'

•

"'v REGiiES..ilOll TEJ;.OOQOE Ill Y.JL'l'I-3'l'Ac:;;: IESIGtl

Diosortation oub::9ittod 1n .t'ulf'lll:lent of tho

roquiromanto for tho nwrd of D1p10Cl 1n

Agricultural nud Atdna] Husbandry

Sto.tiotico of tho Institute of

Agrlcul tural P.scoarch

statiatios (I.C.A.R.),

IZcv Dolhi

1964.

I haw great p1M111zro 1D ezpreae1Dg rq deep sense or

gratitude to Shri D. SiDgb, Dep1lt7 Statistical Adviser, Instituto

of Agricultural Ro£;sarch Statistics (I.c.A.R.), for his val.IZ!lblo

guidanoe, koon iuterest llDd constAnt &IICOuragement throu,gllout tho

course of investlgation and ot propo.ration of the thesis.

I aa also deep]Jr inde~d to Dr. V.G. PllllSIS, ~tistical.

AdriMr, I.C.A.a., tor providing 1a0 vith adeqiZ!lte blcUitJeo and

particularly tor allowing me to uoo tho data collected 1n the pilot

Durve.Y for obtaining Block Level Estitlatos ot Agricultural Procluction.

ko~k K· ~. ( Sl.loanta K. Bzl7 )

'

'

CQN'l'EHTS

2. l!egroes1on EsUmatos In Mult1-stt\go Dos1gzls.

4.. An Application of Doublo S:uupling Proce-dure tor Obtaining Block Level Eetir4ateo of Agrioultural. Production.

· s. MQlt1ple Bserese1on EaUI:ulte In Multi-stage Doo1grl.

6. Summary

'1. Baforencoo

•• •• •• ••

•••••••• 4

••••••••

•••••••• Sl

•••••••• ll9

•••••••• 50

•••••••• 52

IR'i'F!lDUCTIOH.

'rho advances 1n t~ theory of ssmpUng toclud.ques b.a.s rooulted

1n devsloplng dirferent J:IOtbods o! ostil:latloD procedures. 'liB l331n l,lbjoct

of all thooo methodo is bovever tba 881110 i.o. to provide estimatoo which will

bnvo tlOrO precillion at. Din1mun cont. A mmber of Sllllplins t.oclm' qu.es llJ'C con-

ooquontl.y doVeloped for estimating the unlalown values of tho oh.:lracter under

stud,y by caldng use of ouPPle:l!Olltal7 infor=tlon trom correlated cbaractora -which can be obtained at a. lower coot.

ThO ut1Uzation (,)f the supplemonto.ry 1ntol'lll!lt1on fi"otl a COl'l'olated

charncter X tor improving tbo convont10J1Ql unbiased estimator 'Of tba unlalown

cean Y of a population io usunlly dono in fl!llllple eurveys bY.· tha two =n techniqu.eo via. ratio !Uid regression motbod ot estimation. Both these estimates,

as are usually used, a.1'C biased thoutth conaistent apd except whon tho true

rogross1on l1lle p:1ssos tbrol18h tlie origLQ. the regression estimate is leas

b1a:aed and mora prooise than tho ratio oatimato. Tbo mture of bio.s of these

eGtimatce has boon illVCstigated by Vlll'ious authors for large eample bnt no

ss.tisfacto:ry allllWer btls yet been obtained to detBn!ina the size of Simple

that will make the biAs mgUgible in any type of populntion. ' R!r avoldiDg tbio bias, sevoral aatbors bave suggested modified

forma of tho above oot1mates. Bartley- and Boos (1954) bave obt.ninod an

unbiaso4 ratio eot11!1ator in c::~se of sampllDs with equal pl't.'baMUty and vithout

replacement. Murtb,y and Nllnjama (1959), and llill1ElL'ID (1963) considered a

toolmique of ostitl!lt1J!8 unbiasedl,y the biae of a ratio or regression est.imate

by ccans of interpenetrating sub-ssmpleo ruxl used this ontimo.tor tor correctin8

thO ootilnto tor its bio.o. A genertll. method of genero.ting Ul1bi400d ratio and

regresoion ootimator in f:lnito population bD.s beon conoidored by IUcksy- (1959)

-2-

of the CO!lllliOnl.T used selection procedures for obt.a! n1 ng unbinsed oatimates.

Much ot tho o1mpl1oity ot the conventional Z'lltio and regression estimators

are however lost by theso modlticst.ions and as 1n largo Blll!lple tho b1o.e of

the 1W'..m.l eatil:l3tes are very small, the t:!Oditlod forms bave aot ;yet been

trlcd o.xteruli.vo:Q'.

.. Tha ratio

-s-

'ihe developments in tbis wrk aro baeed on largo eaoplo nppro:dm:lt1ons and

tho oDti=tes are obta1nod by cons1dsr1ng a 1d1fteronco eatitw.tor• of popula-

tion mean of tho clru'acter under study~ The diSCW3s1on is coni'1nod to the

casos vhon tho units nt the second and oubsequent stngos are oolected by tm

cethod ot simple raodom SlltiPlins while the pl'1li!f117 stago units arc solectod

either with VIU71ng probabilities with replacement or by simple randol:l ant!p-

l.ing. It uill bo teen bowvver· tl111t tho dsvolop:ncnt lo portectly geoorlll o.nd

a1mU.ar rt.sults hold good for D.ll7 prorob1l1ty ochel!:l) of =t~l11l!J• It has boen

shovn th:lt the eotilllates thW3 obtained aro t!Oro proc11Se than thP ro.tio estimtoo I •

...00 ~ b!.,_ ~~· the estblates decl'e!l~ mora rapidl;r ldth tho tncroae

CHAP'l'm w

-Estimation of tho population cean Y of c chnrantor b7 the atlllldrlrd thoory of 1 t ""Ill" regression requires tho rostrlctivo ani'.IU':Ipo-

tionll tllat tho populo.tion regression ot Yon X is lfnmu.• and tbat

V(Y/X) S.a oonottmt. In samplo IIUl'VOJ'S hovover, thoao c:onditloNJ are

coldoc l!lltiai'1cd tuld tho rogrossion ostim:ltea tllllt aro conrnonq used

are ~cod on lllrgo aamplo thoory vblch 4o not roqulro tho aommptiono

ot Hmarity QQd c:on.sta1lt vari.anco. In thio clrl-ptcr, reaul.to hllvo beon

obtainod for tho regroosion mothod of estimation in ClUlti-stogo dellign

tbcod on largo aamplo npproldmation b7 cons14orin~J Ul\3fl!' 'd1.tfercnco

-ol.lticator• of Y. ibr the eo.ko of oir::lpl1c1ty, tho dipciUWion 1e m1nly confinod to a threo stngo design vharo the pl'1car,y 6:JPPl.il:r8 unite aro

aoleotod with varying•probab111tico with roplaco:DCDt vh1lo the unito

in later etagos aro solectcd b7 a1J:Iple randol:l Slllllpl1ns without roplace>o

cont.

2.2. f'.otlltloq au:! Estimation Procedure.

Lot

P1 =tho probab111t;y of coloctlng tbo ith pr1El8J7 unit.

B = Dll:lbor of pr1mry unite in tho populction.

Ms. = aur:ibar of BOCOndar;v unite in tho 1th pr1.lnar)- unit

(1 = 1, 2, ••• N). f\j = =bor ot torttary units in tho (1, j)th ooco!ldclry

unit. ( .1 = 1, 2. ••• Ms.• 1 = 1, 2, ••• u). 'Yijt:= tho vaJ.uo ot tho character tmdor stud7 for tbo kth

tertiary unit of the Jth cccolld:lry unit in tho 1th

p1'br7 wd.t ( k .. 1, 2. ••• Bs.J).

-5-

lltjk = tho value of tho anJd] 1 t err varinblo tor the kth

tertiary unit of tho jth cocondory unit 1n tJio lth

primrT unit.

