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Successive forest inventories using multistage sampling with partial replacement of units Omule, Stephen Agnew Yen’Emurwon 1981

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SUCCESSIVE FOREST  INVENTORIES USING MULTISTAGE  WITH P A R T I A L REPLACEMENT  SAMPLING  OF UNITS  by STEPHEN A. YenEMURWON OMULE B . S c . , ( F o r . ) ( H o n s . ) , M a k e r e r e U n i v e r s i t y , 1976 M . S c , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1978  A T H E S I S SUBMITTED  I N P A R T I A L FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF  PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES Department of F o r e s t r y We a c c e p t t h i s  thesis  to the required  as c o n f o r m i n g standard  THE U N I V E R S I T Y OF B R I T I S H February  COLUMBIA  1981  ( ^ S t e p h e n A. YenEmurwon Omule, 1981  In p r e s e n t i n g an the  this  thesis  in partial  advanced degree at t h e U n i v e r s i t y Library  shall  make i t f r e e l y  I f u r t h e r agree t h a t permission for  h i s representatives.  of  this  written  gain  shall  permission.  University  of British  2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date  Columbia  I agree  that  copying o f t h i s  thesis  by t h e Head o f my D e p a r t m e n t o r  I t i s understood  thesis f o rfinancial  Columbia,  f o r r e f e r e n c e and s t u d y .  for extensive  Depa r t m e n t The  of British  available  s c h o l a r l y p u r p o s e s may be g r a n t e d  by  f u l f i l m e n t o f the requirements f o  that  copying or p u b l i c a t i o n  n o t be a l l o w e d w i t h o u t my  ABSTRACT Supervisor:  P r o f e s s o r Donald D. Munro  E f f e c t i v e sampling v e r s i o n s of m u l t i s t a g e is. c o s t - e f f e c t i v e itself  partial  sampling.  i n broad  advantageously  of sampling  methods f o r s u c c e s s i v e f o r e s t  designs  forest  on m u l t i s t a g e  sampling  Howpartial  invokes the  sample s i z e or equal v a r i a n c e on s u c c e s s i v e  These assumptions are not u s u a l l y met i n f o r e s t r y . o b j e c t i v e of t h i s  successive forest  of p a r t i a l  study was t o p r o v i d e  inventories using multistage  replacement of u n i t s . nique  through  on s u c c e s s i v e o c c a s i o n s w i t h A l l of the theory  lends  efficiency  u n i t s at the s u c c e s s i v e o c c a s i o n s .  d i s t i n c t i v e assumptions of equal  The  Improved  that  f o r successive i n v e n t o r i e s i s u s u a l l y achieved  replacement of u n i t s has some l i m i t a t i o n s .  occasions.  or subsampling,  a r e a s , and i t i s one technique  t o the use of ' m u l t i l e v e l d a t a .  replacement of sampling  ever, the theory  M u l t i s t a g e sampling,  inventories include  stage  sampling  with  partial the t e c h -  replacement g i v e s r i s e to a number o f a l t e r n a t i v e s .  of the m u l t i s t a g e  the r e s t r i c t i v e  For  i n which p a r t i a l replacement o c c u r s a t design was c o n s i d e r e d .  c o n s i d e r a t i o n was r e s t r i c t e d t o i n v e n t o r i e s on two s u c c e s s i v e o n l y , without  theory f o r  As i s the case w i t h m u l t i s t a g e d e s i g n s ,  p r a c t i c a l purposes, only the case the primary  some g e n e r a l  assumptions o f equal  i a n c e at the two o c c a s i o n s .  ii  In a d d i t i o n , occasions  sample s i z e or equal  var-  Minimum-variance ( b e s t ) l i n e a r and unbiased the c u r r e n t p o p u l a t i o n mean v>y successive occasions  a n c  ^ °f  A, t o g e t h e r w i t h  d e r i v e d f o r two-stage, t h r e e - s t a g e , estimators with The  t  of the r a t i o  n  e  estimators  change i n the mean between two  their  r e s p e c t i v e v a r i a n c e s are  and h-stage  (h > 1)  designs.  form (RE) of Uy and of A are a l s o d e r i v e d  t h e i r r e s p e c t i v e v a r i a n c e s and b i a s e s , f o r a two-stage b i a s e s of REs are n e g l i g i b l e f o r l a r g e sample s i z e s .  that the BLUE had a s l i g h t edge over  An  together  A numerical Y  indicated  the RE; however, f o r e s t i m a t i n g A,  inefficient.  alternative  determining  Biased  design.  comparison of the e f f i c i e n c y of BLUE and RE f o r e s t i m a t i n g y  the RE was very  (BLUE) of  s o l u t i o n approach i s proposed f o r the problem of  the optimum' replacement p o l i c y , that i s , the number of primary  u n i t s t o remeasure and new ones t o take at the c u r r e n t o c c a s i o n . s e q u e n t i a l nature  The  of s u c c e s s i v e i n v e n t o r i e s i s e x p l o i t e d t o c a s t the prob-  lem as a m u l t i s t a g e p r o c e s s  t h a t can be o p t i m i z e d  ming.  are g i v e n  S o l u t i o n procedures  through dynamic program-  f o r determining  the optimum r e p l a c e -  ment p o l i c y f o r a two-stage d e s i g n w i t h the o b j e c t i v e of m i n i m i z i n g the c o s t of the i n v e n t o r y and s u b j e c t t o the s i d e c o n d i t i o n s t h a t the s p e c i f i e d variance The  l e v e l s of i i  y  and A are met.  d e r i v e d theory was i l l u s t r a t e d ,  through a sample f o r e s t  i n v e n t o r y problem.  iii  f o r a two-stage d e s i g n , by working  From a p r a c t i c a l point of view, extension of the theory of sampling with p a r t i a l replacement  from one-stage  to multistage designs i s bene-  f i c i a l , p a r t i c u l a r l y for the inventory of large forest areas.  It would  be useful to extend the theory further to use variable p r o b a b i l i t i e s of selection at the various stages of the multistage design; the cases in which p a r t i a l replacement  and to examine  occurs at other than the primary  stage.  iv  TABLE OF CONTENTS  ABSTRACT  u  LIST OF TABLES  v i i  LIST OF FIGURES  v i i  ACKNOWLEDGEMENT  viii  DEDICATION Chapter  x  :  1  INTRODUCTION  1  2  LITERATURE REVIEW  6  3  THEORY OF MULTISTAGE SAMPLING ON SUCCESSIVE OCCASIONS WITH PARTIAL REPLACEMENT OF UNITS Two-stage SPR Three-stage SPR M u l t i s t a g e SPR Unequal S i z e Sampling U n i t s Other E s t i m a t o r s  4  18 19 31 43 47 53  OPTIMUM ALLOCATION AND REPLACEMENT  68  SPR as a M u l t i s t a g e Model  71  S o l u t i o n Procedure  .  80  5  SAMPLE PROBLEM  83  6  DISCUSSION AND CONCLUSION  94  v  REFERENCES APPENDIX I APPENDIX I I  101 Sample P r o b l e m Data  106  S i m u l a t i o n o f Remeasurement Data  vi  114  L I S T OF  TABLES  Table I  Percent  gain i n efficiency  (Q%) o f y  II  Percent  gain i n efficiency  (Q//0) o f g ^ o v e r  III  Enumeration  over  2  2 j  y  58  2  g _  66  2j  Results  89  L I S T OF  FIGURES  Figure ' • 1 G r o u p s o f s a m p l i n g u n i t s i n SPR on two o c c a s i o n s 2 The s t a g e d i a g r a m f o r t h e o p t i m a l SPR d e s i g n p r o b l e m  vii  ...  73 78  ACKNOWLEDGEMENT I am most g r a t e f u l to Dr. Donald  D. Munro, my major p r o f e s s o r , f o r  h i s d i r e c t i o n and encouragement throughout i e s , and  to the members of my  the p e r i o d of my  t h e s i s c o m m i t t e e — D r s . J . P.  A. Kozak, D. D. Munro, A. J . Petkau, f o r r e v i e w i n g the t h e s i s d r a f t and  Forest  Demaerschalk,  f o r t h e i r v a l u a b l e comments.  I am  and D. H. W i l l i a m s w i t h whom  I had very u s e f u l d i s c u s s i o n s d u r i n g the d e r i v a t i o n of the to Mr.  stud-  D. H. W i l l i a m s , and R. J . Woodham—  p a r t i c u l a r l y g r a t e f u l to Drs. A. J . Petkau  I am v e r y g r a t e f u l  graduate  theory.  David A. Campbell of the B r i t i s h  Columbia  S e r v i c e (BCFS) Inventory Branch f o r the a s s i s t a n c e i n r e t r i e v i n g  the data used  i n the t h e s i s . a n d f o r h i s p a t i e n c e w i t h me  work at Cranbrook, to Ms.  the  field-  Sadie C. Paddock f o r the c o m p i l a t i o n and p r e l i m -  i n a r y a n a l y s i s of the data, and to use  d u r i n g the  to the BCFS I n v e n t o r y Branch f o r p e r m i s s i o n  data.  I extend my  s i n c e r e a p p r e c i a t i o n to the S c i e n c e C o u n c i l of  British  Columbia and the BCFS f o r the f i n a n c i a l a s s i s t a n c e i n form of the GREAT award, to the U n i v e r s i t y of B r i t i s h Columbia f o r the a d d i t i o n a l i n form  of a t e a c h i n g a s s i s t a n t s h i p ,  University  f o r the study f e l l o w s h i p , and  f o r s p o n s o r i n g me at Colorado  to the Ford Foundation  funding  and Makerere  to the Q u a n t i t a t i v e Methods Group  to a t t e n d a workshop on  'Sampling  on S u c c e s s i v e  Occasions'  S t a t e U n i v e r s i t y d u r i n g the summer of 1979.  I extend my  s p e c i a l thanks  some of the d e r i v a t i o n s , to my  to Mr.  F. E. E. Omoruto who  c o l l e a g u e s — D r . Y. viii  El-Kassaby,  helped  check  Mr.  S. S. Chiyenda,  and Mr.  J . I. Mwanje—for  I had w i t h them d u r i n g the course Debogorski  and  Drs. Y.  the f r u i t f u l d i s c u s s i o n s  of w r i t i n g the t h e s i s , and  Z. G. M o y i n i and J . H.G.  to J u s t y n a  Smith f o r t h e i r  encourage-  ment . I am very g r a t e f u l of the  to Mrs.  N. Thurston  f o r the e x c e l l e n t t y p i n g  thesis. Finally,  I am v e r y g r a t e f u l  E r i d a d i O k i r i m a t and from them, gave me  to my mother Agenesi T i n o , my  the r e s t of my  f a m i l y who,  d e s p i t e my  the much needed encouragement and moral  ix  brother  long absence support.  DEDICATION I d e d i c a t e t h i s t h e s i s to late May  my  f a t h e r , E r i y a Emurwon. his soul  peace.  x  rest in eternal  CHAPTER 1 INTRODUCTION In timely  recent years demands have been i n c r e a s i n g  forest  resource s t a t i s t i c s  These s t a t i s t i c s , time, u s u a l l y Inventory  o b t a i n e d with a minimum of e x p e n d i t u r e .  form a b a s i s f o r r a t i o n a l u t i l i z a t i o n  are d e r i v e d .  i n forest  S e v e r a l sampling  have  (occasions) inventories.  v a l u e s are o b t a i n e d .  in  techniques  i n v e n t o r i e s provide information  the s t a t e of the r e s o u r c e at a g i v e n p o i n t i n time.  fied  resources.  i n v e n t o r y designs f o r both s i n g l e - o c c a s i o n and  S i n g l e - o c c a s i o n or "one-shot" on  of f o r e s t  i s f r e q u e n t l y employed to p r o v i d e the data on which  resource s t a t i s t i c s  successive  and  such as timber volume per u n i t area and growth over  sampling  been proposed  for reliable  Sampling  d e s i g n s , such as simple  Only c u r r e n t random, s t r a t i -  random, e t c . , documented i n most sample survey t e x t s , are used  s i n g l e - o c c a s i o n i n v e n t o r y problems.  Successive inventories provide  i n f o r m a t i o n on the s t a t e of the r e s o u r c e at v a r i o u s p o i n t s i n time. Current v a l u e s and changes or average time are o b t a i n e d .  of v a l u e s of the r e s o u r c e  The b a s i c sampling  designs as used  i n v e n t o r i e s t o g e t h e r w i t h methods of l i n k i n g used  over  i n "one-shot"  the designs over time are  i n s u c c e s s i v e i n v e n t o r y problems. S u c c e s s i v e i n v e n t o r i e s may be regarded  as multiphase  which the c u r r e n t phase sample c o n s i s t s of u n i t s observed o c c a s i o n , and i s a subsample of e a r l i e r 1  phase(s)  sample(s)  sampling i n at the c u r r e n t selected  2 on e a r l i e r  occasion(s).  Successive  (1) a new sample on each o c c a s i o n , or  i n v e n t o r i e s may be conducted  using  (2) a f i x e d sample on a l l o c c a s i o n s ,  (3) a p a r t i a l replacement o f sample u n i t s from o c c a s i o n t o o c c a s i o n .  A s e r i e s of independent samples i s simply made without  reference to the others.  temporary marked p l o t s . )  repeated  i n v e n t o r i e s , each  ( T h i s method, f o r example, uses  A f i x e d sample i s a s e t of permanent  u n i t s t h a t a r e observed on s u c c e s s i v e o c c a s i o n s , t r a d i t i o n a l l y continuous  forest inventory  (CFI).  I n sampling w i t h p a r t i a l  sample called  replacement  of u n i t s (SPR), the t o t a l sample i n t h e c u r r e n t o c c a s i o n c o n s i s t s o f sample u n i t s a l r e a d y observed i n e a r l i e r o c c a s i o n s taken;independently inefficient  at the c u r r e n t o c c a s i o n .  Method 1 i s s t a t i s t i c a l l y  s i n c e i t does not e x p l o i t the i n h e r e n t  b e t w e e n p a s t and c u r r e n t o b s e r v a t i o n s . particularly  p l u s new sample u n i t s  Method  correlation  existing  2 i s more e f f i c i e n t  f o r e s t i m a t i n g d i f f e r e n c e s i n v a l u e o f the r e s o u r c e  o c c a s i o n s , but i s more expensive than method  1.  SPR  between  (method 3) combines  the lower c o s t o f independent i n v e n t o r i e s ( f o r o b t a i n i n g c u r r e n t with  the h i g h e f f i c i e n c y of f i x e d samples i n e s t i m a t i n g changes.  f a c t , as we cases  values)  s h a l l see l a t e r , methods  of SPR:  1 and 2 can be regarded  when the p r o p o r t i o n o f u n i t s from p r e v i o u s  i s remeasured i n c u r r e n t o c c a s i o n s t h i s p r o p o r t i o n equals  equals  1, we have method  SPR has been accepted estimating forest resource  as a v a l i d  as s p e c i a l  occasions  0 we have method  that  1, and when  2.  forest  current values  In  inventory technique f o r  and changes i n these  values  over time.  S e v e r a l a r t i c l e s have been p u b l i s h e d  tical  of SPR and, s p e c i f i c a l l y , on i t s use i n t h e e s t i m a t i o n of  theory  f o r e s t a r e a , c u r r e n t timber Large-scale  on t h e g e n e r a l  volume, a r e a change, and timber  a p p l i c a t i o n s o f the t e c h n i q u e  statis-  growth.  have been r e p o r t e d m o s t l y i n  3 the United States and Canada.  However, the design of successive inven-  tories with p a r t i a l replacement  of units i s complex.  As mentioned  e a r l i e r , we require not only the basic sampling designs at given points i n time but also the procedure f o r combining the successive observations. Much of the theory on SPR available i n the forestry f i e l d has been derived assuming simple random sampling successive occasions.  (SRS) as the basic design on the  We know that SRS i s cost-effective i n r e l a t i v e l y  small forest areas, and that the technique i s very rarely used i n forest inventories.  In national and other large forest inventories covering  broad forest areas, SRS  (one-stage) SPR becomes expensive  l e v e l of precision) and highly d i f f i c u l t to apply.  (for a given  Furthermore, straight-  forward SRS does not take advantage of combining remotely-sensed such as s a t e l l i t e and photo-imagery, and ground data.  Multistage  sampling i s c o s t - e f f e c t i v e i n broad forest areas, and i t i s one that lends i t s e l f advantageously  data,  i n the use of m u l t i l e v e l data.  technique This  seems to suggest that multistage sampling schemes would be the more appropriate basic designs for the successive inventory of large forest areas using SPR.  (Other designs such as multiphase and s t r a t i f i e d random  sampling could also be used. other f i e l d s on multiphase SPR,  However, there i s some theory already i n and s t r a t i f i e d SPR i s i n f e a s i b l e i n  sampling forest populations because forest s t r a t a generally change with time.)  There i s , however, no general theory of SPR on a multistage  framework for sampling forest populations.  The theory available so far  a l l invokes the d i s t i n c t i v e assumptions of equal sample size or equal variances on successive occasions. met  i n forestry.  and replacement  These assumptions are not usually  There are no guidelines for the optimal a l l o c a t i o n of sampling units at the various stages of multistage  SPR  4 designs.  Furthermore,  large-scale  forest  m u l t i s t a g e SPR  forest  SPR  so f a r not been a p p l i e d to  inventories.  The o b j e c t i v e of t h i s of one-stage  has  study i s to extend the t h e o r y and  to a m u l t i s t a g e dimension  f o r the purpose  principles  of e s t i m a t i n g  r e s o u r c e c u r r e n t v a l u e s and changes i n v a l u e s over time.  The  r e s o u r c e v a l u e s c o u l d be timber volume, number of stems per ha, e t c . i n the case of a m u l t i s t a g e d e s i g n , the technique of p a r t i a l of sample u n i t s g i v e s r i s e a two-stage  to a number of a l t e r n a t i v e s .  sampling d e s i g n , p a r t i a l (1)  the f o l l o w i n g ways:  replacement  them, on the second o c c a s i o n , ( 2 ) with t h e i r  samples of ssu's and  t a i n a l l the psu's  replacement  For example, i n  of u n i t s can be done i n  r e t a i n a l l primary sampling u n i t s  each psu take a f r e s h sample of secondary  As  sampling u n i t s  (psu's) but  from  (ssu's) within  r e t a i n only a f r a c t i o n of psu's  together (3)  s e l e c t a f r e s h f r a c t i o n of psu's,  from the p r e c e d i n g o c c a s i o n but from each psu  re-  retain  only a f r a c t i o n of the ssu's w i t h i n them and s e l e c t a f r a c t i o n of ssu's afresh,  and  (4)  r e t a i n a f r a c t i o n of the psu's  t a i n o n l y a f r a c t i o n of the ssu's and In t h r e e - s t a g e SPR  t h e r e are about  select  and  from each  a f r a c t i o n of the ssu's  twelve a l t e r n a t i v e s .  stages and o c c a s i o n s i n c r e a s e s , the number of p o s s i b l e i n c r e a s e s too.  For p r a c t i c a l  reasons we  shall  s i t u a t i o n s i n which p a r t i a l  replacement  of the m u l t i s t a g e d e s i g n .  In a d d i t i o n , we  m u l t i s t a g e SPR unequal  such psu r e -  As the number of alternatives  restrict  o u r s e l v e s to  o c c u r s o n l y at the primary shall  afresh.  restrict  stage  o u r s e l v e s to  on two o c c a s i o n s o n l y , assuming v a r y i n g sample s i z e s  and  v a r i a n c e s at the two o c c a s i o n s .  Although some techniques of o p t i m i z a t i o n have been suggested f o r use i n one-stage nature of SPR  SPR,  i n t h i s t h e s i s we  shall exploit  the  sequential  on s u c c e s s i v e o c c a s i o n s to o b t a i n optimum sample  b u t i o n over time by dynamic programming, a mathematical  distri-  programming  5  technique. S p e c i f i c a l l y , we shall ( 1 ) describe the sampling rule for SPR in a multistage framework, ( 2 ) determine suitable (best linear unbiased) estimators of the mean current value, and the changes i n the values, together with their variances, ( 3 ) establish guidelines for an optimal replacement policy for the psu's, and ( 4 ) give an example of the a p p l i cation of the derived multistage SPR  theory to a s p e c i f i c forest inven-  tory problem. F i r s t , we give as a background, the previous work done i n SPR, including multistage SPR  (chapter 2 ) .  of the theory of multistage SPR  Next, we present the derivation  (chapter 3 ) and the optimal replacement  policy construction (chapter 4 ) , together with an application of the derived theory to a forest inventory problem (chapter 5 ) .  F i n a l l y , we discuss  the problems of the multistage SPR theory and s p e c i f i c a l l y , i t s application to forest inventory problems in general (chapter 6 ) .  ''  CHAPTER 2 LITERATURE REVIEW As a method for studying time-dependent populations, sampling on successive occasions has been studied extensively.  In the l i t e r a t u r e ,  sampling on successive occasions i s also sometimes c a l l e d "rotation sampling,"  "sampling  for time s e r i e s , " or "repeated sampling."  In any  case, the method involves successive sampling of the same population with replacement ( p a r t i a l or complete) of the sample from occasion to occasion. We s h a l l review the t h e o r e t i c a l development of sampling  on successive  occasions with p a r t i a l replacement (SPR) i n general and the s p e c i f i c development of theory and application of SPR to sampling tions.  forest  popula-  F i r s t , the general s t a t i s t i c a l theory development. Jessen (1942) was perhaps the f i r s t to r e a l i z e the advantage of  p a r t i a l replacement of the sample to estimate the current population mean i n sampling on successive occasions. of  Jessen was considering the problem  sampling on two successive occasions i n a g r i c u l t u r a l populations.  obtained two estimates:  He  one was the sample mean based on new sample  units only, and the other was a regression estimate based on the sample units observed on both occasions and an o v e r a l l sample mean obtained on the f i r s t occasion.  He then obtained a l i n e a r unbiased estimate of the  population mean on the second occasion by taking the weighted average of the two estimates. variances.)  (The two estimates were weighted inversely by their  Jessen also considered the optimum replacement f r a c t i o n  7 under the assumption that the i n i t i a l sample size was specified and that the t o t a l sample size remained constant over time.  He used simple random  sampling as the basic design. Yates (1949) extended Jessen's result f o r the study of a population on two occasions to more than two occasions under the r e s t r i c t i v e conditions of the same sample size and a fixed replacement f r a c t i o n on each successive occasion.  In addition, Yates assumed the c o r r e l a t i o n between  the same sampling units on two d i f f e r e n t occasions as decreasing i n geometric progression as the time i n t e r v a l between the occasions increased. That i s , the c o r r e l a t i o n between observations one occasion apart as p, anart as p°, p , etc. two occasions apart as p , three occasions apart etc 2z  3  Yates  further ^assumed that the population variance d i d not change with time and that p was known.  He also considered some aspects of the problem of  estimating change from matched observations combined with new independent observations. Patterson (1950), while r e s t r i c t i n g himself to best l i n e a r unbiased estimators, removed a l l the r e s t r i c t i v e assumptions of Yates, except f o r the c o r r e l a t i o n pattern and constant population variance over  time.  Working independently of Patterson, Tikkiwal (1951) also removed the r e s t r i c t i v e assumptions of Yates, but he adopted a s l i g h t l y d i f f e r e n t c o r r e l a t i o n pattern from that of Patterson.  He allowed the c o r r e l a t i o n  between the same sampling units on successive occasions to vary;  the  c o r r e l a t i o n between the same sampling units more than two occasions apart was taken to be the product of the correlations between a l l possible pairs of consecutive occasions occurring i n between (and including) the two occasions i n question. Tikkiwal (1953), using calculus techniques, worked out the optimum  8 p r o p o r t i o n of new and o l d sample u n i t s to take  f o r e s t i m a t i n g the p o p u l a -  t i o n mean on a recent o c c a s i o n , g i v e n the assumptions o f Yates and  Patterson  (1950).  (1949)  T i k k i w a l a l s o gave formulae f o r the optimum  c a t i o n of u n i t s among s t r a t a , when s t r a t i f i e d  random sampling  was used  on s u c c e s s i v e o c c a s i o n s . Narain  allo-  ^  (1953), t a k i n g i n t o account the v a r i a b i l i t y of the r e g r e s s i o n  coefficient  computed from samples, d e r i v e d the b a s i c r e c u r r e n c e  i n sampling  on s u c c e s s i v e o c c a s i o n s .  coefficient  had been i g n o r e d by Yates  The v a r i a b i l i t y  formula  of the r e g r e s s i o n  (1949) and P a t t e r s o n  (1950).  (1954) f u r t h e r extended the r e s u l t s o f Yates (1949) and P a t t e r s o n to that of e s t i m a t i n g c u r r e n t v a l u e s occasions  a p a r t , assuming p a r t i a l  more than  two a p a r t .  Kulldorff  (1963)  Narain (1950)  of a p o p u l a t i o n sampled two or more  c o r r e l a t i o n s were non-zero f o r o c c a s i o n s  d i s c u s s e d the problem of optimum  allocation  of the sample f o r SPR on two s u c c e s s i v e o c c a s i o n s , when there was one v a r i a b l e of i n t e r e s t provided  at a time.  Using  an a n a l y t i c approach, K u l l d o r f f  s o l u t i o n s to the problems o f o b t a i n i n g the sample s i z e on o c c a -  s i o n s one and two when i n t e r e s t  l a y i n e s t i m a t i n g e i t h e r the c u r r e n t mean,  the improved mean on o c c a s i o n one, or the l i n e a r combination  of the two  means under each of the r e s t r i c t i o n s o f minimum c o s t f o r f i x e d  variance  or minimum v a r i a n c e f o r f i x e d c o s t . Eckler  (1955) s i m p l i f i e d P a t t e r s o n ' s  the method of r o t a t i o n sampling b i a s e d estimate a linear  to o b t a i n a minimum v a r i a n c e  developed  l i n e a r un-  of the p o p u l a t i o n mean or t o t a l by s u i t a b l y c o n s t r u c t i n g  f u n c t i o n of sample v a l u e s at d i f f e r e n t  Tikkiwal the study  (1950) approach and  occasions.  (1955, 1956a, 1956b, 1967) extended the theory o f SPR t o  of s e v e r a l c h a r a c t e r s on each of s e v e r a l o c c a s i o n s  under a  9 s p e c i f i e d c o r r e l a t e d p a t t e r n , u s i n g multiphase He  a p p l i e d the d e r i v e d theory  Tikkiwal  (1958a) found  to the survey  that p a r t i a l  e f f i c i e n c y of v a r i o u s e s t i m a t o r s with  wal,  of l i v e s t o c k marketing.  characters In t h i s  replacement f r a c t i o n s were  a l s o extended SPR  (Tikki-  stages  and  units.  composite e s t i m a t i o n procedures of the U n i t e d S t a t e s .  with r o t a t i n g panels, He  presented  i n the r e t a i l  trade  gave composite e s t i m a t o r s , of the  ratio  form, of c u r r e n t p o p u l a t i o n v a l u e s under the assumptions t h a t to  the  to a two-stage design  Woodruff (1959) d i s c u s s e d the advantages of r o t a t i o n , and  survey  the  reaching a l i m i t i n g value.  1958b), assuming a s p e c i f i c c o r r e l a t i o n p a t t e r n at both  e q u a l - s i z e primary  occasions.  replacement of u n i t s improved  study he assumed that the v a r i a n c e and He  on the  of time-dependent p o p u l a t i o n s  i n c r e a s i n g number of o c c a s i o n s  same on each o c c a s i o n .  sampling  o c c a s i o n c o r r e l a t i o n s were the same and  occasion  that v a r i a n c e s were equal  on  each o c c a s i o n . Onate (1960) worked w i t h a f i x e d p a t t e r n of p a r t i a l of  the u l t i m a t e subsample u n i t s i n a m u l t i s t a g e d e s i g n .  total  composite e s t i m a t o r u s i n g data o b t a i n e d  make v a r i o u s e s t i m a t o r s at any f i n i t e p o p u l a t i o n theory p a t t e r n under c e r t a i n Rao sampling finite  current occasion.  He  s t u d i e d the  occasions  to  a l s o developed  a  for his rotation  restrictions.  on s u c c e s s i v e o c c a s i o n s  a u n i f i e d approach to the problem of  employing a f i x e d r o t a t i o n  They c o n s i d e r e d  the p o p u l a t i o n u n i t s at random and, them would be  He  f o r the composite e s t i m a t o r  and Graham (1964) developed  population.  on p r e v i o u s  replacement  a survey  d e s i g n which, f i r s t ,  second, s p e c i f i e d  i n the sample on each of the o c c a s i o n s  E s t i m a t o r s of c u r r e n t v a l u e s and  in a  numbered  i n advance which of (the r o t a t i o n p l a n ) .  changes i n these v a l u e s were  under the assumption that e x p o n e n t i a l  design  developed  and a r i t h m e t i c c o r r e l a t i o n  patterns  10 h e l d over time f o r the c h a r a c t e r i s t i c of i n t e r e s t . g e n e r a l i s e d t h i s work t a k i n g i n t o account that the c o r r e l a t i o n between x  increased.  He  (with s p e c i f i c  not n e c e s s a r i l y t r u e as  j K  Ot  ( x ^ ^ i s the o b s e r v a t i o n on the k t h p o p u l a t i o n u n i t i n  the a t h o c c a s i o n . ) tion  i t was  and x , , would m o n o t o n i c a l l y decrease CX , K  |ot - a'I  that  L a t e r , Graham (1973)  i n s t e a d c o n s i d e r e d the f o l l o w i n g model of  correla-  r e f e r e n c e to c u r r e n t p o p u l a t i o n survey of the  U.S.  Bureau of the Census) p(a, a + 12j + i ) =  p  1 1  p-  1 2  ( i =1,2,...,11;  j = 1,2,3,...)  f o r the c o r r e l a t i o n between o b s e r v a t i o n s i n the same u n i t s e p a r a t e d by (12j + i ) months.  (The index i i s f o r months and  j for years.)  Singh and Singh (1965) c o n s i d e r e d a sampling procedure peated a p p l i c a t i o n of double sive occasions.  sampling  f o r s t r a t i f i c a t i o n on s e v e r a l  on any o c c a s i o n , and that a d d i t i o n of any  d e r i v e d theory was  Raj  Pathak and Rao  p r o d u c t i o n i n the  s e l e c t e d w i 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 to s i z e  over two  and  occasions for estimating current values. sampling was  also considered.  (1967) p r o v i d e d e s t i m a t o r s of the p o p u l a t i o n t o t a l i n  o c c a s i o n s when simple random sampling without  used on both o c c a s i o n s and when p r o b a b i l i t y p r o p o r t i o n a l  efficient  sampling.  (1965) o u t l i n e d the theory of s u c c e s s i v e sampling when sampling  The a p p l i c a t i o n of the theory to double  t i o n was  subtraction  the course of  a p p l i e d to the survey of coconut  the sampling c o n f i n e d to two  was  to or  (India).  u n i t s were c l u s t e r s  sampling  from stratum to stratum  further units  from the p o p u l a t i o n d i d not take p l a c e throughout  s t a t e of Assam  succes-  They gave e s t i m a t o r s of the c u r r e n t p o p u l a t i o n mean and i t s  v a r i a n c e under the assumptions that no u n i t s s h i f t e d  The  involving re-  used on both o c c a s i o n s . than t h a t of  The  Cochran (1977).  replacement  to s i z e  e s t i m a t o r suggested here was However, Ghangurde and Rao  selecmore (1969)  11 showed t h a t , f o r s a m p l i n g simple  random s a m p l i n g  the p o p u l a t i o n t o t a l than  over  without  two  occasions  replacement,  the  c u r r e n t o c c a s i o n had  t h a t o f P a t h a k and  Rao  (1967),  (1967) but  not  Singh  by  Singh  ( 1 9 7 2 ) was  f o r the  (1968) p r e s e n t e d  a theory  equal  and  by  Singh  at  the  t h a t the and  and  Cochran  sampling two  ( 1 9 6 2 ) (RHC)  on  occasions  Sen  g a v e an  unit  simple  random  to provide  ratio  the  linear  size.  on  auxiliary  from the  under s i m i l a r  p r o c e d u r e and The  RHC  correlations auxiliary  values psu's were  later  extended  replacement  occurred  use  of  the  r a t i o method i n  sampling  t h e Rao,  Hartley,  scheme m o d i f i e d  more e f f i c i e n t  than  of s u c c e s s i v e s a m p l i n g b a s e d on  He  same c o e f f i c i e n t  d e p e n d e n t and  v a r i a b l e s are  of  for  that of  the  sampling.  unmatched p o r t i o n .  between the  derived  assumptions.  (two  a m u l t i v a r i a t e double  from the matched p o r t i o n of the  v a r i a b l e s have the  Rao  He  each o c c a s i o n  sample based  a u x i l i a r y v a r i a b l e s w i t h unknown p o p u l a t i o n means, and estimate  esti-  procedures  o n l y at the  T h i s w o r k was  c a s e where p a r t i a l  theory  of  variance The  combinations  u n i t s taken  a combined e s t i m a t e  estimate  sampling  replacement occurred  e s t i m a t o r w h i c h was  (1971b) developed  occasions) sampling  sampling  cost.  three occasions.  (1970) proposed the  successive occasions.  r a t i o method u s i n g  two  (1969) to the  Sukhatme  same e x p e c t e d  and  and  sampling  s e c o n d a r y sample u n i t l e v e l , A v a d h a n i and  the  when p a r t i a l  u n i t s were of e q u a l  Kathuria  (1963) e s t i m a t o r  a s m a l l e r minimum  for successive  scheme f o r two  a s s u m i n g t h a t t h e number o f  under  (1963) e s t i m a t o r .  estimators of current p o p u l a t i o n values several occasions  population,  a l s o s u p e r i o r t o t h a t o f P a t h a k and  as g o o d as K u l l d o r f f ' s  using a two-stage sampling  and  Kulldorff's  on  mator suggested  over  from a f i n i t e  a mean  showed t h a t when of v a r i a t i o n ,  per  the  when  i n d e p e n d e n t a r e e q u a l , and  e i t h e r u n c o r r e l a t e d or are  on  moderately  the when  12 correlated with the dependent v a r i a b l e , considerable gain i n e f f i c i e n c y was achieved over using a single a u x i l i a r y v a r i a b l e .  He assumed that  sample sizes were equal, and population variances were the same, on both occasions.  The e f f i c i e n c y of the multivariate r a t i o estimate was com-  pared to that of the multivariate double sampling regression estimate: the l a t t e r was more e f f i c i e n t i n general.  Sen generalised this theory  from two to several a u x i l i a r y variables using (1) the double multivariate r a t i o estimate (Sen, 1972), and  sampling  (2) the double sampling  regression estimate (Sen, 1973b), under similar assumptions.  Later,  however, the assumption of equal sample size on both occasions removed by Sen (1973a); was  was  and that of equal variance on both occasions  also removed by Sen et a l . (1975), but they considered the case of  the r a t i o estimator with only a single a u x i l i a r y v a r i a b l e .  Sen et a l .  (1975) further extended the theory to use s t r a t i f i e d random sampling. Sen (1971a) successfully applied the theory of SPR iri'a mail survey of water fowl hunters i n Canada.  He observed that the  SPR  estimate of current values was one-third more e f f i c i e n t than the estimate obtained on the basis of current observations only, when the c o r r e l a t i o n between successive observations was high and p o s i t i v e . Avadhani and Sukhatme  (1972) suggested the use of controlled  simple  random sampling with a r a t i o estimator for estimating the population mean in sampling on successive occasions. work (Avadhani and Sukhatme,  1970)  They also extended t h e i r e a r l i e r from two to more than two  Blight and Scott (1973) extended Patterson's  occasions.  (1950) r e s u l t s to  situations i n which the population mean of a time-dependent population followed a l i n e a r Markov process. autoregressive model.  They assumed a simple f i r s t - o r d e r  !3 S c o t t and Smith  (1974) a p p l i e d s t a n d a r d time s e r i e s methods to the  a n a l y s i s of repeated surveys under meters  at each time p e r i o d  the assumption  f o l l o w e d a s t o c h a s t i c model.  used the theory of s i g n a l e x t r a c t i o n Both independent and complete eral  that the p o p u l a t i o n p a r a Their  derivation  i n the presence of s t a t i o n a r y  noise.  remeasurement surveys were c o n s i d e r e d i n gen-  terms w i t h s p e c i f i c r e s u l t s o b t a i n e d fox the time s e r i e s model assumed  i n the work of P a t t e r s o n (1950). applied  The  r e s u l t s were l a t e r extended  and  to surveys of more complex d e s i g n , S c o t t et a l . (1977).  C h a k r a b a r t y and Rana (1974) developed the theory o f sampling on s u c c e s s i v e o c c a s i o n s i n a two-stage  d e s i g n , under  the assumptions  v a r i a n c e s and e q u a l sample s i z e s on both o c c a s i o n s . t i o n s i n which  partial  replacement  o n l y , and of both psu's and s s u ' s . replacement estimating  s i m i l a r assumptions, was  The  of equal  They examined  situa-  o c c u r r e d of the psu's o n l y , of the ssu's Empirical  of both psu's and ssu's was the c u r r e n t mean.  two  r e s u l t s showed that  more e f f i c i e n t ,  partial  i n most c a s e s , f o r  theory f o r a t h r e e - s t a g e d e s i g n , under  d e r i v e d by Rana and Chakrabarty (1976).  Contin-  u i n g the study u s i n g a t h r e e - s t a g e d e s i g n , Rana (1978) c o n s i d e r e d the use of double  sampling r a t i o e s t i m a t o r f o r both s t r a t i f i e d  ing.  He made a n u m e r i c a l comparison,  similar  and  simple random  sampl-  to that of Sen et a l . (1975)  f o r a simple random sampling scheme, of the e a r l i e r  e s t i m a t o r (Chakrabarty  & Rana, 1974)  the former had a s l i g h t  and the one u s i n g a r a t i o e s t i m a t o r :  edge over the l a t t e r .  As also, noted by Sen et a l . (1975), the e s t i m a t o r  u s i n g the r a t i o estimate was purposes  s l i g h t l y b i a s e d , but " f o r most  both e s t i m a t e s seem to be e q u a l l y  Jones  practical  desirable."  (1979) compared the e f f i c i e n c i e s of the approaches  (1950), B l i g h t  and S c o t t  (1973), and of Scott and Smith  of data from r e p e a t e d surveys by computing  of P a t t e r s o n  (1974) to a n a l y s i s  the mean square e r r o r s of the  e s t i m a t o r s of the c u r r e n t mean and of the change i n means on the l a s t  two  14 occasions.  He indicated that there were considerable gains i n e f f i c i -  ency to be made by using the assumed time series relationship between the population means, as assumed by Blight and Scott (1973) and Scott and Smith (1974). Manoussakis (1977) introduced a new  rotation sampling model for  estimating the mean of a time-dependent population. cost, the variance of the derived estimator was  Without considering  less than that of Patter-  son's (1950) but greater than that of Eckler (1955). Good summaries of some of the results c i t e d above can be  obtained  in sampling texts such as Cochran (1977), Sukhatme and Sukhatme (1970), Murthy (1967), and Kish (1965). Most of the l i t e r a t u r e reviewed so f a r has dealt primarily with t h e o r e t i c a l aspects of SPR.  Few  authors  of these have reported a p p l i c a -  tion of the derived theory to actual surveys. (1971a) who  applied SPR  and Tikkiwal (1956b) who the United States.  i n the survey of the waterfowl hunter's i n Canada, used SPR  i n the survey of livestock marketing i n  However, several authors have reported some modifica-  tions to, and application of the theory of SPR tions.  Now  The most notable are Sen  for sampling forest  popula-  we s h a l l review the contributions to the theory of SPR and i t s  applications i n the forestry f i e l d . The concept of SPR was (1956, 1959,  1963).  introduced into forest inventory by Bickford  However, Ware and Cunia (1962) provided a more com-  plete discussion of the p r i n c i p l e and advantages of SPR  i n CFI.  They  treated the s t a t i s t i c a l aspects of the use of remeasured permanent plots and p a r t i a l replacement of the i n i t i a l gave the theory of SPR  sample for forest inventory.  They  for estimating current timber volume and periodic  growth when sampling on two successive occasions, given unequal sample  15 s i z e s and unequal  v a r i a n c e s on t h e two o c c a s i o n s .  used a g r a p h i c a l t e c h n i q u e T h i s w o r k was a r e s u l t of  T.  to determine  Ware a n d C u n i a  the o p t i m a l replacement  of the independent  policy.  w o r k o f Ware ( 1 9 6 0 ) a n d t h a t  Cunia. Building  on t h e w o r k o f Ware a n d C u n i a  (1962), Cunia  (1965)  t h e t h e o r y o f SPR t o u s e m u l t i p l e r e g r e s s i o n m e t h o d s f o r l i n k i n g occasions; two  Cunia  occasions  and C h e v r o u ( 1 9 6 9 ) e x t e n d e d  to sampling  Bickford developed  double  sampling  survey of the Northwest at the f i r s t  successive on  and Newton e t a l .  case.  e t a l . (1963) d e s c r i b e d a t w o - o c c a s i o n  for a forest  extended  the theory of sampling  on t h r e e o r more o c c a s i o n s ;  (1974) c o n s i d e r e d the m u l t i v a r i a t e  field  (1962)  sampling  design  ( U n i t e d S t a t e s ) as a  strati-  o c c a s i o n f o l l o w e d by SPR a t t h e s e c o n d  occasion. Cunia  (1964) gave a b r i e f  historical  a p p l i c a t i o n o f SPR t o f o r e s t r y . explained  from  Frayer test The  an i n t u i t i v e  analysis  of updating  indicated  timber  area a t t r i b u t e s  t i m b e r v o l u m e b y t h e m e t h o d o f Ware  He s u g g e s t e d  the use of weighted  final  (1967) p r e s e n t e d  a method o f c a l c u l a t i n g  on r e m e a s u r e d f o r e s t p l o t s .  and c o m b i n e d w i t h t h e r e s u l t s  estimates of current values.  presented  (1960). of  regression i n  changes  M e t h o d s w e r e a l s o shown  w h e r e b y t h e e s t i m a t e s o f c h a n g e c o u l d be a p p l i e d t o r e s u l t s inventory  data to  volumes.  F r a y e r and F u r n i v a l in  a r i g o r o u s a n a l y s i s of e m p i r i c a l  t h a t the homogeneity of v a r i a n c e assumption  t h e m o d e l was n o t met. updating  SPR was d e f i n e d a n d t h e way i t w o r k s  p o i n t of view.  (1966) undertook  the v a l i d i t y  development o f t h e t h e o r y and  of a previous  of a c u r r e n t i n v e n t o r y t o form  Along  t h e same l i n e s ,  Hazard  (1977)  e s t i m a t o r s o f t h e p r o p o r t i o n o f a r e a and change i n p r o p o r t i o n  16 of area c o n t a i n e d  in a class.  F r a y e r et a l . (1971) r e p o r t e d r e s u l t s of the f i r s t SPR  to timber  States).  i n v e n t o r i e s , i n two  I t was  found  that SPR  working c i r c l e s of Colorado  was  e r r o r as o b t a i n e d w i t h  i n estimating current values.  & Haller,  1964;  See,  1974;  increased e f f i c i e n c y  apply  (United  r e q u i r e d about h a l f the number of sample  p l o t s to o b t a i n the same sampling CFI,  attempt to  Barnard,  i n u s i n g SPR  conventional  E m p i r i c a l s t u d i e s elsewhere 1974)  a l s o i n d i c a t e d that  (Loetsch there  as compared to c o n v e n t i o n a l  inventory  methods. Hazard and  Prommitz (1974) proposed the use of convex mathematical  programming as a t o o l f o r o p t i m a l l y a l l o c a t i n g forest  inventories.  Most of t h i s work was  Dixon and Howitt  resources  d e r i v e d from Hazard  (1979) i n t r o d u c e d the Kalman e s t i m a t o r  n a t i v e to the model developed  by Ware and  Cunia  The  l a t i o n mean (assumed to be u n c o r r e l a t e d by Ware and  (1962) e s t i m a t o r .  found  to be more p r e c i s e than  alter-  Cunia  of the popu-  [1962]).  the Ware and  The  Cunia  However, i t seems the authors were not aware of some  work done i n t h i s d i r e c t i o n by B l i g h t Smith  as an  Kalman e s t i m a t o r  i n t o account the r e l a t i o n s h i p between s u c c e s s i v e v a l u e s  Kalman e s t i m a t o r was  (1969).  (1962), f o r e s t i m a t i n g  the c u r r e n t v a l u e s of a time-dependent p o p u l a t i o n . takes  for successive  and  S c o t t (1973) and by  Scott  and  (1974). From the  l i t e r a t u r e c i t e d above i t i s observed  developments i n m u l t i s t a g e  SPR  have been r e s t r i c t e d  the sample s i z e and/or v a r i a n c e are constant t i o n of m u l t i s t a g e  SPR  i s d e r i v e d without  theoretical  to s i t u a t i o n s  time,  has been r e p o r t e d i n f o r e s t r y .  the theory of m u l t i s t a g e SPR of constant  over  t h a t (1)  and  i n which  (2) no a p p l i c a -  In the next  the r e s t r i c t i v e  sample s i z e or v a r i a n c e on the s u c c e s s i v e o c c a s i o n s .  chapter  assumptions This  17 e s s e n t i a l l y involves extending the work of Ware and Cunia (1962) from the one-stage SPR to a multistage dimension.  CHAPTER 3 THEORY OF MULTISTAGE SAMPLING ON SUCCESSIVE OCCASIONS WITH PARTIAL REPLACEMENT OF UNITS F o r s i m p l i c i t y of p r e s e n t a t i o n , we of one-stage SPR we'shall  to two-stage SPR,  g e n e r a l i z e the e x t e n s i o n  ( b e s t ) l i n e a r unbiased in  estimators  and  shall f i r s t  to t h r e e - s t a g e  to h-stage SPR  Other p o s s i b l e e s t i m a t o r s ,  be d e r i v e d  together  i e n c i e s of RE  to sampling on two  (RE), w i l l  the c u r r e n t mean and shall  successive occasions with p a r t i a l We  the change  variances).  In a l l the d e r i v a t i o n s we  of o n l y the primary sample u n i t s . sampling u n i t s at each stage  Minimum-Variance  be d e r i v e d t o g e t h e r w i t h  t h e i r p r o p e r t i e s ( b i a s e s and  investigated.  Then, next,  (h > 1)-.  r a t i o estimators  r e l a t i v e to BLUE i n e s t i m a t i n g  i n mean w i l l be ourselves  with  SPR.  extension  (BLUE) of the c u r r e n t mean and  the means from o c c a s i o n to o c c a s i o n w i l l  variances.  d i s c u s s the  their also The  effic-  the change restrict  replacement  s h a l l c o n s i d e r the cases where the  are of equal  s i z e and  unequal s i z e i n d e r i v i n g  the BLUE. In g e n e r a l , when sampling l a r g e f o r e s t areas, u n i t s at each stage ered  infinite.  of the m u l t i s t a g e  design  the p o p u l a t i o n of  i s l a r g e enough to be  A l l the d e r i v a t i o n s t h a t f o l l o w w i l l ,  i n f i n i t e p o p u l a t i o n models which, i n t h i s case,  p r o v i d e very  r e s u l t s which would have been o b t a i n e d  t i o n s were done w i t h  f i n i t e p o p u l a t i o n models.  p o p u l a t i o n models g r e a t l y s i m p l i f i e s  the, d e r i v a t i o n s . 18  consid-  t h e r e f o r e assume  mations to the exact  of i n f i n i t e  sample  close  i f the d e r i v a -  F u r t h e r , the the a l g e b r a  approxi-  assumption involved in  19 Two-stage  SPR  C o n s i d e r a p o p u l a t i o n c o n s i s t i n g of N p r i m a r y sample u n i t s and  each psu c o n s i s t i n g of M secondary  suppose t h a t  the sample u n i t s suppose n psu's  replacement  ( s r s w o r ) on t h e f i r s t  from each  s i z e np  (0  a r e s e l e c t e d by  sample psu.  addition,  srswor  for inclusion s e l e c t e d by  indicated  i t will  be  ( s > 0)  srswor  on  on t h e f i r s t  the second  srswor)  f o r the second the f i r s t  o f t h e N-n  from each  o f t h e nm  occasion.  o c c a s i o n as X and  o c c a s i o n as Y, we  by of  occasion  occasion.  In  o t h e r psu's  is  o f t h e ns p s u ' s .  a s s u m e d t h a t N and M a r e i n f i n i t e l y  (p + s ) s s u ' s on t h e s e c o n d  interest  without  i n t h e s a m p l e on t h e s e c o n d o c c a s i o n .  O b s e r v a t i o n s are taken i n each and nm  In  ssu's are s e l e c t e d  ( s e l e c t e d by  i s retained  s s u ' s drawn from  Again, m ssu's are earlier,  m  A random s a m p l e  a random s a m p l e o f s i z e ns  s e l e c t e d by  Further,  s i m p l e random s a m p l i n g  o c c a s i o n and  < p < 1) o f t h e n p s u ' s  together with i t s respective  (ssu's).  (psu's or ssu's) are of equal s i z e .  particular,  srswor  sample u n i t s  (psu's)  As large.  s s u ' s on  the f i r s t  Denoting  the v a r i a b l e  t h e same v a r i a b l e  of  occasion of  interest  d e s i g n a t e the v a r i o u s o b s e r v a t i o n s as  follows: Occasion 1 No.  of unmatched  No.  of matched  No.  of unmatched  No.  of matched  psu's  psu's ssu's  ssu's  Unmatched o b s e r v a t i o n s  nq  ns  np  np  nmq  nms  nmp  nmp  x' . . , i = 1 J  Matched o b s e r v a t i o n s 1 J  l,2, j=l,2,  .nq ,m  np ,i=l,2 j=l,2,...,  y J  y"  i r J  ,1=1,2, j=l,2,.  , ns ,m  i=i,2, j=l,2,  »np ,m  where q = (1 - p ) . We are i n t e r e s t e d  i n e s t i m a t i n g the c u r r e n t p o p u l a t i o n mean  the change i n means A = ix - u Y  assumed t h a t on the f i r s t  of the v a r i a b l e  ij  o c c a s i o n , the o b s e r v a t i o n s  = X U  *  I t w i l l be  A  are d e s c r i b e d by the l i n e a r nested X  of i n t e r e s t .  a n c  +  °li  +  e  (matched or unmatched)  model  l(.i)j  {  . i = 1,2,. .. , N J = 1,2,...,M  where x y  .  = o b s e r v a t i o n on the j t h ssu w i t h i n = o v e r a l l mean of the o b s e r v a t i o n s  X  = effect  °li  e,,. . = effect l(i)j x  and  the i t h psu  a l l the  E^^.^.'S  of the i t h psu of the i t h ssu w i t h i n the i t h psu • are independent random v a r i a b l e s each w i t h  v a l u e = 0 and v a r i a n c e = o 2^ variables,  ,  and a l l the j ^ ' a  s  a  r  expected  independent  e  independent of ( ^ ^ ) j ^ » each w i t h expected e  random  v a l u e = 0 and  v a r i a n c e = o"  2  a Then where here,  cov(x..,x.,..) = n , . , ( a + n...a ij i J i i <*x JJ 2  e  i  and i n what f o l l o w s ,  uv Note, i n p a r t i c u l a r ,  .0 i f u ^ v 1 if u = v  that  cov(x. . ,x. .,) = c o v ( a . .,a. .) = a ij i j l i l i ax 2  cov(x..,x.,.) = cov(o, ,,a, ..) = 0 ij l ' j l i l i In o t h e r words, o b s e r v a t i o n s on d i f f e r e n t correlated,  )  2  for j 4j ' for i ^ i '  ssu's w i t h i n the same psu are  and o b s e r v a t i o n s on the ssu's w i t h i n the d i f f e r e n t  uncorrelated.  Note i n p a r t i c u l a r  t h a t i f i = i ' and j = j ' ,  psu's are then  cov(x..,x..) = V a r ( x . . ) = o Similarly  + o  2  2  on the second o c c a s i o n , the o b s e r v a t i o n s (matched or un-  matched) are d e s c r i b e d by the l i n e a r n e s t e d model. , i = l , 2,. .. ,N ij  y  =  °2i  +  +  £  2(i)j  {  j=l,2,...,M  where y_  = o b s e r v a t i o n on the j t h ssu w i t h i n the i t h psu  P  = o v e r a l l mean of the o b s e r v a t i o n s  Y  ot„ .  = effect  2i  G  2(i) j  =  and a l l the  e  e  ^^  o f the i t h psu °^  t  n  s  r  e  e c t  2(i) j '  a  J  e  t  n s  s  u  w i t h i n the i t h psu  independent  v a l u e = 0 and v a r i a n c e = o  random v a r i a b l e s each w i t h expected  , and a l l the ot  2  c 2  variables,  variance = a Then  random  e  e  a  c  n  w  expected v a l u e = 0 and  ith  2  cov(y..,y.,.,) = n...(o  Note, i n p a r t i c u l a r , cov(y  i  + n.  2  a  )  2  that,  ,y .. ,) = c o v ( a , a . ) = a i  cov(y.^,y In  of ^ 2 ( i ) j ^ '  independent  's are independent  ^ i  2 i  i t  2  2  ^  for j + j  ) = cov(a ,a .,) = 0 2 i  for i ^ i ' .  2  o t h e r words, o b s e r v a t i o n s on d i f f e r e n t  1  ssu's w i t h i n the same psu are  c o r r e l a t e d , and o b s e r v a t i o n s on the ssu's w i t h i n the d i f f e r e n t psu's a r e uncorrelated.  Note i n p a r t i c u l a r t h a t cov(y  In 2,  y  ) = Var(y  ) = o  o r d e r to impose a c o r r e l a t i o n  i twill  be f u r t h e r assumed t h a t  i f i = i ' and j = j ' , then 2  + a  2  .  s t r u c t u r e between o c c a s i o n s 1 and  cov(x y., i r  .,)  = n  l  , ( p  i  1  o  a  i  O  a  j  n ^ . a ^ o ^  +  )  where  Pj = the c o r r e l a t i o n between the e f f e c t s due to the psu's, and p  2  = the c o r r e l a t i o n between the e f f e c t s due to the ssu's within the psu's.  Note, i n p a r t i c u l a r , that corr(a ,a .) = n ^ . P i l i  c o r r ( E  2  Ki)j' 2(i')j' E  corrCa^. c o r r ( B  )  =  .^) = 0  2i' l(i)J e  Furthermore, i f i = i  1  )  s  n  ii' jr n  P !  and  °"  and j = j ' then  cov(x..,y.,.,)=cov(x..,y..)=p a 1  a  +p a 2  a  The c o r r e l a t i o n structure assumed implies that (a)  observations  on the ssu's within the d i f f e r e n t psu's at the two occa-  sions are uncorrelated, (b)  observations  on the ssu's within the same psu at the two occasions are  correlated, (c)  observations  on the different ssu's within the same psu's at the two  occasions are correlated, and (d)  observations  on the same ssu's within the same psu's at the two occa-  sions are correlated.  23  Now, from the sample observations, we obtain simple averages or. preliminary estimators as follows: x'..•=(  nq  m  Z  Z  x'..)/nmq  i=l j = l  ^  np m x".. = ( 2 Z x"..)/nmp i = l j=l 1 J  ns m y' . . = ( £ £ y ••)/nms i=l j=l 1  1 J  np m y".. = ( Z Z y"..)/nmp i=l j=l  and  1 J  where x'.. i s the mean of observations unmatched on occasion 1 x".. i s the mean of observations matched on occasion 1 and 2 y'.. i s the mean of observations unmatched on occasion 2 y".. i s the mean of observations matched on occasion 2 and 1. Of course there are other possible preliminary estimators.  The expected  values of the above estimators are as follows: nq m nq m E(x'..) = — Z Z E(x'. .) = Z Z E(y + a + e ,.-.) = p nmq . . . . i i nmq . . . . X l i l(i)j X 1=1 j=l i = l j=l 1  1  Y  np Z  1  E(x"..) = nmp  i  =  -  E(y'..)=— nms and  i=]  1  ns u s Z _  j= 1  m Z E(x" ) = ^ ij nmp m Z HI  Y  n  i= 1  E(y' ) = — ij nms .  np m Z Z ^ ns m Z Z  = 1 j= 1  E(u + a X l i Y  Y  E(u + a 2i  + e , . ) =y l(i)j X +e, 2U  24 np Z  E(y"..) = — — nmp  w  since a ^ , a  2 i  .  = 1  m Z  np I  E ( y " . .) = xj  x  ,  a  n  d  .  nmp 2(i)j  e  a  1  1  j  = 1  h  a  m 2 =  v  E(y  e  e  + a  Y  1  P  x  e  c  t  +  a  t  i  o  n  e  = U> Y  2(i)j  2i q  u  a  l  t  0  Y  zero.  The v a r i a n c e s of these p r e l i m i n a r y e s t i m a t o r s a r e as f o l l o w s : nq m Z Z x' i=l j=l  Var(x'..) = Var( n  m  q  ) 1 J  nq Z V a r ( ^ Z x', .) m ij m  n  l  (nq)  2 i  =  1  1=  1  m  (nq)m  \-7 (nq)m  x  2  L  m [ 2 j  -r—x—r  =  mm V-ar(x'..) + Z Z ij ,  1  [m(a  (,nq)m  — [a nq and  x'..) ij  Var( Z  2  + a  2  a  + (a  2  + (a  2  Var(y'..) = — [a + (a ns a_  2  Var(y"..) = — [a np a  2  2  cti  2  t  E,  z1  cov(x' . . ,x' . . . ) ] ij ij  ) + m(m  -  l)a  ]  2  a  t  /m)]  similarly, Var(x"..) = — [a np a 2  Ei  x  2  and  + (a  2  /m)] /m)]. E  y  2  /m)]  2  Further, np cov(x"..,y"..) = 7 — T T C O V ( Z (nmp) .  m Z  1  = 1  ^ (nmp)  np Z 2 1  "  x" , ij .  j= 1  I  np Z  m Z  m Z  r  .;  =l  1  ^ np np T rr ^ 2 (nmp) . .,  m I  m Z  2  = 1  = 1  j  =  1  j  t  =  1  np I =  1j  l  m Z =  cov(x" = 1  —  1  y"  )  1  ,,y" , ,,)  I J " i'j'  n..,(p,a a i i ' a, a  + n..,P o a JJ' E, E  )  2  2  2  25 ^  np  (nmp)  .  2  n  P  =  1  x  , , (nmp) N  1  i  i  l  =  =  1  that  2  '  !  be s e e n  a  I  E  a  a  from  current  , a  e  mean  y  y  1  Z j  l  =  + m p a e  1  n .., P ° j j ' 2  o  )  a ) E  2  2  t  2  ] 2  the c o r r e l a t i o n  2  assumed  that  In a l l the d e r i v a t i o n s  i twill  be  , P i , a n d p , a r e known.  2  e  structure  0.  mean.  , a  2  z  =  2  = 0  a  Z  )  E  cov(x'..,y'..)  2  .  2  2  = 0  , a  2  2  on t h e s e c o n d  occasion  i s e s t i m a t e d by a  linear  of the form =  2  be  a  +  + p a a e, e  2  of the current  y £ where  o,  a  cov(x'..,x"..)  l  estimator  a  cov(y' .., y"..) =  Estimator  The  1  + p a  1  and  a a  1  a  r  P l  2  [mpjd a  '"' ''oi 'a  2  n..,(m p,cr a 11' a, a  1  (nmp)  2 z  i t can e a s i l y  assumed  l  m m r,,, , ( m i i '  np £  (mpjO  nmp and  .  1  2  / s (nmp)  ^  =  np. Z  a  x  2  '>«  a , b , c , and d 2  2  2  unbiased,  + b  x".. + c  2  2  y".. + d  are constants.  2  We  2  y'..  require  that  this  estimator  that i s , E ( y  H  )  =  V  Since  and then  E(x'..)  = E(x".•)  =  u  E(y'..)  = E ( y " . .) =  V  t o be u n b i a s e d we a  and  c  Consequently, y  2 n  we  2  2  require  + b  2  = 0  + d  2  =  obtain  = a (x'.. 2  x  y  that  1.  that - x"..) + c  2  y " . . + (1 - c ) y ' . . 2  (1)  26 The  variance of t h i s  Var(y  estimator i s  ) = a|[Var(x"..)  2  + Var(x'..')] + c\ V a r ( y " . . )  JO  + (1 - c since  ) Var(y'..) - 2a c cov(x"..,y"..)  (2)  2  2  2  a l l other c o v a r i a n c e s a r e zero  2  ( g i v e n the c o r r e l a t i o n s t r u c t u r e  assumed). Then by s u b s t i t u t i o n Var(y  2 o  )  = a|{[(a  /np) + ( o  2  + c|[(a  /nmp)] + [ ( a  2  /np) + ( a  2  - 2a c (mp a 2  2  a a  1  a  x  /nq) + ( a  /nmp)] + (1 - c  2  E  0t2  2  2  )  {(a  2  /nmq)]}  2  /ns) + ( a  2  2  2  + p a 2  E  2  2  /nms)}  ^ 2  a )/nmp. j e 2  Letting e,, = ( a  /n) + ( a  2  t»l 9  = (a  2 2  6,  El  /n) + ( a  2  a  2  = (mp.a a a a  i> = B / ( / e . 2  2  2 1  /nm)  2  E  2  + p a 2  :  and  /nm)  2  Ei  2  e  2 2  a  E  )/nm 2  )  we o b t a i n that Var(y  2 j i  )  = a|[(9 /p)  + (e  2 1  -  /q)] + c (6 2  2 1  2 2  / p ) + (1 - c  2  2  2 1  +  +  U  ~  p  C z ) 2  ]e  the v a r i a n c e o f y . i s m i n i m i z e d .  2 2  /s)  - 2a c B /p  2 2  2  (3) w i t h r e s p e c t to (w.r.t.) equal  to z e r o , and then  We o b t a i n , a f t e r a*  c*  simplification,  = '{(p/[s(l -  2  2  2  (3)  2  qipf.)  2  and c  2  each of a  simultaneously  2  and c , s e t t i n g the 2  solving  for a  that + p])B q}/9 i  = p / [ s ( l - q * l ) + p]  i n (3) such  We do t h i s by d i f f e r e n t i a t i n g  2  .  (e  ^  We now choose those v a l u e s o f the c o n s t a n t s a  results  2  2  2  equation  )  2a c B /p  = a ( -+ i ) 6 p q  that  2  2  2  = c- B q/6 2  2  2 l  2  and c . 2  where a * and c"'" are the ' o p t i m a l 2  that minimize  Var(y  S u b s t i t u t i n g these y  values of a  1  2  ), and i>  and c , r e s p e c t i v e l y , 2  = S /( 9 9 ). 2  2 2  2  2  2  2x  22  'optimal' v a l u e s i n t o e q u a t i o n (1) y i e l d s  = (c-'- e q/9 ) ( x ' .. - x"..) + c- y"..+ (1 - c * ) y ' . .  2 j /  2  2  21  2  = c* {y"-- + ( 6 q / 9 ) ( x ' . . 2  2  2  - x"..)} + (1 - c * ) y * . .  2 1  2  But  x.. = qx'.. + (1 - q)x".. = the grand mean on o c c a s i o n 1 or and  x.. - x".. = q(x'.. - x"..) @ /6 2  = cov(x".. , y"..)/Var (x". .) = ^2YX'  2l  3  S  r e  r e s s : L O n  coefficient.  Then yi  %  = c» {y".. + 2  f 3  YX ^" " *"-(  2  Let the q u a n t i t y {y".. +  6.  ) }  +  (  " "2)y'-.  1  c  (x.. - x"..)} = y  , a regression estimator.  2  Then y  = c* (y  H  2  2 r e  )  + (1 - c * ) y ' ..  (4)  2  or, s u b s t i t u t i n g the value of c * i n t o e q u a t i o n ( 4 ) , y  H  = Ue  2 2  /s)(y  2 r e  ) + [(e {( 9  =  CPy  2 2  (i-*  2 2  / ) s  2 2  ))/  + [e  + t e 2  P  2  2 2  (i  ]y-..}/  - <i> )/p + ^ 2  2  + s ( l - q^ )y'..]/[p + s ( l - q * 2  2 r e  2 2  2  2  2  The such  2  e  2 2  )].  We n o t i c e t h a t the c u r r e n t mean e s t i m a t o r i s a weighted u n c o r r e l a t e d estimates y and y'.., re  2  average  ]} (5) of two  and i s unbiased.  v a r i a n c e o f t h i s e s t i m a t o r , which i s the minimum p o s s i b l e f o r  l i n e a r estimators, stated i n equation  o b t a i n e d as f o l l o w s . Var(y ) = c* 2 o  (2) can be otherwise  simply  We can w r i t e t h a t , u s i n g e q u a t i o n ( 4 ) 2 2  Var(y  2  ) + (1 - c * ) 2  2  Var(y'..)  since y, and y'.. are u n c o r r e l a t e d of each other (from the c o r r e l a t i o n re s t r u c t u r e assumed).  A f t e r s u b s t i t u t i o n and f u r t h e r s i m p l i f i c a t i o n we get  28Var(y ) = 6 { [ l - q^ ]/[p + s ( l - q*|)]>  (6)  2  2 2  2jl  2  = 9 {(1 - c- )/s}. 22  2  We shall now examine some special cases. 1.  If q = s, i . e . , equal sample size on both occasions, then y  H  = <P y  2 r e  + q d - q*i)y'..>/ [1 -  UM ] 2  and Var(y ) = { 6 ( 1 - q* )}/ 2  2ji  2.  22  I f , i n addition to (1), 6  21  = 9  [l - (q* ) ] 2  2  = 6 , i . e . , the variances within  2 2  2  the stages are the same on both occasions, then y  2j<  = <p y  + q(i - q^y'..}/ [ i - (q* ) ] 2  2 r e  2  and V a r C y ^ ) = {9 (1 - q*l)}/ [1 - ( q * ) 2  2  2  ]•  It can be seen that for q = 0 or q = 1, i n this special case 2, Var(y  ) = 6.  2  2  This indicates that whether the sample i s completely  Jo  retained or completely replaced by a new sample, the variance of the e s t i mator i s the same.  For a l l values of 0 < q < 1, Var(y  ) < 6 , which  2  2  Jo  indicates that a replacement mean i f I|J ^ 0 ( i . e . , p  1  2  policy w i l l improve the estimate of current  ^ 0 and p  Estimator of the change. sions of sampling, A = u  2  £ 0).  The change i n means between the two occa-  - y , w i l l be estimated by g „ , which i s of 2  a linear form g  2  = e y".. + f x " . . + h y'.. + t x ' . . 2  2  2  2  We require that this estimator be unbiased, that i s , E(g ) = 8 &  M  y  -  U ) (  = A. •  Since E(x"..) = E(x'..) = u  x  29 E(y"..) = E(y'..) = p t h e n t o be u n b i a s e d we r e q u i r e e and Consequently g  f  + h  2  that  = 1  2  + t  2  y  2  = -1.  we o b t a i n t h a t = e y " . . + (1 - e ) y ' . . + f x " . . - (1 + f ) x ' . .  2  2  2  2  (7)  2  36  The  variance of t h i s estimator i s  Var(g  2 o  )  = e  3C-  V a r ( y " . . ) + (1 - e  2  + (1 + f given  2  )  2  the c o r r e l a t i o n  2  )  Var(y'..) + f  2  Var(x'..) + 2  structure  e f 2  2  Var(x"..)  cov(x"..,y"..)  2  assumed e a r l i e r  (8)  i n the d e r i v a t i o n  of the  c u r r e n t mean e s t i m a t o r . By v  ar(g  substituting  ) = e (9 2  2 o  2 2  i n t h e v a r i a n c e s a n d c o v a r i a n c e s , we o b t a i n t h a t  / p ) + (1 - e ) ( e 2  + (1 + f ) ( 6 2  2  where 6  2 l  ,  9  2 2  , 0  2 1  2  2 2  /s)  +  f (6 2  2 l  /p)  /q) + 2 e f B /p 2  2  (9)  2  a r e as d e f i n e d e a r l i e r .  2  We now c h o o s e t h o s e v a l u e s o f t h e c o n s t a n t s e that  the variance of g  equation  i s minimised.  2  ( 9 ) w . r . t . each o f e  2  and f  .and t h e n s i m u l t a n e o u s l y s o l v i n g  w h e r e e'  f*  2  e-  2  = C- p q B  2  2  that minimise  2  2  and f  2  i n (9) such  differentiating  the results .  2  equal to zero,  We o b t a i n t h a t  / K ] - [p(s+ p)/K ]  = [p/K ] +  and f *  v  2 Y X  fore  and f  We do t h i s b y , setting  2  2  2  2  [ps/K ]B 2  2 X Y  are the 'optimal' values of e  the V a r ( g  2  2  and f  2  , respectively,  ) 30  6 /6 2  K and  2  2 2  = cov(x"..,y"..)/Var(y"..) = B  2 X Y  = p + s ( l - q*|)  t h e o t h e r symbols a r e as d e f i n e d p r e v i o u s l y .  , a regression  coefficient  30 Substituting  i n the v a l u e s of e *  (after multiplying 82  a  = (p/K y".. + [ ( p s B V  2 X Y  ) / K ] y ' \ . + [(s(1 - q* ))/K ]y'.. 2  )/K ]y«.. - [(pqB  2 X Y  2 Y X  2  2  )/K ]x'..  2  2 Y X  )/K ]x"..  - [(p(s + p))/K ]x".  2  - [q(s + p -  2  A f t e r f u r t h e r s i m p l i f i c a t i o n , we  = Up/K )y  i n t o e q u a t i o n (8) g i v e s  2  2  2  + [(pq6  t  f*  out)  - [(psB  &2  and  2  2  s*|)/K ]x'.. 2  o b t a i n that  + [ s ( l - qH»|)/K ]y'..}  2re  2  -  {[p(s + p ) / K ] x _ 2  2 ]  + [q(p + s ( l - ^ ) ) / K ] x ' . . }  (10)  2  e  2  where x  2 r e  = x"..  + B  This estimate of change i s seen  2 X Y  [ s / ( s + p)][y'.. -  y"..]  to be a l i n e a r combination  c u r r e n t and p r e v i o u s mean e s t i m a t o r s y  and x  of the  unbiased  , respectively. (Note J6 p r i n c i p a l p i e c e i n e q u a t i o n (10) i s x , the BLUE of the 2  2  Ki  that the second  2  JO  p o p u l a t i o n mean on o c c a s i o n 1, g i v e n the o b s e r v a t i o n s on o c c a s i o n The  v a r i a n c e of g . , 2  2.)  which i s the minimum p o s s i b l e f o r such  linear  e s t i m a t o r s , i s e a s i l y d e r i v e d by c o n s i d e r i n g equation ( 8 ) . Var(g ) H  e  =  [e* / 2  2 2  + (1 - e * ) / s ] + 6 2  p  2  [ l + ( f * + p) /pq] 2  2 1  e* f* \p /e e /p.  + 2 S u b s t i t u t i n g i n the v a l u e s of e * m a n i p u l a t i o n , we Var(g  2 j l  )  =  {[p + s(l - *|)]e + (1 21  s h a l l now  I f q = s,  g  = [p/(l - q ^I)](y  and V a r ( g  2  2 x <  )  -  2  q<l> 2 )  2  2  2 1  2 2  and a f t e r some lengthy a l g e b r a i c  6  2 2  - 2pvp /6 9 2  2 1  }/  2 2  qtpl)].  (10.1)  examine some s p e c i a l  1. H  f*  2  o b t a i n that  [p + s ( l We  and  2  2  2 r e  - x  = [(1 - q * )( 6 2  2 1  2 r e  cases.  ) + [q(l -  + 6  qipl)/(l  -  q *J)](y'.. a  ) - 2p^ /6 , 6 ] / (1 - q ^ ) . 2  2 2  2  2  22  2  -  x'..)  31 2.  I f , i n a d d i t i o n to ( 1), 6  g  = [ p / ( l - q * ) j ( y " - - - x"..) + [ q ( l - 1> )/(1 - q * ) ] ( y ' . . -  2 j l  2 2  = 6,  then 3 ^  2  2  and It  = 6  21  2  V a r ( g ) = 26 (1 2 £  can be  = 6  seen i n equation  (11)  2  (11)  2  2  that f o r (= f i x e d  2  2  = 1, V a r ( g ^ ) = 26  - (q* ) ]  2  q = 0, V a r ( g ^ ) = 29 ( 1 - i> ) 2  x'..)  2  - q*| - p * ) / [ l  2  and  2 y x  sampling  v a r i a n c e of change)  (= independent sampling  2  v a r i a n c e of  change) So that f o r v a l u e s of 0 < q < 1, the v a r i a n c e of the.growth e s t i m a t o r vary  between 26(1  fixed  sampling  2  o  a  t  , a  e  Three-stage  ( t = 1,2)  2  =  [mpjO  and of  z  (i =  1,2)  and  In p a r t i c u l a r ,  o c c a s i o n , m ssu's  r tsu's are s e l e c t e d by  must  a  a  1  + p a  a  2  a  2  E  1  E  2  ]/[/9  2 1  be  6 (nm) ] . 22  sample u n i t s  (tsu's).  ssu's, or t s u ' s ) are of  suppose n psu's are s e l e c t e d by  srswor  on  are s e l e c t e d by srswor from each sample  psu,  srswor from each of the sample s s u ' s .  A  srswor) of s i z e np  (0 < p < 1) of the n psu's  r e t a i n e d f o r the second o c c a s i o n t o g e t h e r w i t h i t s r e s p e c t i v e tsu's- drawn from the f i r s t s i z e ns  q:  as p o s s i b l e s i n c e  each ssu c o n t a i n i n g T t e r t i a r y  random sample ( s e l e c t e d by is  low  depends on i>  q, that i s ,  suppose t h a t the sample u n i t s (psu's,  the f i r s t and  be as  2  a p o p u l a t i o n c o n s i s t i n g of N psu's, each psu c o n s i s t i n g  M ssu's, and  size.  replacement  SPR  Consider  equal  or  t ip  Further,  of growth over p a r t i a l  e f f i c i e n c y of the change e s t i m a t o r g  i n c r e a s e s w i t h i n c r e a s e s i n ty and  h i g h and  of  T h i s i n d i c a t e s that t r a d i t i o n a l CFI  0.  2  it  26.  2  g i v e s improved e s t i m a t e s  as long as ty ^ The  - \|J ) and  will  (s > 0) of the N-n  occasion.  ssu's  In a d d i t i o n , a random sample  other psu's i s s e l e c t e d by  srswor f o r  32 inclusion selected  i n t h e s a m p l e on t h e s e c o n d o c c a s i o n . by s r s w o r  by s r s w o r  be a s s u m e d t o be i n f i n i t e l y  The numbers N, M, a n d T w i l l  again  large.  are taken  i n e a c h o f t h e nmr t s u ' s . o n t h e f i r s t  a n d nmr ( p + s ) t s u ' s o n t h e s e c o n d o c c a s i o n . observations  m ssu's are  f r o m e a c h o f t h e ns p s u ' s a n d r t s u ' s a r e s e l e c t e d  f r o m e a c h o f t h e nm s s u ' s .  Observations  Again,  We d e s i g n a t e  occasion  the various  as f o l l o w s : Occasion 1  No. o f u n m a t c h e d No. o f m a t c h e d  psu's psu's  No. o f u n m a t c h e d No. o f m a t c h e d  ssu's ssu's  No. o f u n m a t c h e d No. o f m a t c h e d Matched  tsu's tsu's  observations  nq  ns  np  np  nmq  nms  nmp  nmp  nmrq  nmr s  nmrp  nmrp  x". ijk  Unmatched  observations  x' . .. ijk  i=l,2,., j=l,2,., k = l , 2 , ..  ,np ,m  1=1,2... j=l,2,. k=l,2,.  ,nq ,m ,r  y  y  "ijk  ijk  i=l,2,. j=l,2,. k=l,2,.  .np ,m  1=1,2,. j=l,2,. k=l,2,  , ns ,m ,r  where q = 1 " P • Recall they  t h a t X a n d Y do n o t r e f e r  refer  to different  t o t h e same v a r i a b l e o f i n t e r e s t ,  Y on o c c a s i o n  v a r i a b l e s of i n t e r e s t : c a l l e d X on o c c a s i o n  2.  We a r e i n t e r e s t e d i n e s t i m a t i n g t h e c u r r e n t p o p u l a t i o n mean the change  1 and  i n means A = u  - u Y  be a s s u m e d t h a t o n t h e f i r s t  of the v a r i a b l e of i n t e r e s t .  It will  A  occasion  the observations  and  (matched o r  33 unmatched) are d e s c r i b e d by the l i n e a r  n e s t e d model i=l,2,...,N  ijk  l i  K i ) j  Kij)k  k=l 2 ... T  where x.., = o b s e r v a t i o n on the k t h t s u w i t h i n the i t h ssu w i t h i n the I jk J  i t h psu y  = o v e r a l l mean of the o b s e r v a t i o n s  A  a, . = e f f e c t li e .,.  r  = effect  w  ^l(ij)k all  o f the i t h psu  the  Y  =  e  ^^  °f  e c t  i(-Lj)k'  o f the i t h ssu w i t h i n the i t h psu  s  a  r  t  n  e  2  Yi  a  2  t  s  u  i  w  independent  e  = 0 and v a r i a n c e = a  independent  ^th  a  i  n  the j t h ssu w i t h i n the i t h psu and  n  random v a r i a b l e s each w i t h expected v a l u e  , a l l the E,,.. ,'s a r e independent random l ( i ) j  of ^ ^ ( ^ j ^ ' i  , and a l l the ^ £ '  t  s a  variables,  each w i t h expected v a l u e = 0 and v a r i a n c e = r  independent  e  random v a r i a b l e s ,  {E, , . . i} and iy,,. . . }, each with expected Ki) l(ij)k ^ Then x  cov(x. ,x. , . ,. ijk l'j'k'  = r\. (a u' o,  independent o f  value = 0 and variance a  2  ct!  +n...o +n...n. , . a ) J J ' E, J J kk' y j  1  2  2  1  where n  uv  0 if u 4 v =1 i i•fr u = v  Note, i n p a r t i c u l a r , that cov(x... , x . ) = cov(a..,a...) + cov(e,,...,£,,...)= a ijk ijk' l i l i ' l(i)j l(i)j a  +a  2  cov(x. ,, ,x. ... ,) = cov(a, . ,a, . ) = a Ijk I j k' h h otj  i f j £ j and k 4 k  2  cov(x.. ,x.., ) k  C 0 V < : x  k  ijk' i'j'k"* X  = cov(a .,a .) 1  =  0  i  f  cov(x. .. , x . , .. ) = 0 ijk l ' j k  1  1  ^  1  1  a  1  J  = a ^.if j 4 j ' 2  nd j ^ j '  i fi 4 i ' .  t  .  1  2  c,  i f k ^ k '  In other words, observations on different tsu's within the same ssu's are correlated;  observations on tsu's within different ssu's on the same psu  are correlated; uncorrelated.  and observations on the tsu's within different psu's are Note i n particular that i f i = i ' ,  cov(x. ., ,x. ., ) = Var(x. ., ) = a + a ijk' ijk ijk a, 2  j = j ' , and k = k', then  + a  2  E ,  2  Yi  S i m i l a r l y , on the second occasion, the observations (matched or unmatched) are described by the linear nested model i=l,2,...,N y  i j k = ^Y  °2i  +  +  £  2(i)j  +  Y  llllly/.^l  2(ij)k  where y. ., = observation on the kth tsu within the i t h ssu within the i t h ijk psu u^. = overall mean of the observations a„. = effect of the i t h psu 2i E  Y  2(i)j  =  e  ^  e  c  t  °f  =  e  ^  e  c  t  °^  2(ij)k  and a l l the ^ 2 ( i j ) k '  S  a  r  J t n  e  t  ns  s  u  W l t  independent random variables each with 2(i)j  2  variables, independent of ^ 2 ( i j ) k ^ ' Y  , and a l l the ot  independent of ^ 2 ( i j ) k ^  a n  expected  , a l l the E , , , . , ' s are independent random  2  Y  2  the i t h psu  i n  e kth tsu within the j t h ssu within the i t h psu  value = 0 and variance = a  variance = a  h  w  's are independent random variables,  ^ ^ 2(i)j^' e  ^ i t h expected value = 0 and  e a c  e  a  c  n  w  i t h expected value = 0 and  A similar correlation structure w i l l be assumed f o r the observations on the second occasion as that on the f i r s t  occasion.  35  In order to impose a c o r r e l a t i o n structure between occasions  1 and  2, i t w i l l be further assumed that cov(x  i j k  ,y  where  i I j I k l  ) =  n . - . L - p ^ o ^  ' p  3  +  ^.(P.a^o^  +  \  k  . > ]  *  = the c o r r e l a t i o n between the e f f e c t s due to the tsu's within the ssu's within the psu's.  Note, i n p a r t i c u l a r , that c o r r ( a , a , ) = n ^ , Pi l i  2 i  /•  corr(  r  r  (  a  li' 2(i)j £  2i' l(i)j £  corr(a ,Y u  2  corr(c c o r r  2 ( i )  .  2 ( i  corr(a ., Y  and  2 (  ., . ) )  = n.^n.-.P,  l  Y  C O r r ( a  o  .,e  l(ij)k' 2(i'j')k'  C O r r ( Y  c  e i ( i )  )  Y  n  P s  = 0  ) >  = 0 = 0  .,Y  £  ii' jj'\k'  °  =  1 ( i j ) k  ( l(i)j'  T 1  = °  )  ) k  =  )  2(ij)k  }  = °  Furthermore, i f i = i ' and j = j ' then C O v ( x  i k' f'j'k' j  y  ) = C O v ( x  i k' ijk y  j  )  =  P  >° °«. 0l  + P , <  Ve,  +  P 3  °Y! Y a  2  The assumed correlation structure implies that (a)  observations  on the tsu's within the different psu's at the two occa-  sions are uncorrelated, (b)  observations  on the tsu's within the same psu's at the two occasions  are correlated, (c)  observations on the tsu's within different ssu's within the same psu at the two occasions are correlated, and  (d)  observations  on the tsu's within the same ssu within the same psu at  the two occasions are correlated. Using the sample data, we obtain preliminary estimators as follows (there are other possible preliminary estimators); nq m r x'...=(I Z Z x'. i=l j=l k=l  )/nmrq  1 J  np m r x". . . = ( I Z Z x". .. )/nmrp i=l j=l k=l J  ns m r y'... = ( I Z Z y'. )/nmrs i=l j=l k=l 1  J  k  37 np = ( E i=l  y"...  m r £ E y". j = l k=l 1  )/nmrp  J  where x ...  i s t h e mean o f u n m a t c h e d  observations  on o c c a s i o n  1  y'...  i s t h e mean o f u n m a t c h e d  observations  on o c c a s i o n  2  x"...  i s t h e mean o f m a t c h e d  observations  on o c c a s i o n  1  y"...  i s t h e mean o f m a t c h e d  observations  on o c c a s i o n  2.  1  The  expected  values  o f the above p r e l i m i n a r y e s t i m a t o r s  a r e as  follows:  E(x'...) = — — nmrq M  nq m r z £ E E(x'..,) = — — . . . , , , i j k nmrq i = l j= l k=l  nq s  m i  r i  E ( i r + a, . X l i i = l j = l k=l  ^ ( D j ^ K i j ) ^  +  =  y  x  Similarly, E(x"....) = p and since tion  a  u  E(y'...) = E(y"...)  c  , a ^ ,  equal  .  x  ,  2 ( i j ) J  Y  1 ( i J ) k  = y , y  ,  and Y  2 (  ..  each  ) k  have  expecta-  to zero.  The v a r i a n c e s  of these p r e l i m i n a r y nq m r  V a r ( x ' . . .) = V a r ( n  -. ^ (nq) 1  m  r  l  =  1  ^ Var(— mr  (nq)  . Var(— mr  (nq)  (mr)  1  + E E  W  are as  follows:  Z Z Ex'...) i = l j = l k=l  q  nq E .  2  estimators  21  m E j  m E k  =  =  1  r E x ' ) ^ i j k  1  E =  k  r Ex:..) i j k  { E R  j  V a r ( x ' ) i j k  1  + r E E ^ ,  cov(x' ,x' , ) i j k i j ' k  E E c o v ( x ! .. ,x!.,,.,)+ m E E c o v ( x ! k^k' k^k' l  j  k  1  J  k  l  j  k  ,x! l  j  k  , )}  38. nq(mr)  {mr(a  2  + a  2  04  E l  + mr(r - l ) ( m - l ) a —{a nq and  Yi  + (a Yi  2  l)[a  - 1) + a  2  ]}  2  /mr)}  2  E l  ) + mr(m  2  + mr(r-  2  + ( a . /m)  2  + a  2  similarly, 1 Var(x"...) = — [  /m)  + (a  2  Var(y'...) = — [a + (a /m) ns ae  + (a  2  Var(y"...) = — [a np a  + (a  np  + (a  a,  2  2  Y  2  and  + (a  2  /m)  2  /mr)]  Yl  El  /mr)]  2  Y  /mr)] 2  .  2  Further, cov(x"...,y"...) =  np (nmrp) ^nmrp,  2  C  O  v  (  Z  np 7 \T (nmrp) ,  .,  = 1  np  (p! a  a  a  ^  = 1  j  1  k  =  " i i 'j k i  1  Z  k  j  m m r S •_ Z •Z Z . j , . • , =  1  =  + n. .,p a a j j ' , e  1  k  =  1  k  E  np  i=l  m  r Z k=l  a  2  1  r Z n k'=l K  + 2  np Z  (nmrp)  2  2  , ,) ; j  k I  y  a Y  ) 2  k k  ,P a Y l  * '° ,V p  +  ^-.P'°  r  a  a  3  K  ( r  )  Y z  2  + r  m Z  2  m Z  1  n..,p a  \  i  m  2  /  y" y  1  Y l  2  I  =  cov(x". .. ,y". , . ,, ,) " U k I'j'k"'  = 1  np np 7 rr Z Z n. . , ( m r p, a a (nmrp) 1=1 i ' = l n' a, a.  J J  l  K 33  2  ,* , ^ , ^ii' i ' = l j=l j'=l  2  +  •r  Z  + n . . . n. . . p a J J ' kk'  2  2  m  Z  3  np  (nmrp)  =  x  np m m r r Z Z Z Z Z TI. i ' = l j = l j ' = i k=l k'=l  Z  i =l  a,  j  I;  2  (nmrp)  = 1  np  r 1  np  2  :  .  m 1  m m r i Z Z Z Z j = l j ' = l k=l k'=l  np Z  ^ l = 1  n. . ,n  , (m r PiO^ 2  A  , , . ill  . . + mrp,a = 1  n..  'n' a ) Yi Y 2  2  UiU  1 M  »  C l  j=l  u u  j'=l  , p o\. a.. ) 3  + mr p a 2  2  u  a.  a  0  39  T  --i-  T r  -(™rp)[mrp «7 l  [mrp,o a nmrp """~ " " a," a L  a  a i  + rp a '' ^ i  2  rp.o^o^  +  a e  2  1  and i t can e a s i l y be seen  a i  P.a^a^]  +  + p a a ' * Yi Y  ]  3  2  2  that  cov(x'...,x"...) = 0 cov(x'...,y'...)=0 and  c o v ( y ' . . . , y " . . . ) = 0.  E s t i m a t o r of the c u r r e n t mean. will  be assumed that a  , a  2  , a  2  ai p  3  a  Again, i n a l l the d e r i v a t i o n s i t , a  2  £1  2  , a  2  e  2  , a , p,, p , and Yi' Y 2  2  2  2  are known. The c u r r e n t mean ji  o  n  t  n  second o c c a s i o n i s e s t i m a t e d by a l i n e a r  e  e s t i m a t o r of the form y  = a x'... + b x"...  3  3  2  where a , b , c , and d 3  We  3  3  3  + c y " . . . + d y'... 2  2  are c o n s t a n t s .  r e q u i r e that t h i s e s t i m a t o r be unbiased, that i s ,  Since  and  E(x'...) = E(x"...) = y  x  E(y'...) = E(y"...) = u  y  then to be unbiased we r e q u i r e  and Consequently we o b t a i n  that  a  3  + b  3  = 0  c  3  + d  3  = 1.  that  y ^ = a ( x ' . . . - x"...) + c y " . . . + (1 - c ) y ' . . . 3  3  3  3  The v a r i a n c e of t h i s e s t i m a t o r i s Var(y  ) = a|[Var(x'...) + Var(x"...)] + c|Var(y"...) + (1 - c ) V a r ( y ' . . . ) 2  3  3  -  2a c cov(x"...,y"...) 3  3  since a l l other covariances  are zero (given the c o r r e l a t i o n structure  assumed). Then by substitution Var(y ) = a {[(a 2  3 n  /np)  2  3  + (a + (a  2  e i  /nmq)  + (a  + (a  /nmp)  + (a  2  /nmrq)]} + c  2  2  2  Yi  2  2  3  3  a  1  Qj  + rp a  a  2  a  2  e  /np)  2  2  + p cr  a  3  H- ( a  /nmp)  2  2  /runs) + (o  2  E  i ^ - 2  /nq)  2  e  2  a  - 2a c (mrp o  [(o  /ns) + ( a  2  3  Y  3  a  /nmrp)] + ( 1 - c ) [ ( a  2  /nmrp)] + [ ( a  2  /nnirs)]  2  12  )/nmrp.  Y 1 Y 2  Letting e,. = ( a  2  + (a  /n)  2  Oi 8,,  B  = =  3  (a  a  +  /n)  2  (a  e  [mrpjO-  a  +  we  obtain  2  3  /nmr)  (a  o  rp cr  a  2  £ l  + E  /nmr)  2  Y  2  Pa  a  3  ]/nmr 12  i ]  2  ,9 )  3  J2  that  Var(y ) = a H  = B /(/6  3  +  /nm)  2  2  1  *  2  Yl  (X 0l2  and  + (a  /nm) Ej  2  3  [(6  3 1  / q ) + ( e / p ) ] +c  2  sl  ( 6 /p) + (1 - c ) ( 9 /s) 2  3  32  3  32  - 2a c B /p. 3  We  now  choose those values of a  i s minimized.  where a*  3  We  and c  3  3  3  such that the variance of  y  3  obtain that + p])B q}/6  a*  3  = { ( p / [ s ( l - q^ )  c*  s  = p / [ s ( l - qi|j ) + p]  2  3  3  3 1  = c* B q/6 3  3  3 1  2  and c*  minimize Var(y  3  3  s  are the  ).  'optimal' values of a  3  and c ,  Substituting in the values of a*  3  3  tion ( 1 2 ) and modifying as shown in the two-stage SPR, y,  (13)  3  z  respectively that  and c* we  3  obtain that  = c * , ( y , ) + (1 - c*,)y'...  (14)  re  =  [  p  y  3  + r  e  s  ^  - q^ )y'---l/tp + s(i - q* )] 2  2  3  into equa-  3  41 • where y  3 r e  = y".-.  B3/631  and  + B  (x  3 y x  - x»...)  = c o v ( x " . . , y " . . ) / V a r ( x " . . ) -• S^xy'  Using e q u a t i o n  (14)  we determine  3  r  e  8  r  e  s  s  i  o  coefficient.  n  t h a t ( g i v e n the c o r r e l a t i o n s t r u c t u r e  assumed) Var(y  ) = c* Var(y  ) + (1  2  3  JO  - c*,) V a r (y' . . ..) 2  LQ = 6 , ( 1 - q+|)/[p + s ( l - qi> )] = 9 2  2  3  Again, we can o b t a i n some s p e c i a l  1.  {(1  - c*,)/s}.  cases.  If q = s y  and 2.  3 2  H  -  = [py,  Var(y  3 j l  +q(i-  r e  ) = 6  3 j l  )  (q* ) l 3  ( 1 - qi|»f)/[l - ( q + , ) ] . 8  3 2  I f , in addition Var(y  q+!)y'.••]/[!-  2  to (1),  6,  = e  3  = 9 (1 - q*f)/[l -  = 6,  32  3  then  (qt ) ]. 2  3  3  S i m i l a r c o n c l u s i o n s can be drawn about V a r ( y  ) i n t h i s case, as i n the  3  s-  JO  two-stage SPR. E s t i m a t o r of change. of sampling,  A = u  - u , w i l l be e s t i m a t e d , as b e f o r e , by I  S,3  l  The change i n means between the two o c c a s i o n s  A  = e y"... + f x " . . . + h y'... + t x ' . . . 3  3  3  3  We r e q u i r e t h a t t h i s e s t i m a t o r be unbiased, E  (g  3 j ;  )  = V  Y  - M  x  = A.  E(x"...) = E(x'...) = n  Since  X  and  E(y"...) = E(y'...)  then to be unbiased we r e q u i r e that e  r  and  + h  3  f  s  + t  = 1  3  s  = -1.  =  u  that i s ,  42 C o n s e q u e n t l y , we  obtain  = e',y"... +  g 3  that (1 - e ) y ' . . .  + f x"... -  3  (1 + f ) x ' . . .  3  3  (15)  36  The  variance  Var(g  of  +  By  the  We  ,  (16)  variances  2  choose the  ,  and  B  values  3 1  are  3  of e  we  get  f (93,/p) 2  + 2e f 6 /p 3  as  3  f  earlier.  i n (17)  3  (17)  3  defined  and  3  covariances,  +  2  /q)  earlier.  the  2  3  3  3 2  and  + (1 - e ) ( 9 3 / s )  (1 + f ) ( 6  9  3  s t r u c t u r e assumed  i n the  3 2  3 1  3  correlation  ) = e ( 9 /p)  9  + ffVar(x"...)  2  s  2  + where  (1 - e ) V a r ( y ' . . .)  3  2  3 j l  is  (1 + f ) V a r ( x ' . . . ) + 2 e f c o v ( x " . . . , y " . . . )  substituting  Var(g  estimator  ) = efVar(y.'-'. . .) +  3 j l  given  this  such t h a t  the  variance  of  g  3  36  i s minimised.  We  f* e*  s  = [=  s  obtain pqB  that /K ] -  3 Y X  [p(s  3  [p/K ] +  [ps/K ]B  3  3  + p)/K ]  .. -  3  3 X Y  where K  = p + s(l -  3  e*  and  3  f*  3  are  qi|if) the  that minimize Var(g  'optimal' values  of  e  3  and  f , respectively, 3  )  3  16  B /6 3  = cov(x".••)/Var(y"..)  3 2  other  symbols are  Substituting a g  lengthy H  =  as  previously  i n the v a l u e s  of  simplification,  ((p/K )y 3  we  e*  obtain  B  f*  3  into  + [ s ( l - q* )/K ]y'...} 3  +  [q(p  = x"...  + 6  = x"...  + ^ 3 Y ^ * ** ~  ,  a regression' c o e f f i c i e n t  equation  {[p(s  +  p)/K ]x  + s(l - ^ ))/K ]x'...  3  r  e  3  3 x y  X  [s/(s  and  after  + p)][y'... y"  , # ,  -l  3  3 r e  (18)  2  where x  (15)  that  2  3 r e  3 X Y  defined.  and  3  =  y"...]  and  43 Then V a r ( g ) = {[p + s ( l - * ) ' ] e 2  H  The  special  3  3 1  + (1 - q * i ) e ,  cases o b t a i n e d i n two-stage  2  - 2pip 3 ^ 9  3 1 9  3  2"}/K,.  SPR and the c o n c l u s i o n s made  t h e r e i n can s i m i l a r l y be o b t a i n e d here t o o . M u l t i s t a g e SPR Following  from the d e r i v a t i o n of the two- and t h r e e - s t a g e SPR, we  s h a l l how g e n e r a l i z e the d e r i v a t i o n  to h-stage  (h > 1) SPR.  Consider  a p o p u l a t i o n c o n s i s t i n g of N psu's, each psu c o n t a i n i n g M s s u ' s , each ssu of the In  c o n t a i n i n g T t s u ' s , and so on, and each p e n u l t i m a t e u n i t W u l t i m a t e sample u n i t s  ( a t the h t h s t a g e ) .  consisting  F u r t h e r suppose  that  sample u n i t s at each stage of the m u l t i s t a g e d e s i g n are of equal particular,  suppose  n psu's are s e l e c t e d by srswor on the f i r s t  size.  occasion,  m ssu's are s e l e c t e d by srswor from each of the sample psu's, r t s u ' s are  s e l e c t e d by srswor from each of the sample ssu's, and so on u n t i l  a random sample u o f the u l t i m a t e u n i t s i s o b t a i n e d by srswor from each of  the nmr . . . p e n u l t i m a t e u n i t s .  of  s i z e np (0 < p < 1) of the n psu's  together with i t s respective  A random sample ( s e l e c t e d by srswor) i s r e t a i n e d f o r the second o c c a s i o n  s u b - u n i t s drawn from the f i r s t  occasion.  In  a d d i t i o n , a random sample of s i z e ns ( s > 0) of the N-n o t h e r psu's  is  s e l e c t e d by srswor f o r i n c l u s i o n  Sub-units are s e l e c t e d occasion. numbers.  It w i l l  i n the sample on the second  occasion.  from each of the ns psu's by srswor as on the f i r s t  be assumed t h a t N, M, T,  W are i n f i n i t e l y  large  O b s e r v a t i o n s are made i n each of the mnr ... u u l t i m a t e  units  on o c c a s i o n 1 and i n mnr ... u(p + s ) u l t i m a t e u n i t s on o c c a s i o n 2. The o b s e r v a t i o n s w i l l be d e s i g n a t e d as f o l l o w s :  44  Occasion  No. unmatched psu's  nq  ns  No. matched psu's  np  np  No. unmatched  nmq  nms  nmp  nmp  No. matched  ssu's  ssu's  No. unmatched ultimate units  us  nm  uq  No. matched ultimate units  nm  Unmatched observations  x' . .. ( i = l , 2 , . . . ,nq) y' I.j k.,. . .w ( i = l , 2 , . . . ,ns) ijk...w  Matched  up  up  1  x". .. ( i = l , 2 , . . . ,np) 1jk.. .w  observations  y". (i=l,2,...,np) I j k . . .w J  where, f o r matched or unmatched o b s e r v a t i o n s , j = 1,2,...,m k  =  l,2,...,r  w = 1,2 ,... , u and  q = 1 - p. Assuming a l i n e a r h - f o l d nested model f o r the o b s e r v a t i o n s at each  o c c a s i o n (matched or unmatched), and g i v e n s i m i l a r assumptions at each stage of the h-stage design and a c r o s s the two o c c a s i o n s , as was i n the two- and t h r e e - s t a g e  d e s i g n s , we can d e f i n e the means of the obser-  v a t i o n s as nq -  ( Z  m  r  Z  Z  u ...  i = l j = l k=l np (  Z  done  m Z  r Z  i = l j = l k=l  x'...  Z  )/nmr  ... uq  w=l u Z  w=l  x". .. )/nmr I l k . . .w  ... up  45 ns =  (  y.  r  u E  E  =  (  np  m  r  E  E  E  i=l  y  w=l  1 = 1 j = l k=l  -  and  m  E  ijk...w  ) /nmr ... us  y". ) /nmr i j k . . .w  E  j = l k=l  up  The v a r i a n c e s of the above means are as f o l l o w s : Var(x'. .  ) = (a  2  Var(x"..  ) = {a  2  Var(y'  ) = {a  2  + (a  «i  + (a a  + (a  /m) +  2  £ I  2  T  /m) + ( a  2  E I 2  e2  2  /mr) + ... + ( ° _ /mr...u))/nq i /mr) + ... +  2  /m) + ( a  Yi  /mr) + ... +  2  Y  2  and Var(y"..  ) = {a  + (a  2  a  2  /m) + ( a  2  E  2  Y  + (a  /mr) +  2  2  2  T,  /mr...u)}/np  Further, c o v ( x " . . .....y"  ) = B,/p = {(mr...up,a a ) + (r...up,a a ) h 'c^cx^ ^ 1 ^ 2 + ... + (p,a a )} / (nmr. .. up ) . h Tj T 2  and  cov(x' ....... ,x"  ) = 0  cov(x'  ,y '  ) = 0,  cov(y'  ,y"  ) = 0.  E s t i m a t o r of the c u r r e n t mean. i n the two- and t h r e e - s t a g e d e s i g n s ,  Using  s i m i l a r assumptions g i v e n  the BLUE of the c u r r e n t mean u f o r Y  the h-stage d e s i g n i s g i v e n by ^h  ^  =  P  ^h  +  s  (  " ^  1  re  h  ^  '  ^  P  +  s  (  1  " ^h^ q  where 7  hl  3  h2 *h  =  B  (o  2  (a  2  h  t*i  / / 6  /n) + ( a  2  /n) + ( a  2  B  h Y X  2  /nm) + ... + [ a  2  T  9  + B  = B /6 . h  /nm) + ... + [ a  hl h2  = y". and  £i  h l  h y x  (x  - x'  i  /(nmr...u)] /(nmr...u)]  46 The v a r i a n c e of t h i s linear  e s t i m a t o r , which i s the minimum p o s s i b l e f o r such  e s t i m a t o r s , i s g i v e n as ) = 9 ^ ( 1 - qip*}/[p  Var(y  + s ( l - q*J)]-  I  If  q = s, then y  = (P y  h  + qd  h  il and  - q^)y'  }/[P -  (q* ) ] 2  h  re  Var(y  ) = 9^(1 - qip*)/[l -  h  (q^) ]. 2  I  In a d d i t i o n , i f 9,, = 9, „ = 9, , then ni hi n Var(9  ) = 9 (l - q^)/[l  h  -  h  (q1> ) ]. 2  h  I  E s t i m a t o r of the change. two o c c a s i o n s  i n means on the  i s g i v e n by  = {(p/Vh y  §h  The BLUE of the change  H  +  [  s  (  1  " ^h q  ) / K  h  ] y  '  }  "  {  [  p  (  s  +  p ) / K  h  ] x  h  re  + Jlqtp + s ( l - *  where  3  re  hXY  =  and  6  h  / 6  2 h  )]/K Ix'.......} h  h2  = p + s(l-q ^ ) The v a r i a n c e of t h i s e s t i m a t o r , which i s the minimum p o s s i b l e f o r  such l i n e a r Var(g  e s t i m a t o r s , i s g i v e n by  ) = {[p  h  s(l- * )]9  (1 - q * ) e  2  +  h l  2  +  h 2  I  ~  2  PV hl h2 e  6  }  /  K  h-  I f q = s, t h e n = { /[i - (q* ) ]>(y  - x  2  §  h  P  h  il  h  re  ) + {q(l - q^ ) ]}(y'  - x'  2  h  h  re  and Var(g  ) = {[1 -  q* «] h  (9^  +  8^) - 2 p  V  l ^ } / [ l  -  (q* )«]. h  47 In  6  and  addition,  hYX  =  B  Var(g  =  6, „ = nz  6,, n  then  hXY ) =  h  i f 6,, hi  29  (1 -  q^  - p* )/[l  -  h  (q*.) ]. 2  I Unequal  S i z e Sampling  In at  each  the  previous  stage  were  populations,  such  encountered. would  be  to  as  I f the  become e q u a l  already  developed  at  each  in  which  the  vary  are  now  SPR  two  occasions.  will  not  o f M\ and  N-n  used  w i t h i n each i n size  e s t i m a t o r s must  in size.  (The  are  (selected  cases  units  second  occasion.  other  psu's  second  occasion.  the  psu's.  n psu's by  ' In  In  in size  the  this  be  developed of  the  that vary  are f r e q u e n t l y analysis  units  within a  the  Often, the  extensive  of  case,  stratum. between  units  i n sampling  method  that  formulae  however,  sampling to handle  units the  c u r r e n t mean and  in size  for three-stage  size  i s selected Again, be  by  m^  of N psu's  selected from np  together with  addition,  It will  are  srswor  srswor) of  occasion  one  sampling  and  in  case the  two-stage  multistage  designs  here.)  selected by  so  Estimators  Suppose  ssu's  say,  n e a r l y so.  ssu's.  the  ns  or  the  that vary  greatly,  size,  a population consisting  nu  first  by  that  However,  units  vary  d e r i v e d f o r sampling  considered  size.  Consider  sample for  be  not  differences  Separate  units  do  units  c o u l d be  assumed  sampling  in size,  substantial  stage.  physical  sizes the  change on  equal  forests,  stratify  exist  s e c t i o n s , i t was  of  stratum  there  Units  ssu's  assumed  are  the  srswor  the on  i t h sample  (0 < p  <  1)  of  i t h psu the  the  sample  of  size  for inclusion selected  that N  and  M.  by  ns  i n the srswor  consisting  first  psu.  i t s r e s p e c t i v e ssu's  a random srswor  by  and  A  occasion  random  n psu's  i s retained  drawn f r o m  the  ( s > 0)  of  the  sample  on  the  from  i=l,2,...,N  the  are  i t h psu  infinitely  of  48 large. m  '  Let =  t i  t  n  number o f u n m a t c h e d s s u ' s  e  occasion and  m"  (Note  (t =  1,2),  = t h e number o f m a t c h e d s s u ' s (t  =  1,2).  t h a t m"  .  =  m"„.  li  on t h e i t h p s u on t h e t t h o c c a s i o n  by m a t c h i n g . )  2i  Observations  on t h e i t h p s u on t h e t t h  .  are taken  "  on t h e  n Z  m, . s s u ' s  1=1  and  n(p+s) Z m„. i =l  on t h e  2  will  be.designated  ssu's  on t h e f i r s t  occasion  U  on t h e s e c o n d o c c a s i o n .  The  observations  1  as  follows: Occasion  Unmatched  observations  Matched o b s e r v a t i o n s  It  will  x'.. ij  {^ }' ' *** ' ? j=l,2,...,m'  x".. ij  {  2  n<  u  np  be a s s u m e d t h a t on t h e f i r s t  y" . . i j  i j  =  °li  +  +  e  n S  i= {  ^  l(i)j  =  } o ,  J — i, Z,  , , , , ,  2 i  J . 2 , ... ,np j = l , 2 , . . . ,m"  o c c a s i o n , the o b s e r v a t i o n s  o r u n m a t c h e d ) a r e d e s c r i b e d by t h e l i n e a r n e s t e d  X  {* } ' ? ' " " * ' j=l,2,...,m'  J  J ' \ **" " ' , j = l , 2 , . . . .m"^ 1=  y' .. i j  2i  (matched  model S  •.•,  where u , X v  The  a,., and e . s . a r e as d e f i n e d l i l ( i ) j w  same c o r r e l a t i o n  stage  SPR w i l l  the  equal  case  the  observations  i n the equal  o f t h e t w o - s t a g e SPR w i l l 2.  case  also  of the two-  on o c c a s i o n  on t h e s e c o n d o c c a s i o n w i l l  m o d e l and t h e same c o r r e l a t i o n  on o c c a s i o n  size  f o r the observations  the o b s e r v a t i o n s  a l i n e a r nested size  assumed  be a s s u m e d h e r e ,  Similarly, by  structure  earlier.  structure  be  1. described  assumed i n  be a s s u m e d h e r e , f o r  49 Furthermore, size  case,  will  t h e same c o r r e l a t i o n  be a s s u m e d  s t r u c t u r e , as t h a t  f o r the o b s e r v a t i o n s  f o r the equal  on t h e f i r s t  and second  occasions. From mators  t h e sample  observations  we  obtain  unweighted  preliminary  esti-  as f o l l o w s : nq x' . . = ( Z  .  m'  Z  .  np =  =  ( £  .  .  y"..  =  expected  E ( x ' . . ) = E{( Z i=l nq =  Z  Z  Z j= l  = E { u + ( Z  p  v  = u . x  Z  m"  .  Z  m'  estimators  + E[( Z  a r e as f o l l o w s :  }  i=l  J  nq (v. + a *  + e. . * , ) ] /  1  U  ,  Z  1  1  1  1  } 1  m ' . + ( Z  1=1 /  nq  1  i=l  m'..a.)/ Z 1  1  nq m ^  1  1  1=1  Z  e  .)/  j=l  m. 1  + ( Z i=l  Z j=l  i  m  i=l  nq m ' j ,  nq  1  m'  i=l  J  nq m'.a.)/Z  1=1  i=l  X  . 2i  i=l  o f the above  nq =  y"..)/  -'..)/ 1  m'  np  j=l  nq  X  Z  . , i=l  m^.  E{[ Z i=l  ij  ns  2  values  j=l  . 2i Z y'..)/  m" .  i=l The  x". . ) / Z m". . IJ . . . . . l i i=l  j=l  ( Z  l i  np  . ,  i=l  np  J  li  ( Z Z ...... i=l j =l m'  m', .  . , i=l  ij  j=l m"  ns y'..  x' . . )/ Z  . ,  i=l  x"..  nq  u  < U  nq c....)/  Z  i=l  m' 1  .] 1  }  50 Similarly,  i t c a n be shown t h a t E(x"..) = u  }  E(y'..) = and We c a n d e f i n e  E(y"..) =  the variances  nq  = a  2  a, 1  = a  2  = a  2  m'a. 1 1  1  +  li V Z Z i = l jJ = l  nq Z m' i=l  1=1  nq  nq  c z,  •(i)j  nq ... Z. . .m"  ( Z tn" . ) / ( . , i i i=l 2  nq  2  . . i=l  m'  2  n  2  a  1=1  1  nq nq ( Z m' . ) / ( Z m< . ) + a a, . . 1l . , 1l 1=1 i=l 2  2  (nqm* )/(nqmj)  2  2  n  2  +a _  nq  I  2  E 1,  i  1 nq  nq  »  m  {  (mp . 2  0  2  a i  1 + —  {u[ a''  a  a  e  2  2  >  }  }  mi  where nq m'j = ( Z m' i=l  m  and  , 2  TT [  )/nq  li  nq = (* Z m' i=l  2  . )/nq  _ -m ]/(m' ) = T  2  1  nq n q ( Z m' i=l  2  nq ) / ( Z m' i=l  ) 1 1  .  Z  .  i=l  /nqm'i  2  1  m'..! l i  nq nq ( Z m' . ) / ( Z m'..) e, . , l i . . l i i=l i=l  2  1  = a  nq  u  . ) + ( Z Z a )/( Z l i . . . e, . , i=l j=l i=l  2  1=1  m'  2  nq nq ( Z m ) / ( Z m' . ) + a , . , . ! ! .-. . li • 1  as f o l l o w s :  nq m'  I i=l  V a r ( x ' . . ) = Var  of the p r e l i m i n a r y estimators  :  m'. . 1l  :  51 Similarly, 1  Var(x"..)  o  •CTI V  np  J_  {TT  ns  a  2  _1_  +  2  «i  " J_  a  2  a  2  a  2  S  2  +  ™2  _1_ np  a  {TT' ' 2  2  a  J_  +  my  2  where TT", = n p (  np 2 m" i=l  np . ) / ( £ m" 1=1  )•  2  ns  ns  i r ' = ns( Z m' . *.)/( z ' m  2  i = l.  „»  )2  a n d  2i  i=l  np np = np( 2 m " . ) / ( S m" .) i=l i=l 2  2  2  (Note  that  Further,  2  2  ny = TTy = IT" s a y , and m"  i t c a n be e a s i l y  c o v ( x " . . ,y". . ) = —  shown  [TT"P,O-  = my  that a  2  2  = m"  in this + —  say.) case of unequal  p a  a  2  size  units  ]  cov(x'..,x"..) = 0 cov(x'..,y'..) = C and  c o v ( y ' . . , y " . . ) = 0. Estimator  occasion  o f t h e c u r r e n t mean.  i s e s t i m a t e d by a l i n e a r  as b e f o r e [ s e e e q u a t i o n ( 1 ) ] . y  The c u r r e n t mean p  on t h e  y  combination of the p r e l i m i n a r y estimators  The u n b i a s e d n e s s  requirement  leads to  = a ( x ' - x") + c y " + (1 - c ) y '  2 j l  2  second  2  (18.1)  2  Then V a r ( y . ) = a [ V a r ( x " . . ) + V a r ( x ' . . ) ] + c V a r ( y " . , ) + (1 2  2  2  - 2a c cov(x"..,y"..). 2  2  c2) Var(y'..) 2  52 If we l e t Var(x"..) =  kh*\> o p n * o,  Var(x-..)  mi  +  4a'  o*  Var(y"..) =  o  —[—(irrt P n 2  ±  2  +— -  2  a  2  Var(y'..) = ^ ( r r ' o s n 2 a  o  -  +  ) ] = i 6",  ) ] = - 9" p  2  E  2  ) ] = i 6«  2  E.  m  z  2  ))=ie-,  ( i  o  2  o  E,  2  '(x"..,y"..)  = -[-(ir"p,a a P n a, a  p  + — p a a ) ] = ~~ & -„ e £2 2  2  t  1  Then Var(y , ) = a [±9'\ + ± 6 \ ] + c ( i 6 " ) + (1 - c ) ( ^ 6' ) :  2  2  2  2j  2  3  We now choose the v a l u e s o f a  2  2  C  2  2  2  p  and c  2  that minimize  Var(y  2  ) as b e f o r e .  We o b t a i n that a*  = c* qB'/(p9'  2  2  I  +  qQ'\)  e' p(pe'!+ qe"!) 2  and (pe' + seyXpe'i-f 2  These v a l u e s of a * and c * 2  qey) - pqsB'  2  can then be s u b s t i t u t e d i n t o  2  equation  to g i v e the BLUE of the c u r r e n t mean f o r the case of unequal units.  size  (18.1) sampling  The v a r i a n c e of the e s t i m a t o r so o b t a i n e d , which i s the minimum  p o s s i b l e f o r such  l i n e a r e s t i m a t o r s i s g i v e n as 9 [9' '(p9| + q y ) - q p B ] ( p 9 ; + qey) , 2  2  Var(y  2  8  )= s[p9{  + q9'/][(p9  + se' ')(p0; + qe*,') - p q s B ' ] 2  2  2  e'p(e; +'qey) 2  = (e /s) { l 2  s[(p9 The  special  P  • + seyHpe; + qey) - p q s e ]  }  12  2  case i n which the sampling  u n i t s w i t h i n stages a r e o f  equal s i z e , c o n s i d e r e d i n the e a r l i e r s e c t i o n , can be o b t a i n e d from  this  53 g e n e r a l r e s u l t by s u b s t i t u t i n g the f o l l o w i n g e q u i v a l e n t  9" 6  and  0  =  9*2  =  2  '1  2 2  6  =  2  2  21  6  9"!  = 92  6'  = B.  2  i -  2  E s t i m a t o r of the change. case of unequal  size  sampling  as b e f o r e [ e q u a t i o n ( 7 ) ] . f  t h a t minimize  Similarly,  In t h i s case, however, the v a l u e s of e 2 o  e* =; {•He j(q8"i + p e ' i ) [ ( s e " ,  a  S2p2q(q9"  {[ICpg 1 !  + q9" )( 9' 1  s p3q(q9" 2  f"  2  2  and  2  f*  1  P  2  2  > respectively,  and  2  are  + p9'2)(qe'\ + pe\) - q s B ' 2 ] 7 P  + p 9 \ ) l ] - [(B'e'^pS'j  1  to estimate change i n t h i s  u n i t s , we combine the p r e l i m i n a r y e s t i m a t o r s  the V a r ( g ) , e *  :  2  forms:  + S9" )[(s9" 2  - qB' 9'2)/p qs]}/  + s9" )  2  2  + 9J )(q9"i P  2  2  2  + p6\)  - pqsB' ]]]/ 2  i)}  + p9'  = - [ e ' j ( p e ' 2 + s 9 " ) + q B ' e ^ C q e " , + pe\)/ 2  q[(s0"  2  + 9' )(q9" P  2  1  + p6'  x  -  qspB' )]. 2  These v a l u e s can then be s u b s t i t u t e d i n t o equation of change f o r the unequal  s i z e , sampling  (7) to o b t a i n the BLUE  u n i t s case.  The v a r i a n c e of the  e s t i m a t o r so o b t a i n e d , the minimum p o s s i b l e f o r such l i n e a r e s t i m a t o r s , i s s i m i l a r l y o b t a i n e d by s u b s t i t u t i n g Other  e*  2  and  f*  2  into equation (9).  Estimators The  d i s c u s s i o n so f a r has been c o n f i n e d to b e s t , l i n e a r and  estimators.  These were o b t a i n e d by  combining  unbiased  a r e g r e s s i o n estimate  the matched p o r t i o n of the sample w i t h a mean per u n i t e s t i m a t e based the c u r r e n t sample.  We  (equal-sized  u n i t s at each stage) u s i n g another  sampling  r a t i o e s t i m a t o r (RE).  s h a l l now  d e r i v e the theory of two-stage  Assume that sampling  from on  SPR  e s t i m a t o r , the  i s done as was  done e a r l i e r  54 for  the two-stage e q u a l - s i z e d Current  an  mean.  Using  improved e s t i m a t o r  sampling u n i t s  a double  y"^ o f y"  case.  sampling r a t i o  estimate,  we c a n o b t a  an f o l l o w s :  = ( y " . • / x " . . ) x . . = R x..  r  where x..  = p x".. + q x'.. = o v e r a l l  We c a n r e w r i t e y"  r  "r  = ^  u  x"..  the usual  expected  -  U  x"..  ) X  the piece  ratio  estimator  p  x  of  and t h a t  (y"../x"..)(x.. is  ^  Z_LL (x..  +  X  (y"../x"..) is  -  +  = £-11  that  occasion.  as f o l l o w s : y  We n o t i c e  s a m p l e mean on t h e f i r s t  t o be v e r y  small  the' q u a n t i t y  - p ) x  (negligible).  T h e n we c a n w r i t e y" Consequently, using Var(y" ) r  = ^  r  = (y"../x"..)x.. -  2 2  v  A  the reasoning (6 -  (y"../x"..) u .  q[ 2 R i | ; / e 2  2  o f C o c h r a n ( 1 9 7 7 : 3 4 3 ) we o b t a i n 1  9  2  2  -  that  R'"6 ]} 2 1  where R = u /u ( e s t i m a t e d Y X An e s t i m a t o r is  given  by R = y " . . / x " . . )  y ^_ o f t h e p o p u l a t i o n 2  mean u  y  on t h e s e c o n d  occasion  b y t a k i n g a w e i g h t e d a v e r a g e o f y " ^ a n d y'.. a s f o l l o w s : y*  r  = w y"  r  where w a n d ( 1 - w) a r e w e i g h t s .  + (1 - w)y'..  (19)  55  The of  minimum-variance e s t i m a t o r of y  w which  with  minimizes  y'..  S i n c e y"^  2  ( g i v e n the assumed c o r r e l a t i o n  Var(y  i s statistically uncorrelated  structure),  then  ) = w V a r ( y " ) + (1 - w ) V a r ( y ' . . ) 2  2 r  (20)  2  r  Differentiating ing  the V a r ( y ^ ) .  i s o b t a i n e d by h a v i n g t h a t v a l u e  y  ( 2 0 ) w . r . t . w,  setting  the r e s u l t s  e q u a l t o z e r o , and  for w gives ' w*  solv-  '  = Var(y'..)/[Var(y" ) + Var(y'..)]  (21)  where w"  i s the v a l u e of w which minimizes  By  substituting  w*  = (6  /s)/H  2 2  = p/{p  i n the v a r i a n c e s i n t o {( 9  - q[2Rip ^9 e  2 2  2  + s [ l - q(2^ A 2  2 l  Var(y ^_). 2  e q u a t i o n ( 2 1 ) we  - R 9 2  2 2  2 I  ])/p}+  obtain that  ( 6 /s)l| 2 2  A )]} 2  where  = the r a t i o of the p o p u l a t i o n c o e f f i c i e n t of v a r i a t i o n  of the  a g e s o v e r t h e s s u ' s on o c c a s i o n 1 t o t h e c o e f f i c i e n t o f of We y  2 r  and  the averages  variation  t h e s s u ' s on o c c a s i o n 2.  then o b t a i n t h a t = <P y "  + s [ l - q(2+ A - A ) ] y ' . . } / { p 2  r  2  + s [ l - q(2* A - A )]} _ 2  2  that  Var(y  2 r  )  = 9 [ l - q( 2i|> A- . A ) ] / {p + s [ l - q ( 2 * A 2  2 2  = 9 We 1.  over  2 2  2  2  A )]} 2  w*)/s}.  {(1 -  c a n o b t a i n some s p e c i a l  c a s e s as  follows:  If q = s ? 2  r  - {p y "  + q [ l - q(2* A - A )]y'..}/[1 - q (2* A 2  r  2  2  2  and Var(y  2 r  )  - 9 [ l - q ( 2 ^ A - A ) ] / { 1 - q (2i|; A 2  2 2  2  2  2  A )}. 2  aver-  A )] 2  56 2.  I f q. = s and y  and  = 9-  2 1  = <P y "  2 r  Var(y _)  2  = 8  2  2  .+ q [ l - q R ( 2 ^  r  = 6 [ l - qR(2*  2 ]  Again,  6  2  here too,  - R)]y'..}/[1  2  - R)]/U  2  i f i> ,  < 6  a replacement  improve the  2 ]  As  was  policy will p o i n t e d out  t h e amount o f t h i s y _ the  expected E(y  2 r  according  We  can  y"..  E(y  2 r  )  y  be  2 ]  B(y We tice,  see  x  given  A = 1,  y  = 530  m / h a , and 3  2 1  - y ]  x  y  /py  -  2 r  - R^ /9 2  x  as +  y'..  +  y  +  E(y'..)  y  2 x  ) - (i|i /9 2  6 )/ y y  2 x  2 2  x  p]  y  2 1  9  2 2  /  y  y  p] +  R(9  y . y  y ) y  2 1  x  /y  2  -  y  2  * /6 2  2 1  9  2 2  y  -  y  y  y  /y )/p y  for w 9  - * / 9 , 9 ] / { (s + p - q s [ 2 ^ A - A ] ) y 2  2l  2  t h a t f o r l a r g e n,  q = 0.7,  determine  by  2 1  ) » q[R  r  i n (19)  shall  (1967:304)  the b i a s i s i n s i g n i f i c a n t  if v  can  2  we  (wq[R6 /y ) - R^ /9,9 /u ]/p + y }  = wq o r s u b s t i t u t i n g w*  Now  0.  T h i s means t h a t  B(y .) = E ( y =  4  2  quantity  2 1  2  i f i>  y  i s biased.  + q E(y"../x"..)y  y  values  of y ,  2]  * w q[R9 /p y  T h e n b i a s o f y ^_ w i l l  f o r a l l other  + q ( y " . . / x " . . ) x ' . . - y'..]  y  1/n.  and  2  + q ( y " . . / x " . . ) x ' .. - y'..}  ) = w E[p  terms of o r d e r  R)>  estimate  rewrite y _  v a l u e of t h i s  t o Murphy  -  R)]  This also indicates that  E(y"../x"..) = ( y / y ) [ l + ( 9 to  = 9  2  2  R > 0.  2  y"..  = w[p but,  2  2  earlier, y ^  bias.  = w{p  2j  Taking  2  2  -  2  - q R(2<f  i f q = 0 o r q = 1, V a r ( y ^ _ )  o f q, 0 < q < 1, V a r ( y _ )  - q R(2^  s = 0.4, y'..  2  the  b i a s becomes n e g l i g i b l e  even f o r moderate samples.  R = 1.1,  = 550  2  i>  z  =  m /ha, B ( y  .9, 6 , = 9 2  3  2  2  2  ( For  2  }.  and,  = 1, n =  ) = 0.00277 f o r y  A  =  i n pracexample,  30, 552.50302.)  57 C o m p a r i s o n o f t h e c u r r e n t mean e s t i m a t o r s . . compare t h e r e l a t i v e e f f i c i e n c i e s  I t w o u l d be u s e f u l t o  o f t h e BLUE a n d t h e RE i n e s t i m a t i n g  t h e c u r r e n t mean on t h e s e c o n d o c c a s i o n  i n a two-stage  The g a i n i n p r e c i s i o n ( e f f i c i e n c y ) o f y ^ o v e r  y _ i s given  2  Q  = [Var(y  2 r  )  - Var(y  ) ]/Var(y  2 j l  ([p + s ( l - g ^ ) ] H i  =  - (q/e  2  (LTP  + s [ l - (q/6  = H{(n  - qs*|)[l  2 j t  2 2  )( 2 R ^ / 6  2 2  2  2j  by  ).  )[2R» /'e  2 1  design.  2  e  2 1  e  - R e 2  2 2  ]}  _  21  - q(2^ A - A )]}/{[fi - qs(2* A - A ) ] ( l 2  2  2  l  q*l)  - R 9 )TJj(l 2  2 2  2 1  2  -  qi>l)}J\-l  where .fi =' s + p = r a t i o We  o f sample s i z e  on o c c a s i o n  2 t o t h a t on o c c a s i o n  c a n f u r t h e r r e w r i t e Q as f o l l o w s :  Q = EE  - qs(p,4>  -C  + P * ) ][l 2  1  { [ f i - qs(2A(p <t> 1  2  2  - q(2A(p «D I  + p 4> ) - A ) ] ( l 2  1  2  2  L  +  P <t> ) - A ) ] } / 2  2  2  - q(Pi«t>,„.+ P <t> )f[] - 1 2  2  where 4> =a 1  and  a  = a  2  We p,,  p,  e  1  a  a  G  /(n/9  a  2 2  )  /(nm/9 ,9 ) 2  2 2  tabulate the values  o f Q f o r each v a l u e  o f fi, A,  T h i s h a s b e e n done f o r some s e l e c t e d v a l u e s  2  2  9  2  <)>!, a n d <t>.  2  2 1  2  c a n now t h e n  A, P i , P , Oj  aj  <t>i, a n d <t>, a n d a r e g i v e n 2  i n Table  I .  o f fi,  Note t h a t i n T a b l e I ,  = <(> = 0.5. 2  From T a b l e  I we c a n make t h e f o l l o w i n g o b s e r v a t i o n s :  1.  f o r f i x e d fi, t h e e f f i c i e n c y g a i n  2.  for fixed  3.  as A t e n d s t o p  4.  efficiency  i n c r e a s e s as p  5.  f o r values  of p  A, t h e e f f i c i e n c y l5  t  i n c r e a s e s as A i n c r e a s e s ,  gain decreases  as fi i n c r e a s e s ,  efficiency declines,  less  x  than  increases, 0.9, e f f i c i e n c y  i s highest  f o r p = .4 o r  1  TABLE  RH1 RH2 0.6 0.6 0.6 0.6  0.6 0.7 0.8 0.9  RH1 RH2  PERCENT  0.2  0MEGA=O.75 P 0.3 0.4 0.5  0.6  0.4 0.9 1 .9 3.4  0.4 1.1 2.1 3.7  0.4 0.9 1.7 2.7  0. 2  OMEGA=0.75 P 0.4 0.5 0. 3  0. 0. 0. 0.  RH1 RH2  I:  8 4 1 0  1 .0 0. 5 0. 1 O. 0  0.5 1.1 2.1 3.5  0.4 1 .0 1 .9 3.2  1.0 0.5 O.1  -o.o  GAIN  0. 6  1.0 0 . 0.5 0. 0.1 0 . - O . O- 0 .  9 4 1 0  IN E F F I C I E N C Y  0.6  0.3 0.7 1 .5 2.7  0.3 0.8 1.6 2.9  0.3 0.7 1.3 2.1  0. 2  OMEGA=1.00 P 0.4 0.5 0. 3  0. O. 0. -O.  6 0. 3 0. 1 0. 0 -0.  O. 2  5. ,4 4 .8 4. 1 3..4  6..5 5..6 4 .6 3 .7  6. 1 4 .8 3.. 7 2 .7  4. 1 3. 7 3..2 2..7  5 .0 , 4 .3 . 3,.6 2 .9  0 .2  0 .2  P  0  .3  15 .4 14 .7 14 . 0 13 .3  0.4 16.9 15.7 14.5 13.2  0.5 17.0 15.4 13.8 12.2  OMEGA=0.75 P 0.4 0.5 0 .3 25.5 25.5 25.7 26.0  29.4 28.6 27.8 27.2  30.6 29.1 27.6 26.1  DELTA=Rje„/e ,;.OMEGA = S+P. i  5.3 4.4 3.6 2.8  5.1 4.2 3.3 2.5  OMEGA=1 . 0 0 p 0 .2  0 .6  9 9 9 9  15 .7 13 . 9 12 . 2 10 .6  .1 .1 .0 .1  0 .6  0 .2  29.2 27.2 25.2 23.2  14.1 14.6 15.4 16.4  RH1= (  0 .3  0.4  0.5  11 .5 1 2 . 6 1 2 . 7 1 .11 1 1 . 81 1 . 6 10 .6 1 1 . 0 1 0 . 5 10 .2 1 0 . 2 9.4 OMEGA=1 , 0 0 P OA 0.4 0 .3 18.6 18.8 19.1 19.5  . RH2=  21.3 21.0 20.6 20.4 £  22.2 21.3 20.4 19.6  y ^ 2  0.50 0.2  OMEGA= 1 .25 P 0.3 0.4 0.5  0.6  0.2 0.6 1.2 2.2  0.3 0.3 0.7 0.7 1.3.1.3 2.4 2.3  0.2 0.6 1.1 1.7  DELTA= 0.6  0.3 0.6 1.2 2.0  0.2  0MEGAM.5O P 0.3 0.4 0.5  0.6  0.2 0.5 1.0 1.9  0.2 0.6 11 2.0  0.2 0.5 1 0 1.7  0.2 0.5 0.9 1.5  0.2  OMEGA=1.50 P 0.3 0.4 0.5  0.6  0.4 0.2 0.1 O.O  0.5 0.5 0.3 0.3 0.1 0.1 0.0 -0.0  0.5 0.2 0.1 0.0  0.5 0.2 0.1 0.0  0.2  OMEGA=1.50 P 0.3 0.4 0.5  0.6  2.8 2.5 2.2 1.9  3.4 2.9 2.5 2.0  3.5 2.9 2.3 1.7  3.1 2.5 2.0 1.5  0.2  OMEGA=1.50 P 0.3 0.4 0.5  0.6  6.1 6.1 6.1 6.2  7.6 7.4 7.2 7.0  8.4 7.8 7.1 6.5  7.8 7.0 6.3 5.6  0.2  0MEGAM.5O P 0.3 0.4 0.5  0.6  0.75 0. 2  0.7. 8 0.8 0.7 4 0.4 0.4 0.3 1 0.1 0 . 1 0 . 1 0 - 0 . 0 -O.O - 0 . 0  0. 6  19.2 19.8 20.6 21.8 NOTE:  0.3 0.8 1.5 2.5  0. 2  12 .2 12 . 0 1 .9 1 1 1.9  RH1 RH2  0.4 0.8 1.6 2.8  OMEGA=1 . 0 0 P 0. 3 0.4 0.5  6.7 5.4 4.3 3.2  DELTA=  0.2  OMEGA=0.75 P 0.4 0.5 0 . ,3 6.9 5.7 4.6 3.5  OF J2-^ OVER  OMEGA=1.00 P 0.3 0.4 0.5  OMEGA=0.75 RH1 RH2  (0%)  0. 0. 0. 0.  5  OMEGAM.25 P 0. 3 0.4 0. 6  3 0. 3 1 0. 1 O -0. 0 1 .0 0  DELTA  0.6  OMEGA=  0.6  0. 2  0..3  0.4  4.6 3.7 2.9 2. 1  3,,3  4 .0 ,  4.3  3.,5 2 .9 2 .4  3.6 2.9 2.3  3..0 2 .6 2 .2  DELTA =  1 ,25 . 0 .2  0 .3  11.7 10.5 9.3 8. 1  7 .3  9 .2  7 .3 7 .3 7 .4  DELTA =  8 .9 8 .6 8 .3  1 .50  0.6  0 .2  21 .2 19.9 18.7 17.4  11.2 11.6 12.2 13.1  I  OMEGA=  0.6  0.6 0.5 0.3 0.3 0. 1 0. 1 0 . 0 -0.0  0.3 0. 1 0.0  0.4  4.2 3.4 2.7 2.0  3.7 3.0 2.3 1 .7  i  10. 1 10. 1 9.5 9.3 8.9 8.3  9.4 8.4 7.5 6.6  OMEGA=  0 .3  0.4  14.7 14.9 15.2 15.6  16.8 16.6 16.4 16.3  17.4 16.8 16.2 15.7  16.6 15.7 14.8 13.9  9.2 9.6 10.2 10.9  12.1 12.3 12.6 13.0  0.2 0.6 1.1 1.9  3.6 3.0 2.5 1.9  8.4 7.9 7.5 7.0  13.8 13.7 13.6 13.6  14.3 13.9 13.5 13.0  13.7 13.0 12.3 11.6  00  TABLE I: PERCENT GAIN IN EFFICIENCY (0%) OF  OVER  DELTA=  RH1 RH2  O. 2  OMEGA=0 .75 P 0. 3 0.4 0.5  0.7 0.7 0.7 0.7  0. 9 1 . 9 3. 4 5 .8  1 .1 2. 1 3 .7 5. 9  RH1 RH2  0. 2  OMEGA=0 . 75 P 0. 3 0.4 0.5  0.7 0.7 0.7 0.7  0. 4 0. 1 0. 0 O. 2  0. 5 0.5 0. 1 O. 1 0. 0 -0.0 0. 2 0.2  O. 6 0.7 0.8 0.9 •  1.1 2.1 3.5 5.5  1 .0 1 .9 3.2 4.9  0.50 0. 2  O. 3  0.4  0.5  0. 6  0. 2  OMEGA =1 .50 P 0. 3 0.4 0.5  0. 6 1 . 2 2. 2 3. 7  0. 7 3 1 . 2. 4 3. 9  0.7 1 .3 2.3 3.6  0.6 1 .2 2.0 3.2  0. 6 1 .1 1 . 7 2 .6  0. 5 1 . 0 1 . 9 3. 2  0. 6 1 .1 2.0 3. 3  OMEGA*1 .00 r  0.5  D  0.6  0. 6  0'.2  0. 3  0.4  0. 9 1 . 7 2. 7 4. 0  0. 7 1 . 5 2. 7 4. 6  0. 8 1 . 6 2. 9 4 .7  0.8 0.8 0.7 1 .6 1 .5 1 .3 2.8 2.5 2. 1 4.4 3.8 3.2  0. 75  DELTA  0.6 O. 7 0.8 0.9  OMEGA=1 .00 0..6  0.5 0..4 0. 1 0,, 1 -0.0 -o. 0 0. 1 0,. 1  0. 3  0. 2  0. 3 0. 4 0. 1 0. 1 -0. 0 -0. C 0. 1 0. 1  r  0.4  0.5  0.6  0.4 0. 1 0.0 0. 1  0.4 0. 1 -0.0 0. 1  0.3. 0. 1 -0.0 0. 1  0. 2  0. 3 0. 3 0. 1 0. 1 0. 0 -o. 0 0.. 1 0.. 1  RH 1 RH2 0.7 0.7 0.7 0.7  4 .8 . 4. 1 3 .4 2 .7  0.6 0.7 0.8 0.9  5..6 4 .6 3 .7 2 .7  5.7 4.6 3.5 2.5  5.4 4.3 3.2 2. 2  1 . 00  4 .8 3 .7 2 .7 1 .8  0. 3  0. 2  f  0.4 . 3 4.4 3..7 4 , 3. 2 3..6 3.6 2.,7 2 .9 2.8 2.. 1 2 .2 2.0  0.5  4.2 3.7 3.3 2.9 2.5 2. 1 1 .7 1 .4  RH'I RH2 0.7 0.7 0.7 0.7  0.6 0.7 0.8 0.9  12 .0 1 .9 1 1 1.9 1 1.9  14 .7 14 .0 13 .3 12 .6  15.7 14.5 13.2 12.0  0 .6  15.4 13 .9 13.8 12 .2 12.2 10 .6 10.7 9 .0  0,. 2 0 .3  0.5  RH1 RH2 0.7 0.7 0.7 0.7  0.6 0.7 0.8 0.9  0 .2  19 . 8 25 .5 20 .6 25 .7 2 1.8 26 .0 23 .5 26 .5  28.6 27.8 27.2 26.6  29 . 1 27 .2 27 .6 25 .2 26. 1 23 . 2 24.6 21 .3  0 .2  0.6  14 .6 15 .4 16 .4 17 .8  NOTE: D.ELTA = Rje /0^.OMEGA = S+P, RH1= ^ 4|  0.5°i  0.6  18 .8 21.0 19 . 1 20.6 19 .5 20.4 20 . 1 20.2  21.3 20.4 19.6 18.7  19.9 18.7 17.4 16.2  RH2 =  0.3 0. 1 0.0 0. 1  0. 3 0. 1 -0. 0 0. 1  0. 2 0. 1 0. 0 0. 1  0. 3 0.3 0. 1 0. 1 0. 0 -O.O 0. 1 0. 1  0. 6  0.2 0. 1 0.0 0. 1  0. 2 0. 1 0. 0 0. 1  0MEGA=1 .50  OMEGA= 1.25  p  0. 2  0..3  0.4  0.5  0. 6  3..0 2 .6 2 .2 1 .8  3,.5 2..9 2 .4 1 .8 .  3.4 3..0 2.7 2 .3 , 2.0 1 .7 1 .4 1 .2  2 .5 . 2 .2 , 1 ,9 , 1 .5 .  2 .9 2,,5 2 .0 1 .5  3.0 2.5 1 .9 1 .4  2.9 2 .3 1 .7 1 .2  2 .5 2. 0 1 .. 5 1 . 0  1 ..25  3.6 2.9 2.3 1 .7  OMEGA=1 .50  OMEGA=1 . 25  p  0 .2  0 .3  0.4  0.5  0 .6  0 .2  0 .3  0.4  0.5  0 .6  7. 3 7 .3 7 .4 7 .5  8 .9 8 .6 8 .3 8 .0  9.5 8.9 8.3 7.7  9.3 8.5 7.7 6.9  8 .4 7 .5 6 .6 5 .7  6. 1 6. 1 6. 2 6 .4  7 .4 7 .2 7 .0 6 .8  7.9 7.5 7.0 6.5  7.8 7. 1 6 .5 5.8  7 .0 6. 3 5.6 4. 9  OMEGA=1 .50 P 0 .3 0.4 0.5  0 .6  13.7 13.6 13.6 13.6  13 .0 12 . 3 1 1. 6 10 .9  1 .50  OMEGA=1 . 25 D  0.4  0 .3  0.3 0. 1 0.0 0. 1  0.5  0..6  OMEGA= 1.00 0 .6  0. 3  o  9 . 1 1 1. 1 11.8 11.6 10.5 9 .0 10 .6 1 1 .010.5 9.3 9 . 1 10 . 2 10.2 9.4 8. 1 9 .2 9 .8 9.4 8.4 7.0  r  0. 2  0.5  DELTA = OMEGA=0 .75 P 0 .3 0.4 0.5  0. 6  0.. 3 0.4  OMEGA=1 .00 0.4  0.5  0. 2  DELTA = OMEGA=0 . 75 P 0 . 2 0 .3 0.4 0.5  0. 5 0. 9 1 . 5 2 .2  P  0.4  0.4  D  0.6  0. 6  OMEGA=1 .50  OMEGA=1 • 25 .  0. 3  OMEGA=1 .00 0 .6  0.6 0.5 1 . 1 1 .O 1 .9 1 . 7 3. 1 2 . 7  D  DELTA = OMEGA=0 .75 P 0.. 2 0..3 0.4 0.5  OMEGA=1 .25  0 .3  0.4  0.5  0 .6  0 .2  11 .6 14 .9 12 .2 15 .2 13 . 1 15 .6 14 .3 16 .2  16.6 16.4 16.3 16.3  16.8 16.2 15.7 15. 1  15 .7 14 .8 13 .9 13 . 1  9 .6 10 .2 10 .9 12 .0  0 .2  12 .3 12 .6 13 .0 13 .6  13.9 13.5 13.0 12.6  Ul VO  TABLE  I:  PERCENT  GAIN  IN  EFFICIENCY  (0%)  OF  DELTA= 0MEGA=0.75 P RH1  RH2  0. 2  0. 3  0.4  0.5  0. 6  0. 2  0.8 0.8  0.6  1 .9 3. 4  2 . 1 3. 7  2. 1 3.5  1 .9 3.2  1 .5 2. 7  0.8  0.8  5. 9  5.5  0.8  0.9  5 .8 9. 6  9. 3  8.3  4.9 7 . 1  1 .7 2. 7 4 .0  0.7  0MEGA=1.00 P 0.5 0.4 0. 3  0.50  0. 2  RH1  RH2  0. 2  0 0 0 0  0.6 0.7 0.8  0. 0. 0. O.  . . . .  8 8 8 8  0.9  0. 1 0. 0 0. 2 0. 8  1 0 2 9  0. - 0 . 0. 0.  1 0 2 7  0. - 0 . O. 0.  1 0 1 6  1.1 1.9 3.1  1.0 1:7 2.7  0.9 1.5 2.2  5.3  5.2  4.7  4 . 0  3.2  =1.50 P 0.4 0 . 5  0.6  1 .2  1 .1  3.8  2.0 3.2  1 .7 2 .6  5.6  4.6  6 .2  6 . 1  5.5  4.7  3 .8  0. - 0 . 0. 0.  1 0 1 5  0. 75  0.6  0. 2  0. 1 - 0 . 0  0. 1  0. 1 0.4  0. 1 0. 6  0. 0  0. 3  0.4  0.5  0. 1  0. 1  0. 1  0.0  0.0  0. 1 0.5  0. 1 0.4  -0. 0 0. 1 0. 5  0. 6  0.2  0. 1  0. 0. O. 0.  -0. 0 0. 1 0. 3  RH1  RH2  0. 2  O.. 3  0.4  0.5  0. 6  0.,2  0.,3  0.4  0.5  0.6  0. 2  0..3  0.4  0.5  0..6  0.2  0.8 0.8  0.6 0.7  4 , 1 3 .4  4 .. 6 3.. 7  4.6  4.3 3 . 2  3..2 2 .. 7  3 .. 6 2. 9  3.6 2.8  3.3 2.5  2. 6  2 .. 9  2.9  2.7  2 . 3  2 . 1  2 . 2  2 .. 4  2.3  2.0  1 .. 7  0.8 0.8  0.8  2 .. 7 1 9  2 .. 7 1 .8  2.5 1 .6  2.2 1 .3  3 ..7 2 . 7 1 .8  2.9  3.5  2.. 1 1 .5  2 ,. 2  2.0 1 . 3  1.7  1 .4  1 .8  1 .8  1 .7  1 .4  1  1 . 1  0.9  1 . 3  1 .2  1 . 1  0.9  0 . 7  0.9  1 . 1  1 .5  DELTA = 0MEGA=O . 75 RH 1  RH2  0 . 2  0 . 3  0.4  0 . 5  0 .6  0.8  0.6 0.7  1 1 .9  14 . 0  12 . 2  12.2  0.8 0.9  1 1.9 12 . 1  13 . 3 12 . 6 1 1. 9  14.5 13.2  13.8  1 1. 9  12.0 10.7  10.7 9.2  10 .6 9 .0 7 .5  0.8 0.8 0.8  0 .2 9 .0 9 . 1 9 .2 9 . 5  RH1  RH2  0 .2  0.8 0.8  0.6  20 .6 21 .8  0.8  0.8  0.8  0.9  NOTE:  0.7  0 .3 25 . 7 26 .0 26 .5 27 .4  23 .5 25 .9  DELTA =R ^ /  e  j  i  2.3  2.0  1.9  1 7  1.5  1.5  1.5  1.4  1.2  1.0  1.1  1.0  0.9  0.8  0.6  0 .6  0.2  OMEGA=1.50 P 0.3 0.4 0.5  0.6  6.1  7.2  7.5  6.2 6.4 6.6  7.0 6.8 6.6  7.0 6.5 6.5 5.8 6 . 0 5 . 1  2  0.5  11.0 10.2  10.5  9.3  7 .3  8 .6  8.9  8.5  7 .5  9.4  8. 1  7 . 4  8 .3  8 . 3  7.7  6 .6  9.4 8.5  8.4  7.0  7 .5  8 .0  7.7  6.9  5 '. 7  7.3  5.9  7 .8  7 .8  7.0  6.0  4 . 9  5.6 4.9 4.2  P  P 0.4  0.5  0 .6  0.2  0.3  0.4  0.5  0.6  15 . 2  16.4  16.2  14 . 8  10.2  12.6  13.6  13.5  12.3  15 . 6  16.3  15.7  13 . 9  10.9  13.0  13.6  13.O  11.6  16.3 16.3  15. 1 14.5  13 . 1 12 . 2  12.0 13.5  13.6  13.6 13.7  12.6 12.2  10.9 10.2  0 .2  0 .3  0.4  0.5  0.6  0 . 2  27.8 27.2  27.6  15 . 4 16 . 4  19 . 1 19 . 5  20.6 20.4  20.4  18.7  12 . 2  19.6  17.4  13 . 1  26.6 26. 1  24.6 23.2  25 .2 23 .2 21 . 3 19 . 5  17 . 8 19 . 8  20 . 1 21 .O  20.2  18.7 17.9  16.2 15.0  14 . 3 16 . 1  16 . 2 17 . 1  RH2 =  7 . 1 6 . 3  OMEGA=1.50  0MEGA=1 . 2 5  0 .6  RH1=  2.5  2.0  0.4  0.5  \ O M E G A =S+P,  2.5  1.9  0 . 3  20. 1  0.3  1 0 1 3  2.2  0 . 2  0.4  26. 1  O. 1 0.4  0. 0. 0. 0.  0.6  0.6  1 .50  0. 1 0.5  0. 1 0 . 0 O. 1  0 . 5  OMEGA=1 . 2 5 p  DELTA =  0.1 - 0 . 0  0.4  1 . 25  OMEGA=1 . 0 0 P  OMEGA=0 . 7 5 P  0.1 0.0  0.3  OMEGA=1 . 0 0 n 0.5 0.4 0 .3 10 .6 10 .2 9 .8 9 .4  0.3  0MEGA=1.5O P  0MEGA=1 . 2 5 P  P  1 0 1 5  1 .00  DELTA  0.6  OMEGA  0MEGA=1 . 25 P  0MEGA=1 . 0 0  0MEGA=O . 7 5 P  1.1 2.0 3.3  1 .3 2.3 3.6  0. 1 -0. 0 0. 1 0 . .7  7  1.0 1.9 3.2  1.3 2. 4 3 .9  0. 1 -0. 0 0. 1 O. 5  1 0 1 6  0.2  1.2  0. 2  0. - 0 . O. 0.  0. 6  2 .2 3 . 7  0. 6  1  0.5  1.3  0MEGA=1 . 0 0 P 0.4 0.5 0. 3  0 1  0.4  2. 1 3.2  4. 6 7. 5  0. -0. 0. 0.  0. 3  0MEGA=1.5O P 0.3 0.4 0.5  1 .5  DELTA 0MEGA=0 . 75 P 0.4 0.5 0. 3  OMEGA=1.25 P  2.5  1 .6 . 1 .6 2.8 2 .9 4.4 4. 7 6.6 7 .4  5. 8  0.6  OVER  Y2±  0 .3  14.4  ON  o  TABLE  RH 1 RH2 0.9 0.9 0.9 0.9  0.6 O. 7 0.8 0.9 .  O. 2  I:  PERCENT GAIN IN E F F I C I E N C Y  0MEGA=0.75 P 0.4 0.5 0. 3  3. 3. 4 5. 5. 8 9 .6 9. 15 . 7 14.  7 9 3 2  3.5 5.5 8.3 12.2  0. 6 2 .7 4 .0 5. 8 8 .0  3.2 4.9 7.1 10.1  o:2  (0%)  OMEGA=1 . 0 0 P 0.4 0.5 0. 3  2. 9 2. 7 4 .6 4 .7 7 .4 7 .5 4 . 12. 5 1 1  2.8 4.4 6.6 9.8  2.5 3.8 5.6 8.0  OMEGA=1 .OO P 0.4 0.5 0. 3  OF  y ^ 2  DELTA=  OVER y  0.6  0. 2  2.1 3.2 4.6 6.3  2. 2 3. 7 6. 2 10. 3  0. 2  0.6 0.7 0.8 0.9  0. 0 0. 2 0 . .9 2.,5  0 . .0 - 0 . 0 - 0 . 0 - 0 . 0 0.2 0.1 0. 1 0. 2 0.7 0 . 6 0 . ,5 0. 8 1 .. 1 1.8 1.5 2. 2  RH 1RH2- '  0 , .2  OMEGA=0.75 P 0.4 0.5 0. .3  0 .6  0. 2  OMEGA=1 . 0 0 P 0.4 0.5 0 . .3  0.6 . 0.7 0.8 0.9  3 .4 2 .7 1 !9 1. 1  3,.7 2..7 1 .8 1.0  2 .7 1 .8 . 1. 1 0 .5  2. 7 2. 1 1 .5 O.,9  2 .9 2 .2 1 .5 0 .8  0 .2  OMEGA=0.75 P 0.4 0.5 0 .3  0. .2  OMEGA=1 . 0 0 P 0.5 0.4 0 .3  0.6  0 .2  9.4 8.4 7.3 6.2  8.1 7.0 5.9 4.9  7 7 7 8  0.9 0.9 0.9 0.9  0.9 0.9 0.9 0.9  RH1 RH2 0.9 0.9 0.9 0.9  0.6 0.7 0.8 0.9-  RH1 RH2 0.9 0.9 0.9 0.9 NOTE:  0.6 0.7 O. 8 0.9  1 .19 1 1. 9 12 . 1 12 . 4  O .2 21 . 8 23 . 5 25 . 9 29 . 6  13 .3 12 .6 1 1. 9 1 1.1  3.5 2.5 1.6 0.8  3.2 2.2 1.3 0.6  0 .6  1 3 . 2 1 2 . 2 10 .6 9 .0 12.0 10.7 7 .5 10.7 9.2 6.1 9.4 7.8  OMEGA=0.75 P 0.4 0.5 0 .3 26 26 27 28  .0 .5 .4 .7  27.2 26.6 26.1 25.6  26.1 24.6 23.2 21.8  0 .6 23 . 2 21 . 3 19 .5 17 .7  DELTA=R^J7g^ .0MEGA = S+P, L  RH1=  0.6  0. 2  9 9 9 9  2.8 2.0 1.3 0.7  . 1 10 .2 1 0 . 2 . 2 9 .8 9.4 9 .4 8.5 .5 .9 9 .O 7.6  0 .2  OMEGA^1 . 25 P 0. 3 0.4 0.5  0. 6  0. 2  OMEGA=1 . 5 0 P 0. 3 0.4 0.5  2 .4 3 .9 6 .1 9. 5  1 .7 2 .6 3. 8 5 .3  3. 2 5. 3 8 .8  1 .9  2. 0  8. 1  0. 2  OMEGA 1 . 5 0 P 0. 3 0.4 0.5  2.3 3.6 5.5 8.1  2.0 3.2 4.7 6.7  OMEGA=1 .25 P 0. 3 0.4 0.5  0. 0 - 0 . 0 0. 1 0. 1 0. 6 0. 5 17. 1 .5  - 0 . 0 - 0 . 0 -o.o - 0 . 0 - 0 . 0 0 . 1 0 . , 1 0.1 0 . 1 0.1 0.6 0.5 0.4 0 . 7 0 . .7 1.5 1 .2 0 . 9 2 . 1 1 .8  I  0 . 75  DELTA=  RH1 RH2  0. 2  .  0 . 50  0MEGA=0.75 P 0.4 0.5 0. 3  O. 6  2  DELTA=  0.0 0. 1 0.5 1.2  0. 2  2.1 2.5 1 .7 1.4 1 . 10 . 9 0.5 0.4  2 .2 1.8. 1 .3. 0..8  2 .4 1 .8 12. 0. 7  DELTA=  2.3 1.7 1.1 0.6  2.0 1 .4 0.9 0.4  , RH2=  .  1 9 . 6 17.4 1 8 . 7 16.2 17.9 15.0 17.0 13.8  0. 0. 0. 1  0 1 5 .5  .4 .5 .8 .3  13 . 1 14 .3 16 . 1 18 .8  3.1 4.7 7.0  1 .7  1 .5  2.7 4.0 5.7  2 .2 3 .2 4 .5  0. 6  0.0  0. 0  0. 1 0.3 0.9  0. 1 0. 3 0. 7  OMEGA=1 . 5 0 P 0. 3 0.4 0.5  0. 6  0. 0 0. 1 0. 5 13.  0. 6  0 . .2  1 .7 1. 2. 0.7 0 .3  1 .9 . 2 .0 , 1 ,. 5 1;.5 1 . 1 1 .0 0. , 7 0.,6  1 .25  -o.o  0.1 0.4 1.1  1.9 1.4 0.9 0.5  1 .7  1 .5  1 .2 0.8 0.4  1 .0 . 0 . .6 0. . 3  OMEGA=1 .25 P 0. .3 0.4 0.5  0 .6  0 .2  OMEGAM . 5 0 P 0 .3 0.4 0.5  8 3 8 .0 7 .8 7 .5 .  6 .6 5 .7 4 .9 4.1  6.2 6 .4 6 .6 7.1  6 .8 6 .6 6 .4  0.2  OMEGA=1 . 5 0 P 0 .3 0.4 0.5  0 .6  13 .0 1 3 . 6 1 3 . 0  1 .6 1  8.3 7.7 7.0 6.4  7.7 6.9 6.0 5.2  1. 5 0 DELTA= 0MEGA=1 .25 OMEGA=1 .OO P P 0 .2 0 .3 0.4 0.5 0.4 0 . 5 •, 0 . 6 0 .3  16 .4 19 .5 2 0 . 4 17 .8 20 .1 2 0 . 2 19 .8 21 . 0 2 0 . 1 23 . 0 22 .4 2 0 . 1  0 6  100 .  0.6  1.9  3  0 . 0 -0. 0 0 . 1 0 .. 1 0.4 0 . .3 1 .0 0. 8  OMEGA=1 .25 P o.3 0 . 4 0 . 5  3. 3 5. 2  O. 6  15 .6 1 6 . 3 15.7 16 .2 1 6 . 3 15. 1 17 . 1 1 6 . 31 4 . 5 18 .4 1 6 . 5 1 4 . 0  0 .6 13 .9 13 . 1 12 .2 1 .3 1  10 .9 12 . 0 13 .5 15 .9  7 .0  7.0  6.5 6.0 5.5  6.5  0. .6 5 .6  5.8 4.9 5 . 1 4.2 4. 5 3. 5  13 .6 1 3 . 6 1 2 . 6 14 .4 1 3 . 7 1 2 . 2 15 .6 1 4 . 0 1 1 . 9  10 . 9 10 . 2 9 .6 ^  62 .5, b u t 6.  f o r P!  a t fi = 1, but  y  A >  f o r A = 0.5,  From t h i s over  and  = .9,  2 r  there  0.5,  efficiency  numerical  s t u d y was  expensive  to compute.  Estimator b e t w e e n two  occasions g  = U  2 r  y"  2,  increase i n  estimated  by  2  in  p .  that y ^  has  a slight  especially  i s obtained  P,  increase 2  seem t o be  2  when Pi  and  equally desirable.  P  edge are  2  This  e s t i m a t o r s were e q u a l l y  o f t h e c h a n g e i n means  as a w e i g h t e d a v e r a g e o f  the  + (1 - a ) y ' . . } - {b x "  r  with  A ratio estimator  occasions  1 and  i t i s noted  efficiency,  the assumption that both  of the change.  successive  decreases  p u r p o s e s and  estimates  done on  decline in  increases with  study,  F o r most p r a c t i c a l  h i g h and fi = A = 1, b o t h  means on  efficiency  limited numerical  «  i s a steady  r a t i o method.  Thus  + (1 - b ) x ' . . }  r  the  (22)  where 0  x"  r  =  — v..  =  p  a,b  are  other The  Var(g  — y"..  7—c—c-  (s+p)  J  and  (x"../y"..)y.. s T  +  (s+p)  y  — y'..  r  s a m p l e mean on  J  occasion  2  v  weights  s y m b o l s a r e as d e f i n e d e a r l i e r . variance  of g ^ 2  i s given  by  ) = a V a r ( y " ) + (1 - a ) V a r ( y ' . . ) 2  2 r  = overall  2  r  +  (1 - b ) V a r ( x ' . . ) 2  - 2 ab  +  b Var(x" ) 2  r  cov(y" ,x" ) r  r  + 2 a ( l - b)cov(y" ,x'..) - 2 b ( l - a)cov(x" ,y ..) (22.1) i s u n c o r r e l a t e d w i t h y'.. and x" i s u n c o r r e l a t e d w i t h x'.. . r r 1  s i n c e y" J  Now,  u s i n g the  reasoning  of Cochran  (1977:343) (and  i n t h e e s t i m a t i o n o f t h e c u r r e n t mean i n t h i s  section)  as  indicated  coy(x"  ,x" > = cov{[y"..'+  R (x. . - x " . . ) ] , [ x " . .  + ( 1 / R ) ( y . . - y". . ) ] }  = cov(y"..,x"..) + cov[y"..,(1/R)y..] + cov[y"..,(-1/R)y"] + cov(Rx..,x"..)  + cov[Rx..,(1/R)y..]  + cov(-Rx"..,x"..).+ +  cov(x" ,y'..) r  cov[Rx..,(-1/R)y"..]  cov[-Rx"..,(1/R)y..]  cov[-Rx"..,(-1/R)y"..]  R(s!p)p  =  +  {  (  S  +  P)f"P^  A  * ^  "  s  }  = cov{[x".. + (1/R)(y.. - y"..)],  y'..}  = (l/R)cov(y..,y'..) = e  2 2  / R ( s + p)  cov(y" ,x'..) = cov{[y".. + R(x.. - x " . . ) ] ,  x'..}  r  =  Rcov(x..,x'..)  = R6  2  j  and V a r ( x " ) = V a r ( x " . . ) + ( 1 / R ) V a r ( y . . - y"..) + ( 2 / R ) c o v [ x " . . , ( y . . - y " . . ) ] 2  r  = 6  2 1  [s(l  - A ) + p(l - A)]/A p(s + p). 2  Then, by s u b s t i t u t i n g ( 2 2 . 1 ) , we o b t a i n Var(g .) = a { 6 2  2 ]  2 2  2  i n t h e v a r i a n c e s and c o v a r i a n c e s a b o v e i n t o  that [l  - q(2* A 2  - A )]/p} 2  + ( 1 - a) ( 6 /s) + b { 6 2  b) (9  2  2ab{[ 6 / R ( s 2 2  + 2a(l - b)R6 Differentiating to  z e r o and t h e n  + p)p][[(s 2 1  2  22  + p ( l - A ) ] / A p ( s + p)} + (1 -  equation  2  2 1  2 I  [s(l  - A ) 2  /q)  + p ) [ - p i | < A - q A ] - sjj } 2  2  - 2 b ( l - a) [ e  e q u a t i o n (22.2) w . r . t . simultaneously solving  2 2  /R(s  + p)]  a a n d b, s e t t i n g  (22.2) the r e s u l t s  f o r a and b g i v e s  a * = (AT - Z<)>)/(AT - <t>) 2  b* = (AZ - <(>A)/(AT - <t>) 2  equal  where A = (9  2 2  /s)  - R6  2 1  T = [6  2 2  / A p q ( s + p ) ] { q [ s ( l - A ) + p ( l - A)] + A p(s + p)}  Z = (6  2 l  /q) + [6 /R(s  2  2  2  + p)]  2 2  A = (e /sp){p + s [ l - q(2^ A - A ) ] } 2  2 2  2  $ = 6 / R p [ p ^ A - qA 2 2  + 1] - R 6  2  2  2 1  and a* and b* a r e t h e ' o p t i m a l ' v a l u e s minimize  Var(g  2 r  and b * i n t o e q u a t i o n  bias w i l l  of g ^ i s obtained 2  ( 2 2 ) , and t h e v a r i a n c e  i s g i v e n by s u b s t i t u t i n g  The amount o f t h i s  2  follows.  B(g .) - E [ g 2]  so o b t a i n e d  (22.1).  that g ^ i s biased.  now be d e t e r m i n e d a s  by s u b s t i t u t i n g a  of the e s t i m a t o r  a* and b* i n t o e q u a t i o n  i t i s expected  2 r  that  ).  The m i n i m u m - v a r i a n c e e s t i m a t o r  Like y >  o f a a n d b, r e s p e c t i v e l y ,  2 r  - (u  Y  - M )] X  = E(y .) - E ( x 2]  2 r  )  -ii  y  + V  X  where x  i s the second p r i n c i p a l p i e c e  2  Now, u s i n g  t h e arguments g i v e n  i n equation (22). i n Murphy  (1967:364) X  E(y  2 r  Then by B(g  ) = aq[(R6 /pu ) - (R^ /6 ,9 2 1  x  2  2  2 2  /pu ]  - u  y  y  substitution 2 r  ) -  {aq[(R6 / M )  (R\|) /6 e )/pu ] + u }  +  (s^ /e !e  2 1  (s6  2 2  P  x  /pRu ) y  2  2  21  2  22  2 2  y  /pRu )] x  y  + u > x  ( b [ y ( s - q) x  Y + yX  65 = aq[(R9 + (s6  2 2  2 1  / p u ) -• ( x  /pRM ) -  / 6  2  2  T h i s b i a s becomes n e g l i g i b l e  useful  t o compare the  change. only  the  a numerical  adopted  2 2  P  be  of  o f n,  estimator. BLUE and  The  gain  Again,  the v a r i a n c e  done.  m.  t h e RE  i t would  f u n c t i o n s of  change,  same p r o c e d u r e  to e s t i m a t i n g the  in precision  of g  over  2  g  current is  2  X/ given  be  in estimating  F o l l o w i n g the  e f f i c i e n c y with respect  follows.  q)  x  of the  complexity  -  x  / RM )].  ratio  comparison w i l l  as  e  y  for large values  efficiency  i n comparing the  mean,we p r o c e e d  2 1  BLUE and  Because of the  /pu ) ] - b [ u ( s  2 2  (sip /e  y  Comparison of  ,6  2  r  by Var(g Qi  ) - Var(g  2  =  2  )  ~  —  Var(g ) H  Using o f Q,  A,  equations  pj, p , 2  ( 1 0 . 1 ) and  <t>i, and  (22.2),  the v a l u e s  <t> w e r e t a b u l a t e d . 2  The  o f Ch  results  f o r some are given  values in  Table I I . It is  seems o b v i o u s  overall  very  from Table  inefficient  as  I I t h a t the  compared to the  ratio BLUE.  estimator  of  change  TABLE I I : PERCENT GAIN IN EFFICIENCY (0,%) OF g • OVER g 1  RH 1 RH2 0.6 0.6 0.6 0.6  0.6 0.7 0.8 0.9  RH 1 RH2 0.6 0.6 O.G 0.6  0.6 0.7 0.8 0.9  OMEGA-O.75 P 0.4 0. 2 -47 .2 2063.7 -33 .3 4257 . 4 - 18 .2 29485.5 -0. 3 -193.0 OMEGA-O.75 P 0.4 0. 2 12 .7 - 1104.6 22 .9 -834 .4 35 .7 -679.5 52 .7 -582 . 7  OMEGA=1.00 P 0.4 0. 2  0.6 549 637 754 918  4 .0 14 .1 26. 2 41 7 :  .3 .1 .3 .7  OMEGA =1.00 P 0.4 0.2  O. 6  0.6 0.6 0.6 0.6  0.6 0.7 0.8 0.9  0. 2  y  0.4  44 .1 - 169.2 53 7 - 137 .0 66 . 3 - 106.7 -75 . 2 83 .8  OMEGA - 1.00 r 0. 2 0.4  O. 6 397 466 558 687  . 1 . .3 ,6 . .8  0.6 0.6 0.6 0.6  0.6 0.7 0.8 0.9  RH1 RH2 0.6 O. 6 0.6 0.6  0.6 0.7 0.8 0.9  0 .2 64 . 1 73 . 9 87 .0 105 . 4  Y  0.4 93 .0 110.4 134 .0 167 .6  OMEGA-O.75 r 0 .2 0.4 77 .9 88 .2 102 .0 121 . 5  254 .4 272 . 1 299 .6 342 .0  0 .6  0. 2  265 .0 315 .8 383 .4 478 .O  72 . 3 81 .0 92 .6 109 . 1  166.3 182 . 2 205.8 240.9  f  0.4 282 .6 298 . 3 324 .0 364 .4  OMEGA-1.00 0 .6 186 .9 226 .8 279 .8 353 .9  NOTE: DELTA = R/e 2 j/8 OMEGA-S+P , RH 1 = 2z  58 .8 67 .1 78 .3 94 . 1  OMEGA-1.00  OMEGA =0.75 RH 1 RH2  1 .6 26.7 54 .5 88.8  38 .5 47 .0 58 .2 73. 5  525. 3 611. 5 726. 7 888 .1  OMEGA =0.75 RH 1 RH2  -361.8 -262.3 -192.9 - 136 .8  px  f  0 .2  0.4  81 . 7 90 .7 103 .0 120 .3  380.7 398 .9 428.8 476 . 2  , RH2- p  2  DELTA= 0.6 498.8 590.4 716.0 897 . 4 DELTA0.6 316.4 375.9 456.6 571 .9 DELTA =  I  r  0.50  OMEGA-1.25 P 0.2 0.4  25 .0 33.4 44 .0 58 .0  0.6 4 10.5 505 .4 650. 8 898 .4  -14.5 14.3 44 . 1 78 . 9  0.75  OMEGA =1.25 P 0..2 0.4  50. 2 57 . 7 67 . 7 81.8  0.6  137 . 2 40. 7 153 . 1 35. 5 175.6 21 .1 208 . 3 -13. 7  1 .00OMEGA- 1 . 25 D  0.6  0.2  0.4  96.5 118.9 148.5 189 . 5  65.9 73.4 83.6 98. 1  245.7 259 . 3 281 .8 317.4  DELTA-  1 .25  0..6 -189.. 7 -232 .0 -295 .4 -395 . 7  OMEGA-1.25  OMEGA-1.50 P 0.2 0.4 36 .3 43 .6 53 .2 66 .2  OMEGA-1.50 P 0. 2 0.4 56. 8 63. 6 72 .8 . 86.0  70..0 76..9 86..4 100  -33.3 -29. 1 -24 .0 -17.6 DELTA-  0.2 76.5 84 . 3 94 . 9 1 10. 2  0. 2  336.7 -268 .3 35 1 . 3 -3 11 .6 376.3 -372 .9 416.8 -464 .0  78 .9 86 . 2 96 .2 1 10.5  1 .50 OMEGA =1.25 D  0.6 -88.2 -90. 1 -93.0 -97 .6  0.2 83.6 91.9 103 . 1 119.1  0.4 417.7 434 .9 463 .9 510. 7  0 .6 -284 .6 -322 .6 -375 .0 -451 .0  0.6  0..6  280. 1 -467 .6 292.5 -596 8 314.0 -798 . 8 349.1 - 1 140. 5  .OMEGA-1.50 1  0 .6  0.4  1  209 . 239 . 8 276 .8 4 16.5  188 . 3 -339 .9 201 .6 -513 .0 222 . 5 -857 .2 254.5 - 1706 .3  OMEGA-1.50 P 0.4 0.2  D  0.6  76 . 9 95 . 8 119.2 150. 5  0.6  P o:4  361 . 3 374 .8 398.8 438.4  0 .6 -463 . 1 -554 . 6 -686 . 2 -886 . 3  OMEGA- 1 .50 P 0.4 0 .6 0. 2 84 .9 92 .6 103 . 1 1 18. 1  434 . 1 450. 3 478 . 1 523.5  -435 . 3 -504 . 8 -601 . 4 -742 . 7  TABLE  I I : PERCENT GAIN  IN E F F I C I E N C Y  (Q,%) OF g ^ 2  DELTA= 0MEGA=0.75 RH1  RH2  8 8 8 8  0.6 0.7 0.8 0.9  OMEGA=1.00  P 0.4  0.2  -18.2 2 9 4 8 5 . 5 -0.3 -193.0 23.1 - 1956.1 58.2 -1706.7  0.6 754.3 918.7 1 165.8 1578.3  .  p  0.2  0.4  26.2 41.7 63.5 98. 1  -192.9 -136.8 -83.5 -21.6  0.6 716.0 897.4 1178.9 1665.2 DELTA=  RH1  RH2  8 8 8 8  0.6 0.7 0.8 0.9  OMEGA=0.75 P 0.2 0.4 35.7 52.7 77.1 116.7  -679.5 -582.7 -521.8 -489.7  0.6 726.7 888.1 1130.5 1535.0  0MEGA=1.00 P 0.2 0.4 58.2 73.5 96.2 133.6  54.5 88.8 135.7 209.5  0.6 456.6 571.9 748.7 1050.4 DELTA=  RH1  RH2  8 8 8 8  0.6 0.7 0.8 0.9  0MEGA=0.75 P 0.2 0.4 66.3 83.8 109.8 153.0  -106.7 -75.2 -38.0 135  0.6 558.6 687.8 881.6 1204.7  0MEGA=1.00 P 0.2 0.4 78.3 94.1 117.9 157.7  205.8 240.9 295.7 389.3  0.6 148.5 189.5 250.4 351.O DELTA=  RH1  OMEGA=0.75 P 0.2 0.4  RH2  87.0 105.4 133.2 179.7  134.0 167.6 218.6 303.9  0.6 383.4 478.0 619.6 855.6  OMEGA=1.00 P 0.2 0.4 92.6 109.1 134.2 176.3  324.0 364.4 429.9 544.4  0.6 -24.0 -17.6 -9.0 3.7 DELTA=  RH1  RH2  0.8 0.8 0.8 0.8  0.6 0.7 0.8 0.9  OMEGA=0.75 P 0.2 0.4 102.0 121.5 151.0 200.4  NOTE : DELTA = R / 6  2  299.6 342.0 409.9 527.8  0.6 279.8 353.9 464.7 649.1  ^ 2 OMEGA = S+P, 2  RH1=pj  OMEGA=1.00 P 0.2 0.4 103.0 120.3 146.6 190.8 . RH2=p  428.8 476.2 553.7 690.0 2  * 0.6 -93.0 -97.6 -105.3 -119.2  OVER  g ^_ 2  0.50 OMEGA - 1 .25 P 0.4 0.2 44.0 58.0 78.4 111.6  44.1 78.9 124.9 195.6  0.6  650.8 898.4 1401.0 2852.3  0. 75 OMEGA=1.25 P 0.2 0.4 67.7 8 1.8 102.9 138.2  175.6 208.3 258.5 343.2  0.6 21.1 -13.7 -100.4 -347.7  1 .00 OMEGA=1.25 P 0.2 0.4 83.6 98. 1 120.2 157.4  281.8 3 17.4 375.1 476.0  1 . 25 0MEGA=1.25 P 0.2 0.4 94.9 110.2 133.4 172.5  376.3 416.8 483.5 601.6  0.6 -295.4 -395 . 7 -567.9 -899.8  0.6 -372.9 -464.0 -609.1 -866.2  1 .50 OMEGA=1.25 P 0.2 0.4 103.1 119.1 143.3 184.2  463.9 510.7 587.6 723.9  0.6 -375.0 -451.0 -568.7 -771.1  OMEGA=1.50 P 0.2 0.4 53.2 66.2 85.6 117.4  119.2 150.5 196.1 270.7  OMEGA=1.50 P 0.2 0.4 72.8 86.0 106.0 139.7  0.6  0.6  398.8 -686.2 438.4 -886.3 504.3 -1214.8 621.6 -1818.8  OMEGA=1.50 P 0.2 0.4 103.1 118.1 14 1.0 179.6  0.6  3 1 4 . 0 -798.8 349 .1 - 1140.5 406.8 -1787.9 508.8 -3255.5  OMEGA= 1 .50 P 0.2 0.4 96.2 110.5 132.4 169.5  276.8 4 16.5 449.0 596.6  222.5 -857.2 254.5 -1706.3 305.3 -3390.1 393. 1 1 1 177. 1  OMEGA =1.50 P 0.2 0.4 86.4 100. 1 120.9 156.3  0.6  0.6  478.1 -601.4 523.5 -742.7 598.7 -964.0 732.5 -1349.4  CHAPTER 4 OPTIMUM ALLOCATION AND  REPLACEMENT  In most applications of sampling, cost i s an important factor since the funds available for sampling are usually l i m i t e d .  It i s therefore  necessary to include methods of optimal sample a l l o c a t i o n i n the o v e r a l l design of a sampling scheme.  In this chapter we discuss the use of  dynamic, programming (DP) i n the determination of the optimum  replacement  policy for multistage sampling on successive occasions f o r several v a r i ables of interest. For sampling on successive occasions, the sampling design i s expected  to be s t a t i s t i c a l l y e f f i c i e n t over the whole series of succes-  sive occasions considered i n i t s entirety. questions must be answered.  In p a r t i c u l a r , the following  Should a l l the sampling units be remeasured  at the successive occasions, and i f not, what proportion of units should be remeasured?  Is the replacement  policy adopted  to each of the several variables of interest  optimal with respect  estimated at the current  occasion, and have the side conditions imposed on the estimation procedure been met? design problem.  These and other questions constitute an optimal For example, the objective of a two-occasion  inventory design may  SPR  timber  be to determine the proportion of sampling units to  remeasure and new ones to take at the current occasion such that the cost of sampling i s minimised  and subject to the side conditions (constraints)  that the specified precision levels are met on several variables of 68  69 interest (such as timber volume and periodic growth). T y p i c a l l y , the objective and constraint functions are non-linear, and the problem of determining the optimal SPR design i s a non-linear decision problem.  Several authors, for example, Rana (1976), Singh and  Kathuria (1969), K u l l d o r f f (1963), Tikkiwal (1953), Patterson (1950), Yates (1949), and Jessen (1942) , have considered the problem of determining the optimum replacement p o l i c y i n SPR.  They were interested  mainly i n the s i t u a t i o n where there was only one variable of interest at a time.  Ware and Cunia  (1962) and Hazard and Promnitz (1974) determined  optimal: SPR designs i n situations where there were several variables of interest at a time.  Rana (1976) and Singh and Kathuria (1969) assumed  multistage sampling and the others simple random sampling on the successive occasions. Ware and Cunia decision problem.  used a graphical technique to solve the non-linear Graphical methods are suitable f o r the case of two-  occasion SPR, where there are no more than two decision variables and the number of side conditions i s r e l a t i v e l y small. Hazard and Promnitz solved t h e i r optimal SPR problem with an algorithm that required that the cost and constraint equations be d i f f e r e n t i a b l e convex functions.  (The assumption of convexity w i l l usually be  met i n optimal SPR problems i n forestry.)  Although a number of i t e r a t i v e  solution techniques have been developed f o r such non-linear programming problems, there i s no assurance that a solution w i l l always be reached i n a reasonable number of i t e r a t i o n s .  Furthermore, a subsequent  sensitiv-  i t y analysis on the derived decisions i s recommended i n order to firmly establish the optimality of the decision v a r i a b l e s . Optimal sample design for SPR involves a sequence of interrelated  70 d e c i s i o n s over time.  P r e v i o u s workers i n t h i s f i e l d  take advantage of t h i s u n d e r l y i n g p r o c e s s replacement  policies.  We  shall exploit  a p p a r e n t l y d i d not  i n determining t h e i r  optimal  the s e q u e n t i a l n a t u r e of the  problem t o c a s t the o p t i m a l sample d e s i g n problem as a m u l t i s t a g e model which can be o p t i m i z e d through  dynamic programming.  F i r s t , dynamic programming i s d i s c u s s e d i n g e n e r a l . dynamic n a t u r e of the o p t i m a l SPR s o l u t i o n procedure  d e s i g n problem i s examined.  of the o p t i m a l two-stage SPR  Dynamic programming d e c i s i o n processes.  DP  (DP)  Next, the Then the  problem i s p r e s e n t e d .  i s an o p t i m i z a t i o n method f o r m u l t i s t a g e  i n v o l v e s s e p a r a t i n g the m u l t i v a r i a b l e o p t i m i z a t i o n  problem i n t o a s e r i e s of o n e - v a r i a b l e o p t i m i z a t i o n problems. r e s u l t a n t o n e - v a r i a b l e problems may  then be  methods of d i f f e r e n t i a l c a l c u l u s or simple of  DP  i s covered e x t e n s i v e l y elsewhere,  Dano ( 1 9 7 5 ) , Briefly,  . Wilde  and  The  solved r e a d i l y using standard search p r o c e d u r e s .  The  f o r example,  B e i g h t l e r (1967), and Nemhauser  the c h a r a c t e r i s t i c s of a DP  theory  problem a r e reviewed  and  (1966). these a r e  as f o l l o w s : (i)  the problem can be d i v i d e d  i n t o s t a g e s , with a d e c i s i o n a t each  stage; (ii)  each stage has  effect  a number of s t a t e s a s s o c i a t e d w i t h i t , and  of the d e c i s i o n at each stage  i n t o a s t a t e a s s o c i a t e d w i t h the next (iii) each  the  i s to t r a n s f o r m the c u r r e n t s t a t e stage;  the p r i n c i p l e of o p t i m a l i t y as s t a t e d by Bellman  (1957) h o l d s a t  stage:  An o p t i m a l p o l i c y has the p r o p e r t y t h a t whatever the i n i t i a l s t a t e and i n i t i a l d e c i s i o n s a r e , the remaining d e c i s i o n s must c o n s t i t u t e an o p t i m a l p o l i c y w i t h r e g a r d to the stage r e s u l t i n g from the f i r s t d e c i s i o n ; (iv)  a r e c u r s i v e r e l a t i o n s h i p can be developed  which i d e n t i f i e s  the  optimal decisions f o r each state with r stages remaining, given the optimal decisions for each state with (r - 1) stages remaining.  This  relationship i s of the general form f ( X ) = optimize { c ( X , d ) + d eD r r r  r  r  r  r  f ^ O ^ ) }  where, D r  i s the constraint set for the decision variable d , r'  f (X )  i s the optimal value when starting i n state X r stages  X  with  remaining,  , = t(X ,d ) i s the transformation of state when decision d r r r  r-1  is used when in state X , r' c^CX^jd^) i s a value function ( p r o f i t , cost, etc.) for using d^ when i n state X (v)  r = 1, 2, . . . ;  the problem i s solved using the recursive r e l a t i o n s h i p .  At each  stage, an optimal solution from a l l previous stages, under any conditions, i s found and carried into the next stage, u n t i l the last stage when the optimal decisions are ound f o r the whole problem. f  Unlike l i n e a r or other non-linear programming problems, there i s no standard mathematical formulation of dynamic programming problems; s p e c i f i c formulations must be developed to f i t individual problems. we examine the dynamic nature of the SPR optimal design problem. \. SPR as a Multistage Model To f a c i l i t a t e this general discussion, we introduce some new notation to include sampling on more than two successive occasions as follows. Let  p .. = n.. /n ij...w i j . . .w be the proportion of sample units at occasion r measured on occasions  Now  72 i , j ,... , and  w  where r  i s a measurement o c c a s i o n r =  i,j,...,v>  are  correspond rent  l,2,...,t  i n d i c a t o r v a r i a b l e s and  their positions  to o c c a s i o n of measurement such that i r e p r e s e n t s  occasion r, j represents  w represents  1st  ( r - l ) * " ^ o c c a s i o n , and  i , j , . . . , and w,  The  n i s the t o t a l  sample s i z e on o c c a s i o n  o c c a s i o n and  1 i f the  sample u n i t was  0 (zero) otherwise.  s i o n s i n f o u r - o c c a s i o n sampling.  For example,  Similarly, p  1 0  c u r r e n t and  p  1 0  io  second  occa-  sampling.  note t h a t the number of groups of sample u n i t s measured at u  occasions  at o c c a s i o n I i s  (  r  m  ru  =  For example, i n two-occasion  ~  sampling  r !  /t(  r  - ) u  ! u !  1  r = 2  2l  = 2!/[(2 - 1)!1!] = 1  m  22  = 2!/[(2 - 2)!2!] = 1 group of sample u n i t s measured o n l y once and  group measured on both o c c a s i o n s i s not  =  m  That i s , t h e r e i s o n l y one  in p  measured  i s the p r o p o r t i o n of  u n i t s measured on c u r r e n t o c c a s i o n o n l y i n two-occasion  one  occasions  1.  i s the p r o p o r t i o n of sample u n i t s measured on both  We  on  and  i n d i c a t o r v a r i a b l e s take on the v a l u e  on c o r r e s p o n d i n g  and  occasion  n. . i s the t o t a l number of sample u n i t s observed 1 j .. . w  >  so on  cur-  an exponent, and  (see F i g u r e 1.).  only  Note a l s o that r  t h a t at o c c a s i o n r , there are r - 1 o c c a s i o n s  remaining. In SPR, will  the t o t a l  sample at each s u c c e s s i v e measurement o c c a s i o n r  c o n s i s t of s e v e r a l groups of remeasured and  new  sampling  units.  For  73-  £ ' 9  u  r  - Groups of sampling  1  e  on  sampling  two  units  in  SPR  occasion*:  occasion 1  P = P. 11 s  = p;  = proportion on  both  10 = proportion  q = p, 01  on the  of  units  measured  occasions of  second  units  measured  occasion  only  = p r o p o r t i o n of u n i t s measured on the f i r s t o c c a s i o n only  74 example, i n two-occasion groups of sampling  sampling  r = 2, t h e r e w o u l d be t h e f o l l o w i n g  u n i t s a s shown i n F i g u r e 1.  a t measurement time  r  For  ...  the example of two-occasion  E w=0  r = l , 2, . . . , t  p.. ) J--' X  sampling,  W  t h e sample s i z e on t h e c u r r e n t  be 2  n  2  = n(p  [ o r as i n e a r l i e r n o t a t i o n DP  size  1  = n( E E E i=0 j=0 k=0  o c c a s i o n 2 would  sample  r is  1 1 1 n  The t o t a l  2 1 0  +  Pu)  = n(s + p ) ] .  i s u s e d i n t h e s o l u t i o n o f t h e o p t i m a l SPR d e s i g n p r o b l e m b e c a u s e  of t h e f o l l o w i n g . (i) by  I t i s recognised  t h a t SPR o p t i m a l d e s i g n p r o b l e m i s c h a r a c t e r i s e d  "time" stages, a stage being  ( r = 1, 2  d e f i n e d as a measurement t i m e  r  t ) .  A t each stage a d e c i s i o n n i s required. I f we . . . r-1 assume t h a t n i s k n o w n , t h e d e c i s i o n then c o n s i s t s of 2 c o m p o n e n t s ( o r " s u b - d e c i s i o n s " ) o f new a n d r e m e a s u r e d s a m p l e u n i t s . We c a n c r e a t e  r-1 "sub-problem"  stages  problem stages  a r e nested w i t h i n the time  nesting (ii)  2  w i t h i n each time  i s given f o r two-occasion  sampling  stage  such  stages. i n Figure  that the sub-  An e x a m p l e o f t h i s 2.  A t e a c h s t a g e r , t h e r e a r e a number o f s t a t e s f o r e a c h s t a t e  v a r i a b l e X^  ( i = 1, 2,  . . ., z ) .  A s t a t e i s d e f i n e d a s t h e amount o f  variance of the v a r i a b l e of i n t e r e s t t o s t a t e v a r i a b l e X^ r e m a i n i n g  ( i = 1, 2,  t o be a c c o u n t e d  z)  corresponding  f o r b y t h e s a m p l e n^_  taken. (iii)  The p r i n c i p l e o f o p t i m a l i t y h o l d s  i n t h e SPR o p t i m a l d e s i g n  T h a t i s , a t a p a r t i c u l a r m e a s u r e m e n t o c c a s i o n r , an o p t i m a l s a m p l e for  the remaining  measurement o c c a s i o n s  model. size  i s independent o f t h e sample  75 sizes  taken  (iv)  A recursive relationship  developed for  the  on  previous  measurement  which i d e n t i f i e s specified variance  occasions In  remaining, order  given  to develop  (b) v a r i a n c e  sion) levels The by and  Z  of  optimal (1)  levels  2  the  optimal  the  the  then  2  Z  problems are  shall  now  then  r e t u r n to the  occasions  remaining.  r e q u i r e (a) a cost  of  restrict  ourselves  t o t w o - s t a g e SPR  stage  two  on  successive  straightforward.  sampling, (preci-  as  a multi-stage stage  process  components,  i s , r e p l a c i n g the  problems.  The  known, t h a t now  i s , we  are  mean a n d n',  way  as  shall  pay  Extension  the  to the other  and  and  n' p s u ' s  the e s t i m a t i o n of  successive  a t t e n t i o n to the  first  aim  sampling m are  that  the  We  already occasions.  i s , to  choose  ssu's)  current  c o s t of  each  designs  population  In choosing the  shall  u n i t s at  ( a n d mn'  occasions. of keeping  3.  second  second o c c a s i o n ,  ssu's)  regards two  i n chapter  w i t h e q u a l - s i z e d sampling  i n v e n t o r y on  change between the we  described  f u r t h e r a s s u m e d t h a t n and  s a m p l e s i z e s n " p s u ' s ( a n d mn" t h e most e x p e d i e n t  problem of d e t e r m i n i n g the o p t i m a l  somewhere i n b e t w e e n t h e  want t o p l a n t h e  r—1  recursively.  plans  occasions. be  2  Z r=l  sampling  It will  cost  decision variables.  one-decision  specific  f o r the  and  r more  specified variance  functions into  r—1  solved  replacement p o l i c y  in  expressed  c o s t and v a r i a n c e  - d e c i s i o n problem w i t h  ^' s )  r=l  We  the  be  to the  be  (p ^  d e c i s i o n v a r i a b l e s to the  the v a r i a b l e s of i n t e r e s t  d e s i g n problem can  one-decision  We  r-1  r e c u r s i v e r e l a t i o n s h i p we  r=l  is  policy with  decomposing the d e c i s i o n problem, t h a t  r—1  problem can  of v a r i a b l e s of i n t e r e s t w i t h  (constraint) functions relating  s e p a r a t i n g the  (2)  u s e d t o s o l v e t h e SPR  the optimum r e p l a c e m e n t p o l i c y  (objective) function relating and  occasions.  the  n"  76 inventory the  as low as p o s s i b l e .  ( n " i s number o f r e m e a s u r e d p s u ' s a n d n' i s  number o f new o n e s a n d assume n " = np a n d n' = n s . ) To f o r m u l a t e  taining  t h e p r o b l e m we s h a l l  t o the second o c c a s i o n  i s given  C = k p + 2  assume t h a t  the t o t a l  by t h e simple  cost  cost  C  per-  function (23)  k i S  where k  k c and  =  t  2  c',n  +  c'2  m  n  = c " , n + c" mn 2  = c o s t o f a new p s u ( i = 1) a n d a s s u ( i = 2 ) , a s s u m e d t o be k n o w n ,  1  c"^ = cost  o f a r e m e a s u r e d p s u ( i = 1) a n d a s s u ( i = 2) a s s u m e d t o  be known. Further,  we s h a l l  assume t h e v a r i a n c e  functions  t i o n s 6 and 10.1) t h a t - r e l a t e t h e v a r i a n c e respectively,  follows.  and  such t h a t  (equa-  o f c u r r e n t mean a n d c h a n g e ,  t o p a n d s.  The t w o - s t a g e SPR o p t i m a l as  developed e a r l i e r  design  d e c i s i o n problem i s then  F i n d p and s such t h a t  the cost  the s p e c i f i e d variance  l e v e l s V! a n d V  stated  of sampling C i s minimized 2  of current  mean a n d  g r o w t h , r e s p e c t i v e l y , a r e met. Expressed  i n a n o t h e r way, f i n d p, s such C = minimum p,s  and  var(y  2 j l  this  case.)  { k j S + k p} 2  < V!  var(g ^)  < V  p,s  > 0.  2 j  (We a r e g o i n g  )  that  2  b a c k t o o u r o l d n o t a t i o n where p = p j , and s = p 2  2 l  0  , in  If  we  set v a r ( y  2  ) < V  and  1  var(g  ) < V ,  2  then  2  equation  (6)  yields . e and  equation  2 2  ([l  - q*|]/[s  qs*i]  + p -  < V,  (24)  (10.1) y i e l d s  [(s + p - sip|)e  + ( l - q\p|)e  2 I  (s  - 2pijj /e e ]/ 2  22  + p - qs*!)  < V  2 1  2 2  (25)  2  where q = (1 First, ponents. C  we  separate  Separation  t h e c o s t and  of the c o s t  i s already accomplished  Separation ever,  and  - p).  after  (25)  {[p(e  algebraic manipulation,  [p/(l  +.p* )] + s V [V, -  - +|  2  2  2 l  2  2 l  2 2  ( i - *|)]}  - s + v  can  now  use  (26)  and  (27)  < V  2  How-  (24)  becomes  ( e / V j - ) ] < V,  (26)  2 2  allocated  variance  to create  s u b s e q u e n t t o an  variance  left  and  + s) ssu's  mn(p  schematic  be  2  2  f u n c t i o n s i s complete;  stage  transition  to occasion 2.  (Note t h a t n(p  2, 2,  and  f u n c t i o n s of  amount o f v a r i a n c e  undertaken at occasion 2).  pi> ) -  (27)  i n v e n t o r y on o c c a s i o n  allocated prior  diagram (Figure  + s) = The  t o be  +  2  2  , = t ( X ,d ) w h i c h c o m p u t e t h e r-1 r r  now  inequation  2 2  s e p a r a t i o n o f t h e c o s t and  may  a d d i t i v e nature.  + ( l - ip* + p * | ) e ] / [ v ( l - ip*  2* /e e )  The  mn(p  and  com-  components  f u n c t i o n s i s somewhat more d i f f i c u l t .  some l e n g t h y  2  e  form X  of i t s l i n e a r  cost  stage  becomes  - v  2 1  functions into  f u n c t i o n C i n t o stage  by v i r t u e  of the v a r i a n c e  variance  left  we the  to  be  as a f u n c t i o n o f and  n(p  + s)  These a r e + s) = n  2  the  psu's  shown i n a and  mn .) 2  expressions regarded  as  on  the  the  left  hand s i d e of  inequations-  " s t a t e s " of the model w i t h Xj =  (26) (X  1 1 5  and X  1 2  )  (27)  78  Figure  2. T h e s t a g e d i a g r a m SPR  for  design  the  optimal  problem.  X  l t  X  Transition  a1  = X, =0 0  = X  M  = 0  functions:  . V -  x** = x „ - J {[p(e ,- v* a  Lv.(i -'Hf • P*;*)  X|f = X X  v  :  X  u  1 t  2%7eI7en) • 0 - ^* • »£ p)e«] / - e„ o - O]} • vj|  - s  = X,* - {[p/(1 -  X*i = X** .• s  • p*l)] • V, - (9*/V,)}  79  and  X  = (X  2  2 1  ,X22)  states  pertaining  Computation o f the  states  t o (26) a n d (27),  o f the  model a t the  2 i s a two-stage process, as i n d i c a t e d first  = X  1 2  and  the  and  mn' s s u ' s i s  - i>\ + p**)]  - {[p/(l  v a l u e o f the  first  state X  Xn  a  tilde  = X  x  + V  2.  v a l u e o f the  The v a l u e o f t h e  mn" s s u ' s i s  x  = x  state X  t o measurement o f n ' psu's  (29)  s t a t e o r o u t p u t a t s t a g e 1 o r 2;  t o s t a g e 1 o r 2.)  - [[{[p(e  tv (i - ^ 2  - v  21  2  Similarly, the  subsequent t o t h e remeasurement o f n " psu's  2  . 2  2 2  (28)  2  - s  1 2  [~] d e n o t e s a n i n p u t  second  and  and  on o c c a s i o n  - (e /Vi)}  x  subsequent  [~] d e n o t e s t h e i n t e r m e d i a t e  state without  22  inventory  s t a t e X j a t o c c a s i o n 2 subsequent t o remeasurement o f n " psu's and  X12  (The  in Figure  respectively.  2  - 2 ^ / e e ) + (1 - ^ + i|; p)e ]/ 2  21  22  2  2  + ^) - e i ( i - * 2 > ] } + v J P  2  22  (30)  2  a f t e r n ' p s u ' s a n d mn' s s u ' s i s X  = X  2 1  (31)  + s  2 2  ( N o t e X.. = s t a t e o f m o d e l a t j * " * s t a g e f o r 1  recall  that  n ' = n s a n d n " = np.)  state variables  X x  X Note t h a t model  and X  x  i 2  :  X  12  :X  theincidence  'Wilde & B e i g h t l e r S e p a r a t i o n o f the  s t a g e model cessive  (Figure  occasions.  2  £ 2 2  are, v  i  a  The i n i t i a l and f i n a l  states  of the  respectively: n  <_ V  s t a t e v a r i a b l e , and  2  d  x  io =  and X  0  2 0  =0  i d e n t i t y X. x(j+l)  = X.., i j  i , j = 1,2 e x i s t s i n t h e  [ 1 9 6 7 ] ) .  cost  and v a r i a n c e  2) o f s a m p l i n g w i t h  functions  partial  has produced a m u l t i -  r e p l a c e m e n t on two s u c -  T h e SPR o p t i m a l d e s i g n p r o b l e m w h i c h h a d t w o d e c i -  80 s i o n s , p a n d s , c a n now be d e c o m p o s e d  into  two p r o b l e m s c o r r e s p o n d i n g t o t h e  two s t a g e s o f t h e m o d e l , e a c h w i t h a s i n g l e  d e c i s i o n v a r i a b l e p o r s.  two s i n g l e - d e c i s i o n p r o b l e m s c a n t h e n be s o l v e d r e c u r s i v e l y simple  search procedures. At  of  stage  new p r i m a r y  1, t h e d e c i s i o n p r o b l e m sampling fi(X  for  units 1 1 5  X  2 1  )  stage  equations  l 2  ,X  2 2  )  equations  2 1  ] , with f ( X i , X 0  s o l u t i o n procedure  1 2  ,X  2 2  i s as f o l l o w s .  into  = 0 = X  10  Similarly,  that  f(X  ) p r e d i c t e d from l 0  ,X  2 0  )  = 0.  At  t h e optimum p r o p o r t i o n o f p r i m a r y that  {k p + f , ( X , X ) } 2  u  1 1  ,X  2 1  (33)  n  ) p r e d i c t e d from the  (28) and ( 3 0 ) .  the optimal d e c i s i o n  function  2 0  ( 3 2 ) a n d ( 3 3 ) c a n t h e n be s o l v e d r e c u r s i v e l y .  condition  o b t a i n s X,, B X is,  case  ] , with ( X  Consider  t h e decomposed d e c i s i o n p r o b l e m . final  0  i n this  min 0<p<l  (32)  2  Procedure  The e q u a t i o n s  the  t h e optimum p r o p o r t i o n  that  i s to find  =  a l l p o s s i b l e values of [ X  Solution  i s to find  t o be r e m e a s u r e d p * o n o c c a s i o n 2 s u c h 2  transition  ,X  1 1  ( 2 9 ) and ( 3 1 ) ;  f (X  for  i s done n e x t a s f o l l o w s .  0  2, t h e d e c i s i o n p r o b l e m  sample u n i t s  s* such  by c a l c u l u s o r  = min {k,s + f (X,„,X „)} s>0  a l l p o s s i b l e values of [ X  transition  of  The d e c o m p o s i t i o n  The  X  2 0  Hence, a t stage  1 2  At stage 1 state  1, X j  - s, t h a t i s , s * =  i s equal t o the input  X  2 x  5 X  S" = - X  2 0  2 1  into  + s  since X  2 2  = X  X  2  Substituting  function  the second 2 2  _  = 0.  stages  1 2  ( 2 9 ) one E X,,, t h a t  state.  = 0 = X  raax(X,i,  and f i n a l  since X  1, w =  0  transition  = 0, a n d s u b s t i t u t i n g  (31) provides  is,  the stage  the i n i t i a l  The  i )  2 s  .  transition  81 and  the d e c i s i o n problem i s s i m p l i f i e d  Proceeding with to f i n d  p* such  2  we know t h a t f (x,)  = k!(x  2  o f w.  the r e c u r s i v e s o l u t i o n , the stage 2 problem i s then  1 2  ) = min { k p + f ^ X j } p>0  from equation (28) 1 2  + P*|)].+ v  - {[p/(i - V  2  condition X  1  = V  2  t  -  t  (e /v )}) 2  (33)  x  i n (33) and s u b s t i t u t e (33)  ( 3 2 ) we o b t a i n t h a t f (X 2  Pj"  (32)  2  Now i f we s e t t h e i n i t i a l into  (31.1)  that f (X  but  that  = min (k,s) s=w  f,(w)  for a l l f e a s i b l e values  t o f i n d i n g s * = w such  1 2  ) = min { k p + p>0  i s equal  2  t o that value  Similarly, obtain  f (X 2  2 2  k . C f l - e2 /V  ) + [ p / ( l - 4<  +  2  t  2  p* )H 2  } .  (34)  2  o f p that minimises ( 3 4 ) .  i f we s e t X  2  = V  2  2  i n ( 3 0 ) a n d s u b s t i t u t e i n t o ( 3 2 ) we  that  )  = m i n { k p + k, p>0  p ( e 2 2 - v2 - 2 * 2 / e 2 2 e 2 1 ) +  ( l - ipj + p\p|)i  2  Again p " i s equal 2  v (i  -  2  t o that value  ^\  + p^l)  -  e  2  1  ( i -  } (35)  i> ) 2  2  of p that minimises ( 3 5 ) .  Hence a t t h e s e c o n d s t a g e 2 z = max(p ",p ") 1  and  the d e c i s i o n problem i s s i m p l i f i e d  2  t o f i n d i n g p* = z such  that  f ( z ) = min { ( k + k j ) p } p =z 2  for a l l f e a s i b l e values  2  o f z.  Once we h a v e f o u n d p-', we now t r a c e b a c k t o s t a g e Using  transition  functions  s t i t u t i n g p = p".  1 t oobtain s*.  ( 2 8 ) a n d ( 3 0 ) , we o b t a i n X j a n d X  Then s * w i l l  1  be g i v e n by  2 1  by sub-  82 s* = This completes the All variances we can  along  max(X j,-X l  solution  i t has  a r e known.  b e e n assumed t h a t n, m and However, i f t h e s e  planning  a future inventory right  be  obtained  by  an  to  be  cost.  of  The  ( n , m)  a l l variances  a r e not  from o c c a s i o n  identifying  and  co-  known, t h a t i s , 1,  then  n and  repeating the procedure  and  variances  and  that pair  the c o v a r i a n c e s  will  m described  which have  estimated. The  in  total  values  i t e r a t i v e procedure:  above f o r a l l f e a s i b l e v a l u e s the  ).  procedure.  are  minimizes  2 1  chapter  use  o f DP  i n optimal  SPR  design  problems w i l l  5 where a sample p r o b l e m i s s o l v e d .  be  better  understood  CHAPTER 5 SAMPLE PROBLEM In  this  developed  c h a p t e r we i n v e s t i g a t e  i n the preceding chapters  Attention w i l l occasions. and  be r e s t r i c t e d  The p r o b l e m  the a p p l i c a t i o n of the general to a specific  forest  t o the use of two-stage  i s t o determine  area.  Inventory data c o l l e c t e d  Columbia's Cranbrook P u b l i c  SPR on two s u c c e s s i v e  a 15-year p e r i o d ) i n  i n recent years  Sustained Yield  Unit  from  be u s e d t o  The s o u r c e  and n a t u r e  t h e d a t a and t h e d e t e r m i n a t i o n o f t h e optimum r e p l a c e m e n t  dynamic programming a r e d e s c r i b e d , and then formed b a s e d on t h e e x i s t i n g Cranbrook  data  British  (PSYU) w i l l  d e m o n s t r a t e t h e s o l u t i o n o f t h e sample problem. of  problem.  t h e c u r r e n t mean v o l u m e p e r ha  t h e p e r i o d i c change i n volume p e r ha ( s a y , over  a forest  inventory  theory  policy  sample c a l c u l a t i o n s  through  are per-  base.  i s one o f t h e 81 P S Y U ' s ^ i n B r i t i s h  Columbia.  I t contains  2 approximately land  506,006 ha o f crown  ( F o r e s t Survey  l a n d a n d 233,032 ha o f n o n - f o r e s t  and I n v e n t o r y D i v i s i o n ,  tree species include: (Tsuga  forest  spruce  1965).  (Picea engelmannii  The p r i n c i p a l  Parry), western  h e t e r o p h y l l a [ R a f . ] S a r g . ) , and s u b a l p i n e f i r ( A b i e s  [Hook.] N u t t )  i n i n t i m a t e m i x t u r e , and s t a n d s  contorta Dougl.). areas.  The u n i t  i s divided  forest  hemlock  lasiocarpa  of lodgepole pine  (Pinus  i n t o 40 c o m p a r t m e n t s o f v a r y i n g  The number o f s a m p l e s e s t a b l i s h e d v a r i e d  from  c o m p a r t m e n t t o com-  partment . Royal  ^Timber R i g h t s and F o r e s t P o l i c y C o m m i s s i o n on F o r e s t R e s o u r c e s , 2  Crown f o r e s t  i n B r i t i s h Columbia, V i c t o r i a , B.C.  land i s land belonging 83  t o the state  V o l . I , 1976.  o r government.  84 Several Forest to  i n v e n t o r i e s have b e e n c o n d u c t e d by t h e B r i t i s h C o l u m b i a  Service  (BCFS) i n t h i s  1964 i n c l u s i v e ,  were e s t a b l i s h e d  basic  inventory  since  462 s a m p l e s ( f i x e d  in all  in all  1952.  During the p e r i o d  1953  a r e a , h a l f - a c r e and t w o - f i f t h  timber types.  samples were e s t a b l i s h e d were e s t a b l i s h e d  unit  D u r i n g t h e 1979 i n v e n t o r y ,  the timber types.  176  (Permanent sample  i n 1968 and t h e f i r s t  r e m e a s u r e m e n t was  technique used over a l l  t h e s e y e a r s was  i n 1978.)  surveys are a v a i l a b l e  plots  stratified  random than the  D a t a s u m m a r i e s b a s e d o n t h e 1964 a n d on t h e  by s a m p l e number and t h e a t t r i b u t e s m e a s u r e d  v o l u m e p e r ha ( t o v a r i o u s per  point  The  s a m p l i n g , w i t h m a t u r e t i m b e r t y p e s b e i n g s a m p l e d more i n t e n s i v e l y immature o r o t h e r t y p e s .  acre)  l e v e l s of u t i l i z a t i o n )  1979  included  and t h e number o f s t e m s  ha. For  our purposes, the compartments w i l l  samples w i t h i n  the compartments, the ssu's.  constitute  the p s u ' s and t h e  I n o t h e r w o r d s , we h a v e a  t w o - s t a g e SPR d e s i g n w i t h u n e q u a l - s i z e d p s u ' s , b u t e q u a l - s i z e d The  1964 s a m p l e d a t a w i l l  and  t h e 1979 s a m p l e d a t a as t h e s e c o n d o c c a s i o n m e a s u r e m e n t s .  tive w i l l ,  therefore,  be a s s u m e d  t o be t h e f i r s t  be t o d e t e r m i n e t h e c u r r e n t  occasion  ssu's. measurements The  objec-  ( 1 9 7 9 ) mean v o l u m e p e r  ha a n d t h e c h a n g e i n mean v o l u m e p e r ha b e t w e e n 1964 and 1979  (that i s ,  over a 15-year p e r i o d ) . T w e n t y - s e v e n o u t o f t h e 40 p s u ' s were of  s a m p l e d i n 1964 a n d t h e number  s a m p l e s p e r c o m p a r t m e n t r a n g e d f r o m 1 t o 25 (mean = 1 1 ) .  c o m p a r t m e n t s were s a m p l e d w i t h an a v e r a g e o f 9 s s u ' s p e r p s u . 35 p s u ' s ,  16 h a d n o t b e e n s a m p l e d i n 1964.)  were a c t u a l l y  r e m e a s u r e d i n 1979.  t o be as f o l l o w s :  (Of t h e  None o f t h e 1964 s a m p l e s  For the purposes of d e t e r m i n i n g the  a p p r o x i m a t e number o f p s u ' s t o r e m e a s u r e , we size  I n 1 9 7 9 , 35  shall  take  the i n i t i a l sample  p s u ' s n = 27 a n d s s u ' s p e r p s u m = 1 1 .  It will  85 be further assumed that m remains constant over the two successive occasions. Before performing  the optimization as described in chapter 4, the  following  information i s required: (a)  As always for planning an inventory, a knowledge of the estimates  of the population parameters for the forest area to be sampled i s required. We  shall assume the following estimates of the population parameters i n  the Cranbrook PSYU: .(i)  average volume per ha u  (ii)  = 475.41 m  3  y  periodic change (over 15 years) of volume per ha + 321.81 m /ha 3  (iii)  variance of volume per ha between psu's a  = 189876.06, (i=l,2)  2  i variance of volume per ha between ssu's within the psu's o  (iv)  2  i  =  1189.56, (i=l,2) (v)  c o r r e l a t i o n between the effects due to the psu's in 1964 Pi  (vi)  =  and  1979  0.95  c o r r e l a t i o n between the effects due to the ssu's within the psu's in 1964 and 1979  p  2  =  0.85.  Using the above information we determine that • and  6 -, 2  \p  2  = 6  2 2  = [(189876.06/27) + (1189.56)/(27 x 11)] = 7420.65  = [(11 x 0.95  x 189876.06 ) + (0.85 x 1189.56 )]/  [27 x 11 x / 6 (b)  2  2 1  6  2 2  ] =  2  0.88  In addition, we require the allowable errors for current mean volume  3 and change.  The BCFS. states  that the standard allowable sampling error  for estimates of gross volume i s ± l07 at the 957» confidence level per o  3 Guidelines for Forest and Range Inventory in B r i t i s h Columbia, Inventory -Branch, Ministry of Forests, B.C.  1980.  )  unit. we  In order  shall  per  86  assume  t o be w i t h i n t h e r a n g e o f t h e d a t a an a l l o w a b l e  e r r o r o f + 30% at the 95% confidence  mean v o l u m e p e r ha o r p e r i o d i c c h a n g e p e r h a .  allowable variance  levels  f o r p e r i o d i c change (over  It will  occasion  i m p l i e s that the  a l s o be a s s u m e d  =  2  5085.116  15 y e a r s ) i s  V = {[0.30 x 3 2 1 . 8 l ] / 2 } 'g (c)  This  level  f o r e s t i m a t i n g c u r r e n t mean v o l u m e p e r ha i s  V- = { [ 0 . 3 0 x 4 7 5 . 4 l ] / 2 } and  a v a i l a b l e , however,  =2330.137.  2  that the t o t a l  (inventory) i s given  by t h e s i m p l e  cost p e r t a i n i n g to the second cost  function  (equation[23])  C = kjp + k s " 2  w h e r e k^  i = 1,2  a r e as d e f i n e d  some new  notation.  earlier  i n equation  (23).  We  introduce  Let . X = k ./k 1  2  and C Then, t h e c o s t  relation  expression  useful,  2  = Xp' + s  t o be m i n i m i z e d .  This  e s p e c i a l l y when t h e a b s o l u t e  Now  we  cost values  relation  F i n d p, s > 0 such t h a t C  problem.  The p r o b l e m i s t h a t  a n d i s f o r m a l l y d e f i n e d as f o l l o w s : i s minimized  and s u c h t h a t t h e  e r r o r s o f c u r r e n t mean v o l u m e p e r ha and p e r i o d i c  c h a n g e i n v o l u m e p e r h a a r e met.  i s more  are not a v a i l a b l e ;  t h e o p t i m u m number o f p s u ' s t o r e m e a s u r e a n d new  on t h e s e c o n d o c c a s i o n ,  allowable  form of the cost  s t a t e the i n v e n t o r y p l a n n i n g  of d e t e r m i n i n g take  C/k .  a b o v e c a n be w r i t t e n as C  the  =  ones t o  Following  diagram f o r t h i s The  sample p r o b l e m i s s i m i l a r  state transition for current  X  l  t o that  0  =  =  2  s  22  (e  2  2  2  2 2  2  C  -  =  X  stage  1  a l l feasible  associated  2 I  ( l - + |)  t h e dynamic programming s o l u t i o n as f o l l o w s .  1, we w i s h t o f i n d C' (X  for  - e  2  + S .  2 1  We now p r o c e e d w i t h At  2 2  ] P  2 0  2  - 2 ^ / e , e ) + ( l - \p| + p ^ ) e  - v  l  v ( l - *f + V ) X  2.  - [ p / ( l - *| + p*|)]  l 2  l i -  X  P  X i  shown i n F i g u r e  functions are:  f o r p e r i o d i c change i n volume p e r ha X  X  the stage  mean v o l u m e p e r h a X j  x,, = x  and  i n c h a p t e r 4,  the o p t i m i z a t i o n procedure developed  with  U )  X  2 1  )  values  s*(X  1 1 9  X  2 1  ) such  that  = m i n {s + C „ (X, ,X ) } s>0 1  0  of ( X  l l  ,X  2 X  the i t h stages.  ),  20  where C .  i = 0, 1 are- t h e c o s t s  Using the t r a n s i t i o n  f u n c t i o n s we s e e  that X  1 0  X Q 2  implying  - s = 0  = X  + s  2 X  0  =  that s  Since  = X„  C' (X 0  1 0  ,X  2 0  )  w  = (Xj j ,  —X j) 2  = 0, t h e s o l u t i o n f o r t h e s t a g e  1 problem i s given  by t h e f u n c t i o n w = and  the associated  raax[X,j,  that  2  cost i s C j(w)  We n o t i c e  - X !]  = min {s} s=w  s* has been d e t e r m i n e d as a f u n c t i o n o f p.  Hence we  88 proceed stage  to stage  2,  1 to f i n d At  stage  to determine  2, we  x and  c o m p u t e C'^w)  (In  this  1 2  p*  = 1.459  and  = 0.67,  the  we  11  see  on  the  C' (X 2  X and  ,X  2 2  ) such  that  + C'jCw)}  We  set  ) for a l l feasible values  2 2  2  = $500.)  I t can p*  be  = 0.67,  t h a t s* = 0.55.  = 15 new  The  seen from Table and  t r a c i n g back to stage  that given  The  giving a total  initially  total  o f 33  c o s t of the  1 using  solution. n = 27  psu's,  = 18 p s u ' s and  psu's,  w i t h an  inventory after  18 p s u ' s was  taken  In a d d i t i o n ,  from the  take  average  the  second  sample ssu's  initial  27  psu's,  15 p s u ' s m e a s u r e d i n 1979  w e r e t a k e n a t random t o g e t h e r w i t h t h e i r  ssu's.  The  i n e a c h o f Che s e l e c t e d p s u ' s a r e  I t s h o u l d be  n o t i c e d that the  (see Appendix I I ) .  First  remeasurement  sumdata for  labelled  Y.  sample data g i v e n  statistics.  o c c a s i o n measurements are  but  volumes  18 p s u ' s w e r e s i m u l a t e d u s i n g t h e e x i s t i n g v o l u m e - a g e c u r v e s  second o c c a s i o n  are  $181,663.  ssu's.  i n Appendix I.  1.  I I I that with  This completes the  imply  of 0 < p <  enumeration r e s u l t s  must r e m e a s u r e ( 0 . 6 7 ) ( 2 7 )  psu's,  per psu.  From t h e sample  l 2  = k  t  III.  at each of the  the area  to  0  = 0,  2 2  together with t h e i r  marized  ).  2 2  =  A random s a m p l e o f  p e r ha  ,X  1 2  22  s e c o n d o c c a s i o n we  i n 1964  t r a c e back  22  occasion i s approximately  not  (X  solutions obtained  ssu's  Up  ) = min p>0  = e /v- = 1.459  and  X  2 2  12  i n Table  (0.55)(27) of  ,X  e x a m p l e , X = 1, k  The on  1 2  2  a l l f e a s i b l e values of  X  X  to f i n d p * ( X j , X  wish 2  summarized  then  s*.  C' (X  for  the v a l u e o f p * and  i n A p p e n d i x I we  o b t a i n the  following  Table I I I .  Enumeration  Results  •59  .58  19.613  2912567.00  3000181.00  .60  ,58  7.176  1065673.00  1154773.00  .61  ,57  4.038  599707.00  690291.81  .62  ,57  2.608  387341.06  479410.87  .63  .56  1.790  265831.18  359386.06  .64  ,56  1.260  187150.50  282190.31  .65  .56  0.889  132045.31  228570.18  .66  .55  0.615  91296.18  189306.06  .67*  ,55*  0.404  82167.50  181662.37*  .68  ,55  0.236  81601.43  182581.25  .69  54  0.100  81047.12  183512.00  Note:  ( 1 ) T h o s e v a l u e s m a r k e d w i t h an a s t e r i s k (*) a r e t h e o p t i m u m solutions. ( 2 ) C j i s t h e c o s t o f m e a s u r i n g t h e new (ns) psu's and t h e i r a s s o c i a t e d ssu's C' i s the t o t a l cost of measuring the 2  n(p + s) psu's  and t h e i r  ssu's.  90 x"..  = 414.64  x'.. = 4 0 6 . 9 0  y"..  = 420.89  y'.. =  a  = [ Z ( x . - x . . ) ] / ( n p - 1) = 1 7 3 1 8 . 0 3 3 6 i=l  2  2  a i  a  124.96  1  np 2 (y. 1 =1  -  = [  2  y..) ]/(np  - 1) =  2  17814.7747  1  np  6  m. i np = [ Z Z ( x . . - x . ) ] / [ Z m.) . . . . li i . , i i = l j=l i=l  2  2  e, 1  - np] = 28859.2144  np  6  m. , . l n(p+s) = [ Z Z ( y . . - y . ) ] / [ ( Z m.) i=lj=l i=l  2  - np] = 31673.3209  2  C z  1  J  1  1  n p  P, = [ Z (x. - x . . ) ( y . - y . . ) ] / [ ( n p 1=1  P  X  1  1  I' We  = 1079.65034  e'  1  = 1116.92573  §'  2  =  ]=0.96  we o b t a i n  that  .= 1 1 5 9 . 7 3 6 4 2 =  1091.27952  statistics  to proceed  with  t h e computa-  o f t h e e s t i m a t e s o f ( i ) c u r r e n t mean g r o s s v o l u m e p e r h a ( 1 7 . 5  cm +) u terval,  y  and ( i i ) p e r i o d i c  change  i n mean v o l u m e o v e r  t h e 15 y e a r i n -  A.  C u r r e n t mean v o l u m e Recall  for  a  1084.22304  now h a v e a l l t h e n e c e s s a r y  tion  (i)  2  ] = 0.99 i  1  And u s i n g t h e a b o v e s a m p l e s t a t i s t i c s  §"  a 1  m. l np Z ( x . . - x . ) ( y . . . - y . ) ] / [ ( z m. - n p ) 8 j=l J ^ i=l  np = [ Z i=l  2  - 1)8  that  the minimum-variance l i n e a r  units' of unequal y  2  unbiased  size i s  = a*(x'..  - x"..) + c * y " . .  + ( 1 - C")y'..  estimator of  91 w h e r e a* and c * a r e as d e f i n e d Using  earlier.  t h e s a m p l e e s t i m a t e s o f 9'^,  9"^  i =1,2  and B', we  obtain that  c * = 0.63183 a* = 0.19947 and  hence  y  = [ 0 . 1 9 9 4 7 x ( 4 0 6 . 9 0 - 414.64)] + ( 0 . 6 3 1 8 3 x 4 2 0 . 8 9 ) + ( 0 . 3 6 8 1 7 x  2  124.96)  36 = 310.39  m /ha. 3  The v a r i a n c e o f t h i s Var(y  2 j l  )  = a* (e',/q 2  estimate  i s given  by  + 9" /p) + c * 9 " / p 2  2  2  730.499. By t a k i n g / V a r ( y ) = 27.0277  + (1 - c * ) 6 ' / s 2  2  -  2a*c*B'/p  =  and u s i n g t h e t - v a l u e a t t h e 957» c o n f i d e n c e  2  level  of approximately  2, we  obtain that  c u r r e n t v o l u m e p e r ha = 310.39 (ii)  Periodic  A for units g2  m. 3  c h a n g e i n mean v o l u m e p e r h a  Again, r e c a l l of  ± 54.055  that the minimum-variance l i n e a r  of unequal s i z e 0  unbiased  estimator  is  = e*y".. + (1 - e * ) y ' . .  + f * x " . . - (1 +  f*)?..  36 w h e r e e* a n d f * a r e as d e f i n e d Using  earlier.  t h e s a m p l e e s t i m a t e s o f 0' ?* =  -0.99664  e * = 0.98514  and 9"^  i = l , 2 a n d B', we  see t h a t  92 and g  2  hence ••= ( 0 . 9 8 5 1 4 x 4 2 0 . 8 9 ) + ( 0 . 0 1 4 8 6 x 124.96) - ( 0 . 9 9 6 6 4 x 4 1 4 . 6 4 ) -  (0.00336 x 406.90)  = 1.878 m / h a 3  The v a r i a n c e o f t h i s  estimate  i s g i v e n by:  V a r t g ^ ) = e* 9" /P + ( l - e * ) 9 ' / s 2  2  2  2  + b- Q"Jp  2e*f*S'/p  + (1 + f *) 6' ,/q +  2  2  = 41.2828. By t a k i n g / V a r ( g level  ) = 6.4251 a n d u s i n g  of approximately change  The  2  results  (living  2, we o b t a i n  the t-value  a t t h e 957- c o n f i d e n c e  that  i n v o l u m e i n 15 y e a r s = 1.878 ± 12.8502 m / h a . 3  of the calculations  trees  only,  indicate  that  the t o t a l  dbh 17.5 cm +) i n t h e f o r e s t  w h o l e - s t e m volume  land area  (506,006 ha)  o f t h e C r a n b r o o k PSYU i n 1979 was (506,006) x (310.39 ± 54.055) = 157,059,202.3  ± 27,352,402.29  m  3  a n d t h e c h a n g e i n t h e v o l u m e b e t w e e n 1964 a n d 1979 was 506,006  x (1.878 ± 12.8502) = 950,279.3  The c h a n g e i n v o l u m e i s a r e s u l t present  etc.),  through n a t u r a l  tut(logging),  measurable s i z e ,  i n this  the g r o w t h components According and t h a t of  lost  (volume o f t r e e s  17.5 c m ) .  (volume o f t r e e s growing  windfall, into  For a complete d e f i n i t i o n of  see Beers (1962).  t o t h e BCFS r e c o r d s ,  through m o r t a l i t y  the t o t a l  volume o f t i m b e r  logged  b e t w e e n 1964 a n d 1979 i n t h e f o r e s t  t h e C r a n b r o o k was e s t i m a t e d a t 2 4 , 0 2 3 , 5 8 4 m . 3  i n g r o w t h and s u r v i v o r g r o w t h volume o b t a i n e d plots  3  causes such as o l d age, i n s e c t s ,  and i n g r o w t h  case  m.  o f s u r v i v o r g r o w t h ( i n c r e m e n t on t r e e s  a t b o t h t h e 1964 a n d 1979 i n v e n t o r i e s ) , m o r t a l i t y  rendered useless fire,  ± 6,502,348.39  An e s t i m a t e o f t h e  from the permanent  i n t h e C r a n b r o o k was 4.39 m / h a / y e a r , g i v i n g  area  a total of  sample  93 (4.39 x 506,006 x 15) = 33,320,495 m  3  between 1964 and 1979.  net change in volume between 1964 and 1979 = 9,296,911 m . 3  Thus the  i s 33,320,495 +(-24,023,584)  This result i s s l i g h t l y higher than the upper 95% con-  fidence l i m i t of the estimate obtained through the sample problem calculation.  No reasonable independent  check was available for the sample  problem current volume estimates, since the results of the BCFS 1979  inven-  tory of the Cranbrook have not been released yet. The confidence l i m i t s on estimates of current timber mean volume and change were constructed based on the central l i m i t theorem that the probability d i s t r i b u t i o n of the SPR estimators was  s u f f i c i e n t l y close to  the normal d i s t r i b u t i o n and for p r a c t i c a l purposes the t-value of 2 was 4 good enough.  The high confidence limits on the estimates of the current  mean and on the change may  be because the sampling fraction of the psu's  was r e l a t i v e l y high and hence i n f l a t e d the variance estimates.  A further  discussion of the various aspects of the sample problem and of the theory derived in chapters 3 and 4 i s given i n the next chapter.  sions.  T. Cunia, Lecture notes, Workshop on sampling on successive occaColorado State University, July 1979.  CHAPTER 6 DISCUSSION AND CONCLUSION The  theory of successive forest inventories with p a r t i a l replacement  of units presented by Ware and Cunia (1962) has been extended to use multistage sampling designs (with p a r t i a l replacement of the primary sample units).  Multistage  designs have many desirable features p a r t i c u l a r l y  for an inventory of large forest areas.  These designs ( i ) provide  ulti-  mate sample units that can be cost e f f i c i e n t l y measured, e s p e c i a l l y when construction of the sampling frame i s d i f f i c u l t or impossible, and ( i i ) cluster the ultimate sample units into larger sample units to reduce the travel cost between measurement  units.  Further, multistage  designs are  useful i n incorporating data from high- and low-altitude and ground level sources simultaneously  for e f f i c i e n c y .  This i s p a r t i c u l a r l y more so i f  variable p r o b a b i l i t i e s of selection are used at the various stages of the multistage  design.  Simple random sampling was assumed at each stage i n  this thesis f o r s i m p l i c i t y of presentation.  A l o g i c a l extension  of the  theory developed here i s to use variable p r o b a b i l i t y sampling at the v a r i ous stages.  This would, for example, involve extending the work of  Langley (1975, 1976) for one-occasion sampling to successive Multistage  occasions.  sampling i s often applied to large regional and national  inventories i n order to reduce cost.  The major potential disadvantage,  however, i s that a small sample of psu's may leave many areas of the target 94  95 p o p u l a t i o n unsampled. divisions  T h i s makes t h e p r o v i s i o n o f i n f o r m a t i o n on  sub-  ( f o r example, compartments) of the p o p u l a t i o n d i f f i c u l t .  If  the s u b d i v i s i o n s areas not may  be  a r e t h e same as t h e p s u ' s , m a k i n g i n f e r e n c e s f o r t h e  sampled  i s usually  p o i n t e d out  t h e t e c h n i q u e o f SPR with different etc., units  gives rise  t o a number o f s a m p l i n g  combinations of replacement  over  the case  time.  i n which  the p r i m a r i e s . prove  apply and  designs  i n c h a p t e r 1, i n t h e c a s e o f m u l t i s t a g e d e s i g n s ,  For p r a c t i c a l  Partial  t o be  of primary, secondary,  reasons,  i t was  tertiary,  decided to consider  tertiary,  replacement  etc. units corresponding to  at a l l stages of a m u l t i s t a g e design  t o o c o m p l e x i n t h e o r y and p r o h i b i t i v e l y  ( P r o f e s s o r T.  alternatives,  o n l y t h e p r i m a r y u n i t s were p a r t i a l l y r e p l a c e d  w h i l e m a i n t a i n i n g the secondary,  may  In such cases o t h e r  employed. As was  only  impossible.  C u n i a — p e r s o n a l communication).  too e x p e n s i v e  In a d d i t i o n ,  to  Singh  K a t h u r i a (1969), assuming  e q u a l sample s i z e  and  both o c c a s i o n s i n a two-stage  design, concluded  that u n l e s s the w i t h i n - p s u  v a r i a n c e and and  correlation,  efficient case  c o r r e l a t i o n were l a r g e r  to between-psu v a r i a n c e  t h e e s t i m a t e o f t h e c u r r e n t mean was,  i n the case of p a r t i a l  of p a r t i a l  in relation  e q u a l v a r i a n c e on  replacement  replacement  of ssu's o n l y .  of psu's  i n g e n e r a l , more o n l y than i n the  ( T h i s c o n c l u s i o n seems  logical  s i n c e t h e r e i s a r e d u c t i o n i n b o t h b e t w e e n - p s u and w i t h i n - p s u v a r i a n c e due  to p a r t i a l  replacement  a n c e component i s a f f e c t e d F u r t h e r , Rana and relative  of the psu's, whereas o n l y the w i t h i n - p s u v a r i due  Chakrabarty  efficiencies  to p a r t i a l  replacement  (1976) c o n c l u d e d  of the  from a n u m e r i c a l study of  of v a r i o u s sampling p l a n s , g i v e n the  o f S i n g h and K a t h u r i a ( 1 9 6 9 ) , t h a t  ssu's.)  i f s a m p l i n g was  assumptions  i n e x p e n s i v e and  the  the  96 p r e c i s i o n of the estimates interest,  partial  replacement of only  other  procedures they  study  the r e l a t i v e  ing  o f t h e c u r r e n t mean a n d c h a n g e was o f m a j o r t h e p s u ' s was more e f f i c i e n t  c o n s i d e r e d i n most c a s e s .  efficiencies  from the d i f f e r e n t  I t w o u l d be u s e f u l t o  of the various  combinations  than  sampling  of p a r t i a l  alternatives  aris-  replacement of the d i f f e r e n t i  stage  units,  f o r e s t i m a t i n g both  current values  and change, under t h e  assumptions o f unequal s i z e and unequal v a r i a n c e The  special  Ware a n d C u n i a here.  c a s e o f o n e - s t a g e SPR as p r e s e n t e d ,  ( 1 9 6 2 ) c a n be o b t a i n e d  In p a r t i c u l a r ,  when a  a.  result  presented  becomes o n e - s t a g e SPR becomes p.  2  This  i s not  I  surprising  since simple  the m u l t i s t a g e Although (best)  t h e t w o - s t a g e SPR d e s i g n  occasions.  f o r e x a m p l e , by  from the g e n e r a l  ( i =1,2) a r e s e t t o z e r o a n d h e n c e *  2  on s u c c e s s i v e  was u s e d w i t h i n e a c h s t a g e o f  design. t h e o b j e c t i v e o f t h e s t u d y was t o p r e s e n t  l i n e a r unbiased  estimators, biased of t h e r a t i o  random s a m p l i n g  estimators  or unbiased,  minimum-variance  i n a m u l t i s t a g e SPR d e s i g n , o t h e r p o s s i b l e  were c o n s i d e r e d .  e s t i m a t o r was i n v e s t i g a t e d .  Specifically,  I f i ti s realized  the use  that the  BLUE f o r c u r r e n t mean i s a w e i g h t e d a v e r a g e o f a r e g r e s s i o n d o u b l e ing  estimate  logical sampling  a n d a mean b a s e d on c u r r e n t o b s e r v a t i o n s  ratio  estimate  a n d a mean o f c u r r e n t  the  values  o r change.  Further,  amount o f t h e b i a s was e x p e c t e d  increased. estimate  However, the w e i g h t s  observations  ( p r e c i s e ) under c e r t a i n  t h e e s t i m a t o r based on t h e r a t i o  current  i t seems  t o p o s t u l a t e an e s t i m a t o r , one w h i c h i s a w e i g h t e d a v e r a g e o f a d o u b l e  BLUE was n e g l i g i b l y more e f f i c i e n t than  o n l y , then  sampl-  estimate the r a t i o  only.  The  conditions  f o r estimating e i t h e r the e s t i m a t o r was b i a s e d ;  t o be n e g l i g i b l e when t h e s a m p l e  o f t h e e s t i m a t o r based on t h e r a t i o  w e r e d e r i v e d by m i n i m i z i n g  the variance  function of the  size  97 e s t i m a t o r of e i t h e r t h a t b i a s was  c u r r e n t mean o r c h a n g e .  zero.  implicitly  assumed  More a p p r o p r i a t e v a l u e s o f the w e i g h t s w o u l d  b e e n o b t a i n e d i f t h e mean s q u a r e e r r o r ( o r c h a n g e ) was  I t was  minimized instead.  (MSE)  have  f u n c t i o n o f c u r r e n t mean  For example,  i n estimating  the  cur-  r e n t mean, t h e f u n c t i o n t o be m i n i m i z e d w o u l d h a v e b e e n ( s t a t e d h e r e out d e r i v a t i o n ) f o r a two-stage MSE(y  2 r  SPR  ) = Var(y  with-  design 2 r  ) + [E(y  2 r  ) - u ]  2  y  However, i n p r a c t i c e and t o t h e o r d e r o f a p p r o x i m a t i o n u s e d , t h e v a l u e s of  the weights d e r i v e d  assuming  b i a s t o be z e r o ( o r n e g l i g i b l e ) a r e  suf-  ficient . Although was  slightly  t h e e s t i m a t o r o f t h e c u r r e n t mean b a s e d on t h e r a t i o e s t i m a t e  less e f f i c i e n t  c o m p u t a t i o n o f £3  YA  ...,h) a r e h i g h a n d  and  3  AY  t h a n t h e BLUE, i t i s s u g g e s t e d t h a t i s costly  the v a r i a n c e s 6 ^  a n d when c o r r e l a t i o n s ( i = 1,2)  p. J  where  (j =  from o c c a s i o n to o c c a s i o n  a r e r o u g h l y t h e same, t h e e s t i m a t o r b a s e d on t h e r a t i o e s t i m a t e may used.  B o t h Sen e t a l . ( 1 9 7 5 ) and W o o d r u f f  Further, Arvanitis  and F o w l e r  (1979: 307)  1,2,  (1959)  be  s h a r e t h e same v i e w .  state:  B i a s e d s a m p l i n g e s t i m a t o r s a r e u s u a l l y s u r r o u n d e d by a v a g u e , c o n t r o v e r s i a l meaning which works a g a i n s t t h e i r acceptance as more e f f i c i e n t t h a n u n b i a s e d o n e s i n c e r t a i n c a s e s . Most of the t i m e , the main e f f o r t of the s a m p l e r s i s t o employ minimum-variance unbiased e s t i m a t o r s . However, b i a s e d e s t i mators have a p l a c e i n s a m p l i n g . What i s o f t e n o v e r l o o k e d i s that t h e o r e t i c a l l y u n b i a s e d e s t i m a t o r s may l e a d t o r e s u l t s w i t h a c o n s t a n t o r b u i l t - i n u n d e t e c t e d b i a s w h i c h c o u l d e x c e e d by f a r the s a m p l i n g e r r o r . The policy used  method a d o p t e d  f o r the d e t e r m i n a t i o n of t h e optimum  o v e r time used dynamic  i n other forestry  programming.  problems,  the optimum t r e e b u c k i n g p o l i c y  f o r example,  replacement  Dynamic programming has  been  i n the d e t e r m i n a t i o n of  ( P n e v m a t i c o s & Mann, 1 9 7 2 ) , a n d  i n the  98 determination  of  optimum l e v e l s of g r o w i n g  U s i n g dynamic programming, the mined f o r s i m u l t a n e o u s l y This and  will  not,  constraint  T h e r e i s no optimal  (see of  functions;  the  this The  and  1 2  equations  the  X  the  2 2  t a k e on and  obtained s u c h as  the  entire  values  obtained  be  deter-  of i n t e r e s t .  objective  on  the  increase.  derived  dynamic programming cost  the  are  known f o r  s o l u t i o n s are  example, i f the t  (cost)  and  V ,  deter-  state  respectively  2  to the  new  values  without necessarily  having  problem. (chapter  5)  c a l c u l u s methods.  For  However, n o n l i n e a r  the  are  instead. problem  more t h a n  m e t h o d s become d i f f i c u l t  ( s u c h as  t o use  c o m p l e t e e n u m e r a t i o n ) may several  be  decision variables,  a f t e r a dynamic programming  was  search methods,  two-decision  H o w e v e r , when t h e r e  problems w i t h  possible  have been u s e d  to solve  d e c i s i o n v a r i a b l e s , these c l a s s i c a l simple search procedures  optimum r e p l a c e m e n t p o l i c y  could  i t w o u l d have been e a s i e r  enumeration i s only  For  other than V  complete enumeration.  alternative.  associated  1968).  however,  [35]), a s o l u t i o n c o r r e s p o n d i n g  Golden s e c t i o n search,  classical  i n t o the  state variables.  sample problem  using  Further,  only  the  analysis  state v a r i a b l e s , since  the  the  computation w i l l ,  built  p o l i c y and  of  [34]  the  I n the  the  of  s t a t e v a r i - a b l e s w o u l d e a s i l y be  to r e - s o l v e  the  cost  variables  s e p a r a b i l i t y of  is automatically  optimal  functions  variables X  more t h a n two  a f f e c t the  f e a s i b l e v a l u e s of  m i n e d as  estimating  in general,  policy;  (Amidon & A k i n ,  optimum r e p l a c e m e n t p o l i c y can  need f o r a subsequent s e n s i t i v i t y  formulation. all  stock  by two and  the however,  decomposition,  N e m h a u s e r (1966). The  major drawbacks of  N e m h a u s e r (1966) a r e sary  f o r the  the  d y n a m i c p r o g r a m m i n g as  s e p a r a b i l i t y and  d e c o m p o s i t i o n of  pointed  out  by  monotonicity conditions  an N - s t a g e p r o b l e m i n t o N p r o b l e m s .  necesIf  99 the number of o c c a s i o n s in general s a t i s f y The  i n v o l v e d i s not  l a r g e , the v a r i a n c e f u n c t i o n s should  these c o n d i t i o n s .  problem of d e t e r m i n i n g  the optimum sample u n i t  s i z e at each  stage of the m u l t i s t a g e d e s i g n has not been t r e a t e d here. has  been d i s c u s s e d to some extent  fices  to mention here  tors,  such as sample c o e f f i c i e n t  each l e v e l ,  and  ....  i n some b a s i c sampling  t h a t the optimum u n i t  subject  texts.  It suf-  s i z e s depend on s e v e r a l f a c -  of v a r i a t i o n , c o s t of measurement at  other p r a c t i c a l c o n s i d e r a t i o n s .  should be determined  The  from the e x p e r i e n c e  The  optimum u n i t  of the i n v e n t o r y manager  size and  a f t e r c o n s i d e r i n g the f a c t o r s i n d i c a t e d above. The  d e r i v e d theory was  by working through  illustrated,  a sample f o r e s t  i n v e n t o r y problem.  s u c c e s s i v e i n v e n t o r i e s d i d not permit t i o n of the d e r i v e d theory. existing  data  The  f o r a two-stage SPR  an i d e a l p l a n n i n g and  sample problem was  s e t , so that s e v e r a l assumptions had  i n order to determine the optimum replacement the number of ssu's per psu was were such existing  timber  to f i t an  to be made. i t was  of  implementa-  For  example,  assumed t h a t  the i n i t i a l p o p u l a t i o n  In the same problem, i n t e r e s t  volume and  v a r i a b l e s of i n t e r e s t stems per ha,  nature  estimates  w i t h i n the range of  the  However, the r e s u l t s o b t a i n e d were w i t h i n the range of  v a l u e s expected. current  e q u a l , and  designed  policy,  that the r e s u l t i n g o p t i m a l p o l i c y was data.  The  design,  c e n t e r e d on e s t i m a t i n g  the change i n volume between o c c a s i o n s .  c o u l d have been estimated,  the  Other  f o r example, number of  b a s a l area per ha, number of deers, e t c .  The  term "change"  as used here means a type of growth which i s the d i f f e r e n c e between s t a n d i n g timber by  volume on o c c a s i o n two  Beers (1962).  The  and  o c c a s i o n one,  termed "net i n c r e a s e "  term c o u l d be a p p r o p r i a t e l y r e d e f i n e d i n order  to e s t i m a t e other components of f o r e s t growth ( f o r example,  ingrowth)  100 as d e f i n e d by B e e r s  (1962).  In the d e r i v a t i o n of the t h e o r y , parameters a  , a  2  Q  i  , and p. ( i , j i -  2  E  lem  a  values , a  2  a  i  E  data.  a r e known;  however, i t i s r a r e l y  t h e y h a v e t o be e s t i m a t e d . = 1,2) w e r e e s t i m a t e d  m are r e l a t i v e l y departs  large.  Furthermore,  effective  from t h e i r sampling  of m u l t i s t a g e sampling.  t o m u l t i s t a g e SPR was  for  the i n v e n t o r y of l a r g e f o r e s t the theory  ous s t a g e s partial  true  methods  SPR  further  that  from the matched  prob-  sample  the c a l c u l a t e d optimum  replacement of  values. f o r resource  t h e r e f o r e of p r a c t i c a l areas.  inventories include  interest,  from  particularly extend  of s e l e c t i o n at the v a r i -  and t o e x a m i n e t h e c a s e s  at other than  one-stage  I t w o u l d be u s e f u l t o  t o use v a r i a b l e p r o b a b i l i t i e s  occurs  of current  i n p r o p o r t i o n as t h e e s t i m a t e s  E x t e n s i o n of the theory  of the m u l t i s t a g e design;  replacement  estimates  The b i a s , h o w e v e r , i s s m a l l i f n a n d  f r o m t h e t r u e optimum  the parameters depart  versions  true  i  mean and c h a n g e a r e n o t u n b i a s e d .  Very  error or  I n the sample  T h i s means, i n g e n e r a l , t h a t t h e c o r r e s p o n d i n g  policy  t h a t the p o p u l a t i o n  = 1,2) w e r e known w i t h o u t  In r e a l i t y  a n d p. ( i , j  2  assumed  1  independent of sampling. these  i t was  the primary  stage.  i n which  101 REFERENCES A m i d o n , E. L., and A k i n , G. S. optimum l e v e l s o f g r o w i n g  1968. Dynamic programming t o d e t e r m i n e s t o c k . F o r . S c i . 1 4 ( 3 ) : 287-291.  A r v a n i t i s , L. G., and F o w l e r , G. W. 1979. Some a s p e c t s o f b i a s e d s a m p l i n g estimators. F o r e s t R e s o u r c e s I n v e n t o r y Workshop P r o c e e d i n g s , V o l . I . W. E. F r a y e r ( E d . ) , C o l o r a d o S t a t e U n i v e r s i t y . A v a d h a n i , M. S., and S u k h a t m e , B. V. p r o c e d u r e s w i t h an a p p l i c a t i o n S t a t . Soc. ( C ) , A p p l i e d S t a t . ,  1970. 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Chas.  APPENDIX I PROBLEM  SAMPLE ID  PSU  SSU  ID  106 DATA  WHOLE-STEM X  (1964)  18  -  8  13  275.8  18  -  8  14  81.4  18  -  8  15  199.0  18  -  8  16  146.0  18  -  8  17  155.8  18  -  8  18  143.3  18  -  8  19  177.1 112.3  18  -  10  5  18  -  19  5  82.9  18  -  21  3  240.1  18  -  21  4  206.9  18  -  25  41  303.1  18  -  25  42  343.5  18  -  25  43  432.5  18  -  25  44  297.7  18  -  25  45  267.2  18  -  25  46  322.8  18  -  25  47  374.5  18  -  25  48  330.6  18  -  25  49  321.2  18  -  25  50  344.7  18  -  25  51  466.4  18  -  25  52  395.0  18  -  25  53  344.4 471.1  18  -  25  54  18  -  25  55  355.4 343.8  VOLUME, Y  — — — — — — — — — — — •— — — -— — — — — — — — — — — — — —  18  -  25  18  -  25  57  355.1  18  -  25  58  411.2  18  -  25  59  391.7  18  -  25  60  549.8  18  -  25  61  527.2  18  -  25  62  420.5  -  25  63  526.4  18  -  25  64  267.4  19  -  7  2  325.1  19  -  7  3  142.0  19  -  7  4  398.1  -  7  5  503.3  -  7  6  704.1  19  -  7  7  254.9  19  -  7  8  748.3  — —  19 19  — — — — — — — — — — —  19  -  7  9  715.5  19  -  15  1  490.4  —  19  -  15  2  714.9  19  -  15  4  632.9  19  -  15  5  454.7  — — —  19  -  15  6  194.1  —  NOTE:  THE  VOLUMES  ARE  FOR  L I V I N G  3  (1979)  56  18  M /HA  TREES  ONLY,  DBH  17.5  CM+  107 SAMPLE PSU  19 19 19 19 19 19 19 19 19 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 18 18 18 18 .18 18 18 18 18  -  NOTE:  ID  15 15 15 15 15 15 15 15 15 5 5 5 •5 5 5 5 5 5 5 7 7 7 7 7 7 7 7 7 7. 7 7 7 7 7 7 7 7 • 7 7 6 6 6 6 6 6 6 6 6 THE  SSU  PROBLEM  ID  7 8 9 10 11 12 13 14 15 43 44 45 46 47 48 49 50 51 52 64 65 66 67 68 69 70 72 73 74 75 76 77 78 79 80 81 82 83 84 58 59 60 61 62 63 64 65 66 VOLUMES  DATA  WHOLE-STEM  X  (1964)  492.8 378.3 574.6 407.5 764.5 436.3 583.6 640.3 644.5 522.9 504.2 510.1 565.8 316.0 484.2 481.5 286.2 313.0 540.2 448.0 131.3 742.7 473.8 437.8 -475.1 253.0 525.9 365.3 625.8 328.6 480.4 352.3 359.8 330.5 465.1 401.8 449.4 551.1 494.0 291.4 479.2 409.7 428.3 431.4 340.5 773.2 572.7 562.0 ARE  FOR  L I V I N G  VOLUME,  M  Y (197S — — — —  -— —  — — —  --— — — —  -— --  •  —  •  — — — — — —  -— — — — —  278.8 401.2 468.3 394.3 442.7 314.4 732.5 537.6 579.2 TREES  ONLY,  DBH  17.5  CM+  108  SAMPLE PSU ID 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18  -  NOTE:  7 7 7 7 7 7 7 7 7 7 15 17 17 17 17 17 17 17 17 17 17 17 17 17 17 20 20 20 22 22 22 22 22 23 23 23 23 23 23 23 23 23 24 24 24 24 24 24  SSU 32 33 34 35 36 37 38 39 40 41 31 45 46 47 48 49 50 51 52 53 54 55 56 57 58 6 7 8 2 3 4 5 6 70 •71 72 73 74 75 76 77 78 15 16 17 18 19 20  T H E VOLUMES  PROBLEM  ID '  DATA  WHOLE-STEM V O L U M E , M / ' X (1964) Y (1979) 3  273.7 614.0 195.2 368.7 120.1 424.3 314.6 317.0 175.0 430.6 650.6 610.2 1.0 527.9 429.1 439.8 392.1 740.5 621.8 374.6 248.3 213.9 498.5 ' 527.2 955.5 107.2 114.2 129.0 201.6 370.8 554.5 183.8 473.6 331.5 406.1 386.2 423.8 619.1 442.0 453.7 328.0 536.0 486.9 403.4 214.3 362.2 354.2 318.9  329.5 598.7 133.4 333.2 74.7 424.8 366.6 326.1 198.4 414.1 662.4 675.5 16.0 546.6 345.1 474.3 242.1 741.2 567.2 401.4 278.9 197.8 517.1 552.8 43.9 74.4 108.2 105.5 248.4 379.2 629.7 261.8 468.9 404.8 495.3 364.6 451.5 663.4 458.7 497.0 376.3 578.1 433.1 350.2 236.4 343.2 324.5 350.5  A R E FOR L I V I N G T R E E S O N L Y ,  DBH 1 7 . 5 CM+  109  SAMPLE PSU 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 •18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18  -  NOTE:  ID 24 24 24 24 24 24 24 24 24 26 31 31 31 .31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 33 33 33 33 33 34 34 34 34 34 34 34 34 34 3434 34 34  SSU 21 22 23 24 25 26 27 32 33 4 10 11 12 13 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 24 25 26 27 28 20 21 22 23 26 27 28 29 32 33 34 35 36  T H E VOLUMES  ID  PROBLEM  DATA  WHOLE-STEM X  (1964)  461.9 426.0 378.7 293.8 387.3 367.7 312.7 150.2 471.9 153.3 505.0 381.1 312.6 378.6 354.5 544.0 139.2 578.5 515.3 593.4 638.8 514.1 . 482.7 301.2 414.7 376.9 304.7 391.0 339.8 307.6 315.2 282.5 428.7 56.0 223.6 509.0 484.9 539.7 385.3 387.2 587.3 510.0 554.5 508.9 249.3 562.8 383.1 325.0  VOLUME, Y  M /HA 3  (1979)  466.7 415.3 327.9 266.5 464.5 369.8 311.3 155.7 531.1 153.2 444.8 340.2 312.7 369.7 389.6 571.7 157.2 579.7 582.6 600.8 646.8 559.6 488.9 285.1 34'4.3 368.7 310.9 332.0 449.3 333.0 380.5 • 330.6 518.4 54.0 199.3 446.6 597.1 640.7 364.7 384.0 584.2 491.8 526.2 499.3 204.5 622.2 384.2 375.9  A R E FOR L I V I N G T R E E S O N L Y ,  DBH 1 7 . 5 CM+  110  S A M P L E P R O B L E M DATA . PSU 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18  -----  -  -----  NOTE:  ID  SSU  34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 35 35 35 35' 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 36 36 36 36 36 36 36 36 36  37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 6 7 8 11 12 - 13 22 23 23 24 25 26 27 29 30 31 32 33 34 35 7 8 9 10 11 12 13 14 15  ID  WHOLE-STEM V O L U M E , X  (1964)  571.2 630.2 268.6 444 . 5 875.8 385.5 449.0 600.3 546.3 449.4 584.3 331.3 507.7 187.4 449.9 711.9 463.3 561.0 246.6 491.1 387.3 325.6 358.1 ' 254.8 381.8 262.2 600.1 413.9 232.9 346.8 485.6 469.8 359.5 388.7 527.8 393.0 600.1 716.0 860.6 419.5 500.0 339.0 488.7 553.1 271.9 782.0 262.9 753.6  Y  M /HA 3  (1979)  677.7 634.3 250.3 487.2 881.4 418.0 454.2 599.8 535.2 453.5 598.8 314.7 488.7 153.4 570.0 736.5 498.4 597.0 271.8 454.6 408.6 310.0 387.6 303.3 "336.3 258.3 700.8 458.1 218.5 334.8 530.1 457.6 402.1 365.5 474.2 362.1 660.7 684.9 797.9 343.1 494.9 355.5 392.9 539.5 26-6.6 814.9 350.3 770.1  THE VOLUMES A R E FOR L I V I N G T R E E S O N L Y , DBH 1 7 . 5 CM+  Ill SAMPLE PSU  ID  SSU  PROBLEM  ID  WHOLE-STEM  X 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 19 19 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 2.1 21 21 21 21 21 21 21  -  NOTE:  36 36 36 36 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 8 8 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 THE  .  16 17 18 20 17 19 20 23 27 29 30 32 33 34 35 36 37 38 39 40 41 42 43 '5 6 12 13 14 15 16 17 18 19 6 7 8 9 10 11 12 13 14 15 * 16 17 18 19 20  VOLUMES  DATA  (1964) 451.0 359.3 173.8 657.2 25.9 695.9 564.8 722.4 479.9 313.1 365.6 816.9 80.1 696.7 51.8 896.1 794.2 346.2 406.5 77.4 165.6 599.5 327.6 330.3 270.4 364.0 423.9 218.4 273.2 117.8 225.9 300.5 404.0 74.0 117.0 493.8 511.4 439.8 432.1 531.6 239.4 222.8 92.4 637.3 647.8 504.7 444.7 569.0  ARE  FOR  L I V I N G  VOLUME,  Y  l  M /HA 3  (1979) 475.7 333.8 202.7 773.1 61.7 785.5 601.2 665.2 446.9 260.5 335.4 872.0 116.3 644.2 39.5 864.5 820.1 288.4 334.1 124.8 89.3 636.7 284.2 356.8 311.4 405.5 366.9 319.3 293.9 108.5 208.0 330.3 403.2 190.1 158.1 565.2 535.0 395.0 401.8 545.5 239.8 132.9 90.0 680.1 632.3 443.1 401.9 638.4  TREES  ONLY,  DBH  17.5  CM+  112  SAMPLE PROBLEM DATA PSU ID  SSU ID  WHOLE-STEM VOLUME, M /HA 3  X 21 21 21 21 21 21 21 21 21 21 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 19 19 19 19 19 19 19 19 19  -  2 2 2 2 2 2 2 2 2 2 18 18 19 19 19 19 19 19 28 28 28 28 28 28 30 30 30 30 30 30 32 32 38 39 39 39 40 40 40 5 5 5 5 5 5 5 8 8  21 22 23 24 25 26 27 28 29 30 8 9 6 7 8 9 10 11 • 1 2 3 4 5 6 1 2 3 5 6 7 39 40 17 13 14 15 10 11 12 17 18 19 20 21 22 23 7 8  (1964) 199.3 240.6 416.8 677.1 490.5 403 . 0 469.3 603.2 151.3 371.2  --—  ---— —  -— — — — — — — — — — — — — — —  -—  -— — — — — — — —  •  Y  (1979) 213.4 282.8 376.2 661.1 498.1 408.6 508.9 579.0 111.3 357.0 138.4 110.3 77.3 119.3 66.4 112.4 169.1 127.6 2.5 78.4 10.3.7 6.4 60.2 47.9 96.4 0.0 91.6 134.9 102.3 61.4 0.0 4.1 381.9 128.3 77.0 85.7 341.9 262.8 597.3 70.5 152.9 132.5 168.9 133.4 200.1 146.5 5.6 4.3  NOTE: THE VOLUMES ARE FOR L I V I N G TREES ONLY, DBH 1 7 . 5 CM+  113  SAMPLE PROBLEM DATA PSU ID  SSU ID  WHOLE-STEM VOLUME, M /HA 3  X 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21  - 8 - 8 -• 8 - 10 - 10 - 12 - 12 - 15 - 15 - 15 - 15 - 15 - 15 - 17 - 17 - 17 - 17 - 17 - 3 - 3 - 3 - 3 - 3 - 3 - 3 - 3 - 6  --  6 6 6 6 6 6  9 10 11 5 6 29 30 17 18 19 20 21 22 1 2 3 4 5 3 4 5 6 7 8 9 10 1 2 3 5 6 7 8  (1964)  --—  -— — —  -—  --•  — — —  ------—  -—  ---  -— —  Y  (1979) 65.3 101.1 67.1 45.2 11.9 506.7 429.5 0.0 277.5 15.6 499.2 92.8 88.4 461.3 3.2 20.7 349.8 386.4 71.2 26.8 41.7 68.1 4.6 28.4 54.5 20.9 87.2 75.2 53.6 25.3 8.5 90.1 191.4  x  NOTE: THE VOLUMES ARE FOR L I V I N G TREES ONLY, DBH 17.5 CM+  APPENDIX II  114  SIMULATION OF REMEASUREMENT DATA Remeasurement data were simulated using the Volume-Age curves (VACs) f i t t e d and currently used by the BCFS.  The curves, Chapman-Richards  generalization of the Von Bertalanffy's growth function, take the form  V = b^l-e  -b.CA-b.) b_ ] 2  4  3  where 3 V = stand volume i n m A = stand age i n years b_£ ( i = 1,2,3,4) are constants e = 2.71828... Estimates of the b^ ( i = 1,2,3,4) are available for each combination of Forest Inventory zone, Site and Growth Type i n B r i t i s h Columbia. The remeasured volume per ha V  (net for decay, with u t i l i z a t i o n  from 30 cm stump height to 10 cm top) at a sample plot was obtained as follows  V  r  = V  n  +. (V  p  - V ) + £ c  where V  n  = volume per ha at- the plot i n 1979 as estimated using the appropriate VAC  V V  P c  = volume per ha at the plot as a c t u a l l y measured i n 1964 ' = volume per ha at the plot i n 1964 as estimated using the appropriate VAC  £  (with A = A^ '= (stand age at the plot i n 1964) + 15)  (with A = A  o  = s t a n d age a t t h e p l o t  i n 1964)  = a random number drawn from a normal population of random numbers with mean V plot  P  and variance.1927•21 (= estimated variance between  volumes).  115  The c a l c u l a t i o n i s shown graphically as below  Volume/ha (m ) 3  Age (years)  The plot volume was allowed to grow at the rate dictated by the VAC, and the random element £ accounted f o r natural disasters, such as w i n d f a l l , etc. For example, plot number 18-015-31 i n good s i t e i n growth type I i n forest inventory zone F: V  = 650.6  V = c P  m /ha 3  479.7227  [1-2.71828" ' 0  0 3 5 7 ( 1 7  °-  0 )  ] ' 8  6 5 8 3  = 470.2 m /ha 3  V = 479.7227 [1-2.71828" '0357(185-0) 8.6583 n 0  = ^7^.1 m /ha 3  and  V  r  = 474.1 + (650.6 - 470.2) + 7.7 662.3 m /ha  The  simulated r e s u l t s were i n agreement with the r e s u l t s obtained from a  remeasurement p i l o t study i n the Cranbrook PSYU (in which the author participated) Twenty-three undisturbed (e.g.  not burnt or logged) sample plots were selected  at random from the 1964 ordinary inventory sample p l o t s , and actually remeasured (according to the 1964 standards) during the summer of 1980.  The c o r r e l a t i o n  between the 1980 measurements and the 1964 measurements and that between the 1979 simulated measurements and the 1964 measurements were not d i f f e r e n t from each other.  significantly  The r e s u l t s of the p i l o t study and the derived  s t a t i s t i c s are summarized below: A. Data Plot ID  Whole stem Volume  (living trees only,dbh 17.5cm+) rffa  1980  1964  18-06-62  450.3  431.4  18-07-34  139.0  195.2  18-07-40  195.1  175.0  18-08-15  286.2  199.0  18-08-18  216.5  143.3  18-20-08  185.8  129.0  18-25-47  413.5  374.5  18-24-62  489.6  420.5  314.6  307.6  18-33-24  390.7  315.2  18-34-28  476.3  510.0  18-34-35  315.5  383.1  18-35-34  684.2  716.0  18-35-35  877.4  860.6  18-36-10  562.6  488.7  18-36-11  596.2  553.1  18-31-31  '  117 Plot ID  B.  Whole stem Volume 1980  ( l i v i n g trees only, dbh 17.5cm+) 1964  18-37-34  692.0  696.7  21-01-14  249.3  218.4  21-01-15  241.1  273.2  21-01-18  307.3  300.5  21-02-06  106.0  74.0  21-02-13  362.4  239.4  21-07-74  646.1  625.8  Statistics  Volume i n 1980  mean 399.90  Volume i n 1964  375.23  standard deviation 201.98 209.08  Correlation between 1980 and 1964 measurements, £ = 0.97.  

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