CHANGING STRATA AND SELECTION PROBABILITIES* Leslie Kish, The University of Michigan. Summary

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1 124 CHANGING STRATA AND SELECTION PROBABILITIES* Leslie Kish, The University of Michign Summry Survey smples re often bsed on primry units selected from initil strt with probbilities proportionl to initil mesures But lter smples could better be served with new strt nd new mesures P. The difference between the initil nd new strt, nd mesures, my be due to chnges in either the popultion distribution or in survey objectives. Efficiency dicttes retining in the new smple the mximum permitted number of initil selections. procedure for chnging mesures within fixed strt is presented first. Then we develop new procedures for retining the mximum number of selected units within chnged strt. Finlly, some problems of selecting units with unequl probbilities re explored. 1. Introduction Unequl selection probbilities re often ssigned to smpling units. Especilly in the first stge of survey smples, primry smpling units re often selected with probbilities proportionl to their size mesures, Nj. Then subsmpling, within the selected primries, of finl units with probbilities inversely proportionl to the N., yields uniform overll selection rte f: x f. (1) The c nd re constnts chosen to yield the desired numbers of primry nd finl units. Typiclly the selections re mde seprtely in mny strt, with one or two (seldom more) primry selections specified from ech strtum. After the initil selection, the primry units my be used for mny surveys. But the initil size mesures my differ considerbly from new mesures tht would better suit the needs of current surveys. The difference between initil nd desired new size mesures my be due to differentil chnges of size mong the smpling units, s reveled by the ltest Census. But it cn lso be due to differences in survey popultions. For exmple, the initil mesures my hve been bsed on persons, but now *This reserch ws done t the Survey Reserch Center, (Institute for Socil Reserch) with the id of grnt USPHS /GM from the Ntionl Institutes of Helth. The uthor is grteful to Irene Hess, Vinod K. Sethi, nd Roe Goodmn for their helpful suggestions. one my wnt to smple popultion of physicins, or college students, or frmers, or vlues of frm products. If mesures of size for the new popultion re vilble, nd if these differ considerbly from those used in the initil selection, then one should prefer to use them. Furthermore, the best strt for the new popultion my be sufficiently different from the initil strt to justify chnging them. The sme chnges or differences in popultion distributions, tht led to chnging selection probbilities, lso will often motivte chnging strt. A brnd new selection would men chnging most of the primry units. But continued use of the initil primry units hs severl dvntges. First, they represent importnt investments in clericl work, nd especilly in selected nd trined field force. The field force in ech county of ntionl smple my represent vlue of mny hundreds of dollrs. Secondly, using the sme primry units decreses considerbly the stndrd errors of comprisons between surveys, (see dt by Kish This problem is cute for pnel smples. To continue using the entire initil set of primry units would restrict the smple to the initil set of selection probbilities nd strt. But one cn shift to the new probbilities nd strt, nd yet retin most of the initil primry selections. To mke tht shift from the initil to new set of probbilities nd strt, with minimum number of ctul chnges of primry units, is the purpose of the methods presented. Section 2 describes simple method for chnging mesures of size within fixed strt, modified in section 5 to introduce further considertions of economy. Section 3 presents methods for retining initil selections from chnged strt, nd is the min purpose of this pper; these re derived nd further discussed in section 4. Finlly section 5 trets some relted problems in multiple selections with unequl probbilities. 2. Chnging Mesures Within Fixed Strt This section presents procedure for chnging from the initil probbilities pj to new probbilities P for fixed set of smpling units within strtum. There re D + I units in the strtum. Of these, D receive decrese in probbility, nd I receive n increse, which my lso be zero. Subscripts

2 125 distinguish the two sets of smpling units, so tht Pd < pd nd Pi pi, where (d = 1, 2,.., D) nd (i = D + 1, D + 2,..., D + I). The initil probbilities used for the selection of smpling units from strtum, nd the new probbilities to which we wnt to switch, both sum to unity in the strtum. D D I The sum of the probbility increses must equl the sum of the probbility decreses: E (Pi Pi) (Pd Pd). The rules for chnging probbilities follow. (2) (3) (3I). If the initilly selected smpling unit shows n increse (or no chnge in probbility), retin it in the smple, s if selected with the new probbility Pi. b. If the initilly selected unit shows decrese, retin it in the smple with the probbility tht is, ssign probbility of 1 - Pd/pd for dropping it. Thus the compound probbility of originl selection nd remining is mde pd x d. c. If decresing unit is dropped from the smple, select replcement mong the incresing units with probbilities proportionl to their increses. The probbility of selection of the i -th unit is I (Pi - pi)/ E (Pi - pi). Thus, the totl selection probbility of n incresing unit consists of pi initilly, but it is properly incresed if ny of the D decresing units hd been selected: pi + (pd x = (4) E (Pi - pi) Often only one smpling unit is selected from ech strtum. But the rules re eqully vlid for two or more fixed numbers of selections from the strtum. This method ws first presented by Keyfitz [1]. It is illustrted, mplified nd modified in section Chnging Strtum Boundries When the smpling units of popultion re shifted cross strtum boundries, from set of initil strt to set of new strt, we fce new problems of prcticl urgency, whose solution motivted our serch. A key step consists in seprting the procedures of this section from those of section 2. First, strtum chnges re solved with section 3 procedures to obtin the preliminry probbilities ; then these re chnged with section 2 procedures to new finl probbilities P. Of course, p. = P. if the finl mesures equl the originl mesures; nd section 3 without section 2 suffices for situtions when strt re chnged but mesures re not. We concentrte here on strting with single initil selection from ech initil strtum nd ending with single finl selection in ech new strtum, nd this represents limittion. But the number nd the sizes of the initil strt my differ from those of the new strt, nd this represents n importnt sitution for smple surveys. For brevity nd clrity, strt will denote the new strt. Ech of these is composed of severl sets, which denote the portions of initil strt it contins. Ech set contins either zero or one initil selection. We must ssume tht the units re sorted into the new strt with procedures tht gurntee tht the initil selection of some units hs no effect on their sorting. This cn be gurnteed either with definitions tht permit no ltitude, or by performer ignornt of the identity of selected units, or both for extr sfety. The gurntee is necessry to ensure tht selection probbilities before sorting re equivlent to selection probbilities fter sorting into the new strt. Except for this gurntee, there is complete freedom in forming the new strt. We shll denote with the probbilities tht the units received in their own initil strt. These mesures were probbilities nd summed to 1 in the initil strt, but their sum E = M in the new strtum is not generlly 1.

3 126 This sum vries from strtum to strtum, but for brevity we shll denote it s M, without subscript for the strtum. The mesures must be converted to the preliminry probbilities = with E 1. Note tht selecting with probbilities proportionl to the mesures M would be equivlent to selecting with probbilities proportionl to the preliminry probbilities p. Then these could be converted, with section 2 procedures, to the finl probbilities P. But this would be only long procedure equivlent to selecting directly with the probbilities P.. Both procedures disregrd our purpose of retining in the new smple the initil selections. First Procedure. Suppose now tht the strtum consists of severl sets, nd denote both these sets nd their mesures s A + B + C +. M. The typicl set A hs the mesure A = E M the sum of the mesures j Mj of the units it contins. First select set with probbilities proportionl to its size, so tht set A hs the selection probbility A/M. If the selected set contins the originl selection, ccept it s the preliminry selection with pj = x - If the selected set does not contin the initil selection, select one with probbilities proportionl to Mj (or to P In either cse, compute pj nd convert,with the section 2 procedures pplied to the entire strtum (not to only the selected set). Understnding this procedure is simple becuse it depends on only two -stge selection for the probbility nd the equiv- lence (gurnteed) of the possible prior