TREATMENT OF UNIT NON-RESPONSE IN TWO-STAGE SAMPLING WITH PARTIAL REPLACEMENT OF UNITS

Size: px

Start display at page:

Download "TREATMENT OF UNIT NON-RESPONSE IN TWO-STAGE SAMPLING WITH PARTIAL REPLACEMENT OF UNITS"

Paul Stokes
5 years ago
Views:

1 STATISTICA, ao LXIV,. 3, 004 TREATMET OF UIT O-RESPOSE I TWO-STAGE SAMPLIG WITH PARTIAL REPLACEMET OF UITS Faba C. Okafor 1 () 1. ITRODUCTIO May authors have worked o samplg o successve occasos; amog them are Jesse (194), Sgh (1968), Kathura, et al. (1971), Okafor (1988) to meto but a few. All of the above researchers assumed total respose from all the sample uts o all the occasos. Sgh, et al (1974) usg Bartholomew (1961) method of treatg ut o-respose proposed a estmator of the populato mea for samplg o two successve occasos. Okafor ad Lee (000) appled Hase ad Hurwtz (1946) techque for treatg ut o-respose to double samplg for rato ad regresso estmato. Okafor (001) dscussed the treatmet of ut o-respose successve samplg over two occasos for elemet samplg. I ths paper we have attempted the exteso of the treatmet of ut orespose to two-stage samplg wth partal replacemet of uts at the secod stage usg Hase ad Hurwtz (1946) techque. We have proposed three dfferet estmators for the populato total whe samplg wth partal replacemet of uts ad wth equal probablty samplg at both stages. These estmators have bee compared emprcally usg actual data from a survey o arable farmg coducted by the Mstry of Agrculture, Botswaa.. SAMPLIG SCHEME Frst Occaso: Select frst stage uts (fsu s) from the populato of fsu s by smple radom samplg (srs). Wth the th selected fsu, select a sample of m secod stage uts (ssu s) from the populato of M ssu s aga by srs. Iformato o the study varable y s sought from the sample ssu s. Suppose m 1 uts respod at the frst attempt ad m of the m ssu s ot respod. Let a smple radom subsample of s (= k m ; 0 1) uts from the m ssu s be revsted. We shall assume 1 k that all the s 1 ssu s gave formato o y at ths secod attempt. () Ths study was carred out whle Uversty of Botswaa, Gaboroe, Botswaa.

2 51 Faba C. Okafor Secod Occaso: All the fsu s selected at the frst occaso are retaed for use at the secod occaso. Select by srs m ( 0 1) ssu s from the m ssu s already selected at the frst occaso. Suppose m 1 ssu s out of the m ssu s respod at the frst attempt ad m fal to respod. Let a smple radom subsample of s1 ( k m ) uts be revsted. We aga assume a complete respose of all the s 1 uts at ths secod attempt. I the th sample fsu, select a fresh sample of m ( + =1) ssu s from M ssu s by srs. For these fresh m uts, let u 1 ssu s respod ad u fal to supply the requred formato o the study varable. A smple radom subsample of sze s ( k u ) from u ssu s s revsted. A complete respose s aga assumed ths tme aroud. 3. ESTIMATIO OF THE POPULATIO TOTAL For smplcty, we make the followg assumptos: m matched ssu s at the secod occaso form a subset of the frst occaso, m1 s1, respodg uts. Samplg s from fte populato at both stages. M ad the populato varablty rema costat at both occasos otatos Let x (y) be the values of the study varate at the frst (secod) occaso. m m1 x1 m x s m x ( ) s the frst occaso mea of the th fsu adjusted for 1 o- respose based o m ssu s. x 1 s the frst occaso sample mea of the th fsu based o the m 1 ssu s. xs 1 s the frst occaso sample mea of the th fsu based o the s1 respodg ut at the secod attempt. z ( m1 z1 m z s ) m 1 ; (z = x, y) s the sample mea adjusted for orespose based o the matched sample of ssu s. z 1 s the sample mea for the matched sample of ssu s based o the m 1 respodets at the frst attempt. s the sample mea for the matched sample of ssu s the th fsu based o the z s 1 u 1 1 s s 1 respodets at the secod attempt y ( u y u y ) m s the secod occaso sample mea for the umatched sample ssu s of the th fsu adjusted for o-respose.

