comparitive efficiency of sampling plans (pages 1-100).pdf

8/9/2019 Comparitive Efficiency of Sampling Plans (Pages 1-100).pdf

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COM PA RA TIVE EFFICIENCY OF

SA MPL ING PL A NS(I LLUSTRA TION - APPLE TREES)

rcECONOMICS, STATISTICS, ANDCOOPERATIVES SERVICE

u .s . DEPARTMENTOF AGRICULTURE


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COMPARATI VE EFF I CI ENCY OF SAMPL I NG PL ANS

( I LL USTRATI ON- - - APPLE TREES)

By

Earl. E. Houseman

EconomicsJ Statistics J and Cooperatives Service

U.S. Department of Agriculture

Septenber 1978


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PREFACE

Thi s publ i c at i on i s r egar ded by t he aut hor as s uppl e-

ment ar y t r ai ni ng mat er i al f or st udent s who ar e f ami l i ar

wi t h, or a r e s t udyi ng, el ement ar y t heo r y of s a mpl i ng i nc l ud-

i ng s t r at i f i c at i on, c l us t er s a mpl i ng, r at i o and r egr es s i o n

est i mat i on, sampl i ng wi t h probabi l i t y pr oport i onal t o s i ze,

and mul t i pl e- st age sampl i ng. Af t er st udyi ng sampl i ng met hods

one at a t i me, i t i s i mpor t ant t o get a uni f i ed v i ew of t he

several met hods and t he condi t i ons under whi ch t hey have

about t he s a me or di f f er ent var i anc es .

I n sampl i ng var i ous popul at i ons we qui t e of t en f i nd t wo

or mor e t ec hni ques t hat ar e r oughl y equal i n ef f i c i enc y and

r educ e s a mpl i ng var i anc e about as muc h as pos s i bl e. Admi n-

i st r at i ve f easi bi l i t y, cost s, and f r eedom f r om pot ent i a l

bi as a r e i mpor t ant c r i t er i a f or s e l ec t i ng a s a mpl i ng pl a n

and become pri mary cr i t er i a when t he choi ce ~s among pl ans

havi ng smal l di f f erences I n sampl i ng var i ance.

Abi l i t y t o pr ej udge ac c ur at el y t he ef f i c i enc y of al t e r -

nat i ve s a mpl e de s i gns wi t h r ef er enc e t o var i ous s ur vey

obj ect i ves and popul at i ons i s i mport ant . Such abi l i t y comes

f r om experi ence and det ai l ed st udy of al t er nat i ve t echni ques

of sampl i ng a popul at i on and of maki ng est i mat es. Qui t e

of t en onl y t wo or t hr ee al t er nat i ves a r e c ompa r e d i n an

anal ys i s bec aus e o f l i mi t at i ons of dat a or onl y a f e w a l t er -

nat i ves ar e of i nt er e s t . I n t hi s publ i c at i on many a l t er nat i ve

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sampl i ng and est i mat i on pl ans ar e appl i ed t o a smal l popul a-

t i on of appl e t r ees and t he r es ul t s ar e r ec or ded i n t abl es

f or c ompar at i ve pur pos es . The f oc us of at t ent i on i s on t he

ma gni t ude of t he di f f er enc es i n ef f i c i enc y i n r el at i on t o

t he pat t er ns o f v ar i a t i on t hat ex i s t .

F or s ome r eader s , par t s of t he pr es ent at i on ar e pr obabl y

t o o det a i l ed. Howev er , i t i s i mpor t a nt t o under s t and f ul l y

t he al t ernat i ves and t o put mat hemat i cal expr essi ons f or

est i mat ors and t hei r var i ances i n f orms t hat are most meani ng-

f ul f or c ompar at i ve pur pos es . Ex er c i s es ar e di s t r i but ed

t hrough t he t ext .

Chapt er I mak es us e o f gr aphi c al , o r geo met r i c al , i nt er -

pr et a t i ons i n t he c ompar i s o n of f our a l t er na t i v e way s of us i ng

an auxi l i ar y var i abl e. Ther e i s a br i ef pr es ent at i on of t he

r el evant t heor y f or eac h pl an whi c h i s f ol l owed by a di s -

cussi on of t he pl ans i ncl udi ng a numeri cal exampl e. Sampl i ng

wi t h pr obabi l i t y pr opor t i onal t o s i z e i n c ompar i s o n t o ot her

met hods i s of s pec i al i nt er es t . F or c ompar i s on, a par t of

each var i ance f or mul a I S wr i t t en as t he s um of s quar es of

devi at i ons f r om a l i ne.

Chapt er I I expands t he compari sons made i n Chapt er I t o

i ncl ude i nt er act i ons i n ef f i c i ency. For exampl e, t he compar-

at i v e ef f i c i enc y o f s a mpl i ng uni t s of v ar i ous s i z e I S r el at ed

t o t he met hod of est i mat i on and t o st r at i f i cat i on. Chapt er I I I

pr ov i des s o me f ur t her c ompa r i s o ns , but t he empha s i s i s on ho w

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t heory and i ngenui t y sol ved an i mport ant pr obl em i n t he samp-

l i ng of f r ui t t r ees. Some compari sons i nvol v i ng t wo- st age

sampl i ng usi ng appl e t r ees as an exampl e are i ncl uded i n

Chapt er I V.

Thi s vol ume was wr i t t en bec aus e i t was a pl eas ur e and

because I al ways l ear n somet hi ng f r om maki ng compar i sons

l i ke t hose cont ai ned herei n.

Ear l E. HousemanSt at i st i ci an

i i i


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CONTENTS

0i APTER I S I MPLE USES OF AN AUXI L I ARY VARI ABLE

1 . 1 I n t r o du c t i o n

1 . 1 . 1 Eq ua l P r o ba b i l i t i e s o f Se l e c t i o n

1 . 1 . 2 Un eq ua l P r o ba b i l i t i e s o f Se l e c t i o n

1 . 2 Re s ume o f T he or y f o r F i v e P l a ns

1 . 2 . 1 P l a n 1 - Me a n Es t i ma t o r

1 . 2 . 2 P l a n 2 - Ra t i o Es t i ma t o r

1. 2. 3 Pl an 3 - Regr ess i on Est i mat or

1. 2. 4 Di s c us s i on of Pl ans 1, 2, and 3

1. 2. 5 P I an 4 - Sampl i ng wi t h PPS

1. 2. 6 P l an 5 - St r at i f i ed Sampl i ng

1. 2. 7 St mmar y

1 . 3 Numer i ca l Exampl e

CHAPTER I I FURl l i ER OBSERVATI ONS ON USES OF ANAUXI LI ARY VARI ABLE


2 . 2 Co mp ar i s o n o f P r i ma r y a n d T en ni n a l Br a n c he sas Sampl i ng Uni t s

2 . 3 St r a t i f i c a t i o n b y T r e e s

2. 3 . 1 P l an 6- - Mean Est i mat or

2. 3 . 2 P l an 7- - Rat i o Est i mat or s by St r at a

2. 3 . 3 P l an 8- - Regr ess i on Est i mat or s by St r at a

i v

Page

1

1

2

4

6

7

9

10

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14

15

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46

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2. 3. 4 Di s c us s i o n o f Pl a ns 6, 7, a nd 8

2. 3 . 5 P l an 9- - Combi ned Rat i o Est i mat or

2 . 3 . 6 P l an 10- - Combi ned Regr ess i onEs t i mat or

2. 3. 7 Pl an l l - - Sampl i ng Wi t h PPS Wi t hi nSt r at a

2 . 3 . 8 Summar y and Di sc uss i on

2 . 4 Fur t he r Compar i son o f Sampl i ng Wi t h PPS ToSt r at i f i ed Sampl i ng Wi t h Opt i mum Al l oc at i on

CHAPTER I I I RANI XM- PAl l i SAMPLI NG OF FRUI T TREES


3 . 2 F ou r Me t h od s o f Sa mp l i n g a Tr e e

3 . 3 Br a n c h I d e nt i f i c a t i o n a n d De sc r i p t i o n o f Da t a

3 . 4 P r obabi l i t y o f Sel ec t i on and Est i mat i on

3 . 5 Va r i a n c es o f t h e Es t i ma t o r s

3 . 6 Di s c u s s i o n o f t h e Me t h od s

a- I APTER I V ThO- STAGE SAMPLI NG


4 . 2 P r i mar y Sampl i ng Un i t s Equa l i n S i ze

4 . 3 P r i mar y Sampl i ng Un i t s Unequa l i n S i ze

4 . 4 Se l e c t i o n o f PSU' s wi t h PP S

4 . 5 Un eq ua l P r o ba bi l i t y o f Se l e c t i o n a t Bo t hSt ages

v

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53

54

58

60

62

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83

84

85

87

96

98

11 1

11 1

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11 7

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CHAPTER I

SI MPLE USES OF AN AUXI LI ARY VARI ABLE

1. 1 I NTRODUCTI ON

P r o f i c i enc y i n t he us e of auxi l i ar y i nf or mat i on t o r educ esampl i ng var i ance i s an i mpor t ant goal i n t he f ormul at i on of

a s ampl i ng pl an. I n t hi s c hapt er we wi l l c ompar e f our al t er na-

t i ve met hods of us i ng an aux i l i ar y var i abl e i n t he des i gn of a

sampl e or i n t he est i mat or and one wi t hout usi ng an auxi l i ar y

v ar i abl e, gi v i ng a t ot al of f i ve al t er nat i ve met hods . The

met hods di scussed ar e commonl y f ound i n t ext books on sampl i ng.

I t i s i mpor t ant t o know whet her an auxi l i ar y var i abl e i s wor t h

us i ng and how t o us e i t mos t ef f ec t i vel y . Ac hi evement of

gr eat er ef f i c i enc y i n t he us e of an aux i l i ar y v ar i abl e i s

usual l y i nexpensi ve compar ed t o i ncr easi ng sampl e s i ze, but

i nc or r ec t us e c oul d c aus e an i nc r e as e r at her t han a dec r e as e

l n sampl i ng er r or .

F or eac h of t he f i ve al t er nat i ves t her e i s an es t i mat or

and t he var i anc e of eac h es t i mat or c an be ex pr es s ed i n a f or m

t hat i s s ui t abl e f or i nt er pr et at i on of t he s ampl i ng var i anc e

as a f unc t i on of devi at i ons of poi nt s f r om a l i ne. The emphas i s

i n t hi s c hapt er i s on s i mpl e dot c har t s as a us ef ul ai d t o

unders t andi ng or j udgi ng t he comparat i ve ef f ect i veness of al t er-

nat i ve met hods i n di f f er ent s i t uat i ons. Speci a l at t ent i on wi l lbe gi ven t o s ampl i ng wi t h pr obabi l i t y pr opor t i onal t o s i ze and

how i t c ompar es wi t h ot her ways of us i ng an auxi l i ar y var i abl e


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i n c l u di n g s t r a t i f i c a t i o n a n d o p t i mu m a l l o c a t i o n . Af t e r a

r e v i e w o f n ot a t i o n , d e f i n i t i o n s , a n d t h e or y , a n ume r i c a l

e x amp l e wi l l b e p r e s e nt e d wh i c h ma ke s u s e o f s o me d at a c o l -

l e c t e d i n a r e s e ar c h pr o j e c t t o d ev el o p t e c hn i q ue s f o r

e s t i ma t i n g a p p l e p r o d u c t i o n .

Co ns i d er a p op ul a t i o n o f N s a mp l i n g u ni t s a nd l e t Y l , . . . ,

YN r e pr e s e nt t h e u nkno wn v al u es o f Y a nd l e t Xl ' . . . ' XN r epr e-

s e nt t h e kno wn v al u es o f a n a ux i l i a r y v ar i a bl e X. A s a mp l e

i s t o be s e l e c t e d a nd t h e v al u es o f Y f o r t h e n s u ' s ( s a mp l i n g

u ni t s ) i n t h e s a mp l e , n ame l y Y l ' . . . ' y , a r e t o b e o bt a i n ed .. n

Th e c o r r e s po n d i n g v a l u e s o f X f o r t h e s u' s i n t h e s amp l e a r e

Xl ' .. , x .n We a s s u me t h a t t h e o bj e c t i v e i s t o e s t i ma t e t h e

p o p u l a t i o n me a n ,

NEY.

1Y = = rr Al s o , i n t h e i n t e r e s t o f k ee pi n g

t h e n ot a t i o n a s s i mp l e a s p os s i b l e , l e t Y a nd X r e pr e s e nt t h eN N

p o pu l a t i o n t o t a l s . Th a t i s , Y = = EY. a nd X = = EX . . Th i s g i v es1 1

" Y" , f o r e x amp l e , a d ua l me an i n g a s i n " t h e c h ar a c t e r i s t i c Y"

o r a s t h e t o t a l f o r t h e p op ul a t i o n. Ho we ve r , t h e me an i n g

s h ou l d b e c l e ar f r o m t h e c o nt e x t .

A r e s u me o f t h e t h eo r y f o r e ac h o f t h e f i v e a l t e r n at i v e s ,

wh i c h wi l l b e c a l l e d p l a ns , i s p r e s e nt e d a f t e r a b r i e f r e v i e w

o f s a mp l i n g wi t h e qu al a nd u ne qu al p r o ba bi l i t i e s o f s e l e c t i o n.

1. 1 . 1 EQUAL PROBABI L I T I ES OF SELECT I ON

A s a mp l e o bt a i n ed b y s e l e c t i n g o ne s u a t a t i me , a t

r a nd om wi t h e qu al p r o ba bi l i t y a nd wi t h ou t r e pl a c e me nt , 1 5

c a l l e d a s i mp l e r a nd om s a mp l e . Wh en t h e v ar i a nc e o f Y 1 n t h e

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p o pu l a t i o n i s d e f i n e d a s

2o =

NE (Y. _ Y ) 2

1

N (1.1)

-t h e v a r i a n c e o f t h e me a n , y , o f a s i mp l e r a n do m s a mp l e o f n

i s g i v e n b yN- n 2V( y ) 0= N =T -n

I f t he v a r i a nc e o f Y i s d e f i n e d a s

N

S2 I : ( Y . - Y)

1= N- 1

a n d t h e v a r i a n c e o f - i sY

V( y ) N- n s 2

= " " '" 'F r -n

(1. 2)

(1. 3)

(1.4)

I n t h e d i s c u s s i o n t h a t f o l l o ws , S2 wi l l b e u s e d a s t h e

d ef i n i t i o n o f t h e v a r i a n c e o f y .

The me a n, y , o f a s i mp l e r a nd o m s amp l e i s a n u nb i a s ed

t h e s amp l e i s an u n b i as e d e s t i mat e o f

2e s t i ma t e o f Y a n d t h e v a r i a n c e , s =

nL (y._ y ) 2

1 , amon g s u ' s i nn- l

S2 . I n c i d en t a l l y , t h ewr i t e r f r o m a p r a c t i c a l p o i n t o f v i e w a d v i s e s u s e o f t h e wo r d

" u n b i a s e d " wi t h s o me c a u t i o n . I n t h e ma t h e ma t i c a l t h e o r y , t h e

me a n i n g o f " u nb i a s e d " i s u s u a l l y c l e a r , b u t i n p r a c t i c e " u n -

b i as e d e s t i mat e " i s o f t e n mi s l e ad i n g t o p e r s on s wh o a r e

i n t e r e s t e d i n e s t i ma t e s f r o m a s u r v e y a n d a r e u n a wa r e o f t h e

r es t r i c t ed mean i ng o f t he t e r m. ! /

! / Se e s e c t i o n s 4 . 4 a n d 4 . 5 o f Ex p ec t e d Va l u e o f a Sa mp l e Es t i -mat e , S t a t i s t i ca l Re p or t i n g S e r v i ce , US DA, S e p t e mb e r 1 9 7 4 .

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Exercise 1.1 Show that either definition of the variance

among the N values of Y leads to the same answer for the vari-

ance of y . That is, show that equations 1. 2 and 1. 4 are the

same.

1. 1. 2 UNEQUAL PROBABI LI TI ES OF SELECTI ON

Some sampl i ng pl ans speci f y t hat sampl i ng uni t s be

s el ec t ed wi t h pps ( pr obabi l i t y pr opor t i onal t o s i z e) . F or

si mpl i ci t y, sampl i ng wi t h r epl acement i s assumed.

I t i s of t en ver y i mport ant t o make a c l ear di st i nct i on

bet ween t he pr obabi l i t y of sel ect i ng t he it h s u of a p opu1a-

t i on when a part i cul ar r andom dr aw i s made and t he pr obabi l i t y

of t he it h s u bei ng i nc l uded i n a s ampl e. To hel p mak e t he

di s t i nc t i on c l ear , t he l et t er " P " or " p" wi l l be us ed t o r epr e-

s ent s el ec t i on pr obabi l i t y and " f " wi l l r epr es ent i nc l us i on

pr obabi l i t y, t hat i s , t he pr obabi l i t y of any gi ven s u bei ng i n

t he sampl e. When s i mpl e r andom sampl i ng i s appl i ed, each su

has a pr obabi l i t y equal t o N of bei ng i n t he s ampl e. That i s ,

t he i nc l us i on pr obabi l i t y , f , i s equal t o ~ f or s i mpl e r andoms ampl i ng.

Wi t h r egard t o sampl i ng wi t h pps and r epl acement , l et Pi '

P Z " " , P N be t he s et of s el ec t i on pr obabi l i t i es f or t he N s u' s

Ni n t he popul at i on. I t i s s pec i f i ed t hat r P. =1. Thus , " s e 1ec t -

1

i ng a sampl e wi t h pr obabi l i t i es pr oport i onal t o X. " means t hat1

X. N

P. = ~ where X=l : X. . Si nce t he sampl i ng i s wi t h r epl acement ,1 A 1

t he sel ect i on pr obabi l i t i es r emai n const ant f r om one r andom

dr aw t o anot her.

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Th e un b i a s ed e s t i ma t o r o f Y f o r a s a mp l e o f n i s

(1.5)

I n t h e e s t i ma t o r , i i s a n i n de x o f t h e n r a n do m d r a ws b e c a u s e

t h e s a me s u mi g h t b e s e l e c t e d mo r e t h a n o n c e . T o i l l u s t r a t e ,

i f o n t h e 4 t h d r a w s u n umb e r 1 5 i n t h e p o pu l a t i o n i s s e l e c t e d ,

Y4 a n d P4 a r e e q ua l t o Y l S a nd PI S An d i f t he 1 5t h s u i s

s e l e c t e d a g ai n o n t h e 1 2t h d r a w, Y 12 a nd P 12 a r e e qu a l t o Y1 S

a n d P I S. I n p r a c t i c e , t e c h ni q ue s f o r a v o i d i n g t h e s e l e c t i o n

o f t h e s u mo r e t h a n o n c e a r e u s u a l l y i n t r o d uc e d b u t s u c h t e c h -

n i q u e s a r e f o r l a t e r c o n s i d e r a t i o n .

y.Ea c h o f t he n v a l ue s o f ~ i n Eq . 1 . 5 i s a n unb i a s e d e s t i -

Pi1n y.

ma t e o f t h e p o pu l a t i o n t o t a l . T hu s , ( - )L ~ i s a s i mpl en Pi

a v e r a ge o f n i n de pe nd en t , u nb i a s e d e s t i ma t e s o f Y , a n d ( ~)

a p pe a r s i n Eq . 1 . 5 s o Y wi l l b e a n e s t i ma t o r o f Y i n s t e ad o f

t h e p op u l a t i on t o t a l .

-The v a r i a n c e o f y , Eq . 1 . 5 , i s

whe r e

a nd

2=E-

n

2 IN Y. 2C J = ( ) LP . ( 1 - Y) =N! 1Pi

NY = LY.

1

(1. 6)

(1. 7)

I s Eq . 1 . 6 r e a s o n ab l e ? St u dy t h e e s t i ma t o r . F o r a nyy.

g i v e n v a l u e o f i , ~ i n r e pe a t e d s a mp l i n g i s a r a n do m v a r i a b l ePi

wh i c h h a s a n e x pe c t e d v a l u e e qu a l t o t h e p o pu l a t i o n t o t a l , Y .

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Y iBy def i ni t i on, t he var i ance of i s

Pi

2 N Y . 2cr t= I P . ( 1-Y)

1r1

wher e i i s t he i ndex t o t he N s u' s i n t he popul at i on. I n t he

i s t he average of n i ndependent est i mat es;2

cr tt her ef or e, t he var i anc e of t hi s aver age i s

n

1ny.es t i mat or , I 1

n p .1

And, s i nc e we

2ar e i nt er es t ed i n es t i mat i ng Y r at her t han Y , c r t mus t be di v i ded

2by N as s hown i n Eq. 1. 7.

1. 2 RESUME OF THEORY FOR FI VE PLANS ~/

As di s c us s ed abov e, we wi l l us e S2, Eq. 1. 3, as t he def i ni -

t i on of t he popul at i on var i ance f or s i mpl e r andom sampl i ng wi t h

r epl acement and cr2

, Eq. 1. 7, i s t he def i ni t i on of popul at i on

var i ance f or sampl i ng wi t h pps and r epl acement . Not i ce t hat ,

when t he Pi al l equal ~, c r2

def i ned i n 1. 7 becomes 1. 1.

F or c onveni ent r e f er enc e, t he es t i mat or s and t hei r var i anc es

f or t he f i v e pl ans t o be di s c us s ed ar e l i s t ed i n Tabl e 1. 1, page

29. The v ar i anc es ar e ex pr es s ed as popul at i on v al ues ( par amet er s )

r a t her t han as s a mpl e es t i mat es of var i anc e. Eac h var i anc e

f or mul a i s wr i t t en i n a f or m whi c h s hows a s um of s quar es of

dev i at i ons of poi nt s f r om a l i ne ( or l i nes ) . Al s o, an al t er nat i v e

' ! ; . / A good r ef erence i s: Cochr an, W. G. , Sampl i ng Techni ques:St r at i f i ed Random Sampl i ng, Chapt er 5; Rat i o Est i mat es,

Chapt er 6; Regr essi on Est i mat es, Chapt er 7; and f or sampl i ngwi t h pr o babi l i t y pr opor t i o nal t o s i ze s e e Sec t i ons 9. 9, 9. 10,9. 11, and 9. 12 of Chapt er 9.

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expr es s i on f or t he var i anc e of t he es t i mat or f or eac h pl an i s

shown. For s i mpl i c i t y, an assumpt i on i s made t hat t he sampl i ng

f r act i ons ar e smal l when t he sampl i ng i s wi t hout r epl acement .

Thus , t he f pc ( f i ni t e popul at i on cor r ec t i on) f a ct or has been

N-nomi t t ed f r om t he var i anc e f or mul as . The f pc , namel y ~, c an

al wa ys be i nc l uded i f needed. Not i c e i n T abl e 1. 1 t hat , f or

a c ons t ant s i z e of s ampl e, i t i s onl y t he s ums of s quar es t hat

di f f er among t he pl ans.

A dot c har t t hat s hows one poi nt f or eac h pai r of val ues

of Xi and Yi pr ov i des s i mpl e, gr aphi c al i nt er pr et at i o ns of t he

s ums o f s qua r e s i n t he v ar i anc e f o r mul as f or t he f i v e pl ans .

Ea ch v ar i anc e f or mul a f o r t he f i r s t f o ur pl ans i nv ol v es t he

dev i a t i ons of Yi f r om a l i ne t hr o ugh t he po i nt ( X, Y) . The

f i f t h pl an i nv ol v es l i ne s egment s . How do t he l i nes f o r t he

f i ve pl ans di f f er and how can one j udge t he sampl i ng var i ance

f or one pl an c ompar ed t o anot her by l ook i ng at a dot c har t ?

1. 2. 1 PLAN 1 - MEAN ESTI MATOR

I n t he f i r s t t hr ee pl ans , s i mpl e r ando m s a mpl i ng i s

as s umed. Thes e t hr ee pl ans di f f er onl y wi t h r egar d t o t he

met ho d o f es t i mat i ng Y . T he f i r s t pl a n i s t o us e t he s a mpl e

nL:y.

a ver a ge y = ~ as an es t i mat or of Y. As a s y mbol f or an~

es t i mat or we wi l l us e y, and a s ubs c r i pt wi l l be us ed t o

di s t i ngui s h t he di f f er ent es t i mat or s . Thus , t he f i r s t es t i -

mat or a nd i t s v ar i anc e a r e

7

nL:y.

1

n (1.8)


15/100

(1.9)

The f or mul a f or t he var i anc e of Yl cont ai ns t he expr es -

si onNE ( y . _ y ) 2

1As shown i n Fi gur e 1. 1, t he ver t i cal di st ance

bet ween a poi nt ( X . , Y . ) and a hori zont al l i ne t hr ough ( X, Y) i s1 1N

equal t o ( Y. - Y) . Hence E ( y . _ y ) 2 may be i nt er pr et ed as t he sum1 1

of squar es of t he devi at i ons of Y f r om a hor i zont al l i ne t hr ough

( X, Y) . The cl oser t he poi nt s ar e t o t hi s hori zont al l i ne, t heA

smal l er t he var i ance of Yl '

I n t he gener al cont ext of r egr essi on est i mat i on, P l an 1 i s

a speci al case. Cochr an, i n Chapt er 7, Sampl i ng Techni ques,A

di scuss es r egr essi on est i mat i on wher e Y i n t he f ol l owi ng equat i on

i s t he r egr essi on est i mat or :

Y ~ Y + b( X- x) (1.10)

The val ue of t he r egr es s i on coef f i ci ent , b, mi ght be pr eas s i gned

or i t mi ght be comput ed f r om t he sampl e dat a. I t i t i s pr e-

assi gned, b i s a const ant when one consi der s t he expect ed

val ue of y. I f b i s c ons t ant , i t i s c l ear f r om t he t heor y of A

expect ed val ues t hat E( y) = Y because t he expect ed val ue of y

i s Y and t he expect ed val ue of t he second t er m, b( X- x) i s zer o~/ .

Thus , t he expec t ed val ue of y i s Y r egar dl ess of t he val ue t hat

i s pr eas s i gned t o b. Ther e ar e c as es wher e a pr eas s i gned v al ue

of b equal t o 1 i s of i nt er es t but t hat i s not per t i nent t o t he

~/ E[ b( X- x) ] = E( bX) E( bx) = bX

8

bE( x) = 0 because E( x) = X


16/100

p r e s e nt d i s c u s s i o n. T he p o i n t o f i n t e r e s t i s t h at P l a n 1 ma y

b e r e g a r d e d a s a s p e c i a l c a s e o f r e g r e s s i o n e s t i ma t i o n wh e r e

b i s g i v e n a p r e as s i g ne d v a l u e e qu al t o z e r o . I n P l a ns 2 a nd

3 , t h e v a l u e o f b i s c o mp u t e d f r o m t h e s a mp l e .

1 . 2 . 2 P L AN 2 - RAT I O E ST I MATOR-Wh en we l e t b e qu al ~ , t h e r i g ht s i d e o f Eq . 1 . 1 0

x-

bec omes X ~ wh i c h i s t h e e s t i ma t o r f o r P l a n 2 . T hu s ,x

y = X ~2 -

x( 1. 11 )

Th i s e s t i ma t o r i s c a l l e d a r a t i o e s t i ma t o r s i n c e i t i s t h e r a t i o

o f t wo r a n d o m v a r i a b l e s y a n d x . For s i mp l e r ando m s ampl i ng t hev a r i a n c e o f Y 2 i s o f t e n wr i t t e n a s f o l l o ws :

S22

n (1. 12)

whe r e

a n d

NE (Y . _ Y) 2

1

N-l

NE (X . - X) 2

1

N- l

NE (X. - X) (Y. -Y)

1 1

N- l

N

E Y i YR = =W- xEX.1

9


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The var i anc e f or mul a f or Pl an 2, Tabl e 1. 1, shows t hat

t he devi at i ons, ( Y. - RX. ) , ar e squared and summed.1 1

Exeraise 1. 2 Wi th re fer ena e to t he v ar ian ae o f Y2' Eq. 1.12,

s ho w t ha t

2E ( Y . - R X . )

1 1

N- l

Cons i der a l i ne t hr ough t he o r i gi n and t he poi nt (X , Y ) ,

F i gur e 1. 1. The s l ope of t hi s l i ne R Y

The ver t i cals ee l S = -X

di st ance bet ween t hi s l i ne and a poi nt ( X. , Y . ) i s ( Y. - RX . ) .1 1 1 1

A

bet ween t he v ar i anc es of Y l and Y2 i s t he di f -

f e r e nc e bet we en E( y . _ y) 2 and E( y . - RX. ) 2. T he po i nt s f o r t he111

Ther ef or e, t he s um of s quar es , E( y . - RX. ) 2, i n t he var i anc e1 1

i s t he s um of s quar es of t he devi at i ons of t he

f r om t he l i ne t hr ough t he or i gi n and ( X, Y) .

A

f ormul a f or Y2

PI)ints (X., Y. )1 1

onl y di f f er ence

The

as s umed popul at i on i n F i gur e 1. 1 ar e s o mewha t c l os e r t o t he

l i ne t hr ough t he or i gi n and ( X, Y) t han t o a hor i z ont al l i neA

t hr ough ( X, Y) . Ther ef or e, one woul d expec t Y2 t o have a

smal l er sampl i ng var i ance t han Y

l

.

Exeraise 1. 3 Verify that Y. - RX. ~s the vertiaal distanae1 1

b et we en a p oi nt ( X . , Y . ) a nd a s tr aig ht l in e t hat p as ses t hr oug h1 1

t he o ri gi n a nd ( X, Y) .

1. 2. 3 PLAN 3 - REGRESSI ON ESTI MATORA

The es t i mat or , Y3' i n Pl an 3 i s cal l ed a r egr es s i on es t i -

mat or . I t mak es us e of a l i ne t hat i s der i ved by appl yi ng t he

l e as t s quar es met hod 1n f i t t i ng a l i ne t o t he s a mpl e val ues of

x and y. The equat i on f or t he l eas t s quar es l i ne ( f i t t ed t o

10


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t he s a mp l e da t a ) ma y be wr i t t e n a s f o l l o ws :

y . = y + b( x. - x)l. l.

( 1. 13 )

whe r e b =E (x.-i) (y. -y)

l. l.

- 2E (x. -x)

l.

i = 1,... , n

a nd y . i s t he po i n t o n t he l i ne whe r e x i s e qua l t o x . . Thel. l.

es t i mat or of Y i s obt a i ned by s ubs t i t u t i ng X f o r x . i n 1 . 13l.

wh i c h g i v e s

Y3 = Y + b( X- i ) (1.14)

To unde r s t a nd t he v a r i a nc e f o r mul a f o r Y3 ' s uppo s e a

l e a s t s q u ar e s l i n e i s d e t e r mi n e d f o r t h e p o pu l a t i o n o f p o i n t s

s h own i n F i g ur e 1 . 1 . I t i s

"Y. = Y + B( X . - X)l. l.

(1. 15)

whe r e B =E (X. -X) (Y. -Y)

l. l.

E (X . - X) 2l.

, i = 1, ... , N.

"a nd Y. i s t he p oi n t o n t he l i ne whe r e X i s e qua l t o X . . Th i sl. l.

l i n e h a s b e en d e t e r mi n e d s o t h e s u m o f t h e s q ua r e s o f t h e

N "2d e v i a t i o n s o f Y . f r o m i t i s a mi n i mu m. T ha t i s , E( Y . - Y . )

l. l. l.

i s l e s s t h an t h e s u m o f t h e s q ua r e s o f t h e d ev i a t i o ns f r o m a ny

o t h e r s t r a i g h t l i n e . Th e s u m o f s q ua r e s o f t h e d ev i a t i o ns o f Y.l.

f r o m t he l e a s t - s qua r e s r e g r e s s i o n l i ne c a n be wr i t t e n a s

f o l l o ws :N "2E (Y . - Y . )

l. l.

N= E{y . - [ Y+B( X. - X) ] 12

l. l.

11

(1.16 )


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The expr es s i on on t he r i ght s i de of 1. 16 appear s i n t he var i anc e~

f or mul a f or Y 3 i n Ta bl e 1. 1 whi c h i s

NSZ E{Y. - [ Y+B( X. - X) ] }Z

3 = ( ~ ) 1 1n n N- l

(1. 17)

Exercise 1. 4 Show that the right side of 1. 16 reduces to

2 - 2( l - r ) E( Y. - Y) where r is the coefficient oj' eorrelation between

1

X and Y.

1. 2. 4 DI SCUSSI ON OF P LANS 1, 2, a nd 3~ ~ ~

The var i ances of Yl ' Y2, and Y3 hav e been r el a t ed t o t he

s ums of s quar es o f dev i a t i ons f r om t hr ee l i nes r es pec t i v el y :

( 1) a hor i z ont al l i ne t hr ough ( X, Y) , ( 2) a r at i o l i ne ( t hat i s ,a l i ne t hr ough t he or i gi n and ( X, Y) , and ( 3) a r egr es s i on l i ne

( whi ch i s a l i ne det ermi ned by t he met hod of l east squares) .

Si nc e t he s um of s quar es of devi at i ons f r om t he r egr es s i on l i ne~

i s l eas t , t he v ar i anc e of Y3 wi l l gener al l y be l es s t han t he~ ~

v ar i anc es f or Yl and YZ' The c ompar at i ve var i anc es c an be

j udged f r om vi s ual exami nat i on of how cl os e t he poi nt s ar e t o

eac h of t he t hr ee l i nes .~

The var i anc e of Y2 i s not al ways l es s t han t he var i anc e

of Yl ' Mor eover , t he c or r el at i on c oef f i c i ent i s not a r el i abl e~ ~

meas ur e of how t he v ar i anc es of Yl and Y Z c ompar e. Ac c or di ng t o

Z 2Eq. 1. 12, 2RSXY mus t be l ar ger t han R Sx or t he var i anc e of YZ

wi l l be l ar ger t han t he var i anc e of YI ' I n ot her wor d s, us e

of an aux i l i ar y var i abl e i n a r at i o es t i mat or c oul d r es ul t i n

an i nc r e as e r at her t han a dec r e as e i n v ar i anc e.

l Z


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Th e v a r i a n c e f o r mu l a s d i s c us s ed a b o v e a r e p o p u l a t i o n

va r i anc es ( pa r amet e r s ) wh i ch mus t be es t i mat ed f r om t he s ampl e .

F o r a l l t h r e e p l a n s , f o r mu l a f o r e s t i ma t i n g t h e s a mp l i n g v a r i -

a n c e s a r e o f t h e s a me f o r ma t a s t h e p o p u l a t i o n v a r i a n c e f o r mu l a .

Th e o n l y d i f f e r e n c e i s t h a t t he s um o f s qu a r e s i s c o mpu t e d f r o m

s a mp l e d a t a i n s t e a d o f d a t a f o r t h e e n t i r e p o p ul a t i o n . T he

v a r i a n c e f o r mu l a s f o r P l a n s Z a n d 3 a r e l a r g e s a mp l e a p p r o x i -

ma t i o n s , wh i c h a r e c o mmo n l y u s e d i n p r a c t i c e . ( S e e Co c h r a n ' s

b o o k s e c t i o n s 6 . 4 a n d 7 . 4 . )

I n a s u r v e y i n v o l v i n g ma n y v a r i a b l e s a n d t a b ul a t i o n s b y

v a r i o u s c l a s s i f i c a t i o n s , t h e f i r s t t wo e s t i ma t o r s ( p l a n s ) a r eA

c o mmo n l y u s e d . Al t h o ug h t h e v a r i a n c e o f Y 3 i s , t o s o me d e gr e e ,A A

g en e r a l l y l e s s t h an t h e v a r i a nc e o f Y I o r YZ' i t s u s e i s g e n-

e r a l l y l i mi t ed t o s pec i a l s i t uat i ons wher e l ow e r r o r i s ve r yA

i mp o r t a n t a n d t h e v a r i a n c e o f Y3 i s a p p r e c i a b l y l e s s t h a n t h eA

var i anc e o f Y I o r YZ F o r ex ampl e, i t mi g h t b e u s e d t o e s t i -

ma t e t h e p r o d uc t i o n o f a p a r t i c u l a r c o mmo d i t y o r wh e n i t i s

v e r y i mp o r t a n t t o ma k e e s t i ma t e s wi t h a h i g h d e g r e e o f a c c u r a c y

f o r a f e w s e l e c t e d c h a r a c t e r i s t i c s .

Al l t h r e e o f t h e e s t i ma t o r s ma y b e u s e d wi t h s a mp l i n g

p l a n s o t h e r t h a n s i mp l e r a n d o m s a mp l i n g ; f o r e x a mp l e , r a t i o

es t i mat or s and s t r a t i f i ed r and om s ampl i ng ar e qui t e c ommon.

Exercise 1. 5 F or t he s pe ci al c as e w he re t he r eg re ss io n

line is the same as the ratio line, show that the variance

A A A

of Y3 is e qual to t he variance of YZ' Can V(y3 ) e ve r beA

larger than V( yZ) ?

13


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Exercise 1. 6 Compare Plans 1~ 2~ and J with regard to

the following three dot charts representing three different

relations between X and Y.

Y Y y

, t o

'. c . ~-

\. . '. t ' " 4 ,

'" .' ',' t

'. o . . . . . . .

, ... .

..... . .

. . '. t

. . .

... . \ ~ ." . . ,

. to .

. . . .. ". .. . .

r ~ ' ... - ..

. .. ...Cas e 1 Cas e 2

XCas e 3

X

For each of the three cases rank the three plans from largest

to smallest sampling variance.

1. 2 .5 P LAN 4 - SAIvfP LIN G W ITH PP S

P l ans 2 and 3 us e d t he auxi l i ar y var i abl e i n e s t i mat i o n

and not i n t he des i gn or s el ec t i on of a s ampl e. Pl an 4 i s t o

s e l ec t a s a mpl e of n el e ment s wi t h r e pl ac ement a nd t o us e

pr oba bi l i t i es of s e l ec t i o n pr opor t i onal t o X . . By s ubs t i t ut i ng1

x. x.~ f or Pi i n Eq. 1. 5 and ~ f or Pi i n 1. 7, t he f ol l owi ng ex-

pr es s i ons ar e obt ai ne d f or t he es t i mat or a nd i t s var i anc e:

(1. 18)

and (1. 19)

The f or mul a f or t he var i anc e of Y4 s hows t hat ( Y. - RX. )1 1

ar e t he dev i at i ons whi c h ar e s quar ed. Thus , t he l i ne i nv ol v ed

14


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i n P l a n 4 i s t h e s a me a s t h e l i n e f o r t h e r a t i o e s t i ma t o r .

No t i c e t h a t t h e s q u a r e s o f t h e d e v i a t i o n s , ( y . - RX . ) 2 , a r e1 1

we i g h t e d b y ~ o wi n g t o t h e u n eq u a l p r o b a bi l i t y o f s e l e c t i o n .1

F o r t h e r a t i o e s t i ma t o r , t h e s q u a r e s o f t h e d e v i a t i o n s we r e

we i g h t e d e q u a l l y . I n c i d e n t a l l y , t h e a p p r o p r i a t e f o r mu l a f o rA

e s t i ma t i n g t h e v a r i a n c e o f Y 4 f r o m s a mp l e d a t a i s n o t o f t h e

s a me f o r m ( a n d wi l l n o t r e d u c e t o t h e s a me f o r m) a s E q . 1 . 1 9 .

I n p r a c t i c e o ne o f t e n f i n ds t h at t h e v a r i a n c e o f t h e

d e v i a t i o n s , ( Y . - RX . ) , i n c r e a s e s a s X i n c r e a s e s . T ha t i s , t h e1 1

v a l u e s o f Y ar e u s u a l l y mo r e wi d e l y s c a t t e r e d f o r l a r g e v a l u e s

o f X t h an f o r s ma l l v a l u es o f X. I f t h e r e l a t i o n b et we en X a nd

Y i s l i ke t h e d o t c ha r t i n F i g ur e 1 . 2 , P l a n 4 wi l l h a v e a l o wers a mp l i n g v a r i a n c e t h a n t h e f i r s t t h r e e p l a n s . A l i n e t h r o u g h

( X, Y) a nd t h e o r i g i n f i t s t h e d a t a a b ou t a s we l l a s a n y l i n e .A

Bu t , Y 4 wo u l d h a v e t h e l e a s t s a mp l i n g v a r i a n c e b e c a u s e , a s s h o wn

i n t h e f o r mu l a f o r i t s v a r i a n c e , t h e l a r g e s t v a l u e s o f ( Y . - RX. ) 21 1

r e c e i v e t h e s ma l l e s t we i g h t s i n t h e s u m o f s q u ar e s . J u d gi n g t h e

e f f e c t i v e n e s s o f P l a n 4 i s mo r e t h a n a ma t t e r o f o b s e r v i n g h o w

we l l t h e d at a f i t a l i n e t h r o u gh ( X, Y) a nd t h e o r i g i n . I n f a c t ,

i t i s e a s y t o mi s j u d g e t h e e f f e c t i v e n e s s o f s a mp l i n g wi t h p p s .

We wi l l r e t u r n t o t h i s p o i n t a f t e r p r e s e n t a t i o n o f P l a n S.

Exercise 1. 7 Start with 2

defined in 1.7 and showa as

1 N X 2 X. Ythat it reduces when P . 1to (N) E r ( Y ' - RX. ) =r and R = X11 1

1

1. 2 . 6 P L AN 5 - STRATI FI ED SAMPL I NG

Th i s p l a n ma ke s u s e o f t h e v a r i a b l e X a s a b a s i s f o r

s t r a t i f i c a t i o n . S u p p o s e t h e s a mp l i n g u n i t s i n t h e p o p u l a t i o n

15


23/100

have been l i s t ed i n or der f r om s mal l es t t o l ar ges t val ues of X.

The l i s t i s t hen di vi ded i nt o L s t r at a. Let

Nh = t he popul at i on number of s u' s i n s t r a t um h,

nh = t he sampl e number of su' s,n

hfh = ~ = t he sampl i ng f r act i on,

h

Yhi and Xhi = t he val ues of Y and X f or t he it h

s u i n s t r at um h,

2t he var i anc e of Y wi t hi n s t r at um h,SYh =

Yh = t he a ver age val ue of Y i n st r at um h, and

Xh = t he average val ue of X i n s t r at um h.

We ar e pr i mar i l y i nt er es t ed i n pr opor t i onal al l oc at i on of t he

s ampl e t o s t r at a f or c ompar i s on wi t h Pl ans 1, 2, and 3, and i n

opt i mum al l ocat i on f or compar i son wi t h Pl an 4.

Wi t h pr opor t i onal a l l oc at i on t he s a mpl i ng f r ac t i ons , f h

,

ar e al l equal and i t i s appr opr i at e t o us e t he unwei ght ed s ampl e

mean as an es t i mat or of Y. Henc e,

Ys = y (1.20)

Assumi ng si mpl e r andom sampl i ng wi t hi n st r at a and t hat t he

f pc' s ar e negl i gi bl e,

(1.21)

wher e

and

16


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Wi t h r e f e r e n c e t o a d o t c h a r t f o r s h o wi n g d e v i a t i o n s t h a t

a r e s q u a r e d i n t h e v a r i a n c e f o r mu l a s , i n s t e a d o f o n e l i n e , we

n o w l a v e a s e r i e s o f l i n e s e g me n t s , o n e f o r e a c h s t r a t u m, a s

s h o wn i n F i g u r e 1 . 3 . E ac h l i n e s e g me n t i s a h o r i z o n t a l l i n e

t h r o u g h t h e s t r a t u m me a n . T h e s a mp l i n g v a r i a n c e , S ~, i s a n

a v e r a g e o f t h e s q u ar e s o f d e v i a t i o n s f r o m t h e s e h o r i z o n t a l l i n e

s e g me n t s . I f t h e p o i n t s a r e c l o s e t o t h e l i n e s e g me n t s , t h e

s a mp l i n g v a r i a n c e wi l l b e s ma l l f o r s t r a t i f i e d r a n d o m s a mp l i n g .

Co n s i d e r wh a t h a p p e ns t o t h e s u m o f s q u a r e s f o r s t r a t i f i e d

r a n d o m s a mp l i n g a s t h e n u mb e r o f s t r a t a i n c r e a s e s , t h a t i s ,

a s t h e d i f f e r e n c e b e t we e n t h e l a r g e s t a n d s ma l l e s t v a l u e o f

X f o r e a c h s t r a t u m d e c r e a s e s . I f t h e r e l a t i o n b et we e n X a n d Yo v e r t h e wh o l e p o p u l a t i o n i s a p p r o x i ma t e l y l i n e a r , t h e s u m o f

s q u a r e s o f t h e d e v i a t i o n s f r o m t h e l i n e s e g me n t s wi l l b e c o me

a p pr o x i ma t e l y e q ua l t o t h e s u m o f s q ua r e s o f t h e d e v i a t i o n f r o m

a r e gr e s s i o n l i n e a s i n P l a n 3 . Un de r t h o s e c o n di t i o n s P l a ns 3

a n d 5 wo u l d h a v e a p p r o x i ma t e l y t h e s a me s a mp l i n g v a r i a n c e . I f

t h e r e l a t i o n b e t we e n X a n d Y i s n o t l i n e a r , t h e s a mp l i n g v a r i a n c e

f o r P l a n 5 mi g ht b e l e s s t h a n t h e s a mp l i n g v a r i a n c e f o r P l a n 3 ,

d e p e n di n g o n t h e wi d t h o f t h e s t r a t u m i n t e r v a l s , t h e d e g r e e o f

n o nl i n e a r i t y , a n d h o w c l o s e t h e p o i n t s a r e t o a c u r v e d l i n e .

S up p o s e t h e r a t i o l i n e ( t h a t i s , a s t r a i g h t l i n e t h r o u g h

C X , Y ) a nd t h e o r i g i n ) f i t s t h e p oi n t s a bo ut a s we l l a s a n y l i n e.

I n t h i s c a s e , t h e s a mp l i n g v a r i a nc e s f o r P l a n s 2 , 3 a n d 5

( as s umi ng t he s t r a t um i n t e r va l s a r e s ma l l ) wou l d be appr ox i mat e l yequa l .

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P l an 4 mus t be j udged wi t h r egar d t o how wel l t he pr ob-

abi l i t i es of s el ec t i on f i t t he s i t uat i on as wel l as t he c l os e-

nes s of t he poi nt s t o t he r at i o l i ne. I t i s hel pf ul t o c ompar e

i t wi t h us i ng t he aux i l i ar y v ar i abl e f or s t r at i f i c at i on and

opt i mum al l ocat i on of t he sampl e t o st r at a. We know t hat t he

opt i mum s i z e of s ampl e f r om s t r at um h i s pr opor t i onal t o NhSyh'

Or , i n t erms of sampl i ng f r act i ons, t he opt i mum sampl i ng

f r ac t i on, f h, i s pr opor t i onal t o SYh'I n st r at i f i ed sampl i ng, t he opt i mum sampl i ng f r act i ons ar e

pr opor t i onal t o X h when SYh I S pr opor t i onal t o X h. I n t hi s cas e,

t he sel ect i on pr obabi l i t i es i n sampl i ng wi t h pps woul d be appr oxi -

mat el y i n pr oport i on t o X

h pr ov i ded t he s t r at um i nt er v al s ar e

s mal l . I n ot her wor ds , when SYh i s pr opor t i onal t o X h t he

opt i mum sampl i ng f r act i ons i n st r at i f i ed sampl i ng ar e i n cl ose

agr eement wi t h t he sel ect i on pr obabi l i t i es i n sampl i ng wi t h

pps . I t i s ver y i mpor t ant t o r ec ogni z e t hat t he s i t uat i on mos t

f avorabl e f or sampl i ng wi t h pps occur s when ( 1) t he dat a f ol l ow

t he r at i o l i ne, and ( 2) t he c ondi t i onal s t andar d dev i at i on of

Y i s pr opor t i onal t o X. ( " Condi t i onal s t andar d devi at i on"

r ef er s t o t he s t andar d devi at i on of Y f or a gi ven val ue of X. )

The dot char t , Fi gur e 1. 2, meet s t hos e condi t i ons . Not i ce t hat

t he ver t i cal di st ance bet ween t he t wo dot t ed l i nes i s pr oport i onal

t o X; henc e, t he c ondi t i onal s t andar d devi at i on of Y i s , at l eas t

r oughl y, pr opor t i onal t o X.

Rec ogni t i on of a r el at i on l i k e t he one i n F i gur e 1. 2 as a

good case f or sampl i ng wi t h pps pr ovi des gui dance when maki ng a

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c h ~i c e a mo n g a l t e r n a t i v e s i n c l u di n g t h e p o s s i b i l i t y o f ma k i n g

a t r a n s f o r ma t i o n o f X t h a t wo u l d p r o v i d e a b e t t e r me a s u r e o f

s i z e . Somet i mes a s i mp l e t r ans f o r mat i on l i ke X : = X. + C,1 1

wh e r e C i s a c o n s t a n t , wi l l p r o v i d e a me a s u r e o f s i z e , X ~,

s u c h t h at t h e c o n di t i o n al s t a n da r d d ev i a t i o n o f Y wi l l b e i n

p r o p o r t i o n t o X ~. I n s o me c a s e s , a s i mp l e t r a n s f o r ma t i o n c a n

c h a n g e s a mp l i n g wi t h p p s , c o mp a r e d t o P l a n 1 , f r o m a s u b s t a n t i a l

i n c r e a s e i n s a mp l i n g v a r i a n c e t o a n i mp o r t a n t r e d u c t i o n . Wi t h

p p s s a mp l i n g i t i s i mp o r t a n t t h a t t h e ma x i mu m v a l u e s o f Y a p p r o a c h

z e r o a s X a p pr o a c h e s z e r o .

On e mi g h t f e e l t h a t s a mp l i n g wi t h p r o b a b i l i t y p r o p o r t i o n a l

t o X d o e s n o t f u l l y r e mo v e , f r o m t h e s a mp l i n g v a r i a n c e , v a r i a t i o na mo n g s t r a t a wh e n X i s t h e c r i t e r i o n f o r s t r a t i f i c a t i o n . L o o k

a t t h e p p s e s t i ma t o r . I t i s v a r i a t i o n i n t h e s t r a t u m r a t i o s ,

1 YhiR - - - E r a t h e r t h a n v a r i a t i o n i n Y h t h at n ee ds t o b eh - Nh i Xh i

c o n s i d e r e d. I t wi l l b e e a s i e r t o d i s c u s s t h i s p oi n t i n t h e

n e x t c h a p t e r wh e n s t r a t i f i c a t i o n i n c o mb i n a t i o n wi t h d i f f e r e n t

me t h o d s o f e s t i ma t i o n i s c o n s i d e r e d .

So me n u me r i c a l r e s u l t s a s we l l a s d o t c h a r t s wi l l b e

p r e s e n t e d l a t e r i n t h i s c h a p t e r a n d i n Ch a pt e r I I .

Exercise 1. 8 Ref er t o Fig ur e 1. 2 a nd v er if y f re m t he or ems

pertaining to similar triangles tha t the range in values of Y

i s p ro po rt io na l t o X. I n t hi s c as e, a s a r ou gh a pp ro xi ma ti on ,

w e m ay r eg ar d t he s ta nd ar d d ev ia ti on o f Y as b ei ng i n p ro po rt io nto X. Is i t p os si bl e i n sa mp li ng w it h p ro ba bi li ty p ro po rt io na l

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to X to h av e a lo we r s am pl in g v ar ia nc e th an s am pl in g w it h

s tr at if icat io n b y X a nd o pt im um al lo ca ti on ? Wh en ?

Exercise 1. 9 ( a) Re fe r t o F ig ur e 1. 1 and rank Plans 1,

2, 3, an d 5 fr om le as t v ari an ce t o hi gh es t. An s. 3, 5, 2, 1

with 3 and 5 be in g c lo se de pe nd in g on th e nu mb er o f s tr at a.

(b ) It a pp ea rs th at t he va ri an ce fo r P la n 4 w ou ld b e

m uc h l ar ge r t ha n t he v ar ia nc e f or s tr at if ie d r an do m s am pl in g

wi th o pt im um al lo ca ti on . Wh y? Lo ok at th e co nd it io na l

s ta nd ar d d ev ia ti on o f Y.

( c) S in ce t he r an ge i n t he o pt im um s am pl in g f ra ct io ns

f or s tr at if ie d ra nd om s am pl in g is sm al l, w ou ld y ou ag re e t ha t

Plan 4 wo ul d h av e a m uc h la rg er v ar ia nc e t ha n Pl an 1?( d) Con si de r t he s impl e t ra ns fo rmat io n x : = X. + C where

1 1

C is a constant. Is there a value of C su ch t ha t X~ w ou ld b e

a n e ff ec ti ve m ea su re o f s iz e.

E xe rc is e 1 .1 0 R efe r t o E xe rc is e 1. 6 an d f or ea ch c ase

r an k a ll f iv e p la ns wi th re ga rd to sa mp li ng v ar ia nc e.

Exercise 1.11 Pr ep ar e a d ot c ha rt s ho wi ng a r el at io n

between X an d Y su ch t ha t s tr at if ie d r an do m s am pl in g w it h

a ll ocat io n p ro po rt io na l t o Nh, Plan 5 , w i ll h av e a s ma ll er

s am pl in g v ar ia nc e t ha n t he r eg re ss io n e st im at or , P la n J .

Exercise 1. 12 Pr ep ar e a d ot c har t su ch th at th e v ar ia nc e

f or P la n 5 w it h p ro po rt io na l a ll ocat io n w il l b e a pp ro xi ma te ly

eq ua l to th e v ar ia nc e fo r P la n 1 and (at the sam e time) the

v ar ia nc e f or P la n 5, wi th o pt im um a ll oc at io n w il l b e mu ch

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les3 than the variance for plan 1. This would be a case where

gain from stratification would be entireZy attributable t o

varying sampling fractions rather than stratification t o remove

variation associated with differences among stratum means.

1. 2. 7 SUMMARYI f t her e i s no r el at i on bet ween X and Y, i nc l udi ng a

r el at i on bet ween X and t he condi t i onal st andard devi at i on of

Y, i nf or mat i on abo ut X of f er s no pos s i bi l i t i es f o r r educi ng

s a mpl i ng v ar i a nc e; i n f ac t , t he s a mpl i ng v ar i anc e c oul d be i n-

c r e as ed by us i ng X. I f t her e i s a r el at i on, s ome al t er nat i v e

ways t o t ak e advant age1 of i t have been s hown. Cl ear l y, t he

mo s t ef f ec t i v e way of us i ng a n aux i l i ar y v ar i abl e depends on

wha t t he r el at i o n i s l i k e.

I n t he sampl i ng and est i mat i on speci f i cat i ons f or a

par t i c ul ar s ur vey, an aux i l i ar y v ar i abl e woul d gener al l y be

us ed i n o nl y o ne wa y. F or ex ampl e, a t t empt i ng t o us e a r el at i o n-

s hi p bet ween X and Y as a bas i s f or s t r at i f i c at i on and al s o i n

es t i ma t i o n i s gener al l y no t adv i s a bl e. T r y t o f ul l y ut i l i z e

t he pot ent i al c ont r i but i o n o f a n a ux i l i a r y v ar i a bl e i n one

way . Whet her a n aux i l i ar y v ar i a bl e i s us ed i n s t r a t i f i c at i on

or i n est i mat i on mi ght depend on t he nat ur e of ot her auxi l i ary

var i abl es t hat ar e avai l abl e. F or ex ampl e, s ome k i nds of

auxi l i ar y var i abl es ar e r eadi l y us ef ul i n s t r at i f i c at i on but

not est i mat i on. Consi der usi ng quant i t at i ve measur es i n est i -

mat i on or i n sampl i ng wi t h pps and usi ng nonquant i t at i ve measur es

i n s t r a t i f i c at i on. T hi s poi nt wi l l r ec ei v e f ur t her at t ent i o n.

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1. 3 NUMERI CAL EXAMPL E

Al t hough our i nt er es t i s i n t he pr ac t i c al appl i c at i on of

sampl i ng t heory, a maj or obj ect i ve i n t he pr esent at i on of

numer i c al i l l us t r a t i ons i n t hi s and l at er c hapt er s i s t o

i mpr ove one' s compr ehensi on of pat t erns of var i at i on t hat

ex i s t and t o dev el op one' s s k i l l a t j udgi ng t he ef f ec t i v enes s

of al t er nat i ve sampl i ng and est i mat i on met hods i n speci f i c

s i t uat i o ns . I t i s i nf or ma t i v e t o a ppl y s ev er a l al t er nat i v es

t o t he same popul at i on even t hough some of t he al t er nat i ves

a r e no t pr ac t i c al l y f ea s i bl e.

The dat a f or t he f Ol l owi ng exampl e wer e t aken f r om a

r es ear c h pr oj ec t t o dev el op t ec hni ques f or s a mpl i ng appl e t r ees

t o f or ec as t a nd es t i ma t e appl e pr oduc t i on. The pr i mar y pur pos e

was t o make an i nt ensi ve i nvest i gat i on of ways of sampl i ng a

t r e e r a t her t han how t o s el ec t a s ampl e of t r e es . As a par t

o f t hi s pr o j ec t , t he br anc hes o n s i x appl e t r ees wer e mapped.

I ncl uded among t he measurement s t hat were t aken are t he cr oss-

s ec t i onal ar ea of eac h br anc h and t he number of appl es on eac h

br anc h. Ther e was a t ot al of 28 pr i mar y br anc hes on t he s i x

t r ees . A pr i mar y br anc h, whi c h i s a br anc h f r om t he t r ee t r unk,

pr obabl y woul d not be us ed as a s ampl i ng uni t i n pr ac t i c e. How-

ever , dat a f or t hes e 28 p r i mar y br anc hes ar e us ef ul as a

numeri cal exampl e of al t er nat i ve ways of usi ng an auxi l i ary

v ar i abl e. Al s o t he r es ul t s wi l l be us ef ul i n l at er di s c us s i ons

and compar i sons of met hods of sampl i ng wi t hi n t r ees.

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F o r p u r p o s e s o f t h i s n u me r i c a l e x a mp l e , t h e 2 8 p r i ma r y

b r a n c h e s i s t h e p o p u l a t i o n o f s a mp l i n g u n i t s . We a s s u me t h e

p u r p o s e o f s a mp l i n g i s t o e s t i ma t e t h e t o t a l n u mb e r o f a p p l e s

o n t h e s i x t r e e s . T he a u x i l i a r y v a r i a b l e X i s t h e c s a ( c r o s s -

s e c t i o na l a r e a) o f a b r a nc h . Th e f r u i t c o un t s , Y , a nd t h e

c s a ' s , X, f o r t h e 2 8 l i mb s a r e p r e s e n t e d i n T ab l e 1 . 2 . L e t

u s c o mp a r e t h e f i v e p l a n s o u t l i n e d a b o v e b y r e f e r r i n g t o a d o t

c h a r t . F i g u r e 1 . 4 s h o ws t h e p o i n t s ( X. , Y . ) a n d t h r e e l i n e s :1 1

( 1 ) t h e h o r i z o n t a l l i n e f o r P l a n 1 , ( 2 ) a r a t i o l i n e t h r o u gh

t h e o r i g i n a n d ( X, Y) wh i c h p e r t a i n s t o P l a ns 2 a n d 4 , a nd

( 3 ) t h e l e a s t s q u a r e s r e g r e s s i o n l i n e f o r P l a n 3 . T o o r d e r

t h e s a mp l i n g v a r i a n c e s f r o m s ma l l e s t t o l a r g e s t , o n e wo u l d

u nd o ub t e d l y r a n k t h e f i r s t t h r e e p l a ns i n t h e o r d er 3 , 2 , a n d

1 , wi t h 1 h a v i n g a mu c h l a r g e r v a r i a n c e t h a n t h e o t h e r t wo .

Si n c e t h e s c a t t e r o f t h e p o i n t s i n c r e a s e s a s t h e c s a i n c r e a s e s ,

o ne mi g ht e x pe c t P l a n 4 t o b e b et t e r t h an P l a n 2 , b ut P l a n 4

i s s o me wh a t d i f f i c u l t t o j u d ge . I n Ch a p t e r I I , s i mi l a r c o m-

p a r i s o n s o f t h e p l a n s wi l l b e ma d e u s i n g t e r mi n a l b r a n c h e s

( a n d h e n c e mo r e p o i n t s ) a s s a mp l i n g u n i t s .

Th e t o t a l nu mb e r o f s amp l i ng u n i t s , 2 8 , i s t o o s ma l l t o

p r o v i d e a g o o d e x a mp l e o f s t r a t i f i e d r a n d o m s a mp l i n g i n c o m-

p ar i s o n t o t h e o t h e r f o ur p l a ns . Ho we v e r , f o r p ur p os e s o f

i l l u s t r a t i o n , a c o mp a r i s o n wi l l b e ma d e . S i n c e 2 8 i s d i v i s i b l e

b y 4 , i t i s c o n v en i e n t t o d i v i d e t h e b r a nc h e s , a f t e r b ei n g

o r d e r e d b y c s a , i n t o f o u r s t r a t a o f 7 b r a n c h e s e a c h a s p r e s e n t e d

i n T a b l e 1 . 2 .

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The s t r at um boundar i es ar e i ndi cat ed by ver t i cal dot t ed

l i nes i n F i gur e 1. 4. I t i s evi dent t hat l i ne s egment s , f or

t he st r at i f i ed r andom sampl i ng as speci f i ed i n t he pr ecedi ng

par agr aph, do not f i t t he dat a as wel l as t he r egr es s i on l i ne,

Pl an 3. Al t hough t he s ampl i ng var i anc e f or Pl an 5 i s c l ear l y

muc h l es s t han t he v ar i anc e f or P l an 1, i t i s undoubt edl y gr eat er

t han t he var i anc e f or Pl an 3. I t s r ank c ompar ed t o Pl ans 2 and

4 i s uncer t ai n.

We wi l l now compar e t he j udgment s f or med f r om l ooki ng at

F i gur e 1. 4 wi t h numer i c al r es ul t s . The r el at i ve var i anc es of

t he f i ve est i mat or s, assumi ng n = 1 ( t hat i s , a s ampl e of one

br anc h) , ar e pr es ent ed i n Tabl e 1. 3. Rel at i ve var i anc es ar e t he

- 2var i ances di vi ded by Y . Al t hough i t i s not pos si bl e t o s el ec t

a s t r at i f i ed r andom s ampl e of one br anc h, i t I S a ppr opr i at e t o

l et n = 1 f or pur pos es of c ompar i ng P l an 5 wi t h t he ot her pl ans .

I n t hi s ex ampl e, t he r el at i ons hi p bet ween X and Y i s s uc h

t hat al l f our Pl ans 2, 3, 4, and 5 pr ovi de l ar ge r educ t i ons i n

s ampl i ng v ar i anc e. St r at i f i c at i on, as appl i ed, r educ ed t he

s ampl i ng v ar i anc e by mor e t han 80 per c ent c ompar ed t o P l an 1

but not as muc h as Pl ans 2, 3, and 4 bec aus e i t di d not ut i l i z e

as f ul l y t he i nf or mat i on pr ovi ded by X. I f i t wer e f eas i bl e t o

di v i de t he popul at i on i nt o mor e s t r at a, per haps 8 or 10 i ns t ead

of 4, t he r el at i ve var i anc e f or Pl an 5 woul d have been l es s t han

0. 307 and per haps near l y as l ow as t he var i anc e f or t he r egr es s i on

es t i mat or , P l an 3. Howev er , f r om t he r es ul t s t hat we hav e s een,

i t appear s t hat t he auxi l i ar y var i abl e X c an be us ed t o r educ e

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t h e s a mp l i n g v a r i a n c e f r o m 1. 1 1 7 t o a b o ut 0 . 2 0 0. S ome o f t h e

p r a c t i c a l c o n s i d e r a t i o ns i n t h e c h o i c e o f a p l a n wi l l b e d i s -

c u s s e d l a t e r . I n t h e n e x t s e c t i o n o u r u n de r s t a n d i n g o f s a mp l i n g

wi t h p p s wi l l b e e x t e n d e d b y c o mp a r i n g i t t o s t r a t i f i c a t i o n wi t h

opt i mum a l l oc at i on.

1. 3 . 1 VARYI NG THE SAMPLI NG FRACTI ON WI TH SI ZE OF SAMPLI NG UNI T

F r o m F i g ur e 1 . 4 i t i s c l e a r t h at t h e v a r i a nc e o f t h e

n u mb e r o f a p p l e s i n c r e a s e s wi t h t h e s i z e o f b r a n c h . T he s t a n -

d a r d d e v i a t i o n wi t h i n s t r a t a a n d t h e a v e r a g e c s a p e r b r a n c h a r e

p r e s e n t e d i n T a b l e 1 . 4 .

S i n c e t h e l a r g e s t S Yh i s a b o ut 1 0 t i me s l a r g e r t h a n t h e

s mal l es t , t he l a r ges t s ampl i ng f r ac t i on ( wi t h s t r a t i f i ed

s a mp l i n g a n d o p t i mu m a l l o c a t i o n ) wo u l d b e a b o u t 1 0 t i me s

l a r g e r t h a n t h e s ma l l e s t . T h i s r a n g e o f v a r i a t i o n i n s a mp l i n g

f r a c t i o n s i s l a r g e e n o u g h t o e x p e c t o p t i mu m a l l o c a t i o n , c o mp a r e d

t o p r o p o r t i o n a l , t o g i v e a s u b s t a n t i a l r e d u c t i o n i n v a r i a n c e .

Th e r e l a t i v e v a r i a n c e f o r o p t i mu m i s 0 . 2 1 1 c o mp a r e d t o 0 . 3 0 1

f o r p r opor t i ona l .

Wi t h r e f e r e n c e t o s a mp l i n g wi t h p p s , n o t i c e t h a t t h e

c o n d i t i o n a l s t a n d ar d d e v i a t i o n o f Y i s r o u g hl y i n p r o p o r t i o n

t o X. Th i s i s i n di c a t e d b y t h e f a c t t h at t h e r a t i o o f SYh t o

X h, T ab l e 1 . 4 , i s n ea r l y c o n s t a nt . Al s o , t h e p oi n t s i n F i g ur e 1 . 4

f o l l o w, a p p r o x i ma t e l y , a l i n e t h r o u g h t h e o r i g i n a n d ( X, Y) . Ther e-

f o r e , i t i s r e a s o n ab l e t o f i n d t h at 0 . 2 11 , t h e v a r i a nc e f o r

s t r a t i f i e d s a mp l i n g wi t h o p t i mu m a l l o c a t i o n , i s c l o s e t o 0 . 1 9 4 ,

t h e v a r i a n c e f o r s a mp l i n g wi t h p p s .

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Si nc e SYh i s appr ox i mat el y i n pr opor t i on t o Xh, cs a i s a

good meas ur e of s i z e. Howev er , i t i s i nf or mat i v e t o c ompar e

t he f i v e pl ans when c i r c umf er enc e i s us ed or a meas ur e of s i z e

of br anch. To exami ne t he r el at i on bet ween number of appl es

and c i r c umf er enc e, s ee F i gur e 1. 5. Not i c e t hat t he l eas t s quar es

l i ne ( Pl an 3) depar t s f ar t her f r om t he or i gi n t han di d t he l eas t

s quar es l i ne f or c sa, Fi gur e 1. 4. Thi s i s r ef l ec t ed i n t he

var i anc es whi c h ar e pr es ent ed i n Tabl e 1. 5. The r el at i ve var i -

anc e, 0. 256, f or Pl an 3 i s c ons i der abl y l es s t han t he r el at i ve

var i anc es f or Pl ans 2 and 4. Al s o not i c e t hat c i r c umf er enc e

i s l es s ef f ec t i ve t han c sa f or al l t hr ee Pl ans 2, 3, and 4.

Exercise 1. 13 R ef er t o T ab le 1. 2 an d c om pu te th e fo ur

v al ue s o f Xh ta ki ng t he c ir cu mf er en ce a s t he a ux il ia ry v ar ia bl e.

C om pa re t he se va lu es o f Xh with the values of SYh gi ve n i n

Table 1. 4. W ha t d o es t hi s c o mp ar is on i nd ic at e r eg ar di ng t he

us e o f c ir cu mf er en ce a s a me as ur e of si ze in p ps sa mp li ng ?

Not i ce t hat csa i s a mat hemat i cal t r ansf or mat i on of c i r cum-

f er enc e. The ques t i on mi ght be as ked, " I s t her e a bet t er

t r ansf ormat i on?" Thi s quest i on wi l l be gi ven f ur t her at t ent i on

i n t he next c hapt er . F or t he r es ear c h s t udy, a c s a meas ur ement

was made by wr appi ng a t ape ar ound t he bas e of a br anc h. The

t apes had been c al i br at ed t o gi ve a di r ec t r eadi ng of t he c s a

as s umi ng t he br anc h i s c i r c ul ar . F i gur e 1. 4 s ugges t s t hat c s a

i s a good meas ur e of s i z e f or s ampl i ng wi t h pps , but br oader

ex per i enc e i s needed. I n a l at er i l l us t r at i on i t wi l l bec ome

evi dent t hat sampl i ng wi t h pps i s a good pr act i cal met hod of

sel ect i ng a sampl e of br anches.

2 6


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Exercise 1. 14 By c ar ef ul pl an ni ng on e ca n co mp ut e su b-

22 Yi

to ta ls an d to ta ls o f r Y . , r x . , r Y . , r x . Y . , and r - - X that1 1 111 .1

provide intermediate results from which the variances for

s ev er al a lt er na ti ve p la ns a re e as il y o bt ai ne d. F or p ur po se s

o f c om pu ta ti on s ho w th at th e va lu es o f s 2 fo r t he fi ve pl an s

C 1 )[rY~N=T 1

m ay b e w ri tt en as follows:

C r Y . ) 21 ]

N

2 2 r Y .2Rr x. Y. + R r x . ] where R ~

1 1 1 r A .1

s ; = Si C1- r2) where r is the correlation coefficient

S in ce t he re a re 7 b ra nc he s i n e ac h s tr at um t he e xp re ss io n

in Table 1.1 for S~ r ed uc es t o

Ey 2~] where Yh i s the total of Y for

stratum h .

From Table 1. 2 t he f ol lo wi ng i nt er me di at e r es ul ts a re

obtained:

r Y . = 7, 199 r X .1 1

2

3, 844, 283

2

r Y . = r X .1 1

r X. Y. = 67 , 633. 47 Y~1 1

E 1r

1

27

=

=

=

157. 76

1, 329. 98

392, 247. 3


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Y1 = 202, Y2 = 923, Y3 = 1, 594, and Y4 = 4, 480

22222Compute the values of Sl ' S2' S3' 04' and S5 .

Answel' : S2 = 73, 8281

S2 = 16 , 3392

S2 = 12, 2923

212, 826

4 =

S2 = 20, 2745

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Ta bl e l . l - - Es t i mat o r s a nd Th e i r Re l a t i v e Va r i anc e s ! /

S2 o r 02 e xpr e s s e d a s a n An a l t e r na t i v ea v e r a g e o f s q u ar e d d e v i a t i o n s : e x p r e s s i o n f o r S 2 o r 0Pl an Es t i mat or

Var i anc eof es t i mat or

1 - ( ! ) S2Y

1Y n 1

-( ! ) S22

Y2 X~

- n 2x

3 Y3 y + b( X- x) ( ! ) S2

n 3

1 y. 14 Y = X( - ) E- 2. (-)02

1 + n x . n 1 +1

N 5ENhYh - ( ! ) S2y = Y\0

5 N n 5

1/ Upper cas e l et t er s r ef er t o popul a t i on val ue s ,

S21

S2

2

S2 =3

E(y._y)21

N- l

E( Y. - RX. ) 21 1

N- lE{Y. - [ ' I + B( X. - X) ] }2

1 1

N- l

N -1: (! . . . . . ) ( Y . - R X . ) 2

X. 1 11

N

s 2 S21 Y

S2 S2 + R2S2 - 2 RS

2 Y X

S2 S2( 1- r2 )3 Y

02

1 N Y.= - E P. ( - 2 . _ Y ) 2

1 + N2 1Pi

S2 1 25

NENhSh

N = t ot a l nunt >erof uni t s i n popul at i on.n = t ot al nunt >erof uni t s i n s ampl e.

L = nunt >erof s t r at a.Nh = nunbe r of uni t s i n t he popul at i on i n s t r at umh.

L

N = ENh

~ = nunt >erof uni t s i n s ampl e f r an s t r at umh.

L

n= E~

N

Y = EY .1

EY .va l ue o f Y f o r t he i t h ~'I=

1 Yh i =rr r = ( Sy ) ( ~)uni t i n s t r at un h.

N E Yh

. Sx yX = EX . . 11 - 1 B=Y = -- = r r eano f Y i nh N

~EX . h s t r a t u mh .

X 1

N E( Y. _Y ) 2 S~=

1..E-N- I b =

X. s 2P. = 1 E( X _ X) 2

x1 X

~_ i - 2- N- l ~( Yh i - Yh )

Y S2 = 1R = Nh- lX E( Yi - Y) ( X. - x) h

~_ 1

N- l


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Tabl e 1. 2- - Dat a f or Pr i mar y Br anc hes on Si x Appl e Tr ees

( Ar r ay ed by c s a)

.' .... c r J J ' Y ~ No. of ' . JJ ' Y' ~ ' No f St r at un: Br an : c s a : Cl r . : a l es : : St r at um: Br anc : c s a : Ci r . : ' 10 . , . pp . , . . . app es

1 1- 4 . 87 3. 3 5 3 6- 3 4. 84 7. 8 1831- 5 1. 03 3. 6 34 1- 2 5. 09 8. 0 405- 6 1. 34 4. 1 4 2- 4 5. 75 8. 5 3961- 3 1. 83 4. 8 59 6- 2 5. 89 8. 6 2505- 4 1. 83 4. 8 18 2- 3 6. 16 8. 8 1575- 5 1. 83 4. 8 17 5- 1 6. 16 8. 8 1796- 4 1. 99 5. 0 65 4- 2 7. 18 9. 5 389

2 2- 5 2. 68 5. 8 89 4 2- 2 8. 94 10. 6 3334- 4 2. 86 6. 0 238 4- 1 9. 28 10. 8 6964- 5 2. 86 6. 0 81 2- 1 9. 63 11. 0 4734- 3 3. 57 6. 7 254 3- 1 11. 60 12. 1 7621- 1 3. 68 6. 8 76 3- 3 12. 84 12. 7 5175- 3 4. 48 7. 5 97 3- 2 13. 45 13. 0 6225- 2 4. 72 7. 7 88 6- 1 15. 38 13. 9 1, 077

'IUl'AL 157. 76 221. 0 7, 199

Y Tr ee ( f i r s t di gi t ) and br anch wi t hi n a t r ee ( sec onddi gi t ) .Y Cr oss- sect i onal ar ea of br anch i n squar e i nches.~ Ci r cumf er enceof br anch i n i nches.

Tabl e 1. 3- - Rel at i ve Var i anc es of Es t i mat or s

Pl an : Rel at i ve var i ance of y

1 1. 1172 0. 2473 0. 186

4 0. 1945 0. 307

30


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Tabl e 1. 4- - Mean cs a and St andar d Devi at i on of Y by St r at a

Mean csa, SYhSt r at um

Xh SYh Xh

1 1. 53 24. 8 16. 22 3. 55 78. 4 22. 1

3 5. 87 129 22. 0

4 11. 59 240 20. 7

Tabl e 1. 5- - Rel at i ve Var i anc es When t he Auxi l i ar y Var i -abl e i s Ci r cumf er ence

Pl an

1

2

3

4

3 1

Rel at i ve var i anc e of y

1. 117

0. 559

0. 256

0. 438


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y

F i g u r e l . l - - De v i a t i o n s i n Va r i a n c e F o r mu l a s f o rP l a ns 1 , 2 , a nd 3

/

y

//. .

/ ./.

. --- - - -- - .

/

// ./.

/ .../'

/

- -- -- --

/

/ ../.

: ~ ':{': .~~i'.Yi)

(Y~ -~~) ' : . .. , .I

---

x

P l a ns 2 a nd 4

F i g u r e 1 . 2 - - De v i a t i o n s i n Va r i a n c e F o r mu l a s f o rP l a ns 2 a n d 4

32


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y

I

St r at um I St r at um 2 St r at um 3 St r at um 4 St r at um 5 x

F i g ur e 1 . 3 - - De v i a t i o n s i n Va r i a n ce Fo r mu l af o r P l a n 5 , S t r a t i f i e d Ra n d o m Sa mp l i n g

33


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1100

1000

900

800

700zCE l0'

600ltl"1

0

H'l

> 500'-'l '1j.;;.. '1j

I-'

ltlU l

400

300

200

100

5 6 7 8 9 10 11Cr o s s - s ec t i o na l a r e a i n s qua r e i nc he s

12 13 14 15

Fi gur e 1. 4- - Rel at i on bet ween Number of Appl es and CSA


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1000

900

800

700z~g .

600r o'1

0I-tl

VI > 500V1 't:l

't:l~r oC ll

400

300

200

100

1 2 3 4 5 6 789Ci rc umf er ence i n i nches

10 11 12 13

Fi gur e 1. 5- - Re1at i on bet ween Number of Appl es and Ci r cumf er ence


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CHAPTER I I

FURTHER OBSERVATI ONS ON USES OF AN AUXI LI ARY VARI ABLE

2. 1 I NTRODUCTI ON

The ef f ect s on s ampl i ng var i anc e of var i ous f act or s i n

sampl e desi gn and est i mat i on are not i ndependent . For exampl e,

t he di f f er ence i n t he sampl i ng var i ance bet ween a mean est i -

mat or and a r at i o est i mat or mi ght var y wi t ~ t he def i ni t i on of

t he s ampl i ng uni t or wi t h t he c r i t er i a us ed f or s t r at i f i c at i on.

I n t hi s chapt er some numer i cal exampl es t hat di spl ay such i nt er -

ac t i ons wi l l be gi ven. The obj ec t i ve i s t o f ur t her devel op a

per c ept i on of pat t er ns ( or c omponent s ) of v ar i at i on and a bi l i t y

t o j udge how al t er nat i ve met hods r ank wj t h r egard t o sampl i ng

var l anc e. As you s t udy and ac qui r e exper i enc e i n s ampl i ng t r y

t o vi s ual i z e t he pat t er n of var i at i on i n a popul at i on t o be

s ampl ed a nd t es t y our s k i l l at pr ej udgi ng t he ef f ec t i v enes s of

al t er nat i ve sampl i ng pl ans.

The dat a f or t he exampl es l n t hi s chapt er ar e t ak en f r om

t he r esearch pr oj ect on met hods of est i mat i ng appl e pr oduct i on

whi c h was r ef er r ed t o i n Chapt er I . The s a mpl i ng al t er nat i v es

t hat ar e c ons i der ed r equi r e a map of eac h t r ee t hat i s s ampl ed.

That i s , a map of a t r ee whi ch def i nes t he s ampl i ng uni t s

( br anches) i s t he sampl i ng f r ame. Met hods of pr obabi l i t y

s ampl i ng ar e av ai l a bl e whi c h do not r equj r e pr epar i ng a c om-

pl et e map of a t r ee. Thi s wi l l be di s cus s ed i n Chapt er I I I .

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As b a c kg r o u nd , r e f e r t o F i g ur e 2 . 1 wh i c h i s a ma p o f o n e

o f t h e s i x a p p l e t r e e s u s e d f o r t h e n u me r i c a l e x a mp l e p r e s e n t e d

i n Ch a pt e r I . T he ma p s h o ws t h e s c h e me t h a t wa s u s e d f o r i d e n-

t i f y i n g b r a n c h e s . F o r e x a mp l e , 3 - 1 - 4 r e f e r s t o t h i r d - s t a g e

b r a n c h n u mb e r 4 f r o m s e c o n d - s t a g e b r a n c h n u mb e r 1 a n d f i r s t -

s t a g e b r a n c h n u mb e r 3 . B r a n c h e s f r o m t h e t r e e t r u n k we r e

map p e d u n t i l " t e r mi n a l " b r an c h e s we r e r e ac h e d . " Te r mi n a l

b r a n c h " r e f e r s t o t h e l a s t s t a g e o f b r a n c h i n g wh e r e t h e ma p p i n g

o f b r a n c h e s wa s t e r mi n a t e d . T he c s a ' s ( c r o s s s e c t i o n a l a r e a s )

o f t h e t e r mi n a l b r a n c h e s r a n ge d f r o m a b o u t 3 / 4 t o 2 s q u ar e i n c h e s

wh i c h s e e me d t o b e a b o u t t h e s ma l l e s t p r a c t i c a l s i z e o f b r a n c h

t o c o n s i d e r a s a s a mp l i n g u n i t . T h e r e we r e 2 8 p r i ma r y b r a n c h e s

a n d 1 3 5 t e r mi n a l b r a n c h e s o n t h e s i x t r e e s . T he a v e r a g e n umb e r

o f a pp l e s o n a t e r mi n a l b r a n c h wa s a b ou t 5 0.

Wh e n f o l l owi n g a t r e e t r u n k t o p r i mar y b r an c h e s , t o s e c on d -

s t ag e b r an c h e s , e t c . , s ma l l b r an c h e s a r e s ome t i me s f ou n d wh i c h

a r e n o t l a r g e e n o u g h t o b e c l a s s i f i e d a s t e r mi n a l b r a n c h e s .

F o r e x a mp l e , s i x a p p l e s we r e f o u n d o n s ma l l b r a n c h e s o n p r i ma r y

b r a n c h n u mb e r 2 b e f o r e t h e 4 s e c o n d - s t a g e b r a n c h e s 2 - 1 , 2 - 2 , 2 - 3 ,

a n d 2 - 4 we r e r e a c h ed . Ap pl e s o n s u c h b r a n c he s h a ve b ee n c a l l e d

" p a t h " f r u i t , me a n i n g f r u i t o n t h e p a t h o f a t e r mi n a l b r a n c h .

P a t h f r u i t p r e s e n t s ome s p e c i a l p r ob l e ms wh i c h wi l l b e d i s c u s s e d

i n Ch a pt e r I I I . T he a mo u n t o f p a t h f r u i t i s r e l a t i v e l y s ma l l

a n d wi l l b e i g no r e d i n t h i s c h a p t e r .

F o r e a c h o f t h e f i r s t f o u r p l a n s t h a t we r e d i s c u s s e d i n

Ch a p t e r I p r i ma r y a n d t e r mi n a l b r a n c h e s wi l l b e c o mp a r e d a s

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s ampl i ng uni t s . Then, us i ng t er mi nal br anc hes , t he f i r s t f our

pl ans wi l l t hen be appl i ed wi t hi n st r at a ( t r ees) f or compari son

wi t h eac h o f t he f our pl a ns when t her e i s no s t r at i f i c at i o n.

2. 2 COMPARI SON OF PRI MARY AND TERMI NAL BRANCHES AS SAMPLI NG

UNI T S

The number of appl i es on each of t he 28 pr i mar y br anc hes

and t he c s a of eac h br anc h wer e pr es ent ed I n Tabl e 1. 2. Dat a

f or t he 135 t er mi nal br anc hes ar e pr es ent ed i n Tabl e 2. 1. The

number of appl es on pr i mary br anches i ncl uded pat h f r ui t whereas

t he number s on t er mi nal br anc hes do not . The di f f er enc e I S pr e-

s umed t o be negl i gi bl e f o r pur po s es of an ex er c i s e i n v ar i a nc e

c ompar i s ons . F i gur es 1. 4 and 2. 2 ar e t he dot c har t s f or pr i mar y

and t er mi nal l i mbs r espect i vel y.

Tabl e 2. 2 pr es ent s r el at i ve var i anc es f or t er mi nal and

pr i mar y br anc hes . The r el at i v e var i anc es f or pr i mar y br anc hes

a r e t a ken f r o m T abl e 1. 3 i n Cha pt er I , and r el at i ve var i ances

f or t ermi nal br anches were comput ed usi ng t he same var i ance

f or mul as .

When i nt er pr et i ng v ar i a nc es i t i s es s e nt i a l t ha t t he

di mens i ons of t he v ar i anc es be c l ea r . What v ar i a t i on does

a pa r t i c ul ar v ar i anc e meas ur e a nd i n what uni t s i s t he v ar i anc e

ex pr es s e d? Ar e t he r el at i v e v ar i anc es i n T abl e 2. 2 c ompa r a bl e?

L et us ex ami ne t he f o r mul a f o r t he r el a t i v e v ar i anc e ( RV) o f

Yl ' whi ch i s

R IV (h ) = L (l) (S 2 )e ar Yl y2 n 1

38

(2.1)


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A quant i t y l i ke s i i s s o met i mes c al l ed " uni t var i anc e" as i ti s a meas ur e of var i a t i on among i ndi v i dua l s a mpl i ng uni t s . The

S 2

1; : : r Yl

.

wher e

quant i t y

L (Y. _Y) 21

N-l

may be c al l ed " uni t - r e1at i ve var i anc e" whi c h i s

t he s quar e of t he c oef f i c i ent of var i at i o n a mong i ndi v i dualA

uni t s . I n Eq. 2. 1, when n = 1 t he r el at i ve var i anc e of Y1'

i s t he uni t - r el at i v e v ar i anc e. A s i mi l ar i nt er pr et at i on of

t he var i anc e f o r mul a

comparitive efficiency of sampling plans (pages 1-100).pdf

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