Mr

is. = 1/I\ 7- S,.j•

i'hon

-Q =

-It=-------

PUrthor, lot

M" ' -

= 11!\ !t. "ij~

v1jk c(ns/IlPi l lltJk lllld Dijk a (ut/IlPt) Yi$k

- .... - -'ilion V = ~P1V1 =X - ... -and Z = fF.P1Z1 c t

vhoro i 1 IIJld Zi nrc definod o1D1larl.y co abovo. kll aasuoe that Xijk IIDd Tijk arc e!Ulb obta' ned tor CVOJ.7

un1 t in the sample dro.vn aOCol"dillg to tho apoclfiod s:lillpl.liia ccl!em lllld

-tbat tho poplllo.tion ~:!~Gall X of ziJk io known. For obtn1n1ng unbiased ootil:ntoo of X aiXI Y, o. G!ltlplo of n pr1m:l.17 tmite 1o scl.octod out of

-Goo

ll tlith ropl.o.ccumt with probab111t1os Pr Fro::l tho 1th colactod pri%1:117 unit, a. l'lllll:!o:l sample of ~ eocondD.17 tm1to lo eoloctcd out

ot M1 Wllto llUd t= tho (1, j)th oolcctod cocond::lr,r unit, a nmdOil

&m~Plo of bJ.j tort1D.17 UD1ts 1s drown out of Bij units for obrorvt.Zl3

both 1- and Y.. e!laractor. i'hon lt lo easy to sbDv that

-v=

-and Ill =

•

and r1 = (1/~~~t) ~ vu f1l = (1/Jlls.) j; (v1~/bij) ~. :11Jk sro 'lmb1accd ootimatea of it am t,.. -fb3 linear ditf'OreiUlG ootii:Jate of Y 1D lllllt1-otago dooign 1s then given b;y

• •••••• (2.2.1) .

whel'o 1\z 1o a. know conotont. i'hel ostii:Jato i 1 1o tmbinccd for a givon val.uo of I\:a alii ito wrianoo 1a given by

vc~> .. vci > • 1\a 2 vc; > - alb eov c;, ; > ••• • c2.2.2>

2.3. Minlnua Varinnco Est1n:fltp.

to obtain tho lllnlm1n vnrllulco ootimato, Bm 1o to be oo dstorl:lilled that V( Zt ) 1o D1n1.ttu::!.

_,_

run1mts1Ds (a.a.a) 'lld.th roapect to lin• tho optimum value ot 1\:a 1D obtJ'I!ned llD

1\m = aov (v , i >I v(v > - - 2 lliJd honco V(Zl) "' V(a ) ( 1 - rm )

whero r. = aov c ;, a,, c vc; >· v cii' >.i ' Jibr a giwn vnlus ~t Ita , eay B a

•

eonsequenUy, 1t' the proportional increnco 1n VIU"ian.ce .tor a given wlue

ot 1\a ·is to be lono thaD DltlB prce.Stl1gned qu::mtitj C, vo CIW'.!t ho.vo I

B'll 1 -- ..:: C c C 1.- f9>Jlr f'.,

Cbsngf ng II into IT tm:l IT into a 1D the abovo upl"'Ssions, oiailar

ro8lllt.a can al.GO bo obtatned tor v (v ) IUid v(i). Th:lroibre \10 have

·~········( 2.~1J

•

Since ,,;i ). 11(i1 ) and Cov(;i, si) fU'O f'lmct1ons or Dt. lllld bij' it follovo that Bm IUI4 pm' obtained b1 cons1dor1Ds tho lll1n1mtD:l vnrlance

estimtc will dopGild not only on tlla population vulueo of tho S1I::I of

oqWll'Oa o.t .,nell nt:J.gc but. o.loo upon the mnbor of unito colocted in

. tb:s s:llilplo at the second IJ.Ild tlllbaequent otagas. ?:b1o lo boVOITOr not

so in cace of o. uni-sto.ge design, Wel'e the minimlD ~e estil:la.to

of" the fol'III { 2.2.1) will givo 1\:t or P131 indepe.m!ent or C8l:lpl.o oiso. 'i'ru, o;qJression ror lb can o.lso be dez!ivc4 b.v con:d.dering a

rolo. tion of' tho t'01'Ill

91 = z • ~ • 0s. ••••••••••• (2.4.8) B .. o.

C - -:::. - - Jf E (Bi - Z1 - 1\o (v1 - V )

with roapeot to 1'\n 1lhich gives

EC

-!l-

suppoce that a oamplo or n pr1t!ary Wlito 1o colectod with varying

prooob111t1eo vi~ rcpltlcet!OJ1t and vhezzevor the 1th pr1zl:ary un1t is

ooloctod a a11:Jplo rando:a =plo of 135, eocondary UDits in drawn thoro-

f'rom w1 thout ropla.cc:!l9nt.

coo.c1dor a qu:mtity fvz, given by

f. = vn

It vo aS!Iu:llO

F['Cv1J -v > C zik - Z) .1 .E ,{"(v1,J -i> (~J ... Z,J .

-

~· 1st Ul3

1o conllt:lnt tor all 1 llfld oquo.l. to Swz, t!len it can ba show thtlt

fw = -o-;.-:bvlil:.;.:;..-_(\vz=_l_\_> - , •••••.•••••• (2.5.1)

wharo

Alco ············~

IruminatiJis ~ botvoon ( 2.5.1) and (2.5.2) we obtain~

o-bW = < Qm A\ > C 1 • < i\a- 1) fvo-7

'

'

-10..

SillCO - ...,. 1 r - - 1 - -1 (vik -i> J -fv "' .. "' ..

- 2 B ( viJ- V)

E L(!11J - i) ( -elk - 1JJ

I

I 2

'

; I

.,

-111-

where f= ~I crv oe· •

Evf.don\17 I" v !llld . fa an the Intra claDs coft'elat!oml to1" V and Z

~ the ~uallty f V a t'111 10 fv. v111 hold when " a 1. f'ID vUl t!ma dopolld upon f , tbe oVftllll eorreltlt!on cootf1clont and an O%pl'eas1oD

dopondizlg apon tbe 1ntrll clAss col'IN1at1on cooftlclents of VIs aD!

z• s. fhe equaUt7 f 111 a f' 1a satUf'le4 when lilt a Ubr G1l. 1 or 'llh8n

f'v""fa=fv.•

Gem~, for ll high value of f' , !", f'_. ana f',. w111 not ~ V817• CoDSOqlr.Sntl;p', when \bo md.ts 1D the popalat!on

ore supposod to be hi gbl7 col'l'Ols.ted, an t1ppl'Old.mat1on to tbe value of f'

can fa1l'ly be made b7 f'11• 2. 6 Eatilllate of I)

Slnae S. ls not llS118l.l7 J1:x1ow. lt ls to be oatimated 1'l'olll the I!IGIP1e ltsaU.. A corud.stent olltlcato of S.. can bel obta' aect 'b7 conaidel'lJl8 tbe rat.1o of t.hD conaietent ·ostimteo of Cn(V,i) to that of V(V) •

If 110 doftlltt

8twa .. ..J.... ~

'

..

-12-

2.'1 &poGtatioll t.Uid Vorlnllce of it•

Do ostJ.z:ate ~ is 'b!aaed o1Dco lb 1D cst1mte4 br tm mt!o of Qll oatimtor of aov (V, ii) to tba t of V(i). 'lakl.D3' czpoctction

B = B(e3) a o.

fbm

-Cov(b ,V) a •B •a -~~-......:::;._ __ i za eov(V',i) +o1 } II D V(V) + 9a a •f\a B ~ o3 CJ. +

01

· ~ D Cov(V, 1r) --=o-_,•1 ~ 7 C1• 2 .J

D V(i)

•

' !"'13-

Dov .,} - ) -3

{n-1)~ v = n(n-1 v., / 1~ ""-2-.. - 2 L v1 v.• + D \~\' ~

faking expectations tom b;y torm tuld e1cpll.f11118 ve obtain

B(e2 ,;) a n E(;..i)3

• n V. V(V} bV .

••••••••••• (2.7.2)

S1 mU nrl.y1 it C1ll1 be ebow tb:l.t.

E(~,V) a n E Cfv- i)2 (;.i>J. n i IJov(v,B) •••• (2.7.3) &mm.ttut.1Dg tb9 vo.luss of E(~ V> and. E(~,;)1n (2.'1.1), vo h:Lvo attar Glmpll.f1oot1on

-eov (be' V>

=1\n~-So tho bSAo vU1 bo aoro lf

Bm- Cov ( v,_B) v ( v)

Ebr ll tw otap deaS.gn

.. r - -2 - '::I.J = "'k ( v - V) (z- Z1 E c"V- 'V)5

'

N

• 2 f;,P 1 (i; • i) f;(•\1•

•

•

-14.-

p. = 0 ( g ::. Jii1 ) , U' ~t lo avon l(at.) 1

-~ a 0 (o

1 ) , J.1' ~t. 18 o4d.

'lald.Dg 1:11

=. a, wa hnvo theroforo

S(iF-V)3= O(n-2) C i+o(m-; +O(m-2)J ~r 2 J -2 C -1 -a SSnUnrJ¥, ~ (i-V) ~Z') a O(n .) 1+0(~:~ ) +O(m )J

S1rica both V(V) 8lld Cov)(V,i) an ot oMor O(n-1~ C L+O{lm-S.,j,

1t f'ollovo t.bat vlth the 1ncreruJe 1n am:Jp].e me, E(V-i)3 and

B C (~V) ~'CJJ will more l'I!.P1d17 4ocroaso 'th!1r1 V\v) lllld Cov(v,s):

lu:J (2.7.4) ls of ordor O(n-1), tbB b1ao 1n a1

VUl be rosllglblo fbr

large n and v1l1 .rurtbor rodace~ it tbo population ls D.J!Ciltr1cal. ' To obtain tho var1llnoo ot ~~ vo htlvo

DJ. - i .. e:.f • bEl (i-V)

.,. ci1- i) • (b -a > ci-V>

1!1 D

ADSIC1Dg t.bat fbr large oample, the sampUug error 1n \. 1a DOgl1glblo,

wobtaln:

..

-15-

2. 8 .BatimaUon of' Vllriance•

l3norlbg the b1as 111 ihe esUmate of B tull! tn'k1 ng the m

eatimo.te ~ bm "' s~o:V, a constatent oat!mate of v(i1) la given b.1

.Bat V(i1

) a Bat V(i)+~ Bat( V(i) - 2b81

!$at Cov(V,'I)

GiDce ~a, ~n e.n4 '\v/n are unb1aoed ost11:14tes of V(i), V(i)

flDI! Cov(V,'I), we obtaln:

Est v(l].> .. l C at. ·~ ~ - eba -.,..J !> sL "' (1 - r. h

I

It baa been 1118llt1oncd earllel' tluit the regresa!on estimate

1r:L is always more o.fticient than tho o1mple eat1111ate i . 'fo oae

'llbether regreos!on estllllate results 111 any gain 111 precls1on OVOl" '

the mUon estimate, we C!011p8re tile two aetWs by atw211ng tllrl

ditterance betveen tile correspouU ng VIIJ'1ancatU fhe vari.ance expretJS1on for tho ratio estimate I' "' I V r '1

ot the ~pulo.tion IIIII3D f l8 g1ven b.1 vra: ) "' v(l) • a2 v(V) - 2a Cov('l, ll')

7f

v1utre a .. i/V. 'lhe l"'greasion estimate ii .Ul tberetol"' be mdre otf'1c1ent than tho mtio estimate w it vc;: )- v(J:1) '> o 7f 7f i.e. 1t V(V)

i.e. 1t V(i)

2 2 (ll -B.) •2Cov(i, i) ~B81) ,. o CR-s.J2 70

-16-

- both w111 bo equal. If"' the unlto bo of 1181:10 elze and lf tl=

D8I:IJI}..lrls la Cll'Z'ic4 oat at o'ach stage vltb equal probabWty tdth

replace=nont than the tw motbodo of estllilatloD wl1l bo ~ precise

vhcm • ~ 1: • ~

c f.= ,.a aD4 f. = 1IX fb., l1b;. -v Cwy v (fVT

-a-m O""VJC.Y .. o-1001 vhoro f'b a ~= f, = • a-ad'-vy • - a UbzO""by "

' I

- 3 ~ '\ L 3( Y't-Vt) t-~· V(Vt)

·•

L ~val. "~' t

arJd vil1 be sero u tM roJ.atJ.on

eo ... rv,;at.> B a-.....,;!---lllt v(Vt.)

is satisfied 1n each stratum.

Let

A11811:111Dg 811111ple slQ v1th1n each stratum II.Sl.al'ge,

V(ils) a z: { V(ilt.) = ~ v: v(it) (t- f!t>

t f mt "' CtovCii~tll C v\vt) • v(i,l J

B. Collbimd Remaston Est1"te•·

\

J

I

-18-

'lhen 't.ho C01J!h1ned r&gl'CBillon oiM.mate 1B g1wn b.7'

ii'lc = tr. b1110 ('-V) Bat eov rv.i>

b =------1110

It vo roplac:e ('\ -1) 1n the above ezpreaaion b.7' Dt ~ aosune that

tlx3 otnLt1flcnUon ts p:oportloml. b will: ruduce to the pooled 1110

!'he b1nB ot tliB ootlmto ii'lc • to-n ftrat !lppl'OJd.mat1o is g1von b.7'

- Cov(bmo'V)

r:ov

B(Vt-it)a .. t '\•

-

c 2 ~ . ..... t

vtllml a,. a,. Itt nn4 st v111 each contain teftll3 tnvolvlDa ax:J:~nto

ot tho maana at each at.ago tor t.bo t.-th otmtum. If f'or all t.

Gt = a. ift = H, \ = It and St = S and 1t 11e a ore propozot!oml to '\• than the b1na 1n the above G8Uc:!.te becc:Doo

:c~-1> vboro na *-~v Undor such assumptio!ID, the biao will depend upon n, tho totQ1

mriber ot pr1l::!lry units l.n tllo eamplo. Thuo, nnUkD oop3mto

ost11::3te, tho bio.G in CtlCO ot n comM ned rogrollDion estf.lnte c>n

bo I:Uide pogUs1bl.o b)' mnklng ila the total oamplo c1ao largo.

f!rJ ~ ot 'ilc is given b)' the oxprolltl1on

2 VCii.c) a V(i3) + smo V(V) • 2Bco Cov(V,ii)

., ± v2 C v(i.) • o2 v(i'.) - 28 Cov(V;t,iit).J ~=-~· t .. '"l:IC .. ao

L.. 2 )2 V(i. ) - V(ii._) a :E vt V:(9t) (~ -Jllilt , ubich 1D alwe.YS poaltivc.

lc .... t-••

It t.horef'oro fbllovo tllnt the seporato rogrosaion ostim:lto vU1 nlw;vs

h:lvo a mal)or var1anco than that ot CO'JbSmd eotil::ato.

2.11 Rogropn E!Jt1!!'?to lffltm !:"" Pr1.m:uz Uplt!! aro S01actc!l Bz Silnpl.t? Rrul!!og Bsmplipc.

For oicplo l'lllldol:a Ba1llp11ng '(fithout replacement ot tho

prilllnry 'IIDits, Wlbin=d eotlmates of X and Y o.ro sivon by

- !""'-X an 1f"1X1

on4 y ., ~ % uf. y1 Conaidorl.!!s tho diftoronco ost1mte V1 = y + 80 (I-i), tho OP'izlpz:l vnluo of B

0 1B then obtalncd ns

B = QgyC &f) m V(i)

1 N 2 -( u - b> ~· dt7 ~ u1 Cov~,y1)

= ( ~- ~ )~ + ~ * u: V(i1) N -

where s~ .. rk ~ c'\ \-i> cu1 t 1-n

~:~. ... vhich tollovtJ froa tbo rolat!oD ..

s('a1> = o.

Alco tho ~ of ~ 1o gS:von b7

f' = D

vcv,_> = vcf) (1- r!>

• fa= Cov(~y)/ C V(x) V(T)J is approldmatcd by 1 "1 2 ~-sblliv • 11 ~ ui CoY\&i,,.i) ,

1 c ~. n *' { ""')j jC {.. ·n t ·~ '"·l} r Lot obr¥ = ~ ;f;. (u

1x1-i) (u.;i1-Y>

B(o ) 4 ~ 2 bq = ~ • 11 ~ u.1 Cov(il,71)

Also llD osUmto of f. io obtdned o.o D

r = D /o o • D ~ ~ \w

UBin« tbo ont.tma.ted value of Bo' the regreos1on caUmte ot Y la aiwn 11¥ V: = y + b (~i)

1 D

-21-

'lb1a 'IIUl. bo b1n=S, tho or:10unt of bio.a bl)ing- Cov(b .i). Proeoodi113 ll

·111 t.bB G:llllO 1:111Dn01' as s1von 1n (2.'1), it Ciln bG obovn that to tho ts.rot -1

older of o.ppl'Ollil:lat1on, B{Y1) = 0( n ) and vill bo l:ll!lgl.l.clblG tor

lo.rgo sanpl.o.

~slectilla tho ll8lllpl.1Ds Ol"l'Ql' 1n bm' tho vor1anco ot

71 is s1110n b1 V(71) = VG) (1- f~)

and its oatimato 1o obtained as

Bot v(T.1

) = Bat vG> (1-r2)' ·'1:1

2 '01~1 ., '

D "' 2: (v1j 5,y 1· j= ~ b\)

- - )2 713- 71

82 "' ~ >a '2.:::

CHAP'mt 'fHRE8

Ibublo ~ ln !fultl-Stago Jlesigna.

8.1 1int1'odnsti&

b oat1r:!lt1on ot tbD population cea.n ot a chara.ctcr by

tho rogroOD1on cothod roqairea o. kmVl.Eidae of tb3 popnlnt.Jon csan

or total of tho onaflUary chttro.ct&r. C. lb::o (1943) first cona1dorc4

the probl.otl of est11!ntion in un1-atogo des!gn vhon th1o lrlfOl'lllfltion

is laa!d "8 by adoptllla th3 tochniquo Bf double Olll!lpliog b:lood on

1D2epolli!Dnt DIIOplos cb'aw:n i'rol:l tbo popalatiozl. ~m l'Osults woro

t'Urtl:r3r oxton&ld by Saal (1951) to tho Cllll~ ot oovorul amrUltory

vnr1e.blea Wldor 411'toront prob:lbUlty oahco:!ls of lltl1llpliDs. 'i'bo caoo

ot 4ro.td.ng a prollmill!l1'f lArgo c:u:Jple !rom the population Blld then

tald ng a rando::a sul>-sacpl.o £:rom tho ~ SSl!lp].o wae f'irst

cona1dorod by Coc!lran (1958). Bio roBUlto o.ro hovover based on

etnndord theory of U=ar rogrooaion and ae ouch tt•o o.ppUcation

1n oenple suuoyo ie rather l1cited o~ whon n aulti-staso

doalgn io adopted.

b cathod of ootil:lation developod in Chapter iVo ean

bo DimUarly extended fbr tho double co.aplJ.JI:J oche::llo tn aul.t1•otaeo

dotlisn when I io oot !mow. In thio c!J!lpter. 1'6oulto 81'e obtained

for tho caao vhon tho pr1J:tlry UD1to aN oslccted with VIU'J"l,.rlg

probabUitJ&e with roplscocollt vb.Uo tho UD1ts at later otaeoa lli'O

coloctod by lW:Iplo randol5 tlal:lp1i.ns v1 tllout roplllCCil!Qnt.

IJoi!lg tho mtatlou of tho prov1oue cbaptor, cons.idor tho

follovi.ng BChe:le of double ~ .t'Dr tbm1ng Dllitablo l'OBI'Oooion

oottl!l:lte of i vl1oD X 1s not know.

I

·• -23-

Out of U ¢l!l!l.l7 units, a B:U:Jplo of n units is celeotod

v1th replfloc=mt v1th problb1llt1es P1 for observtDg both.!- o.nd Y-

chl:u'acter whlle an add1t1oD31 Sltlple o.f (n• - n) units ie ooleotcd '

v1tb tbl snme probab1Ut1os v1th ropla.ccr::ant to obsorve the .5-

chnrnoter alone. i'rol3 ths lth selected pr1l:lnr7 unit, a I'8Ddom

am:rple of ml ~colld!u'y units is soleotod out of lis, vhilo f'1'om each

o.f a;_ unite contained 1n n units a ra!ldoD ll!l!liPlo of ~ 1m1ts 1s

solcotod tor obt:.orviilg both X o.nd r. .li'l'olll oach of the cele~ a;. eocondary ~to. a 1'!lDdom sample of biJ tertinry units is sOloctod out of a

13 while f'r'oln oa.oh of tho 'bJ.J units ~ntainod in m

1 units a

l'8DI!am ~eamplo of biJ units 18 solected tor obmrv1ng both X•

and Y- oha.raoter. 'lbm

- l. .,. - 1 .... u.. v• = - -:2::: v• = - ~ -A.. i' n• ,.. 1 n' '"11'1 1

ulll provide unblnmd estilaatos of \_ whore Zj_ lUll! ~ aro

unWoDCd ostiJ:lates of is, maed on (m;_ • bj, ) and (~, bs.s> units. Sinflttrly. an unbiaosd estlclate of Y is givon b7

vholO ¥1 is an unbla&ed o~to of \ blsed on C~~~t• b1J) U!llta. In vhat follows wo shall doZJOto 11!!,1 olltlcate baeod on (n', 131• blJ) v1th a dosb vhile llll ostblato vitbout a dallh will ba baaed on

Cu. a1

, b13

) •

•

s.s. Minii!!Ul:! Vnr'.anoe Est1cnte.

Conaldor the osthlo.te

i• = ;: • B' c ;. - ; > 1 D ••••••••

(8.3.1)

Tldo 1.o unbiased i'or a give11 valllt! or B~ • Rinitll~~atlon of tbo variance

or Zi with aspoot to B~ will glvo

::a eov(v , i > - eov(v• , 8• >

••••••• (S.B.2) vc'V >- vc ;. >

V(Zi ) = (1- p~2) LV( i ) - V(s' >J • V (z' ) •••• (8.3.3)

where Cov(v ' 8> - Cov(v' , e• ) {r ~ CvcV> - vcv•>J Cvc;>-v J j

•

and will bo c:.w:tly oqtllll to Bur vhen ml = lilt and bjJ = bij• S1mtlnrly, tor largo n', f~ can sa.t1nfaoto1'1~ .be tnkon ns

.. P.'" m

'l'hua, for large 8!ll:lp1e the eati.mato z;_

zi=i·!h (v•-v> ••••••••••• (3.3.5)

-25-

Using tho olltil:u:.t.ted valoo ot % f'rotl tho ca.mplo. tho ostll:lato of I 1o then obtaiaod ao

' 1=;. he J

V(i1) - V( 'il } = ( *- 1Cil >CBu,2• fit*·"tl

Alco,

Bo.aco, Q cons1atont oat.1cato ot vCBi,) 1s giwn b.1

Bat ·v(i') .. ~ - .Jl ~ C~- J..> ~ • -A... ~ tJ..- ~ )Ji 1 D D ~ nt - D3 iJnl f;:: "'"'1 c

1 b

• ~ }: ~ E .a(~-~ rl l zm• ;.. a1 a1 j» 1J blJ blJ 1Ja ~

3.5 Bft1o1ona7,

fD mmnSm whothor doublo 83l:lp1.IDs resulto 1D mv soJ.n

ot of'f1c1cncy, vc oomp!lZ'O lto Vll1'1aJICO vi\h th!lt ot the aSnplo oat.1Dato ire

-tv-

Tb::l oO\imato bnosd on doubllo lll!!lpling v1ll bo coro otf1ciont

tllsn tho o.iaplo oetimto i 1t

v(i') - v~> /' o 1

Loo 1t P! /: V(Jr) - V(r)J > 0 which io al~o true 01Dce V{i) - V (i') > 0 aDd f': 1o posit1vo.. fhc gain 111 officieD07 wUl houwOP dopmi on fa !llld hlgilar tho vnl.uo ot f. , tho gnntor the gain 1n ettio10DC7 t!lnt ctiD bo nchiowa a . b.1 doubkl Slll:lp11Ds procoduro.

I "Q -It vo COD01c!or a ratio ootlm:lto '\. a V x V' • tit)

V£11'1Q11Co oxprooooion of ~ tdl.l bo g1von b.J

V(ll') a V(i)+ filL V\v) - V(V1)J lr

-m£ eov(V,i) - aov(V'.,it)J _. ' 2r, J

lbv V("J,) a V(i) + Bill t.' V(i) - V(V'I)

-2ar;{ eov(i,i) - eov(V'I;i•>J

iborotoro

v(\J- v(i1)

a(R-1\n)LV\v)•V(VI)J ~(R+B )- ifeox

•

Hence double s:mpllng 1n II!Ulti-stage da~ bl' the rogroSD1on mDthoa

of eotil:lat1on w1U bG al~ ICON efficient th:ln the corrcspord'ng ratio

3. 6. Optilllllll Allncation.

• In our prGVioua discuaDions the vnl~PD of (n', tli.• btj) end (u, ~· bt_j) aro cbooon Bl'bltr~ and tho astlc!Ate 'Bi to obt.oSne{j ~ m11l1mililing tho vo.riance tor given (n•, •1• b!J) and (n, Dt• btj)• Bllt those (n•, ml• b'lJ) and (n, ~· ~.1) OI1D be chooonin an opt1mu:D way

so ns to '"'"'"'so t11s vartanoo for a given coot or m1Df.c1zc the cocst

for a given variance. In this IICCition we will at~ tho problec ot

optimmll allocation of sample siee tor a /!).ven cost. lohen double sampl-

1118 is ad.opted ln a two-stage dBDigD.

Lot 118 auppoac that the contribation, to tll!l total cost tor

n double oompling oohe!:lo QOIIIOO i'l'ol:l the follov11:18 three SOUl'CO:IJ

1) tho cost of enu::el't.lting a pril!lsry un1t ( o OJ,)

2) t.be cost per OJEPOl"iment of observing the enc1111ar:v clulrncter ( = "2) and 8) tbo cost par experiment of observing tbo ch:lract.er under st~

( = es>· In a particular cample tll!l total coot 1a theroi'ore given b7

..,.,. "' C = c1 n' + c:a ~ mJ. + Cs· 2:: fat• ''lo' . f:.•

• Closrly, C will '11117 from llSiliPle to eamplo. lib shnll, therotore,

considsr the evorago instead of tho actual cost ot ~ a ea~:~ple. •

i'hls is g1 von b.Y

"" ~ C"' CJ. nt + "2 ll1 'f; Pi •1 + "3 D ?;. 'Pi 111, ••••••••••• (8.6.1).

~ opt.tmlllll valuea ot (n1, ml,) and (u, l:lt ) CSD then be dotol'IJined b7

mSnimising •

V(Bj_ ) a ( 1- fa2) ('V( a ) - V ( a•)J • V ( a• ) 2

tor a gt,von cost Co• !l1nco ~ 1B a fUnction of ~· 1t lo DOt

poiWiblo to obtain oxplic1t mlut.tona tor (n'• ml ) and (u, at_} by tbiB aothod oxcopt wh:ln m1 = 1 lb which caeo f'a = f' • vll01'e

p = GW If ho1IOVIIl' w QOiill:IO that tho l!!ll:lpliDs unite !U'O vv- ae highly correlated so thnt ·~ can be a~ ted ~ P, than

V(il) a (1 - #> ['v (i) - IT(a' ) J + V(i•) = (1- ~) c ( ~- ~.> »! + ~ * ~ ~- :. ~ ~

;; b

•••••• (S.6,2)

... l'l -?:::-. •. !lit

•

vboJ'o >.. lo n constant tllllt1p11or.

DiffbroxztW1113 cf td.th roopcct to D1 .o.a;, aDd a1

G.lld c~ to 001'0

••••••.• (3.6.8)

=0 ••••.••• (8.6.4)

cO ••••••• (s.e.s)

=0 ........ (8.6.6)

•

-so-

• ..... (3.6.'1)

••••••• (S.G.B)

••••••• (8.609)

An ~liCit. O>lution fol' n1 is !lOt poosi~e to obtain f'ro:l this mothod.

Ou:~ can bovovor dotormino tho wluoo ot n by o.ssign1Da difi'Gront vo.l.uos

to 1:1. in ( 3.6.9) and tbo opt.imtrll wrianco can oo obtained 117 aubatitut.-

ina tho opt.imu:n vnluas ~ n•, nand ll{ in (3.6.2).

For "s. q 1, we haVe ~ = f' and t.be abovo vnluos td.ll tben roproscnt oact. opt.itlu;a oolutions. Tho optimum variance obtained 117

-oubst1tut1ng tbeso valuos when Su = 1\n, is o.osu:ned to be coneto.nt tor all 1 1s given b,v

CRAFi'ER FOUR

An Applicntlon ot n,ublo 5::l1:lpl1ng Pro03duro t'br Obtld.nlng Block Lcftll. Estiiiutos ot Agr1.culturnl Prodtmtion.

4.1. Introduction.

An aJ)pl.icat1on of the tlothod of est1ElAt1on diccussod 1n

eaaptor 'lbno t411 ~ mado in this chapter tor obtalning rollablo

ootimatoo of ytold or principal coreal. crops grown 1n each Co:::rnumty

Devolop:lCllt block or a district, thToug!l the eot~birllld &!'preach or crop

cutUns ~a nntl oatimtlon ot yields by e;yo nppmisal.

Eotitlatos or crop production lll"'l nt pnn;ent available o~

at tJlo district l.Dvol in rospoct of llQl!IO importa.nt food and DOn food

crops and are blood on o.n oxtensivo' ,!:Or1oo ot crop-cutting OJI:J"lrit!ontO.

If eatl=tes are now to bo provided with stlff1ciont roUnbility at tho

lovol or tho block, the present ol'dor ot .crop-cutting oxporioonto will

bavo to bo 1ncrenaed manifold, perhaps 10 to 15 til:wa. Tllo ort;nn1ee-

t1oDill nnd f'Jnlncial implications of suah a htfSO prograr.me of crop

oatimation 1o obv1ouoly goittg to bo a hcaV7 and ditr.t.ault ch:lrge,

pract1cal.l7 lcpossi.blo to tlllltl80 on tho stAto adrdnfatrntlw Dnch1nol7·

To lied a Sllitablo aolution or thio problc::J, a pilot

GUl'VUJI we lllucchcd during tho year 1902-631 by I.I .• R.S.(I.C.A.R.), to

atudy whothar a joint usa ot crop-cutUng ozporiz:lonte 8!ld ost1cat1on of

yield rntoo by eyo appraisal will f\n'tdllh reUo.blc estJ.matoo ot erop-

pm>dw:Uon at t.hl3 block lovel without onduly onh!inoi ng tho prosent uork

load of tho tiold mchin317 avn1lablo in tho bloclto. 'lito curvey W3.D

t.Ltkcm ap ill tbnte d1atr1cts in tho firat 1DOtallCO aZid tho d1atr1cts

cbollOll tor tho SUMoy wore Pntm, llhuJI1.o aDd f.":;~artlt 1n tlt:l atntoo ot

Bihar, M:lhal'tlGbtrn nnd uttar Pro.deth raapect1vol7•

•

-ll2-

'i'11c &ud.gn of tho Olll'VOY adopted vtl.ll ono of two phaood . otratificd randal~~ Glll:lpl.1nm the villngo-lovol.-lolDl'kor o1rcla (V .L.ta.

Circle) within a block coJWtitlltiJig the otro.tll:ll1 o. village within a

c1rclo boing tho pr1m!lry unit of oatlpl1Jl8, a field '111th1n a vilJ.a8o

boiDB the sub-unit of stllllplJ.rJ,g ana a plot within a field boing tho

ultil:Jato unit of ll:lllplJ.ng. All ttlriJ a:1lllpl1ug urd.t.o at. dittorent otageo

woro oolectcd '14th cqll.:ll probability vitbollt ropl.rlcc:lont. ProD each

otra.tm:~ 4 to 5 villages voro soloctcd and. in allch village thuo

solectod 4 fS.olds wro aoloctcd for obtaining ~ost.!mo.tes of Jiold.

Frol:1 tho vlllngoo oolectod tor obtaining ostimd.teo of Jioltl b;v oye

approllltl4 o. oub-BOmplo or 2 villages, and fl'oll tho tiolds sol.eoted for oyo ootbntlon ln oach village, a aub-eamplo of 2 f1el.ds voro

•

soloctorl at rn.ndo::l for conducting crop-c¢tin8 ezporlconts. Tho

colloctlon of do.ta was ontrustod to tliD oosmato f180nclOS. In all

casoo, oyo-eoti~:~ntlon wo dono by tho V .L.l:l' o and tho Cl'OpooCUt.t1ng

oxpol'laonto by Patvarle/RAramcharl.oo.

4.3. Ngtqtlog an!! EgtlL!!e.tlon Prgcodnro~

Lot

L = llll:lbor of V.L.U. cirol.es ( i.o. strata) in a block.

H = I1U!Ilber of villagoo in a V.L.V. c1rclc,

"' n'= number of villagos BOlectod tor ayo eotlD!:It1on 1n V.L. W.c1rclo. ~

n = nlll:lbor of vlll.agos oolected tor crop-cuttins lD a v.r..w. cirCle (o. sub-lmlplo ot n')

c'= IUllllbel' of fS.oldo soloctod for cyo ootillllltion in 11 vUlago

n = mnbor of f1oldo celoct.ed for crop.-cuttina ln a villagO (a oub-llatlplo of m1 )

II

J II

II

'I

~

I II

•

1't13 = tb3 dry )'iolcl (in kUogrono por hocto.ro b:lolD) fro:J ~ jth

tlold ot ith vUlaao ot t-th v.t.W. c1rc1o obtn'nod by crop-cuttirJ&

~ = tbo oyo oatiE!ntad dry )'ielcl fnJ!l tho jth field ot ith 91l1.a8o

or tho t.-th v.t. w. circle. llithout I10!ng oyo ootimateo. o. olcplo ost11late b:wcd on

orop-cuttiDg rooult o1onJ io given by

('\) = nt)

A.oll1l:lins t.bo DII:Ibor or fiolds 1dthln a vUlaso to bo lo.rso. tho

Vll1"1o.Zico ot 7 Dlll2 1t'o ooti=te llZ'O given by -2

whoro

v(j) = ( t- t> ~ • ~ Est v(y) = ( .!.. - .A.) rf_ + ~

Do % 111' cfb

=

Cfb = m.)

conM I'IEid ~egroDDSon oatit:lato or tho avorago )'iold p3r block COil bo

tnkon tiD

7' = 'I • b (it -2.> 1 a

•••••.• (4.3.5)

'I

lj II I II

II II

I ·I

I I

" r

-S4-

vhct'C f'or largo n' , .P 1o takon ao 0 1c

f'= 1:1 ccs.!.: s!> j•

An ootlc:!.to of v&;_> 1s tb8n obtn2"'9d ao 'i2

Bat vm') .. C "if+ wz J - "'l. Do lio -D7 Dt\,

S1nco tho work of condWJtlrlg cro~ oxpar1conto and

oyo olitic:ltion wo aardgnod to Pnt113J'io/Koramcbaris and v.r..wo, to bel

doD' 1n tho1J' roopoctivo JuricUctloD. coot on travolUng fl'o:n ollll

vU.1ago to amtbor wo.s nssu::od to bo I!Og11glblo. 'lhct tlll1n item of

expon

-ss-

tho cost tor 11 given vnrianco. Colllli&lrin8 tho vnrillnco oxprollllion

glvon 1n (4-li.S), 1t 1s d1tticult to obtn1n expl1e1t solutillns of

(Dot m) Ellld (1\\t a•). It hovovor1 thll variables oro h13hi.Y corre-lated m tbat ft:l can bo llPpl'O:dootcd by tho ovornll corrolatilln

coofticio~ f' tbon t~ opt11!m:l val.~s of (~, c) IUid (U0. c') that r:d n1mho tho Vlll'innco for a given cost or m1n1o1zo tho cost tor

F:lx1ng tho val.uoo ot a, "1• "2 and f ono OQJl theroforo dotomlno n am n' to obtain a doo1l'ed pl'0c1o1on of tho oot1J;mto by ~ above relations.

4.5. Nu!:pricnl Ill!l8\£nt1on.

libr tho purposo of illustration we \l1ll collDidor baro only tho

data on Rab1 ~!heat colloctod .froa tb:l Patua D1atriot of B1h9r.

ThO oaticatos of average yiold of whc:lt 1n Kilograt:QOo por

hoctara for each Block 1n tho district obtai nod both by cn~p-ctlttizls

ozperitlonta and c;ve-nppraiaals aro preoontod 1n Tablo 1 togotmr

111tb t.ho1r percontago otandard orroro. In tbo oaaa table, tho gain

1n oftio1onc1es of tho regross1on cstil:latoo ovor sil:lplo ostil:latao and

tho values of .,. 0

are lll.so given.

It can bo soan from tho tablo thllt tor bigber vuluoo of ,.. m

thoM is cons1dorablo go.in in ofi'icioncy of the regression osticato

ovnr staple ostil:lnto. i'bllroforo, it Clll1 be safely B:l1Q that 'llith

1ncl"C:lcod acoUl'llcy in obts1n1ng oyo ostim:J.too lllOro proo1o1on could

bo obtnined 1n the osti.mntoo of avorago Jiold at Blook l.ovol.

libr dUtorent vnluoa of the ovorall colT'Cllltio» cootficient

and tb3 ratio ot ~c1, the saJll?lo oizo required tor obtaini!lll tho ootil:lete or a.vorago yield with o. prenalon correspon4ing to tho

!J'tQndo.l'd error of 5~ baa bcon worked out IUid givan in Table 2. It

~ bo aeon tbat tho highol' the correlation coefficient or the r>llor

tho coot r~t1o of oyo-oaUoation to cropo-cutt1J:I& tho t!l!!nllol' is the

aizo of iho aarlpl.o rcqairod fbr conductlng Cll'Op-Cuttlng experiments.

Howovor, if thB llUillb!lr or cropoocuttiaga IUld oy()oo()stil:!atlons per villngo is increased there ld.U be a reduction in the llU:lbaf qt villo.soa. It

!:ley aloo bo eeen f.hat once o. villago is coleoted, ~ lm1!lber or crop.

cuttings or oyo-ootim:lt!ons need ba conducted 1n ·tho e!CO fields and

110 oubstantial gain in etfic1oncy io to bo oxpocted in l:laking r:JOl'O mz:~bar

or ~ticat1ons than cropoocutting in tho samo vU.lngq. It will bo

rath\!r ucoful to aproo.d tho ayo..cst!oatos ovm- 1araor 1:3l;ple or ~a. 'rhis i'ollova as a rooult ot th0 assu:~ption that expondlturo for tho

ourvoy

'lnblo 1.

EsUmto of nvora.go dry J1old per hectsro 1n IW.ogrot~o f'or oncb block.

Fro!:l cropo.cut,. Collb1111na oyo-o.pprainal ~ oxporiconta i~ cropo.cuttiDs rooult.

mock. o.lono Ro~oOton ostll:lo.~) !Avo J1ol4~ S.E. Av.Jioldl ~ S.B.l ll5 Ga1n 1n

Pol.igan,1 828 5.46 823 3.82 1D'1

Barb 54\J 2.18 153 1.86 85

lboraar1 'IS4 '1.96 '119 6.66 58

RBJsh' 567 s.'iO 563 3.18 3'1

HallUl'h1 987 10.25 994 a.6s 87

Punpun 5~ '1.90 583 ·G.~ 93

Fatwcll 957 1.56 976 1.26 as SamDrn M5 4.94 447 4.43 a lWltn 902 4.l2 859 s~.u. 01

Blharooriff' 749 7.63 862 5 .. 15 66

Etflc. J'tl

o.os 0.65

0.'1'1

0.68

0.66

0.63

~1)1

o.ss Oo81

o.oa

-ss-

Iablo 2.

Saoplo o1Bo roqu.ircd tor oat.11:mt1D8 t.ho avorago dry 7lold o.t t.he Block level vith 5~ standard error. (For diffOl"Cnt vnl:aoe of tho cocffloiont of corrolation and coat-rat1oo).

eg_/CJ. I e>·:Z ... I 0•1 I o·o; I D I n• I 1:1 I Jl I n• • 1:1 I D I n•• D I I I I I I • 2~ 31 2 22 40 2 21 ~ 2

o.s 20 26 3 19 85 9 18 46 8 19 24 4 18 83 4 'I 44 4

22 37 2 20 4'1 ~ 19 64 2 0.6 19 32 8 17 (() 'a 16 54 s

1'1 as 4 16 38 4 1S 50 4

19 42 2 17 53 2 16 '10 2 0.'1 16 35 3 1S 46 3 14 61 3

1S 33 4 13 40 4 12 53 4

15 45 2 13 55 2 12 'l2 2 o.a 12 36 3 u 46 3 JO 60 11

u sa 4 JO 42 4 9 54 ~

II

5.1. Introduction.

In forl:liag tho rcgroooion oGtimato of tho population =n

of o. cbaractor un.t'lor study in multi-otnge dcalgno, it bao bean

aOilUlJOd 1n tho provioWJ cbaptcro th:lt ndditioDal information lo

aVIlil.o.blo £l:'o::l a a1nglo corrolatod cmu.uotor X tor o:lch Wlit 1n

tho population. Suppoco nov t!lat inatoad oi' ono o.ux111.1cry va.rinblo,

tor each Wlit 1n tho population thoro IU'O p corrolatod chamctoro

(.X,• J12• •. • xJ o.vnllo.blo tor coalliU'UIOat.S. each of wbich lo b1ebl3' COl'l'Olated with tm vo.r14blc Ulldor study. 'Tho problc::l tbon 'OOcomao

to 1nvotltigcto llov best ono can utilizo thoao lnfo1'1:13t10D8 1n f'cm:d.Ds

tho ooticate of the populo.tlon oaan or total of tho cb:lrnctor lmlior

stllltr co th:lt ~ preclolon of tho csticato can be o'bta.lma.

In tbis cllaptor :rosulto aro obte1nod tor tho rogression

cotbod of ost11z:lt1on 1n nultl-stago deaigD vhon oovural attx1lllnt7

varto.bloo Bl'8 o.wUable for mcasur=at.. by foi'IJi.Dg a nult1plo l.J.mcr

rogrooalon o~to oi' tho population EIOilJl of tho cllaro.ctor Ulldar

otudy. i'b.o caeo vhon tho population c.oans of tho ancllllAlzo;v chaructoro

IU'O not know is alco com)jdorod by odoptiDg tho tochn1quo o£ doublo

saopllDg on oo.ch of tho onxt ns 1117 vo.rillblo. •

5.2 Rotation Md Egtlmntion Prpcod'lll'EI.

SUppooo th:lt onch unit 1n the population ho.a (p ~ 1)

obSOl'VIlblo charo.ctoristlco :fo and ( 1.1• ~ • • • Xp). ConoJ.dor1l:ls a throe otago design arx1 us1!18 tho notation of tho provious chapter~

let

ll'oijk a tho val= of tho charo.oter andor otud;y tor tho kth tort11117

-«>-

unit of th3 jth socolldary uuit in tbo 1th pr:lJ3!U7 unit.

zti31t = tho valuo of ~ t-th nwd.ll11117 variable tor tho kth tortlary unit of tho Jth socolldln7 unit 1n tho 1 th ~ tmit.

(t = 1. 2, ••• p).

- c....\.. s·· Xoi3 Bij ~ Zoljk - M; -l01 .. ..J.... :?: '\J lmJ Ms. J:O - N ~ = A ~~ io1 Ill - .. -t,p1 -zo zo1

.. 11~ - -=L: pi ( Xo1) = Jb f ~· m-1

SimUar~, tho qU2DtU1os luJ• i"u• ~ and~ can bo dof11131 for t = 1. 2, ••• p.

Lot tho population IJIO()JliJ ( ~. ii• . . . ~ ) of tho p COJTOoo lAtod ehsrnctors bo kDI)vn. ibr bu1l.IU.ng a sllitablo tl11lt1plo linea?

- ' rcgreoalon oatiriUltc of JO• a GSDplo of n ~ uuito 1B eo1.octed out of U Vi~h roplacc:nont with probo.b1.11t1ee P1• ii'o:ll each of tho

n solectod p.r1l:zary lmits, 11 Dimple ·l'llDilotl S31llple ot r1:1. ~

UDlts 10 ealoctod out ot MJ. units ami troc each of tho soloctod l!lJ.

socondar7 UDlto, a s1r.lplo rnndoD fllllDPlo ot bj_J tol'tlary units lo

selected out of B.l.j wd.to tor obBOl"Vlng th!l ( p • 1) characters lfo

o.lld ( xl., x.z, ••• Xp ) • 'J.'htu1 an unbillscd ostiaato ot " 1e gS.wn b.'7

-4].,-

(t a O, 4 2,. • • p) •

whoro

-ia an uubiaocd oet~t.e of Zu· Tho miltiplo linear d1t£oronce ostit!lato or X., in milti-stago

dooign is tho11 stven by

••••••••••

io unbiasod for given valuoa or 1\:rt 0 am lt 1 e~ .varianco 1o given by p p

V(i'ol) a ~ C '\,o • ~ dtt. ~ 2 - 2 t=; 4\t Bet p

• 2 L: 1\-t 1\ar 1\J ....

-42-

Ir wo

a.n4 co the leaot Bqallre eotimateo of fbt o obtalno4 b1 nSnin1~ (5.4.1) with ~·ospoct to 1bt vill lead to tho IIOJlO oolutioD ao obtained oarllor.

-5.5. For obta1n1ng tho nin1rnu:~ vnrianco or Zal Ulldor tha optimu::a

- - \:> Zo1 - z .. 8 - L: 8t 1\rt;· 0 0 )-:I

SUbotitu,UDs tho opt11:1tD values or Bat ott.bia tiitforonce beeot~Go

- - --&_ p zol - zo .. l»oof E St. !Dot I.

~roforo

B(~-z0 )= o. B £ ffs.C~ - zo } J = it I lin /Dot I

l n.l .. • t'ozo r = 0 •••• •••• (5.5.1) 1»00 I

= o. .for r = 1. 2 ••. p.

- ) - - 2 HelllJO v( zol = E ( z01 - Z0 )

= B C< ~ - zo> ( 8o - ~ ~ ~ ) J z: L!.! a doe' L .!..!..I by ( 5.5.1) ....... (S'·S'· ~)

""'»oo I " h oo I

-M-

-i'.hUD tho min1mll:ll vnr1eJlCo of Z01 obtained by c111lulng tho wrio.nco -ot ZoJ. with rospect to lht s 1s given by (5.L2).

E(Etg (/) a E L {\, {80

- ( Zo1- Z0)J J

= dOo!,..- ~ D j by ( 5.5,1) 00

=d·{,-/D/ • oo ..,/Doc/

Cono1der mv o. qunutity fa o.l2 ••• p' (l1von by

,.~c!»cfL fa o.12 ••• p LL f!.o2• B i2Jlz

TI:IOn f'rom tho o.bovG roGUlts it follows that

/P/ J~ /P00 I

om s1nco /D/ :f, .t /D 1, wo hllve 0 £: f. l2 "- ~ t!ms -oo "od' · a o. •••P -

vc z_. > = .. . H 1- r,2 12 > ..... "'00 ., 1:3 o. • •• p •••••• cs.s.s)

-45-

s.e. ~ dotnmtnation of tile optiau:n vnluo of Bat requlreo tbat tho Vll1'1nnco - coVIll'ianco mntrb of aaaplo ceaDD or tho (p + 1)

vario.bloo be 'lmo1m. Ao this is ~not known 1n o.otu!ll llltuation,

one v1ll havo to ostil:ls.te ~ s from the sample ltsolf. Let bat. bo an oBtimato ot Bat from the Olllllplo suCh t.ha.t tho c:ll:lpl1Jl8 orror 1n it lo n&gligiblo. Conoidor thon tho eatil:lo.to

- - p D l a II - 2: tL b_..

0 0 !-'>• , w .. ••••••••••• (5.6.1)

This cnn bo vr1 tten aa

Binco the lllll:lpl1ng errors in bm s oro negl181blo, ve obtlll.na

E ( sol ) = E ( ~1 ) c

v(i~, ) = v (~ ) = 4 ·1;( 1- f'.2 ) ••• (5.6.1) - oo 1:1 o.12 •••P

- ~ -t;, Cov (".nt.' "t ) • Tho 6llpl'Ossion for tho varlanco glvon in (5.6.1) wlll then 1"'pZ''CC::l\

M nppro.Jd.llato valuo of V ( ~1>• For lorga sample size bowevor,. tho -biao is DeBllglble IUid the approxillation to the. var1anoe or Bol 'by the

above forculn 1o quite o:~tiotactory.

S.V. Eetill!ation Of Vp.rinnco,

To obtain o.n eBtillate of V (~ ), an ost11lo.te ot tbe

ootrix D 1o rsqtlil'fid •.

-46-

S1DCO tho first GtQgo unito aro oel.octod with vuryinB proba-

bilities with ropl"c~CSnt, it follova that

la an unbiased eotimato ot drt• B>aco an ostimste ot D io civon ~ a

tho mtrh (( .. brt )) (r, t = 01 1. ... p) • .. Lot ( ( D )) ::(st,rt) vhero

f1taot = fbtr = 8UI:l ot products botveen pr!.mar,y units of tbe rth oml ith dlaractor in tho GaZJple.

ibm the estimatos of lbt a are obttdned tv solvi~~g the equtLtioDD

'

fbtl ~. ~ ~ ••••• ~ ~"' ~ ( t lC 1, 2p ••• p).

2 f'm o.12 •••P =

IDI 1- a;;o ~»oo I

= C'\,1 ~ • 4o2 ~ • • • • • aop 9mp> I 1\,o•

i'harotore, an estimate of f..tl 0 12 is given b7 the pooitivo • •••P

sqll31'6 root ot 2

f li; o.12 •• • p = C"m!stm • ~ Bt:lo2 •·· ~ • ~ 'i»p)l Dtloo

5.8. Pemble §am)ling.

When the popul.$t1on means ~~ Ia • .. ip 111"0 mt krlDim. the

1UI11:Il procodure ot double c:l!llpling can bo adopted. Uo108 tho doublo

atll!lpUag oc:ha"e given in ( 5.2), tho obllOI'V'Qtion on tho p c!!amctol'o

~· ~ •••. Xp aro cnde tor e:lCh 1llllt 1n t.ha omnplo (n', Ill• biJ ) tor - - -obto1n1ng unbte.sod oet.il:!s.tea ot ~~ ~ • •• ~ vbUo fl'O!l each llllit 1o tho ll:ltlpl.o Ca. ~· 'l_j), ths observnUollS on tbo ( p + 1) cblractoro

lfo• "1• ... Xp o.ro taken tor hntldiOS up a IRlitablo regrosllion ootil:!ato. l::vident~

~ ~

~ .. :.-~itt cn4 it .. ~~.it1 - - -nrc UEibia.sod oatimo.tos ot ~ vbBrs at,_ and '\1 aro unbitlcsd

ostltlstes ot it1 baDad on ( oi, bi.t) IUid ( Dt• b1 j) units.

-Aloo an unbiased ostillato of X0

1s given by

Conoider now the oatittato

- - - - .... -Z1 c B + B• ( & 1 - B ) + • • • + B' ( & 1 - B ) ol o ill1 1 1 ap p P

~hlo is unbiased tor given vnlueo ot ~· Tho opUl:lu:J valuas of B~ o -tor vhic:h tho vnriallce ot z~1 1o D1DS.mu:a nre then obtained by eol"'1Dg

tho 131flta:l of oqaat1oDS

dh B~ + d•l"2 Btk + ••. + drt_ B:at + ••. + d:.S, B~p c ~

(r = 1, 2, ••• p)

drt_ = o Cov ( ~ ~) - IV Cov (;';_. et ) a d' tt).

II

•

-48-

ProcoediDg as bet'Ol'O wo bll.ve then

B' a-Dt

I DJt I I »• I

00

~ro ID~ I is tbo cotactor or d~ in tho =trix D, given by

D' = (( d~ )). EIJld the mlirl.mma variance io s1ven by

- 2 r.- - - 7, ~) V(Z1 ) = (1-f.' ) L. V(a) - V(z') - V(a') J• V,a• 01. m o.12 •••P o o o o

where r>' ["!-I D' I :J=['l- /P• I Jil

1ll o.12 ••• p d' ID' I /P• I 00 00 00

pt = (( p~))

and f'' .. d' 1 (dt ct• >* 111rt rt rr tt

fbv

11 1Nr, -- -->J d' s (--) 4_.- -::£:''- Cov(s .... ,at1>- Cov(s!.••u' rt n n• .-., n' t•• ,..... .-...

nnd tberotoro -for large preUminary cample the opt.1mm "Vnluea oE

'\nt a can be obtained by conaidoi'S.ng the Ba1IIO oa.trs.ceo D and P •

-- 1 p /D/ Z! .. =a • -::8 g' ..... 0 /D I ... t ot

00

••••••••••••••••• (5.8.2)

•••••••••••• (5.8.8)

•

-ootlm.ato ot lQ is glvon by • •••••••••••••• (~3.4)

Io senoral. tho estimate vlll bo bjaced, tho amount oE

• ...ud SCl'O:lOOG ao tho cmrp1o alae 1ncrcaooD. lklgloct1aa the campl1ns

orroro 1n btlt ~ tbs 'VIll'1anc8 or i'~ v11l the ll:lCO o.s givon 1n

(5. ''I.S) and ·• o&Ucato 1o glvon by

Eat v (a~ > = ~

-~ {.{ A..- ..A ) ~·.4- i: ( !.. - ..A.. ) a'· 11 a• w• ... Dt al 1oo

•

In the prooont work, tho rogreoa1on aothod~oDtillation based on lnrso o::u:JP1o thcoey h:lo boon extended to wlti-stQge deo1gno. 'fho

dioeuso1on bile been confinod to a thrco-Dtllgo doo1an and the rooulto

aro obtained by cons1dering 11 Unoar 1dit£01'011Ce oot1cator1 of tho

popullltion 111000 i of 11 charo.ctor UD:ior DtlJI37. 'lho torcul.ao tor ooticateo 11M their vario..Dcos are all expresso

-5!.-

~mUY. gonsral.1aat1on of tho rooults to smreral auxflUary

vnritLblas ~ aloo token into ~nsidomtion by tom1Dg a l!lult1plo

rogross1on o stirolato and codiftco. tions of tJw tol"'::Ul.ao who:n tho popuh:.-

Uon ceano ot tho unc1ll.inry ~otero tll'6 not known. nro obtnS.md

bJ conllidoring the tea!miquo of clouble B3l3pl1ng for oovoral anzUlSary

vnrillblo.s.

I II

, I

-saal, LC.

-

Sukhntm, B. v. end Itoshal,R.S.

'H.W.'\IQ].., B.ll..

Goswm, .r.u.

REY~IlEl!CBS

(1943)

(1951)

(1951)

(1953)

(1953)

(1953)

(1959)

(1959)

(1961)

(1962)

Q)to on the o::mpling error in tbs< cethod of doUble nampllng1 ~ 6, 329- 330.

Sol:le further rooalte on o:rrors in double sampling techn1qus; s..,nkhya, u, 191.-194.

On errors ot ootimntion in vnrlous types ot doable Blllllpling procedUl'ODJ S::uit;bya, u, 1.25-144.

On cortain extelldod casas or doublo G!!.lllplinaJ ~ !21 S:.i't·-~-- •

Sacpllns Tachniquoa-John WU1o7 & Sons, Now York.

&mpllng Theory of Survoyp vitb t.wli-co.tions, 'l'ha Iova Btllte Pollego i'reoo, Acoe, ~ova, u.s.A. Some t1n1to population unbi.a!lOd ratio aDd ragreDilion esticatoro; Jour. ~. stat. Assn., 54, 596-612.

A contribution to double a:unpliDgJ Jr. Ind. Soc. Agri. Stot., 11, 128-144.

Qn t.b.& t.heoll'f of cl.aao1co.l re~na1.on J and double samplins estimatioDJ Jr. Royal. stat. Soc., Sorice B, 22, 181·138•

On como oxtcnsiono or doublo c:unplinsJ J unpubllehad theeie tor Diploma, I.C.A.R. l RegroesiOn AM~lJis in Sample ~DJ Jour. &:cr. stat. A.een., 5'1, 590-606.