1962.This procedure consists of systematic sampling with selection probabilities pro-portional to the value of the attribute adopted as the criterion of the samplingunit size, after a prior randomly ordering of these units. In this case, the num-bers of housing units in census units are the size criterion. In each stratum, thesampling procedure was as follows:a) Census units were randomly ordered;b) First order probabilities; Πhj - were determined:Πhj = n∗hMhjMh(2)c) A cummulative series of probabilities was created:Π∗hj =kh∑j=1Πhj(3)where:Π∗h,0 = 0 and Π∗h,Nh= n∗hj = 1, 2 . . . , kh kh = 1, 2, . . . , Nhd) The sampling interval t = 1 was determined and a random sampling startingpoint a was determined from the (0; 1〉 range.e) The jth census unit was included in the sample, if the following inequalitywere satisfied:Π∗h,j−1 < a + (b− 1) ≤ Π∗h,j(4)where:j = 1, 2 . . . , Nh b = 1, 2, . . . , n∗h.Census units from each stratum h were selected using the same procedure.94 W. Daba la, B. Lednicki4. Sampling of second-stage units, i.e. sampling of housing unitsHousing units (second-stage units) were selected from the previously sampledcensus units using simple sampling scheme without replacement. Six housingunits were sampled from each census unit.5. Sampling of third-stage units, i.e. selecting adults to be in-terviewed from a housing unitsAdults were selected from previously sampled housing units (secondary samplingunits). One adult was sampled from each previously sampled housing unit. Theindividuals were selected using the Kisch’s method. It is a procedure of simplesampling of one person from each dwelling. Under this method, one random num-ber is generated for the expected number of persons in a dwelling within the 1 to9 range. These numbers are assigned to the number assigned to the housing unitduring the sampling and cannot be changed. An interviewer visiting a dwellingreceives these numbers together with the address, generates the sampling framein the dwelling and selects an individual using the random numbers. In this way,the selected individual can be uniquely identified.The table with random numbers for one housing unit is as follows:Expectednumberof adultsin dwelling:c = 1 2 3 4 5 6 7 8 9Randomnumber RN1 RN2 RN3 RN4 RN5 RN6 RN7 RN8 RN9i.e sampledperson’snumberinthe samplingframewhere:RNc – random number for the expected number of adults in the housing unitfrom the 1–c range.c = 1, 2, . . . , 9Estimation of Fraction of Distinguished Elements in Population . . . 95Let:Lhjl – real number of adults in the l-th dwelling (sampled), the j-th census unitin the h-th stratum (Lhjl = 1, 2, 3, 4, . . . , 9).6. Estimation and evaluation of estimatorvarianceLet:U – Finite population (adult population of the country),U = {u1, u2, . . . , u30000000},i = 1, 2, . . . , 30, 000, 000 – the number of an individual in the population,Ud – non-empty subset of U separated for analysis, or, in special case, the wholepopulation. In practice, an analysis of population subsets selected for analysispurposes after the survey execution is frequently required. Due to the broad scopeof the matter, in this paper we shall only consider the situation where Ud = U ,d = 1, 2, . . . , D,D – number of subsets in the analysisX ′, Y ′ – functions defined on U with values 1 or 0.The estimated parameter is a fraction of units with distinguished property. It isthe ratio of global values of functions X ′ and Y ′.R =XY(5)where:X =∑XiXi ={1 for ui ∈ U with distinguished property0 for other units}Y =∑YiYi ={1 for ui ∈ U0 for other units}Xi, Yi – values of functions X′, Y ′ for units ui ∈ U .Let:s∗ – the total sample sampled from the population U ,s – the total sample after realization from the population U (s ⊂ s∗ ⊂ U)96 W. Daba la, B. LednickiIf s∗ = s then estimator R is the following statistic:r =xy(6)where:x =∑h∑j∑l∑iMhjΠhjLhjl6xi(7)y =∑h∑j∑l∑iMhjΠhjLhjl6yi(8)xi – values of X′ for ui ∈ sxi ={1 for ui ∈ s with distinguished property0 for other units}yi – values of Y′ for ui ∈ syi ={1 for ui ∈ s0 for other units}Since the realized sample s is smaller than the selected sample s∗, the formulas7 and 8 are transformed as follows:x =∑h∑j∑l∑iWhjlixi =∑h∑j∑l∑iMhjΠhjLhjlmhjxi(9)y =∑h∑j∑l∑iWhjliyi =∑h∑j∑l∑iMhjΠhjLhjlmhjyi(10)where:Whjli – weight of the ith person in the realized sample s, sampled from the lthhousing unit in the jth census unit, the hth stratum.Whjli =MhjΠhjLhjlmhj(11)mhj – number of sampled housing units in which interviews were conductedmhj ≤ 6 (6 housing units were sampled from each census unit).Other symbols as previously.Estimation of Fraction of Distinguished Elements in Population . . . 