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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 SAMPLING WITH PARTIAL REPLACEMENT OF UNITS by STEPHEN A. YenEMURWON OMULE B.Sc., ( F o r . ) ( H o n s . ) , Makerere U n i v e r s i t y , 1976 M . S c , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n 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 t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA F e b r u a r y 1981 ( ^ S t e p h e n A. YenEmurwon Omule, 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o an advanced degree at the 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 , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r 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 the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date ABSTRACT Supervisor: Professor Donald D. Munro E f f e c t i v e sampling methods for successive forest inventories include versions of multistage sampling. Multistage sampling, or subsampling, is. c o s t - e f f e c t i v e i n broad forest areas, and i t i s one technique that lends i t s e l f advantageously to the use of 'multilevel data. Improved e f f i c i e n c y of sampling designs for successive inventories i s usually achieved through p a r t i a l replacement of sampling units at the successive occasions. How-ever, the theory on multistage sampling on successive occasions with p a r t i a l replacement of units has some l i m i t a t i o n s . A l l of the theory invokes the d i s t i n c t i v e assumptions of equal sample size or equal variance on successive occasions. These assumptions are not usually met i n f o r e s t r y . The objective of t h i s study was to provide some general theory for successive forest inventories using multistage sampling with p a r t i a l replacement of un i t s . As i s the case with multistage designs, the tech-nique of p a r t i a l replacement gives r i s e to a number of a l t e r n a t i v e s . For p r a c t i c a l purposes, only the case i n which p a r t i a l replacement occurs at the primary stage of the multistage design was considered. In addition, consideration was r e s t r i c t e d to inventories on two successive occasions only, without the r e s t r i c t i v e assumptions of equal sample size or equal var-iance at the two occasions. i i Minimum-variance (best) l i n e a r and unbiased estimators (BLUE) of the current population mean v>y a n c ^ °f t n e change i n the mean between two successive occasions A, together with t h e i r respective variances are derived for two-stage, three-stage, and h-stage (h > 1) designs. Biased estimators of the r a t i o form (RE) of Uy and of A are also derived together with t h e i r respective variances and biases, for a two-stage design. The biases of REs are n e g l i g i b l e for large sample s i z e s . A numerical comparison of the e f f i c i e n c y of BLUE and RE for estimating y Y indicated that the BLUE had a s l i g h t edge over the RE; however, for estimating A, the RE was very i n e f f i c i e n t . An a l t e r n a t i v e solution approach i s proposed for the problem of determining the optimum' replacement p o l i c y , that i s , the number of primary units to remeasure and new ones to take at the current occasion. The sequential nature of successive inventories i s exploited to cast the prob-lem as a multistage process that can be optimized through dynamic program-ming. Solution procedures are given for determining the optimum replace-ment p o l i c y for a two-stage design with the objective of minimizing the cost of the inventory and subject to the side conditions that the s p e c i f i e d variance l e v e l s of i i y and A are met. The derived theory was i l l u s t r a t e d , for a two-stage design, by working through a sample forest inventory problem. i i i From a practical point of view, extension of the theory of sampling with partial replacement from one-stage to multistage designs is bene-f i c i a l , particularly for the inventory of large forest areas. It would be useful to extend the theory further to use variable probabilities of selection at the various stages of the multistage design; and to examine the cases in which partial replacement 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 v i i i DEDICATION x Chapter : 1 INTRODUCTION 1 2 LITERATURE REVIEW 6 3 THEORY OF MULTISTAGE SAMPLING ON SUCCESSIVE OCCASIONS WITH PARTIAL REPLACEMENT OF UNITS 18 Two-stage SPR 19 Three-stage SPR 31 Multistage SPR 43 Unequal Size Sampling Units 47 Other Estimators 53 4 OPTIMUM ALLOCATION AND REPLACEMENT 68 SPR as a Multistage Model 71 Solution Procedure . 80 5 SAMPLE PROBLEM 83 6 DISCUSSION AND CONCLUSION 94 v REFERENCES 101 APPENDIX I Sample Pr o b l e m Data 106 APPENDIX I I S i m u l a t i o n of Remeasurement Data 114 v i LIST OF TABLES T a b l e I P e r c e n t g a i n i n e f f i c i e n c y (Q%) o f y 2 o v e r y 2 58 I I P e r c e n t g a i n i n e f f i c i e n c y (Q//0) of g 2 j^ o v e r g 2 j_ 66 I I I E n u m e r a t i o n R e s u l t s 89 LIST OF FIGURES F i g u r e ' • 1 Groups of 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 73 2 The s t a g e diagram f o r the o p t i m a l SPR d e s i g n p r o b l e m ... 