3 Treatmet of ut o-respose two-stage samplg wth partal replacemet of uts 513 y 1 s the secod occaso sample mea for the umatched sample ssu s the th fsu based o the u 1 respodets at the frst attempt. y s s the secod occaso sample mea for the umatched sample ssu s the th fsu based o the s respodets at the secod attempt. 3.. Estmators of the populato total Let us cosder the frst estmator of the form: T1 M y x x y [ { ( m )} (1 ) ] u (3.1) 1 s a kow costat of proportoalty the th fsu s the weght, the th fsu, chose so as to make the varace of T 1 a mmum. The estmator T 1 s ubased (see appedx for proof). I (3.1) the frst term sde the square brackets ad eclosed gothc brackets s a double samplg regresso estmator, sce the frst occaso observato, x s used as a auxlary varable to mprove the curret estmate of the populato mea each fsu. Ths regresso estmator s sutably weghted wth the basc estmator, y u obtaed by usg the fresh sample selected o the secod occaso to form oe estmator of the populato mea wth each fsu. The same procedure s adopted obtag the other remag two estmators. It may be cumbersome to compute the weght, especally whe the umber of sample fsu s large. Therefore, t may be advsable to have a commo weght for all the fsu s. Hece the estmator T M y x x M y { ( m )} (1 ) u 1 1 (3.) A thrd estmator of the populato total s obtaed by usg a commo weght ad a commo regresso coeffcet, c. Ths estmator s of the form T3 M y x x M y T T [ ( )] (1 ) c m (1 ) u m u 1 1 (3.3) 3.3. Varace of the proposed estmators To derve the varace of T 1, we adopt the codtoal approach ad follow the procedure used Hase ad Hurwtz (1946) estmator. The dervato of the varace s gve the appedx.

4 514 Faba C. Okafor V ( T1 ) S M S m m where, (1 ) (1 ) b 1 + V (1 ) (1 ) m m (3.4) (1 ) V W k S k S b s the populato varablty betwee the fsu s. S s the populato varablty wth the th fsu. S s the populato varablty of the o-respodets wth the th fsu. s the populato correlato coeffcet betwee the frst ad secod occasos for the th fsu. s the wth fsu populato correlato coeffcet betwee the frst ad secod occasos for the o-respodets. W s the populato proporto of the o-respodets. By dfferetatg V(T 1 ) wth respect to ad solvg the dervatve, we have the optmum value of as 0 V S V [(1 ) {1 ( )}] (3.5) V S V Substtutg 0 (3.5) ad smplfyg, the mmum value of V(T 1 ) becomes 0( 1 ) M b (1 0 ) 1 m V T S V (3.6) 0 ( S V ) V The optmum value of s We shall use the same procedure for obtag the varace ad the optmum weght for the estmator, T 1 to derve the varace of T ad ts optmum weght. Hece the optmum value of s M M 0 V S V 1 m 1 m [ (1 ) {1 ( )}] Whle the mmum varace of T s M o( ) b (1 o ) 1 m V T S V (3.7)

5 Treatmet of ut o-respose two-stage samplg wth partal replacemet of uts 515 Fally by mmzg the varace of T 3 wth respect to, the optmum value of s { V( T ) Cov( T, T )}/{ V ( T ) V ( T ) Cov( T, T )} (3.8) o u m u m u m u Whle the mmum varace s o 3 u m m u m u m u V ( T ) { V( T ) V( T ) Cov ( T, T )}/{ V ( T ) V ( T ) Cov( T, T )} (3.9) Where, m u b Cov( T, T ) S ; M u b 1 m V( T ) S V (3.10) M M m b c 1 m 1 m V( T ) S V { S V } M c S V 1 m { } (3.11) The optmum value of c obtaed by mmzg the varace of T m, V(T m ) s M M c { S V } { S V } 1 m 1 m 4. COMPARISO OF THE ESTIMATORS 4.1. Theoretcal comparso To ga sght to the performace of the proposed estmators ad for smplcty, we shall compare oly the estmator T wth the estmator, T 0 obtaed whe there s o partal matchg of uts. T 0 M ym (4.1) 1 wth varace 1 M V ( T0 ) Sb V m (4.) For the purpose of comparso, we shall make the followg assumptos:

6 516 Faba C. Okafor k = k, W = W ; Based o these assumptos ad for all frst stage uts. W(1 k) 0 V ( T ) Sb Sw S w k (4.3) S 1 M w S 1 m ; S 1 M w S 1 m The effceces of T wth respect to T o have bee calculated for dfferet values of Sw Sb, Sw Sb,,, k ad W. The results are preseted tables 1 ad below. We have to ote that sce varaces of T 0 ad T volve W ad k, the dvdual varaces wll be hgh f W s hgh or k s small. So the effcecy of T wth respect to T o s based o the gve values of W ad k. From tables 1 ad, we observe that T s ot ecessarly more effcet tha T o whe k = 0.4 because here the effcecy of T over T o s less tha oe whe =0.5. Ths may be partly due to the assumptos used the comparso ad partly due to the ature of the estmator. But most cases there s much to be gaed usg the estmator T place of T o For stace uder the gve assumptos, there s a ga effcecy of T over T o whe k= 0.1. The effcecy s hgher for W =0.7 tha for W =0.3. For a gve value of, the ga effcecy creases for creasg value of. Also for a gve value of other tha 0.1, the effcecy creases for creasg value of. Further more, the effcecy of T o- ver T o creases for creasg value of ; but decreases for creasg value of. TABLE 1 Effcecy of T wth respect to T o for W = k

7 Treatmet of ut o-respose two-stage samplg wth partal replacemet of uts 517 TABLE Effcecy of T wth respect to T o for W = k We have to be extremely cautous terpretg these effceces sce the fact that the effcecy s hgher for k= 0.1 or W =0.7 does ot mea that the estmate obtaed from the estmator T whe k s small s more precse tha the oe obtaed whe k s large, as see from table 5. The reaso for ths crease effcecy s that the rate of crease of the varace of T s lower tha the rate of crease of the varace of T o as k or W chages. 4.. Emprcal comparso The data used for the emprcal comparso are from a survey coducted 000 o arable farmg Botswaa. The orgal desg was a stratfed twostage desg. The strata cossted of geographcal dstrcts. I each dstrct a sample of agrcultural exteso areas (lad area covered by a exteso worker) was selected. Wth each selected exteso area, a sample of farmers from the lst of regstered farmers the area was selected. Iformato was the collected o hectares of lad owed ad lad plated 1999/000 agrcultural seaso ad past four seasos from the selected farmers. Iformato o other varables of terest was also obtaed from the farmers. For the purpose of our emprcal comparso, we have used the sample data from oly fve dstrcts (gwaketse West, South, Cetral ad orth ad Barolog) as our populato data. Frst occaso s the 1998/1999 platg seaso; whle the secod occaso s the 1999/000 platg seaso. We shall use the dstrcts as our frst stage uts ad farmers wth dstrcts as the secod stage uts. Usg these data we calculated the populato parameters preseted tables 3 ad 4 below. We have hypothetcally defed our o-respodet categores usg the sample data as follows: I gwaketse West ad Cetral dstrcts, we assumed that the o-respodet stratum s made up of those farmers owg lad of more tha or equal to 16 hectares; gwaketse South, t s those havg more tha or equal to 14 hectares of lad. o-respodet stratum gwaketse orth ad Barolog dstrcts cosst of those farmers wth lad equal to or more tha 10 ad 1 hectares respectvely. The large o-respodet varaces show

8 518 Faba C. Okafor table 4 s due to the fact that there s very wde dsparty the sze of lad owed by ths category of uts accordg to our categorzg procedure. TABLE 3 Populato parameters (overall) for 1999/000 Dstrct Total (Y) M S gwaketse West gwaketse South gwaketse Cetral gwaketse orth Barolog TABLE 4 Populato parameters (o-respodets) for 1999/000 Dstrct Respose Category W S gwaketse West Respodet o-respodet gwaketse South Respodet o-respodet gwaketse Cetral Respodet o-respodet gwaketse orth Respodet o-respodet Barolog Respodet o-respodet S b = ; S s the populato varablty wth each dstrct. We have chose wth fsu samplg fracto of 0.1 for the purpose of computg the samplg varaces of the proposed estmators. The varaces of the proposed estmators for a matchg fracto of 0.6 are preseted table 5 below. We observe that the estmator T 1 s the best a- mog all the other estmators. Ths estmator s obtaed by usg dvdual wth fsu weghts ad regresso coeffcets. TABLE 5 Varaces of the proposed estmators k= 0. k= 0.8 Estmators Varace Effcecy Varace Effcecy To T T T Departmet of Statstcs Uversty of gera, sukka FABIA C. OKAFOR

9 Treatmet of ut o-respose two-stage samplg wth partal replacemet of uts 519 Proof The estmator T 1 s ubased. APPEDIX E( T ) E E E E ( T ) (A.1) 1 1 s 1 E 1 = expectato over the frst stage sample. E = codtoal expectato over the secod stage sample gve the frst stage. E = codtoal expectato over matched sample the th fsu. E s = codtoal expectato over the subsample of the secod stage orespodets the th fsu. Takg expectatos oe by oe E ( ) [ { ( )} (1 ) ] (A.) where, s T1 M y x x yu 1 1 z z j, m j 1 z 1 zj m ; z =x, y ad j 1 y u 1 m j 1 y j E E ( T ) M [ y (1 ) y ] (A.3) s 1 u 1 E E E ( T ) M Y (A.4) Fally, s 1 1 E E E E ( T ) Y Y 1 s 1 1 Ths completes the proof. The ubasedess of other estmators s proved s a smlar fasho. Proof of varace of T 1 Proof V( T ) V E E E ( T ) E V E E ( T ) E E V E ( T ) E E E V ( T ) 1 1 s 1 1 s 1 1 s 1 1 s 1