selection with the M. Second Procedure. In the first procedure the sets re selected proportionl to their initil probbilities (E nd the conversion from pj to is pplied to the entire strtum. The second procedure selects with probbilities proportionl to the finl probbilities (E in the sets, nd the conversion from is confined within the selected sets. The Census Bureu switched from 333 re smple bsed on 1950 dt to 357 re smple bsed on 1960 dt, nd described its procedure s follows [5]: to from more thn one strtum in the 333 re design. In principle, one of these prts cn be selected with probbility proportionte to 1960 size. If, then, the selected prt does not include PSU previously selected in the 333 re design, selection cn be mde from PSU's in tht prt with probbility proportionte to 1960 size. On the other hnd, if the selected prt contins smple PSU from the 333 re design, selection from the prt cn be mde by the Keyfitz method [1], mximizing the chnge of retining the previously - selected PSU." Third Procedure. This depends on identifying uniquely ech of the initil strt with only one of the new strt. Then section 2 procedures re pplied for chnging mesures. All units in the new strtum tht did not belong to the specified initil strtum re treted s newly creted, with 0; nd some new strt my consist entirely of such new units. On the contrry, units of the initil strtum tht did not trnsfer to the specified new strtum re treted s eliminted from it, with Pi 0. Rules re needed to specify the unigyie identifiction of the initil strt with the new strt; they must void selection bis nd preferbly should mximize the retention of selected units. The first two procedures hve the dvntges of simplicity, tht my fcilitte their dpttion to designs with further demnds. Two of these deserve mention. First, some designs cll for two or more selections from ech strtum. Second, in some designs the selections from different strt re not independent, becuse some form of "controlled selection" or "multiple strtifiction" is used. However, those two simple procedures hve the disdvntge of not utilizing initil selections present in other sets, when the selected set fils to contin one. This disdvntge is moderte if one set predomintes within ech of the strt. But the disdvntge increses rpidly with the number of sets of the sme generl mgnitude present in the strt. Fourth Procedure. This procedure is optiml in the sense tht it retins ll initil selections, - subject to conversions with the section 2 procedures. It retins ll initil selections, whenever one is present in ny set of the strtum. It requires the ssumptions of single selection from ech strtum, nd independence of selections between strt. These points re developed, together with the justifictions for the following selection rules in section 4. "For generlity, consider revised strtum in the 357 re design mde up of prts

4 . If no initil selection occurs in ny set in the entire strtum, select one unit directly with the new probbilities P.. b. If single initil selection occurs in the entire strtum, ccept it s the preliminry selection with pj = M.M. Then convert pj to P. with section 2 procedures pplied in the entire strtum. c. If two or more sets contin initil selections, select one set with probbility y. From the selected set ccept the selected unit, nd convert from the p. to P. with section 2 procedures pplied to the entire strtum. 1. When there re only two sets in the strtum, with totl mesures A nd B, select one set with odds inversely proportionl to its mesure; tht is, select set A with probbility y = B /(A + B), nd set B with probbility yb = «A + B): 2. When the strtum contins more thn two sets, order ll sets into series of dichotomies, with n objective ordering. Whenever both brnches of the dichotomy contin selections, choose one with the rules symbolized with: where nd b A' B + ( - b) A+B C + D C' + D' - C'D' E' E + F - E'F' (5) To compute the fctors A', B', then C', D', nd E', F', etc., follow nd continue these rules: If A contins only single set, A =C nd D =0, then A' =A nd = 1. Similrly for B. If both A nd B contin single sets, we hve = b = 1, nd the simple cse of two sets in the strtum. If A contins two sets, A = C + D, nd C nd D contin only single set ech, then C' C nd D' D nd C + D - CD. Similrly for B = E + F. If C contins two then C' = G' + H' contins single but if G contins nd G' = T' + S' for H nd for D = sets C = G + H, - If G set then G' = G; two sets, G = T + S, - T'S'. Similrly I + J. These rules would become cumbersome if crried fr. But in prctice, rrely will two sets contin selections, when the strtum hs mny sets. An objective rule is needed to prevent personl bis in ordering the dichotomies. One resonble rule is: () A set contining more thn hlf of the strtum estblishes the first split. (b) Order ll others by size; divide them successively into two prts with equl numbers of sets, putting odd numbers into the second hlf. For exmple, strtum composed of mjority set X, nd of 9 other sets (denoted with their rnks) would be divided s follows: [(1,2)(3,4)][(5,6)(7,819)]} 4+. Justifiction nd Expnsion of the Optimum Procedure To derive the optiml fourth procedure, regrd it s the modifiction of the first procedure, whose justifiction is obvious. The selection of one of the sets from the strtum with probbility proportionl to its mesure is mde to depend on the probbility of its contining n initil selection. Bsed on those probbilities the rules must chieve the equivlent of selecting the set A with probbility proportionl to A,M. Let us begin with strtum contining two sets only, with the mesures A + B = M. The presence of only one set would be merely the specil cse of B O. We shll mintin the odds of choosing between the two sets t A /B, becuse this rtio of the mesures is the bsis for the selection within the sets. We use the importnt fct tht these mesures A nd B lso represent the probbility for ech set tht it contins the initil selection from the initil strtum. Hence, if we cn ssume independence between the two probbilities A nd B, we hve: (1 - A)(1 - B) = probbility tht neither (6) set contins selection A(1 - B) = probbility tht set A, but not set B contins selection

5 128 B(1 - A) = probbility tht set B, but not set t contins selection AB = probbility tht both sets contin selection. Our strtegy shll be s follows: When neither set contins selection we choose between the two sets with the desired odds A /B. When selection is present in only one set, we choose it with certinty s desired. But this introduces n imblnce in the odds. When selection is present in both sets, choose set A with odds y/01.- y ), computed so s to redress the imblnce introduced before. In other words, the desired odds A/B re chieved over ll possibilities by chieving them seprtely for the cse of no selection, nd jointly over the cse of one or two selections. We need to solve for y in the expression tht equtes the desired odds A/B to the compound probbilities of the occurnce of cse nd the selecting of the set: A(1 - B) x 1 + (1 - A)BxO+ABxy A A(1 - B) xo+(1-a)bx1+abx (1 -y)b (7) re not simply A C + D nd B E + F; they re A' C + D - CD nd B' E + F - EF respectively. But we cn del more conveniently with the odds A/B, proportionl to the sums of mesures, thn with A'/B'. Hence we must djust ccordingly the probbilities y nd (1 - y) for choosing between the two groups, when both contin selections. We set these so s to ttin, jointly for double selections nd single selections, the desired odds A/B: A'(1 - B') + A'B'y (1 - A')B' + A'B'(1 - A'(1 - Bb') + A'Bb'y (1 - A')Bb' + A'Bb'(1 - y) A where A' = A' nd B' Bb'. Then '(1 - Bb') + B'b'y (1 - A')b' + A'b'(1 - y) nd y (A'b' + B'b') = (b' - ') + B'b'. (9) Thus when both groups A nd B contin selection group A should be chosen with probbility y = [B + (1/' - llb')j/(a + B), or nd (1 - B) + By = (1 A) + A(1 y). y = B +A(+ b), where = nd b (10) Thus when both sets contin selections, set A should be chosen with probbility 3' A+ or odds = y Direct extension of the bove method proved too complex even for three sets. But one cn rrnge lrger number of sets into series of dichotomies, nd then pply in sequence procedures similr to those for two sets. For exmple, strtum contining four sets cn be divided into group A contining sets C nd D, nd into group B contining sets E nd F. We cn choose group A with the odds A/B (C + D) /(E + F), then select C with odds from A C + D, nd finlly unit within C with the probbility /C. The overll probbility of selecting the unit would be A /(A + B) x /(C + D) x M /C (8) M/M. We wnt n equivlent procedure in which the odds A /B, C/D, nd E/F re preserved. A new problem rises in the probbility tht group of two sets contins either one or two selections. The probbilities for the two groups Thus ( - b) represents n djustment of the probbility of choosing group A over group B, when both contin selections, nd when ech is composed of two or more sets. Of course, group B would be chosen with (1 - y) [A + (b - )] /(A + B). Note tht vlue of ( - b) > A would result in the impossible choice of y > 1; similrly (b - ) > B would men y <0. But this cnnot occur if A nd B re both not greter thn 3. Since this requirement will be met esily in prctice, we desist from pressing further solutions for this problem. The.presence of only three sets in the strtum mens tht group A contins only one set A = A' nd 1. The presence of only two sets mens b 1, nd (10) becomes (8). When there re only four sets, nd A C + D nd B E + F, then A' C + D - CD nd B' E + F - EF (11)

6 129 Here C nd D, nd E nd F ll represent the probbilities for ech of the four sets, tht it contins one or more selections. When there re more thn four sets in the strtum, the formtion of dichotomies cn be continued s needed. Any nd ll of the four groups, C, D, E, nd F, cn be considered s hving been composed of two sets. In generl then, insted of (11) we write A' C' + D' - C'D' nd B' E' + F' - E'F' (12) where ech of C', D', E', nd F' must denote the probbility tht the group contins one or more selections. For exmple, if C contins single set, C' = C; if it contins exctly two sets, C = G + H, then C' G + H - GH; if both G nd H re groups of sets then C' G' + H' - where G' nd H' re similrly defined. These procedures for single selections per strtum would become too complex if pplied to multiple selections from strt. More direct nd generl solutions seem to be possible if some strong symmetries re imposed on the strt nd on the selections. But generliztions in this direction re not helpful if they impose conditions tht re typiclly bsent in prcticl work. The simplest exmple would be the initil selection with simple rndom smpling of n units from the entire popultion of N units; tht is, ll (N) combintions eqully probble. These cn be sorted into rbitrry strt with procedures tht gurntee no effect from the initil selections on the sorting. The n selections cn be ccepted s equivlent to rndom selections within the strt. The strtum smples then cn be incresed or decresed t rndom to obtin the numbers of rndom selections needed from ech strtum. The symmetries of simple rndom selections re sufficient, nd no more complicted proofs re needed. If the initil selection ws comprised of simple rndom selections within the initil strt, then they re lso rndom selections within the corresponding sets of the new strt. Treting the sets s substrt within the new strt, some tretment cn be evolved to yield smple pproximting strtified rndom smple within ech new strtum. But for selecting units with unequl probbilities this is likely to become too complicted to be prcticl. The problem of selecting two (or more) units with unequl probbilities from strtum cn be simplified by dividing the strtum into two (or more) rndom substrt of equl size. This cn be done with rndom sorting of units, s in the first procedure of section 6. Selecting one unit from ech substrtum reduces the problem to tht of single selection per strtum. The initil strt nd the new strt cn both be treted this wy. We should dd tht fter the single selections in the new strt re secured, other selections cn be dded either with replcement, or by using one of the procedures of section 6 for selecting without replcement. Single selections per strt chrcterize much prcticl work, nd to these the optiml procedures re directly pplicble. The requirement of independence between strt is violted when controlled selection is used. I fer tht it would be too difficult to chieve forml tretment of the joint probbilities for the severl sets within strt; but this seems most unlikely s source of genuine bis. However, imposing controlled selection on the new strt seems difficult to chieve. 5. Modifictions nd Illustrtion of Chnging, Mesures ;'ithin Fixed Strt. A simpler procedure thn tht of section 2 cn hndle problem confined chiefly to lrge growth in smll proportion of the primry units. For exmple, suppose we hve smple of blocks selected with the initil probbilities p., proportionl to the initil sizes N.. 3Suppose lso tht for new smple one is willing to retin the originl probbility for ll blocks, except for those tht hve t lest doubled in size. Tht is, if the new size Nj < 2N., we ccept the initil pj; but for the growth blocks, with > 2N, we wnt new probbilities P proportionl to N'. Generlly, plce into new growth strt the portion (N' -N), denoting the size increse of primry units designted s growth units. Then select from these growth units smple with probbilities proportionl to the vlues (N' - Ni). In most situtions, the constnt of proportionlity will be the sme (c /f) s for the initil selection, to preserve the uniform overll smpling frction f; but different frction cn be introduced for the growth strtum.

7 130 Thus the growth units re selected with the sum of two probbilities in the growth strt nd in the initil strt: N' -Ni Nj cf + (13) If selected, these primries should be subsmpled with the rtes c/ni so tht the overll selection probbility becomes: + x f. (13') Note tht, s desired, the plnned subsmple is c when unit becomes selected. This cn occur both in the initil strt nd in the growth strt. The procedure mounts to splitting the growth units into two prts, consisting of n initil size N. plus growth size (N - Nj), nd subjecting them to seprte selections. The ordinry units, tht do not qulify s growth units, remin subject to the initil selection rtes Nj /(c/f) x =f. b. Into the procedures of section 2 one cn introduce further considertions of efficiency, knowing tht generlly it is neither necessry nor possible to hve precise mesures of size. It is desirble to hve probbilities pproximtely proportionl to size mesures. And it is necessry nd sufficient for pplying the procedures of section 2 tht the sum of chnges in probbilities be zero in the strtum, (so tht 1 cn be stisfied). Within tht requirement we cn djust the selection probbilities to stisfy some criteri of chnge lrge enough to be recognized s importnt, nd to delibertely neglect smller chnges. For mny smpling units the chnges in probbilities re smll enough tht they cn be delibertely neglected. If these units re numerous, nd if neglecting their chnges dds little to the overll vrince, then one cn reduce the number of chnges significntly with little scrifice in the vrince. To these units one cn ressign the initil probbilities; they constitute set of units for which pi = Pi is rbitrrily ssigned. This "flexible" procedure reduces the number of units one must switch; it lso elimintes the tsk of revising office records for selected units with no chnge in probbilities. But this "flexible" procedure requires more computtions for blncing the probbilities within the strt thn the rigid "strict" pln does. The overll dvntge of the flexible pln is importnt when it confines expensive switching to smll proportion of units. This my require tolerting rther wide limits of differences in ctul mesures for the units to which Pi pi re rbitrrily ssigned. We used the flexible procedure in chnging from the 1940 to 1950 Census mesures for the counties, representing s mny strt in the ntionl smple of the Survey Reserch Center. In choosing criteri we blnce the costs of chnging counties ginst the increse in survey vrinces due to smll distortions in the selection probbilities. These were considered reltive to other sources of vritions in smple size. We decided on the following procedure: () Define n importnt increse s 10 per cent or more (P1/p. > 1.10). Compute the sum of the increses in the strtum. (b) Add enough of the lrgest decreses (smllest vlues of Pfp) to blnce exctly the sum of the increses; tht is, E (Pi - pi) - E (Pd - pd) 0. (c) Consider ll other counties s not hving chnged probbilities, with Pi = The detils re in [2]. Two other modifictions deserve brief mention. Fced with rther smll probbilities of chnge (1 - in ech of 16 strt, we did not drw independently within ech strtum. Insted we controlled the number of chnges by cumulting the probbilities of chnge from one strtum to nother, then pplied n intervl of one, fter rndom strt. Thus the ctul number of chnges, which ws three, ws controlled within frction of the expected number of chnges. Second, we need not merely ccept the chnges from one Census period to nother; we cn lso project them forwrd into the middle of the period of the use of the Census frme. This my be worth doing for the fstest nd stediest growing counties. For exmple, the Cliforni counties tht hve shown unusul growth in one decde my lso be expected to show similr growth in one decde my lso be expected to show similr growth in the next decde. The projection my improve the design of smple of counties for use during tht decde.

8 Some Relted Problems When selection probbilities nd strt re chnged, the sitution my lso require chnges in the boundries of smpling units. Some my be split, nd others combined. The initil selection probbilities of unit my be divided mong severl portions ccording to specified rules; similrly severl of the my be combined into one new unit. The cretion of entirely new units is noted with pj = 0, nd the elimintion of initil units with = 0. But we must void detiled tretment of this problem. Selecting more thn one unit with probbilities proportionl to unequl mesures is difficult to do "without replcement ". (It is even more difficult to find simple formuls for computing the lower vrinces tht selection without replcement yields.) This problem hs been the subject of severl investigtions, most recently by Fellegi [6] nd Ro [4]. He discusses the first procedure below, but the other four procedures re new, I think. For brevity two selections per strtum re discussed, but the methods cn be extended to more seleolons. I. Sort the units of the strtum t rndom into two equl substrt nd select one unit from ech, with probbility proportionl to P.. This cn lso be ccomplished by putting the units in rndom order nd selecting two with systemtic intervl. 2. Select from the strtum with equl probbility nd without replcement 8 units nd put them in the sme rndom order in which they were selected. Obtin the sum of their mesures, E. Now select rndom number from 1 to /2; then dd E/2 to it. These two numbers re the two selection numbers from the strtum. The number 8 is suggested merely s desirble compromise: we wnt to sve lbor, but we lso wnt to void situtions in which double selection is possible, becuse unit is lrger thn E /2. 3. When selecting units with equl probbility, reselection of ny unit is voided by substituting some unselected unit. This is simple becuse of the complete symmetry of equl probbilities, which is generlly lcking in units with unequl probbilities. But we cn use the presence of prtil symmetries, when t lest two units pper with the sme size, for ny size, in the sme strtum. If one unit is selected twice, select t rndom one of the others with the sme size. If exctly equl sizes do not exist, generlly one cn estblish strict rules for "best mtches" of sizes. When two sizes Pj nd Pk re mtched, the uniform subsmpling rte f my be obtined by subselecting in both with (Pj + Pk)/2, whenever both units pper in the smple. 4. First select K = 6 units using the desired mesures nd with replcement, which is generlly esy. Then sort these t rndom into three pirs, but without llowing the sme unit twice into pir. Now select one of the pirs t rndom. The probbility for unit with Mj 0.10 of ppering twice in two drws is 1/100; but its ppering more thn three times in K 6 drws is reduced to bout For Mj 0.05 the reduction is from 1400 to bout 1/12,000. (If K smpling units re selected with probbilities proportionl to mesures Pj, subselection of k units with equl probbilities yields smple of k units with probbilities proportionl to the Pj.) 5. An initil selection with equl probbilities would be specil cse of the procedures in section 2, with the pj for single selection, nd p, 2/N for 2 selections; then convert these to probbilities proportionl to the Pj. This procedure reduces but does not eliminte the probbility of duplicte selection of the sme unit. References [1] Keyfitz, Nthn, "Smpling with Probbilities Proportionl to Size ", Journl of the Americn Sttisticl Assocition, 46(Mrch, 1951), pp [2] Kish, Leslie nd Hess, Irene, "Some Smpling Techniques for Continuing Survey Opertions ", Proceedings of the Socil Sttistics Section. Americn Sttisticl Assocition, Wshington (1959). [3] Kish, Leslie, "Vrinces for Indexes from Complex Smples ", Proceedings of the Socil Sttistics Section Americn Sttisticl Assocition, (1961), pp [4] Ro, J. N. K., "On Three Procedures of Unequl Probbility Smpling Without Replcement ", Journl of the Americn Sttisticl Assocition, 58(Mrch, 1963), pp [5] U. S. Bureu of the Census, "The Current Popultion Survey - A Report on Methodology ", Technicl Pper No. 7, (1963). U. S. Government Printing Office, Wshington, p.67. [6] Fellegi, Ivn P., "Smpling with Vrying Probbilities Without Replcement ", Journl of the Americn Sttisticl Assocition, 58(Mrch, 1963), pp