78 v i i ACKNOWLEDGEMENT I am most g r a t e f u l to Dr. Donald D. Munro, my major professor, for his d i r e c t i o n and encouragement throughout the period of my graduate stud-ie s , and to the members of my thesis committee—Drs. J . P. Demaerschalk, A. Kozak, D. D. Munro, A. J . Petkau, D. H. Williams, and R. J . Woodham— for reviewing the thesis draft and for t h e i r valuable comments. I am 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 and D. H. Williams with whom I had very useful discussions during the d e r i v a t i o n of the theory. I am very g r a t e f u l to Mr. David A. Campbell of the B r i t i s h Columbia Forest Service (BCFS) Inventory Branch for the assistance in r e t r i e v i n g the data used i n the thesis.and for h i s patience with me during the f i e l d -work at Cranbrook, to Ms. Sadie C. Paddock for the compilation and prelim-inary analysis of the data, and to the BCFS Inventory Branch for permission to use the data. I extend my sincere appreciation to the Science Council of B r i t i s h Columbia and the BCFS for the f i n a n c i a l assistance i n form of the GREAT award, to the University of B r i t i s h Columbia for the a d d i t i o n a l funding in form of a teaching assistantship, to the Ford Foundation and Makerere University for the study fellowship, and to the Quantitative Methods Group for sponsoring me to attend a workshop on 'Sampling on Successive Occasions' at Colorado State University during the summer of 1979. I extend my special thanks to Mr. F. E. E. Omoruto who helped check some of the derivations, to my c o l l e a g u e s — D r . Y. El-Kassaby, v i i i Mr. S. S. Chiyenda, and Mr. J. I. Mwanje—for the f r u i t f u l discussions I had with them during the course of writing the thesis, and to Justyna Debogorski and Drs. Y. Z. G. Moyini and J. H.G. Smith for t h e i r encourage-ment . I am very g r a t e f u l to Mrs. N. Thurston for the excellent typing of the th e s i s . F i n a l l y , I am very g r a t e f u l to my mother Agenesi Tino, my brother Eri d a d i Okirimat and the rest of my family who, despite my long absence from them, gave me the much needed encouragement and moral support. ix DEDICATION I dedicate t h i s thesis to my late father, E r i y a Emurwon. May his soul rest in eternal peace. x CHAPTER 1 INTRODUCTION In recent years demands have been increasing for r e l i a b l e and timely forest resource s t a t i s t i c s obtained with a minimum of expenditure. These s t a t i s t i c s , such as timber volume per unit area and growth over time, usually form a basis for r a t i o n a l u t i l i z a t i o n of forest resources. Inventory sampling i s frequently employed to provide the data on which resource s t a t i s t i c s are derived. Several sampling techniques have been proposed i n forest inventory designs for both single-occasion and successive (occasions) inventories. Single-occasion or "one-shot" inventories provide information on the state of the resource at a given point in time. Only current values are obtained. Sampling designs, such as simple random, s t r a t i -f i e d random, etc., documented in most sample survey texts, are used in single-occasion inventory problems. Successive inventories provide information on the state of the resource at various points i n time. Current values and changes or average of values of the resource over time are obtained. The basic sampling designs as used in "one-shot" inventories together with methods of l i n k i n g the designs over time are used i n successive inventory problems. Successive inventories may be regarded as multiphase sampling in which the current phase sample consists of units observed at the current occasion, and i s a subsample of e a r l i e r phase(s) sample(s) selected 1 2 on e a r l i e r occasion(s). Successive inventories may be conducted using (1) a new sample on each occasion, (2) a f i x e d sample on a l l occasions, or (3) a p a r t i a l replacement of sample units from occasion to occasion. A s e r i e s of independent samples i s simply repeated i n v e n t o r i e s , each made without reference to the others. (This method, f o r example, uses temporary marked plots.) A fixed sample i s a set of permanent sample units that are observed on successive occasions, t r a d i t i o n a l l y c a l l e d continuous forest inventory (CFI). In sampling with p a r t i a l replacement of units (SPR), the t o t a l sample i n the current occasion consists of sample units already observed i n e a r l i e r occasions plus new sample u n i t s taken;independently at the current occasion. Method 1 i s s t a t i s t i c a l l y i n e f f i c i e n t since i t does not e x p l o i t the inherent c o r r e l a t i o n e x i s t i n g b e t w e e n p a s t and current observations. Method 2 i s more e f f i c i e n t p a r t i c u l a r l y for estimating d i f f e r e n c e s i n value of the resource between occasions, but i s more expensive than method 1. SPR (method 3) combines the lower cost of independent inventories (for obtaining current values) with the high e f f i c i e n c y of fixed samples i n estimating changes. In f a c t , as we s h a l l see l a t e r , methods 1 and 2 can be regarded as s p e c i a l cases of SPR: when the proportion of units from previous occasions that i s remeasured i n current occasions equals 0 we have method 1, and when t h i s proportion equals 1, we have method 2. SPR has been accepted as a v a l i d forest inventory technique for estimating f o r e s t resource current values and changes i n these values over time. Several a r t i c l e s have been published on the general s t a t i s -t i c a l theory of SPR and, s p e c i f i c a l l y , on i t s use i n the estimation of forest area, current timber volume, area change, and timber growth. Large-scale applications of the technique have been reported mostly i n 3 the United States and Canada. However, the design of successive inven-tories with partial replacement of units i s complex. As mentioned earlier, we require not only the basic sampling designs at given points in time but also the procedure for combining the successive observations. Much of the theory on SPR available in the forestry f i e l d has been derived assuming simple random sampling (SRS) as the basic design on the successive occasions. We know that SRS i s cost-effective in relatively small forest areas, and that the technique is very rarely used in forest inventories. In national and other large forest inventories covering broad forest areas, SRS (one-stage) SPR becomes expensive (for a given level of precision) and highly d i f f i c u l t to apply. Furthermore, straight-forward SRS does not take advantage of combining remotely-sensed data, such as s a t e l l i t e and photo-imagery, and ground data. Multistage sampling is cost-effective in broad forest areas, and i t is one technique that lends i t s e l f advantageously in the use of multilevel data. 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 st r a t i f i e d random sampling could also be used. However, there is some theory already in other fields on multiphase SPR, and st r a t i f i e d SPR is infeasible in sampling forest populations because forest strata 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 distinctive assumptions of equal sample size or equal variances on successive occasions. These assumptions are not usually met in forestry. There are no guidelines for the optimal allocation and replacement of sampling units at the various stages of multistage SPR 4 designs. Furthermore, multistage SPR has so far not been applied to large-scale forest inventories. The objective of t h i s study i s to extend the theory and p r i n c i p l e s of one-stage SPR to a multistage dimension for the purpose of estimating forest resource current values and changes in values over time. The resource values could be timber volume, number of stems per ha, etc. As in the case of a multistage design, the technique of p a r t i a l replacement of sample units gives r i s e to a number of a l t e r n a t i v e s . For example, i n a two-stage sampling design, p a r t i a l replacement of units can be done in the following ways: (1 ) r e t a i n a l l primary sampling units (psu's) but from each psu take a fresh sample of secondary sampling units (ssu's) within them, on the second occasion, (2) r e t a i n only a f r a c t i o n of psu's together with th e i r samples of ssu's and select a fresh f r a c t i o n of psu's, (3) re-tai n a l l the psu's from the preceding occasion but from each psu r e t a i n only a f r a c t i o n of the ssu's within them and select 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 and from each such psu re-t a i n only a f r a c t i o n of the ssu's and select a f r a c t i o n of the ssu's afresh. In three-stage SPR there are about twelve a l t e r n a t i v e s . As the number of stages and occasions increases, the number of possible a l t e r n a t i v e s increases too. For p r a c t i c a l reasons we s h a l l r e s t r i c t ourselves to situ a t i o n s i n which p a r t i a l replacement occurs only at the primary stage of the multistage design. In addition, we s h a l l r e s t r i c t ourselves to multistage SPR on two occasions only, assuming varying sample sizes and unequal variances at the two occasions. Although some techniques of optimization have been suggested for use in one-stage SPR, in this thesis we s h a l l e x p l o i t the sequential nature of SPR on successive occasions to obtain optimum sample d i s t r i -bution over time by dynamic programming, a mathematical programming 5 technique. Specifically, 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 in 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 appli-cation of the derived multistage SPR theory to a specific forest inven-tory problem. First, we give as a background, the previous work done in SPR, including multistage SPR (chapter 2 ) . Next, we present the derivation of the theory of multistage SPR (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 ) . Finally, we discuss the problems of the multistage SPR theory and specifically, its applica-tion 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 literature, sampling on successive occasions i s also sometimes called "rotation sampling," "sampling for time series," or "repeated sampling." In any case, the method involves successive sampling of the same population with replacement (partial or complete) of the sample from occasion to occasion. We shall review the theoretical development of sampling on successive occasions with partial replacement (SPR) in general and the specific development of theory and application of SPR to sampling forest popula-tions. 