10 50 Faba C. Okafor V1, V, V ad V s are varaces aalogous to the expectatos defed earler. From (A.4) 1 s = V E E E ( T ) Y ( Y Y ) ( 1) Sb (A.5) From (A.3) E1V E E s ( T 1 ) E 1V M[ y (1 ) yu ] 1 = S S M (1 ) 1 m m (A.6) From (A.) E1EV E s ( T 1 ) M S (1 ) 1 m (A.7) Further more, (1 ) E1EE Vs ( T 1 ) M V (1 ) 1 m m (A.8) ow combg (A.5), (A.6), (A.7) ad (A.8) the varace of T 1 s (1 ) (1 ) V( T1 ) Sb + M S 1 m m + V (1 ) (1 ) m m (A.9) REFERECES BARTHOLOMEW, D.J. (1961): A method of allowg ot-at-home bas sample surveys, Appled Statstcs, 10, pp HASE, M.H. ad HURWITZ, W.. (1946): The problem of o-respose sample surveys, Joural of Amerca Statstcal Assocato,41, pp JESSE, R.J. (194): Statstcal vestgato of a sample survey for obtag farm facts. Statstcs Research Bullet, 304, Iowa State College of Agrculture.

11 Treatmet of ut o-respose two-stage samplg wth partal replacemet of uts 51 KATHURIA, O. P. ad SIGH, D. (1971): Relatve effceces of some alteratve procedures two-stage samplg o successve occasos, Joural of Ida Socety of Agrcultural Statstcs, 3, pp OKAFOR, F.C. (1988): Rato-to-sze estmators of mea per subut two-stage samplg over two occasos, Australa Joural of Statstcs, 30, 3, pp OKAFOR, F.C. (001): Treatmet of o-respose successve samplg, Statstca, ao LXI,, pp OKAFOR, F. C. ad LEE, H. (000): Double samplg for rato/regresso estmato wth subsamplg for o-respose, Survey Methodology, Caada, 6,, pp SIGH, D. (1968): Estmates successve samplg usg a multstage desg, Joural of Amerca Statstcal Assocato, 63, pp SIGH, D., SIGH, R. ad SIGH, P. (1974): A study of o-respose successve samplg. Joural Ida Socety of Agrcultural Statstcs, 6,, pp RIASSUTO Trattameto delle macate rsposte total el campoameto a due stad co sosttuzoe parzale delle utà Vegoo presetat tre stmator per l totale d popolazoe basat su u dsego campoaro a due stad co sosttuzoe parzale solo delle utà del secodo stado. Gl stmator propost soo costrut teedo coto dell potes d preseza d macata rsposta totale, e l aggustameto per la macata rsposta è basato sulla tecca proposta da Hase e Hurwtz (1946). Da u aals emprca emerge che gl stmator per l totale d popolazoe che usao pes dvdual per le utà d prmo stado hao mglor prestazo degl stmator che usao pes comu. Sul versate teorco vee dmostrato che maggore è l rapporto fra la varabltà etro le utà d prmo stado relatve a o rspodet e la varabltà fra le utà d prmo stado, maggore è l guadago che s ottee, term d effceza, usado gl stmator propost vece degl stmator che s ottegoo quado o c è matchg parzale delle utà. SUMMARY Treatmet of ut o-respose two-stage samplg wth partal replacemet of uts Three separate estmators, for the estmato of the populato total, based o twostage samplg desg o two successve occasos wth partal replacemet of secodary stage uts oly have bee preseted. For these estmators t s assumed that there s ut o-respose. Hece, Hase ad Hurwtz (1946) techque has bee used to adjust for the o-respose. Emprcally, t has bee foud that the estmator that uses dvdual weghts wth frst stage uts perform better tha the other estmators that use commo weghts estmatg the populato total. Theoretcally, t has also bee show that the larger the rato of the wth frst stage ut varablty of the o-respodets to the betwee frst stage ut varablty the hgher the ga effcecy of the proposed estmators over the estmator obtaed whe there s o partal matchg of uts.

{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:

{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution: Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed