ON SOME MODIFICATION OF THE SUM-QUOTA SAMPLING SCHEME

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1 M A T H E M A T I C A L E C O N O M I C S No. 7(14) 2011 ON SOME MODIFICATION OF THE SUM-QUOTA SAMPLING SCHEME Abstract. A smple extenson of the well-known sequental sum-quota samplng scheme s proposed. The modfcaton facltates the smultaneous examnaton of several samplng unts between checks of the cost lmt. Ths may speed up the data gatherng process whle some degree of control over the varable sample cost s stll retaned. It s proposed to estmate the populaton total of the studed characterstc under such a samplng scheme usng emprcal estmates of ncluson probabltes evaluated n a smulaton study. It appears that at least n some stuatons the emprcal Horvtz-Thompson estmator s approxmately unbased. Keywords: fxed-cost samplng, sum-quota samplng, emprcal ncluson probabltes. JEL Classfcaton: C Introducton The costs of carryng out the sample survey are usually dvded nto two broad categores, namely: fxed costs and varable costs (Groves, 1989, pp. 51). Fxed costs do not depend on the partcular sample whch s actually drawn whle varable costs do depend on t. Varable costs arse due to varous crcumstances assocated wth the survey, such as the geographcal localzaton of sample unts, the extent of effort needed to gather the data from ndvdual unts or the effect of ntervewer tranng, and they are generally harder to control. As an example let us consder the stuaton where the populaton under study conssts of 100 unts u 1,..., u 100 such that per unt cost of gatherng nformaton s equal to 5 for u 1,..., u 10, and t s equal to 2 for the remanng unts. If the sample of ten elements s drawn from such a populaton usng smple random samplng wthout replacement (SRSWOR), then the expected cost of examnng all sample unts s proportonal to the average of per-unt cost n the whole populaton and so t equals = 23. However, a much less lucky sample contanng unts u 1,..., u 10 Department of Statstcs, Unversty of Economcs n Katowce, Boguccka Street 14, Katowce, Poland. E-mal: wojcech.gamrot@ue.katowce.pl

2 84 yelds the total cost of 50, whch s over two tmes greater than the expected cost. Obvously, such an extreme event s unlkely. However, even the theoretcal possblty of a dramatc ncrease n the cost may render the analyss based on the assumpton of equal ncluson probabltes questonable when the nsttuton carryng out the survey does not have the capacty to handle such unexpected costs and would be forced to modfy the sample or cancel the survey completely. The well known sum-quota samplng scheme (Pathak, 1976; Kremers, 1985, 1986; Lehmann, Cassella, 1998) s dedcated to controllng the survey cost n unfavorable stuatons as descrbed above. The scheme does not rely on the requrement of fxed sample sze. Instead t s attempted to keep the total sample cost constraned. Ths s acheved by sequentally drawng ndvdual unts one by one to the sample wth equal probabltes untl the total cost of the sample reaches some predefned lmt. Hence the total cost of the sample s guaranteed not to exceed the fxed budget of the survey. A sgnfcant drawback of the sum-quota procedure manfests tself when per-unt samplng costs are not known n advance. Then the procedure requres examnng all unts sequentally and to test f the cost lmt has not been breached before every examnaton. Ths practcally prevents the smultaneous examnaton of more than one unt. Hence the survey may become very tme-consumng and ts organzaton (e.g. dvson of feld work) problematc. In ths paper a modfcaton of the sum-quota samplng scheme s consdered. It makes possble the smultaneous examnaton of more than one populaton unt whle stll mantanng some control over the total cost of the sample. 2. Sum-quota samplng scheme and ts extenson Let the fnte populaton of sze N be represented by a set of unt ndces U = {1,..., N}. Let c 1,..., c N represent ndvdual per-unt costs of examnng ndvdual populaton unts. Let some pre-determned maxmum cost lmt L be gven. The orgnal sum-quota samplng procedure of Pathak (1976) s carred out by drawng unts randomly untl for some say M-th unt the total cost of M unts already drawn s greater or equals L. The M-th unt s dscarded and precedng M-1 unts form a sample. The jont cost of the sample s guaranteed not to exceed L. The sample sze s random, and ncluson probabltes may n general vary, so the sample s non-smple. The nclusons of any two dstnct unts n the sample are not necessarly nde-

3 On some modfcaton of the sum-quota samplng scheme 85 pendent events. Let us now propose the followng extenson of ths orgnal procedure: K-sets of unts are sequentally drawn from U usng smple random samplng wthout replacement. Each drawn k-set s removed from the populaton so that -th k-set s drawn from among just N k (-1) unts ( = 1, 2,...). If N-k (-1) < k for some, then the whole populaton s taken as a sample (a trval specal case). The procedure stops when the cumulatve cost of all the elements n all drawn k-sets exceeds L. The last k-set for whch the lmt L s exceeded s ncluded n the sample. Denote the sample obtaned ths way by s. The proposed procedure dffers from Pathak s algorthm n two ways. Frstly, t allows for the smultaneous examnaton of k unts so that testng f the cost lmt s breached may be done less frequently. Secondly, the last k-set for whch the cost lmt s breached s ncluded n the sample, whch elmnates the need to assess the cost of examnng all unts n the current k-set before they are sampled. The proposed procedure dffers from the k-fnte populatons samplng scheme consdered by Kremers, Robson (1987) because all the unts wthn the k-set are examned and any partcular unt may be drawn wthn many k-sets (samplng wthn k-sets s not ndependent). As a result of the ntroduced modfcatons the guarantee not to exceed the cost lmt exactly s lost. However some control over the random total cost of the sample s gven by: C s s retaned. Let c (1),..., c (N) be a sequence of ndvdual costs sorted n decreasng order. The maxmum excess C L of sample cost over the lmt L may be computed as: c max ( ) 1,..., k Moreover, snce C s not lower than L, and consequently C [L, L + max ], the varance of C s not greater than: c 2 ( ) max V. max C 4

4 86 When the populaton s large and ts dstrbuton not extremely skewed, both max and V max (C) may be reasonably small n comparson to the expected cost. It may also be possble to estmate both on the bass of external knowledge of the survey subject, and to set the lmt L n such a way that suffcent funds are avalable to examne any possble sample. The frstorder ncluson probabltes defned as: P( s) P( s) for = 1,..., N and characterzng the proposed procedure may vary. Ther exact calculaton from the defnton s prevented by the combnatoral exploson effect even for rather modest sample szes. The followng example presents ther estmates for a certan specfc stuaton. Example 1. Let N = 40 and c = for = 1,..., 40. Thus, per unt costs are dstrbuted qute unformly on the (1, 40) nterval. Let L = 80. A total of samples were drawn ndependently usng both sum-quota procedure and the modfed sum-quota algorthm wth k = 2. The observed frequences of each populaton unt appearng n the sample that may be treated as estmates for frst-order ncluson probabltes are shown n Fgure 1. Somewhat surprsngly, ncluson probabltes for a modfed scheme dffer very modestly from each other and even tend to grow wth ncreasng per-unt cost. Ths s n contrast wth the orgnal sum-quota samplng scheme where they clearly decrease when the per-unt cost ncreases. Let us also defne a vector of sample membershp ndcators I = [I 1,..., I N ] n such a way that 0 for s I 1 for s for = 1,..., N. Its observed correlaton matrx for the orgnal scheme as well as for the modfed scheme wth k = 2 are shown n Fgure 2. It appears that populaton unts whch are relatvely cheap to examne correspondng sample membershp ndcators are postvely correlated whle for relatvely expensve unts correlaton tends to be negatve. Ths effect manfests for both schemes, and the correlaton appears to be rather weak. In general for both schemes the sample sze may vary. However ts expectaton whch s equal to the sum of all frst order ncluson probabltes s clearly greater for the modfed scheme. Ths s not surprsng snce the modfed scheme allows for drawng between one and k extra unts above the cost lmt L. Dstrbutons of sample sze are compared n the followng example nvolvng real data. s

5 On some modfcaton of the sum-quota samplng scheme Sum-quota Sum-quota(k=2) Fg. 1. Frst order ncluson probabltes characterzng sum-quota samplng and modfed sum-quota samplng scheme wth k = 2 for the same cost lmt Source: author s own work. Fg. 2. Correlatons c j = cor(i,i j ) between sample membershp ndcators for sum-quota samplng and modfed sum-quota samplng scheme wth k = 2 for the same cost lmt Source: author s own work.

6 88 Example 2. Consder the data obtaned n the agrcultural census carred out n 1996 by the Polsh Central Statstcal Offce (GUS). The data descrbes a populaton of 695 farms n the Gręboszów borough. Assume that the cost for obtanng data from any ndvdual farm s roughly proportonal to ts area (ths mght be justfed n the case of a survey nvolvng the assessment of geo-botanc assets, veternary nspecton of the cattle or some specalzed examnaton of crops). The hstogram of the farm area n the whole populaton s shown n Fgure 3. The farm area exhbts a strong postve skew wth a mean equal to and a medan equal to 651. Denote G = c c N. A total of samples were drawn for L = 0.1 G usng orgnal sum quota samplng scheme and a modfed sum-quota samplng wth k = 2. The dstrbuton of sample cost relatve to G computed as λ = C/G s shown n Fgure 4. Frequency area Source: author s own study. Fg. 3. Dstrbuton of farm area As expected, the observed values of λ for the orgnal sum-quota samplng scheme never exceed the desred value 0.1 correspondng to the lmt L, whle for the modfed scheme they always do. However, absolute devatons (n plus or n mnus) from L never exceed 10% and n most cases they do not exceed 5% (for orgnal scheme 99.83% of absolute devatons dd not exceed 5%, for the modfed scheme 98.12% of absolute devatons dd not exceed 5%). The maxmum possble devaton of max = correspondng to λ = was never observed for the modfed scheme. Hence one mght conclude that both samplng schemes provde a reasonable degree of control over samplng costs.

7 On some modfcaton of the sum-quota samplng scheme 89 Sum-quota L/G L max G Frequency Sum-quota(k=2) L/G L max G Frequency Fg. 4. Dstrbuton of sample cost under sum-quota samplng and modfed sum-quota samplng scheme wth k = 2 for the same cost lmt Source: author s own study. 3. Estmaton based on emprcal ncluson probabltes Let y 1,..., y N be fxed values of some populaton characterstc of nterest. If frst-order ncluson probabltes were known, one would easly estmate the populaton total t = y y N wthout desgn bas usng the well-known Horvtz-Thompson statstc: ˆ y tht. s When unknown, these probabltes may be estmated from a smulaton experment nvolvng massve numbers of ndependently drawn sample replcatons s 1,..., s H. Let X represent the number of tmes -th unt s

8 90 ncluded n H sample replcatons. It was shown by Fattorn (2006) that the statstc: ˆ X 1 H 1 s a consstent estmator for. It may be used to form an emprcal Horvtz- -Thompson estmator for t n the form: tˆ EHT s y ˆ. The above statstc always takes fnte values snce ˆ 0 for = 1,..., N and t s asymptotcally unbased for the populaton total t (n terms of H growng to nfnty for all ncluson probablty estmates). 4. Smulaton study To shed some lght on the propertes of EHT tˆ, a smulaton study was carred out usng the populaton from Example 2. A total of samples were drawn usng modfed sum-quota samplng scheme wth k = 2, and L = λg wth λ = 0.1, 0.2, 0.3, 0.4. For each sample smulaton, experments were ndependently executed for r = 100, 200,...,500 replcatons. Hence, = sample replcatons were drawn n the whole study. Relatve bas and relatve root mean square error of emprcal Horvtz- -Thompson estmates for the varable representng total farm sales are shown n Fgure 5. The observed relatve bas seems to behave n a very rregular way wth no systematc tendency. Ths s because bas values are very small n comparson to the lmted accuracy of ths smulaton study. The absolute value of the observed relatve bas was never greater than , whch means that the bas was extremely small n comparson to the estmated parameter. The mean square error (MSE) of estmates decreases slowly when number H of the sample replcatons grows, and seems to stablze around some postve value, dependent on the cost lmt L. By tself, the cost lmt seems to nfluence the MSE more strongly (the greater the cost lmt, the lower the MSE). The share of squared bas n the MSE never exceeded , whch ndcates that the bas was neglgble n comparson to the MSE.

9 On some modfcaton of the sum-quota samplng scheme 91 Relatve bas -1e-03-5e-04 0e+00 5e-04 1e-03 L=G/10 L=2G/10 L=3G/10 L=4G/ Relatve root mean square error Number of replcatons Number of replcatons Fg. 5. Relatve bas and relatve root mean square error for emprcal Horvtz-Thompson estmator under modfed sum-quota samplng scheme wth k = 2 Source: author s own study. 5. Conclusons The proposed smple modfcaton of the sum-quota samplng scheme facltates a smultaneous examnaton of several unts and may lead to a serous speedng-up of the data gatherng process when ndvdual per-unt costs of data acquston are not known n advance. On the other hand, some degree of control over the total varable cost of the survey s stll retaned. The proposed method for estmatng the fnte populaton total through the Horvtz-Thompson formula usng Fattorn s emprcal estmates of ncluson probabltes seems to provde an approxmate desgn-unbasedness at least n some stuatons, apart from asymptotc consderatons.

10 92 Acknowledgement The work was supported by grant No: N N from the Mnstry of Scence and Hgher Educaton. Lterature Fattorn L. (2006). Applyng the Horvtz-Thompson crteron n complex desgns: A computer-ntensve perspectve for estmatng ncluson probabltes. Bometrka. Vol. 93(2). Pp Groves R.M. (1989). Survey Errors and Survey Costs. John Wley & Sons. New York. Kremers W.K. (1985). The Statstcal Analyss of Sum-Quota Samplng. Unpublshed PHD thess. Cornell Unversty. Kremers W.K. (1986). Completeness and unbased estmaton for sum-quota samplng. Journal of the Amercan Statstcal Assocaton. Vol. 81(396). Pp Kremers W.K., Robson D. (1987). Unbased estmaton when samplng from renewal processes: The sngle sample and k-sample random means cases. Bometrka. Vol. 74(2). Pp Lehmann E.L., Casella G. (1998). Theory of Pont Estmaton. Sprnger. New York. Pathak K. (1976). Unbased estmaton n fxed-cost sequental samplng schemes. Annals of Statstcs. Vol. 4(5). Pp