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 realize the advantage of partial replacement of the sample to estimate the current population mean in sampling on successive occasions. Jessen was considering the problem of sampling on two successive occasions in agricultural populations. He obtained two estimates: 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 overall sample mean obtained on the f i r s t occasion. He then obtained a linear unbiased estimate of the population mean on the second occasion by taking the weighted average of the two estimates. (The two estimates were weighted inversely by their variances.) Jessen also considered the optimum replacement fraction 7 under the assumption that the i n i t i a l sample size was specified and that the total sample size remained constant over time. He used simple random sampling as the basic design. Yates (1949) extended Jessen's result for the study of a population on two occasions to more than two occasions under the restrictive condi-tions of the same sample size and a fixed replacement fraction on each successive occasion. In addition, Yates assumed the correlation between the same sampling units on two different occasions as decreasing in geo-metric progression as the time interval between the occasions increased. That i s , the correlation between observations one occasion apart as p, 2 three occasions anart as p 3, etc. Yates two occasions apart as p z, p °, etc further ^assumed that the population variance did 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 restricting himself to best linear unbiased estimators, removed a l l the restrictive assumptions of Yates, except for the correlation pattern and constant population variance over time. Working independently of Patterson, Tikkiwal (1951) also removed the restrictive assumptions of Yates, but he adopted a slightly different correlation pattern from that of Patterson. He allowed the correlation between the same sampling units on successive occasions to vary; the correlation 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 in question. Tikkiwal (1953), using calculus techniques, worked out the optimum 8 proportion of new and old sample units to take for estimating the popula-ti o n mean on a recent occasion, given the assumptions of Yates (1949) and Patterson (1950). Tikkiwal also gave formulae for the optimum a l l o -cation of units among st r a t a , when s t r a t i f i e d random sampling was used on successive occasions. ^ Narain (1953), taking into account the v a r i a b i l i t y of the regression c o e f f i c i e n t computed from samples, derived the basic recurrence formula in sampling on successive occasions. The v a r i a b i l i t y of the regression c o e f f i c i e n t had been ignored by Yates (1949) and Patterson (1950). Narain (1954) further extended the res u l t s of Yates (1949) and Patterson (1950) to that of estimating current values of a population sampled two or more occasions apart, assuming p a r t i a l c o r r e l a t i o n s were non-zero for occasions more than two apart. K u l l d o r f f (1963) discussed the problem of optimum a l l o c a t i o n of the sample for SPR on two successive occasions, when there was one vari a b l e of i n t e r e s t at a time. Using an an a l y t i c approach, K u l l d o r f f provided solutions to the problems of obtaining the sample siz e on occa-sions one and two when in t e r e s t lay in estimating e i t h e r the current mean, the improved mean on occasion 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 of minimum cost for fi x e d variance or minimum variance for fixed cost. Eckler (1955) s i m p l i f i e d Patterson's (1950) approach and developed the method of ro t a t i o n sampling to obtain a minimum variance l i n e a r un-biased estimate of the population mean or t o t a l by suitably constructing a l i n e a r function of sample values at d i f f e r e n t occasions. Tikkiwal (1955, 1956a, 1956b, 1967) extended the theory of SPR to the study of several characters on each of several occasions under a 9 s p e c i f i e d correlated pattern, using multiphase sampling on the occasions. He applied the derived theory to the survey of livestock marketing. Tikkiwal (1958a) found that p a r t i a l replacement of units improved the e f f i c i e n c y of various estimators of time-dependent populations characters with increasing number of occasions reaching a l i m i t i n g value. In t h i s study he assumed that the variance and replacement fract i o n s were the same on each occasion. He also extended SPR to a two-stage design ( T i k k i -wal, 1958b), assuming a s p e c i f i c c o r r e l a t i o n pattern at both stages and equal-size primary units. Woodruff (1959) discussed the advantages of r o t a t i o n , and presented composite estimation procedures with rotating panels, i n the r e t a i l trade survey of the United States. He gave composite estimators, of the r a t i o form, of current population values under the assumptions that occasion to occasion c o r r e l a t i o n s were the same and that variances were equal on each occasion. Onate (1960) worked with a f i x e d pattern of p a r t i a l replacement of the ultimate subsample units i n a multistage design. He studied the t o t a l composite estimator using data obtained on previous occasions to make various estimators at any current occasion. He also developed a f i n i t e population theory for the composite estimator for his r o t a t i o n pattern under c e r t a i n r e s t r i c t i o n s . Rao and Graham (1964) developed a u n i f i e d approach to the problem of sampling on successive occasions employing a fixed r o t a t i o n design in a f i n i t e population. They considered a survey design which, f i r s t , numbered the population units at random and, second, s p e c i f i e d in advance which of them would be in the sample on each of the occasions (the r o t a t i o n plan). Estimators of current values and changes in these values were developed under the assumption that exponential and arithmetic c o r r e l a t i o n patterns 10 held over time for the c h a r a c t e r i s t i c of i n t e r e s t . Later, Graham (1973) generalised this work taking into account that i t was not necessarily true that the c o r r e l a t i o n between x and x , , would monotonically decrease as CX , K Ot j K |ot - a'I increased. (x^ ^ i s the observation on the kth population unit i n the ath occasion.) He instead considered the following model of c o r r e l a -tion (with s p e c i f i c reference to current population 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,...) for the c o r r e l a t i o n between observations in the same unit separated by (12j + i ) months. (The index i i s for months and j for years.) Singh and Singh (1965) considered a sampling procedure involving re-peated a p p l i c a t i o n of double sampling for s t r a t i f i c a t i o n on several succes-sive occasions. They gave estimators of the current population mean and i t s variance under the assumptions that no units s h i f t e d from stratum to stratum on any occasion, and that addition of any further units to or subtraction from the population did not take place throughout the course of sampling. The derived theory was applied to the survey of coconut production in the state of Assam (India). Raj (1965) outlined the theory of successive sampling when sampling units were c l u s t e r s selected with p r o b a b i l i t y proportional to size and the sampling confined to two occasions for estimating current values. The a p p l i c a t i o n of the theory to double sampling was also considered. Pathak and Rao (1967) provided estimators of the population t o t a l i n sampling over two occasions when simple random sampling without replacement was used on both occasions and when p r o b a b i l i t y proportional to size selec-t i o n was used on both occasions. The estimator suggested here was more e f f i c i e n t than that of Cochran (1977). However, Ghangurde and Rao (1969) 11 showed t h a t , f o r s a m p l i n g o v e r two o c c a s i o n s from a f i n i t e p o p u l a t i o n , under s i m p l e random s a m p l i n g w i t h o u t r e p l a c e m e n t , K u l l d o r f f ' s (1963) e s t i m a t o r of the p o p u l a t i o n t o t a l on the c u r r e n t o c c a s i o n had a s m a l l e r minimum v a r i a n c e t h a n t h a t of Pathak and Rao ( 1 9 6 7 ) , f o r t h e same e x p e c t e d c o s t . The e s t i -mator s u g g e s t e d by S i n g h (1972) was a l s o s u p e r i o r t o t h a t of Pathak and Rao (1967) but not as good as K u l l d o r f f ' s (1963) e s t i m a t o r . S i n g h (1968) p r e s e n t e d a t h e o r y f o r s u c c e s s i v e s a m p l i n g p r o c e d u r e s u s i n g a two-stage s a m p l i n g scheme f o r two and t h r e e o c c a s i o n s . He d e r i v e d e s t i m a t o r s o f c u r r e n t p o p u l a t i o n v a l u e s and l i n e a r c o m b i n a t i o n s of v a l u e s o v e r s e v e r a l o c c a s i o n s when p a r t i a l r e p l a c e m e n t o c c u r r e d o n l y a t the psu's and assuming t h a t the number of s a m p l i n g u n i t s t a k e n on each o c c a s i o n were e q u a l and t h a t the u n i t s were of e q u a l s i z e . T h i s work was l a t e r e x tended by S i n g h and K a t h u r i a (1969) t o the case where p a r t i a l r e p l a c e m e n t o c c u r r e d at the s e c o n d a r y sample u n i t l e v e l , under s i m i l a r a s s u m p t i o n s . A v a d h a n i and Sukhatme (1970) p r o p o s e d the use of the Rao, H a r t l e y , and Cochran (1962) (RHC) s a m p l i n g p r o c e d u r e and the r a t i o method i n s a m p l i n g on s u c c e s s i v e o c c a s i o n s . The RHC s a m p l i n g scheme m o d i f i e d f o r two o c c a s i o n s gave an e s t i m a t o r w h i c h was more e f f i c i e n t t h a n t h a t of the r a t i o method u s i n g s i m p l e random s a m p l i n g . Sen (1971b) d e v e l o p e d the t h e o r y of s u c c e s s i v e s a m p l i n g (two o c c a s i o n s ) t o p r o v i d e a combined e s t i m a t e based on a m u l t i v a r i a t e d o u b l e s a m p l i n g r a t i o e s t i m a t e from the matched p o r t i o n of the sample based on two 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 a mean pe r u n i t e s t i m a t e from the unmatched p o r t i o n . He showed t h a t when the a u x i l i a r y v a r i a b l e s have the same c o e f f i c i e n t of v a r i a t i o n , when the c o r r e l a t i o n s between the dependent and independent a r e e q u a l , and when the a u x i l i a r y v a r i a b l e s a r e e i t h e r u n c o r r e l a t e d or a r e m o d e r a t e l y 12 correlated with the dependent variable, considerable gain in efficiency was achieved over using a single auxiliary variable. He assumed that sample sizes were equal, and population variances were the same, on both occasions. The efficiency of the multivariate ratio estimate was com-pared to that of the multivariate double sampling regression estimate: the latter was more efficient in general. Sen generalised this theory from two to several auxiliary variables using (1) the double sampling multivariate ratio estimate (Sen, 1972), and (2) the double sampling regression estimate (Sen, 1973b), under similar assumptions. Later, however, the assumption of equal sample size on both occasions was removed by Sen (1973a); and that of equal variance on both occasions was also removed by Sen et a l . (1975), but they considered the case of the ratio estimator with only a single auxiliary variable. Sen et a l . (1975) further extended the theory to use st 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 in Canada. He observed that the SPR estimate of current values was one-third more efficient than the estimate obtained on the basis of current observations only, when the correlation between successive observations was high and positive. Avadhani and Sukhatme (1972) suggested the use of controlled simple random sampling with a ratio estimator for estimating the population mean in sampling on successive occasions. They also extended their earlier work (Avadhani and Sukhatme, 1970) from two to more than two occasions. Blight and Scott (1973) extended Patterson's (1950) results to situations in which the population mean of a time-dependent population followed a linear Markov process. They assumed a simple first-order autoregressive model. !3 Scott and Smith (1974) applied standard time series methods to the analysis of repeated surveys under the assumption that the population para-meters at each time period followed a stochastic model. Their derivation used the theory of signal extraction i n the presence of stationary noise. Both independent and complete remeasurement surveys were considered i n gen-era l terms with s p e c i f i c r e s u l t s obtained fox the time series model assumed in the work of Patterson (1950). The r e s u l t s were l a t e r extended and applied to surveys of more complex design, Scott et a l . (1977). Chakrabarty and Rana (1974) developed the theory of sampling on two successive occasions in a two-stage design, under the assumptions of equal variances and equal sample sizes on both occasions. They examined s i t u a -tions i n which p a r t i a l replacement occurred of the psu's only, of the ssu's only, and of both psu's and ssu's. Empirical results showed that p a r t i a l replacement of both psu's and ssu's was more e f f i c i e n t , in most cases, for estimating the current mean. The theory for a three-stage design, under s i m i l a r assumptions, was derived by Rana and Chakrabarty (1976). Contin-uing the study using a three-stage design, Rana (1978) considered the use of double sampling r a t i o estimator for both s t r a t i f i e d and simple random sampl-ing. He made a numerical comparison, si m i l a r to that of Sen et a l . (1975) for a simple random sampling scheme, of the e a r l i e r estimator (Chakrabarty & Rana, 1974) and the one using a r a t i o estimator: the former had a s l i g h t edge over the l a t t e r . As also, noted by Sen et a l . (1975), the estimator using the r a t i o estimate was s l i g h t l y biased, but "for most p r a c t i c a l purposes both estimates seem to be equally desirable." Jones (1979) compared the e f f i c i e n c i e s of the approaches of Patterson (1950), Blight and Scott (1973), and of Scott and Smith (1974) to analysis of data from repeated surveys by computing the mean square errors of the estimators of the current mean and of the change in means on the l a s t two 14 occasions. He indicated that there were considerable gains in 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. Without considering cost, the variance of the derived estimator was less than that of Patter-son's (1950) but greater than that of Eckler (1955). Good summaries of some of the results cited above can be obtained in sampling texts such as Cochran (1977), Sukhatme and Sukhatme (1970), Murthy (1967), and Kish (1965). Most of the literature reviewed so far has dealt primarily with theoretical aspects of SPR. Few authors of these have reported applica-tion of the derived theory to actual surveys. The most notable are Sen (1971a) who applied SPR in the survey of the waterfowl hunter's in Canada, and Tikkiwal (1956b) who used SPR in the survey of livestock marketing in the United States. However, several authors have reported some modifica-tions to, and application of the theory of SPR for sampling forest popula-tions. Now we shall review the contributions to the theory of SPR and i t s applications in the forestry f i e l d . The concept of SPR was introduced into forest inventory by Bickford (1956, 1959, 1963). However, Ware and Cunia (1962) provided a more com-plete discussion of the principle and advantages of SPR in CFI. They treated the s t a t i s t i c a l aspects of the use of remeasured permanent plots and partial replacement of the i n i t i a l sample for forest inventory. They gave the theory of SPR 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 the two o c c a s i o n s . Ware and C u n i a (1962) used a g r a p h i c a l t e c h n i q u e to d e t e r m i n e t h e o p t i m a l r e p l a c e m e n t p o l i c y . T h i s work was a r e s u l t of the independent work of Ware (1960) and t h a t of T. C u n i a . B u i l d i n g on the work of Ware and C u n i a ( 1 9 6 2 ) , C u n i a (1965) e x t e n d e d the t h e o r y of SPR t o use m u l t i p l e r e g r e s s i o n methods f o r l i n k i n g s u c c e s s i v e o c c a s i o n s ; C u n i a and Chevrou (1969) extended the t h e o r y of s a m p l i n g on two o c c a s i o n s t o s a m p l i n g on t h r e e o r more o c c a s i o n s ; and Newton e t a l . (1974) c o n s i d e r e d the m u l t i v a r i a t e c a s e . B i c k f o r d 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 s a m p l i n g d e s i g n d e v e l o p e d f o r a f o r e s t s u r v e y o f the No r t h w e s t ( U n i t e d S t a t e s ) as a s t r a t i -f i e l d d o u b l e s a m p l i n g at the f i r s t o c c a s i o n f o l l o w e d by SPR a t the second o c c a s i o n . C u n i a (1964) gave a b r i e f h i s t o r i c a l development of the t h e o r y and a p p l i c a t i o n o f SPR t o f o r e s t r y . SPR was d e f i n e d and the way i t works e x p l a i n e d from an i n t u i t i v e p o i n t o f vie w . F r a y e r (1966) u n d e r t o o k 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 d a t a t o t e s t the v a l i d i t y of u p d a t i n g t i m b e r volume by the method of Ware ( 1 9 6 0 ) . The a n a l y s i s i n d i c a t e d t h a t the homogeneity of v a r i a n c e a s s u m p t i o n of the model was not met. He s u g g e s t e d the use of w e i g h t e d r e g r e s s i o n i n u p d a t i n g t i m b e r volumes. F r a y e r and F u r n i v a l (1967) p r e s e n t e d a method of c a l c u l a t i n g changes i n a r e a a t t r i b u t e s on remeasured f o r e s t p l o t s . Methods were a l s o shown whereby the e s t i m a t e s of change c o u l d be a p p l i e d t o r e s u l t s o f a p r e v i o u s i n v e n t o r y and combined w i t h the r e s u l t s of a c u r r e n t i n v e n t o r y t o form f i n a l e s t i m a t e s o f c u r r e n t v a l u e s . A l o n g the same l i n e s , H a z a r d (1977) p r e s e n t e d e s t i m a t o r s o f the p r o p o r t i o n of a r e a and change i n p r o p o r t i o n 16 of area contained i n a c l a s s . Frayer et a l . (1971) reported r e s u l t s of the f i r s t attempt to apply SPR to timber inventories, in two working c i r c l e s of Colorado (United States). It was found that SPR required about half the number of sample plot s to obtain the same sampling error as obtained with conventional CFI, in estimating current values. Empirical studies elsewhere (Loetsch & H a l l e r , 1964; See, 1974; Barnard, 1974) also indicated that there was increased e f f i c i e n c y i n using SPR as compared to conventional inventory methods. Hazard and Prommitz (1974) proposed the use of convex mathematical programming as a tool for optimally a l l o c a t i n g resources for successive forest inventories. Most of t h i s work was derived from Hazard (1969). Dixon and Howitt (1979) introduced the Kalman estimator as an a l t e r -native to the model developed by Ware and Cunia (1962), for estimating the current values of a time-dependent population. The Kalman estimator takes into account the r e l a t i o n s h i p between successive values of the popu-l a t i o n mean (assumed to be uncorrelated by Ware and Cunia [1962]). The Kalman estimator was found to be more precise than the Ware and Cunia (1962) estimator. However, i t seems the authors were not aware of some work done in t h i s d i r e c t i o n by Blight and Scott (1973) and by Scott and Smith (1974). From the l i t e r a t u r e c i t e d above i t i s observed that (1) t h e o r e t i c a l developments i n multistage SPR have been r e s t r i c t e d to si t u a t i o n s in which the sample size and/or variance are constant over time, and (2) no a p p l i c a -tion of multistage SPR has been reported i n f o r e s t r y . In the next chapter the theory of multistage SPR i s derived without the r e s t r i c t i v e assumptions of constant sample size or variance on the successive occasions. This 17 essentially 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 For s i m p l i c i t y of presentation, we s h a l l f i r s t discuss the extension of one-stage SPR to two-stage SPR, and to three-stage SPR. Then, next, we'shall generalize the extension to h-stage SPR (h > 1)-. Minimum-Variance (best) l i n e a r unbiased estimators (BLUE) of the current mean and the change in the means from occasion to occasion w i l l be derived together with t h e i r variances. Other possible estimators, r a t i o estimators (RE), w i l l also be derived together with t h e i r properties (biases and variances). The e f f i c -iencies of RE r e l a t i v e to BLUE i n estimating the current mean and the change in mean w i l l be investigated. In a l l the derivations we s h a l l r e s t r i c t ourselves to sampling on two successive occasions with p a r t i a l replacement of only the primary sample u n i t s . We s h a l l consider the cases where the sampling units at each stage are of equal size and unequal siz e i n deriving the BLUE. In general, when sampling large forest areas, the population of sample units at each stage of the multistage design i s large enough to be consid-ered i n f i n i t e . A l l the derivations that follow w i l l , therefore assume i n f i n i t e population models which, i n t h i s case, provide very close approxi-mations to the exact r e s u l t s which would have been obtained i f the deriva-tions were done with f i n i t e population models. Further, the assumption of i n f i n i t e population models greatly s i m p l i f i e s the algebra involved in the, d e r i v a t i o n s . 18 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 ( p s u ' s ) and each psu c o n s i s t i n g of M s e c o n d a r y sample u n i t s ( s s u ' s ) . F u r t h e r , suppose t h a t the sample u n i t s ( p su's or s s u ' s ) are of e q u a l s i z e . I n p a r t i c u l a r , suppose n psu's a r e s e l e c t e d by s i m p l e random s a m p l i n g w i t h o u t r e p l a c e m e n t ( s r s w o r ) on the f i r s t o c c a s i o n and m ssu's a r e s e l e c t e d by s r s w o r from each sample psu . A random sample ( s e l e c t e d by s r s w o r ) of s i z e np (0 < p < 1) of the n psu's i 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 s s u ' s drawn from the f i r s t o c c a s i o n . I n a d d i t i o n , a random sample of s i z e ns ( s > 0) o f the N-n o t h e r psu's i s s e l e c t e d by s r s w o r f o r i n c l u s i o n i n the sample on the second o c c a s i o n . A g a i n , m s s u ' s a r e s e l e c t e d by srswor from each of the ns p s u ' s . As i n d i c a t e d e a r l i e r , i t w i l l be assumed t h a t N and M are i n f i n