97Estimator r variance is as follows:D2(r) ≈D2(x) + r2D2(y) − 2COV (x, y)y2(12)In the case where the Hartley-Rao scheme was used for first-stage sampling,and simple sampling without replacement at the second-stage, the xHR and theyHR estimators variances: D2(xHR), D2(yHR) and COV (xHR, yHR), are given inliterature (Bracha,1996). The approximate estimator for variance D2(xHR) waspublished by Sarndall (1992).7. Adjusted estimatorThe estimation method described in literature (section 6) has the following short-comings:- It does not account for non-response sufficiently;- In the case of basic demographic distributions, it allows relatively largediscrepancies between the survey results and the ongoing estimates of theCentral Statistical Office. The main reasons for discrepancies include ran-dom errors, non-responses and incomplete sample execution.Therefore, an adjusted estimator rA was used:rA =xAyA=∑iwixi∑iwiyi(13)where weights wi, in addition to the varying probabilities of respondent selection(as in formula 11), also take into account the differences in response rates betweenthe six locality categories and the population structure according to the CentralStatistical Office by gender, age group (five age groups are distinguished) andplace of residence (urban / rural).Step 1Weights for second- stage units, i.e. housing units, were determined:W1hj =Mhj6Πhj(14)98 W. Daba la, B. LednickiLet us also remind that:Mhj – number of housing units in the jth census unit in the hth stratum; Πhj –first-order probabilities calculated using formula 2.6 is the number of housing units sampled from the j th census unit.Step 2Determining response rates for the six locality categories. The following catego-rization of localities has been adopted:k = 1 – Warsaw (the capital)k = 2 – other cities with more than 500,000 inhabitantsk = 3 – towns with 100,000 to 500,000 inhabitantsk = 4 – towns with 20,000 to 100,000 inhabitantsk = 5 – towns with up to 20,000 inhabitantsk = 6 – villagesThe response rate for survey Ek for the kth locality category was determinedusing the following formula:Ek =BkBk + Dk(15)where:Bk – number of housing units surveyed, i.e. dwellings in which a respondent wassuccessfully sampled and interviewed.Dk – number of housing units in the kth locality category, in which interviewswere not conducted for different reasons.Dk does not include sampling frame errors, i.e. non-existent housing units,unoccupied dwellings and housing units used for non-residential purposes,e.g. offices, warehouses etc. Bk and Dk values were determined using thefollowing formulas:Bk =∑h∑jmbkhjW1hj(16)Dk =∑h∑jmdkhjW1hj(17)where:Estimation of Fraction of Distinguished Elements in Population . . . 99mbkhj – number of housing units surveyed in the jth census unit of the hthstratumand the kth locality category;mdkhj – number of housing units not surveyed in the jth census unit of the hthstratum and the kth locality category, excluding housing units included inthe sampling frame due to errors.The calculated Ek values allowed to determine the W2 weight taking into accountthe response rate in the kth locality category.W2khj =W1hjEk(18)Step 3During the third step, differences in the demographic distributions between thesample and the official Central Statistical Office data were eliminated. For thispurpose, the sample was divided ex post into 20 groups distinguished by the placeof residence (rural/urban), gender, and subsequently into 5 age categories. Basedon data from the sample, gp values were determined, i.e. the number of personsby gender, place of residence and age category, using the following formula:gp =∑k∑h∑j∑l∑iW2khjLhjlzkhjli(19)where:p – group number after the sample division – by place of residence, gender andage categoryp = 1, 2, 3, . . . , 20zkhli ={1 for for persons included to the group p0 for other persons}(20)Subsequently, the following values were determined for each of p demographicgroups:Gp – the number of persons in the population belonging to the p group – basedon the Central Statistical Office data.Step 4Ultimately, weight Wpkhjli of the ith respondent, sampled from the I th housing100 W. Daba la, B. Lednickiunit in the jth census unit,the kth locality category and the pth demographicgroup was as follows:Wpkhjli = W2khjGpgpLhjl(21)For the sake of simplification, let us adopt the following formula:Wi = Wpkhjli(22)wi = naWi∑iWi(23)where:Wi – “big weight” of the ith respondent in realised samplewi – “litttle weight” of the ith respondent in realised samplena – size of realized sample of persons, i.e. the number of interviews conducted.The final adjusted ratio estimator, after simplifying the symbols, shall be ex-pressed with the following formula:rA =∑iWixi∑iWiyi=∑iwixi∑iwiyi(24)8. Evaluation of rA estimator varianceThe complex nature of the rA estimator resulting from its adjustment and thecomplex sampling scheme caused problems with finding a formula for its vari-ance. Therefore, in order to evaluate the variance, the Jacknife method (Wolter1985) was used, in the version appropriate for a compound estimator of randomvariables ratio, along with a stratified sampling scheme.Under this method, “pseudo-values” of the parameter being estimated are cal-culated based on the sample less one unit. If the first-stage units were stratifiedbefore sampling, the estimator “pseudo-values” are estimated by eliminating theunits from the first stratum, the second stratum etc. until all possibilities areexhausted in the last stratum. Each time, after reducing the sample by one first-stage unit (census unit), estimator “pseudo-values” are determined.Variance estimator:Estimation of Fraction of Distinguished Elements in Population . . . 101d2(rA) =∑h[nh −1nh] ∑j[rhj − rh]2(25)where:rhj – evaluation of parameter R based on the sample reduced by the jth censusunit of the hth stratum;rh – average evaluation of parameter R based on the rhj evaluations from the hthstratum;rh =1nh∑jrhj j = 1, 2, . . . , nh;nh – number of first-stage units in the hth stratum, in which the survey wasexecuted.nh ≤ n∗h9. ConclusionThe presented three-stage sampling scheme and estimation methods were usedin practice in the years 1990-2003. The response rate was falling gradually fromover 83% to just over 70%.It became possible to use the sampling frame which allows two-level sampling.The change of the sampling scheme did not, however, cause an increase of theresponse rate and simplified methods of estimator variance evaluations still hadto be used.As the response rates are falling, a question arises about the ”threshold” responserate below which the parameters are no longer correctly estimated and not closeenough to the actual ones in the population. What other methods could be usedinstead of the ones based on adjusted estimators and estimations of their varianceusing simplified methods based on resampling (e.g. bootstrap, jacknife etc.)?R E F E R E N C E S[1] Bracha Cz. Wykorzystanie informacji o cechach dodatkowych w badaniachreprezentacyjnych. ZBSE GUS i PAN (1987) Warsaw, Poland.[2] Bracha Cz. Teoretyczne podstawy metody reprezentacyjnej. WydawnictwoNaukowe PWN (1996) Warsaw, Poland.[3] Bracha Cz. Metoda reprezentacyjna w badaniu opinii publicznej i mar-ketingu. Efekt (1998) Warsaw, Poland.102 W. Daba la, B. Lednicki[4] Bracha Cz., Lednicki B., Wieczorkowski R. Wykorzystanie z lor˙onychmetod estymacji do dezagregacji danych z badan aktywnos´ci ekonomicznejludnos´ci w roku 2003. ZBSE. GUS (Central Statistical Office) (2004) War-saw, Poland.[5] Daba la W. How to estimate population parameters using partially real-ized random sample? XXV International Seminar on Stability Problems forStochastic Models, Maiori/Salerno, Italy. Series of the Journal of Mathe-matical Sciences (2005) (Kluwer-Plenum, New York-London).[6] Efron B., Tibshirani R.J. An Introduction to the Bootstrap Monographson Statistics and Applied Probability, 57. Chapman and Hall, New York(USA) - London (Great Britain).[7] Hartley H.O., Rao J.N.K. Sampling with unequal probabilities and with-out replacement. AMS 33 (1962).[8] Hunsen M.H., Hurwitz W.N., Madow W.G. Sample Survey Methodsand Theory. John Wiley & Sons, Inc. (1993) New York-Chichester-Brisbane-Toronto-Singapore.[9] Madow W.G. On the theory of systematic sampling II. AMS 20 (1949).[10] Sarndall C.E., Swensson B, Wretman J. Model Assisted Survey Sam-pling. (1992) Springer Verlag.[11] Tryfos P. Sampling Methods for Applied Research. Text and Cases. (1996)John Wiley & Sons, Inc. New York, USA.[12] Volter K.M. Introduction to Variance Estimation. (1985) Springer Verlag.Wies lawa Daba laPublic Opinion Research Centre, Warsaw, Poland115, Universitetska Str.e-mail: w.dabala@cbos.plBronis law LednickiCentral Statistical Office, Warsaw, Poland

Estimation of Fraction of Distinguished Elements in Population Based on Partially Realized Random Sample

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Estimation of Fraction of Distinguished Elements in Population Based on Partially Realized Random Sample

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