i t e l y l a r g e . O b s e r v a t i o n s are t a k e n i n each of the nm ssu's on the f i r s t o c c a s i o n and nm (p + s) s s u ' s on the second o c c a s i o n . D e n o t i n g the v a r i a b l e of i n t e r e s t on the f i r s t o c c a s i o n as X and t h e same v a r i a b l e of i n t e r e s t on the second o c c a s i o n as Y, we 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 f o l l o w s : O c c a s i o n No. of unmatched psu's No. of matched psu's No. of unmatched ssu's No. of matched ssu's Unmatched o b s e r v a t i o n s Matched o b s e r v a t i o n s 1 nq np nmq nmp x' . . , i = l , 2 , 1 J j = l , 2 , , i = l , 2 1 J j = l , 2 , . . . , .nq ,m np ns np nms nmp y ,1=1,2, J j = l , 2 , . y " i r i = i , 2 , J j = l , 2 , , ns ,m »np ,m where q = (1 - p). We are interested i n estimating the current population mean a n c * the change i n means A = ix - u of the variable of i n t e r e s t . It w i l l be Y A assumed that on the f i r s t occasion, the observations (matched or unmatched) are described by the l i n e a r nested model . i = 1,2,. .. , N X i j = UX + °li + e l ( . i ) j { J = 1,2,...,M where x . = observation on the j t h ssu within the i t h psu yX °li = o v e r a l l mean of the observations = e f f e c t of the i t h psu e,,. x. = e f f e c t of the i t h ssu within the i t h psu l ( i ) j • and a l l the E ^ ^ . ^ . ' S are independent random variables each with expected value = 0 and variance = o 2^ , and a l l the a j ^ ' s a r e independent random va r i a b l e s , independent of ( e^^)j^» each with expected value = 0 and variance = o"2 a Then cov(x..,x.,..) = n,.,(a 2 + n...a 2 ) i j i J i i <*x J J e i where here, and i n what follows, .0 i f u ^ v uv 1 i f u = v Note, i n p a r t i c u l a r , that cov(x. . ,x. .,) = cov(a. .,a. .) = a 2 for j 4 j ' i j i j l i l i ax cov(x..,x.,.) = cov(o, ,,a, ..) = 0 for i ^ i ' i j l ' j l i l i In other words, observations on d i f f e r e n t ssu's within the same psu are correlated, and observations on the ssu's within the d i f f e r e n t psu's are uncorrelated. Note i n p a r t i c u l a r that i f i = i ' and j = j ' , then cov(x..,x..) = Var(x..) = o 2 + o 2 S i m i l a r l y on the second occasion, the observations (matched or un-matched) are described by the l i n e a r nested model. , i = l , 2,. .. ,N y i j = + ° 2 i + £ 2 ( i ) j {j=l , 2,...,M where y _ = observation on the j t h ssu within the i t h psu P Y = o v e r a l l mean of the observations ot„ . = e f f e c t of the i t h psu 2 i G 2 ( i ) j = e ^ ^ e c t °^ t n e J t n s s u within the i t h psu and a l l the e 2 ( i ) j ' s a r e independent random variables each with expected value = 0 and variance = o 2 , and a l l the ot 's are independent random c 2 ^ i v a r i a b l e s , independent of ^ e 2 ( i ) j ^ ' e a c n w i t h expected value = 0 and variance = a 2 Then cov(y..,y.,.,) = n...(o 2 + n. a 2 ) Note, i n p a r t i c u l a r , that, cov(y i ,yi.. ,) = c o v ( a 2 i , a 2 . ) = a 2 ^ for j + j 1 cov(y.^,y i t ) = c o v ( a 2 i , a 2 . , ) = 0 for i ^ i ' . In other words, observations on d i f f e r e n t ssu's within the same psu are corre l a t e d , and observations on the ssu's within the d i f f e r e n t psu's are uncorrelated. Note i n p a r t i c u l a r that i f i = i ' and j = j ' , then cov(y y ) = Var(y ) = o 2 + a2 . 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 c o v ( x i r y . , . , ) = n l i , ( p 1 o a i O a j + n ^ . a ^ o ^ ) where Pj = the correlation between the effects due to the psu's, and p 2 = the correlation between the effects due to the ssu's within the psu's. Note, in particular, that corr(a l i,a 2.) = n ^ . P i c o r r ( E K i ) j ' E 2 ( i ' ) j ' ) = n i i ' n j r P ! corrCa^. .^) = 0 and c o r r ( B 2 i ' e l ( i ) J ) s ° " Furthermore, i f i = i 1 and j = j ' then c o v ( x . . , y . , . , ) = c o v ( x . . , y . . ) = p 1 a a + p 2 a a The correlation structure assumed implies that (a) observations on the ssu's within the different 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: nq m x'..•=( Z Z x'..)/nmq i=l j = l ^ np m x".. = ( 2 Z x"..)/nmp i = l j=l 1 J ns m y' . . = ( £ £ y1••)/nms i=l j=l 1 J np m and y".. = ( Z Z y"..)/nmp i=l j=l 1 J where x'.. is the mean of observations unmatched on occasion 1 x".. is the mean of observations matched on occasion 1 and 2 y'.. is the mean of observations unmatched on occasion 2 y".. is 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: 1 nq m 1 nq m E(x'..) = — Z Z E(x'. .) = Z Z E(y Y + a + e ,.-.) = p nmq . . . . i i nmq . . . . X l i l ( i ) j X 1=1 j=l i=l j=l 1 np m np m E(x"..) = Z Z E(x" ) = Z Z E ( u Y + a + e , . ) = y nmp i = 1 ^ i j nmp i = 1 ^ X l i l ( i ) j X ns m n ns m - u s H I Y E ( y ' . . ) = — Z Z E(y' ) = — Z Z E(u + a + e , nms i = ]_ j = 1 i j nms . = 1 j = 1 Y 2i 2U and 24 np m np m E(y"..) = — — Z Z E(y". .) = I 2 E(y + a + = UY> w nmp . = 1 x x j nmp . = 1 j = 1 Y 2i 2 ( i ) j Y since a ^ , a 2 i , a n d e 2 ( i ) j a 1 1 h a v e e x P e c t a t i o n e q u a l t 0 zero. The variances of these preliminary estimators are as follows: nq m Var(x'..) = Var( Z Z x' ) n m q i=l j=l 1 J n nq l m Z V a r ( ^ Z x', .) 1= (nq)2 i = 1 m i j 1 m Var( Z x'..) (nq)m 2 x i j m mm \-7 [ 2 V-ar(x'..) + Z Z cov(x' . . ,x' . . . ) ] (nq)m 2 L j = 1 i j , i j i j -r—x—r [m(a 2 + a 2 ) + m(m - l ) a 2 ] (,nq)m a t z1 a t — [ a 2 + ( a 2 /m)] nq cti E, and s i m i l a r l y , Var(x"..) = — [ a 2 + ( a 2 /m)] np a x E i Var(y'..) = — [ a 2 + ( a 2 /m)] ns a_ and Var(y"..) = — [ a 2 + ( a 2 /m)]. y np a 2 E 2 Further, 1 np m np m cov(x"..,y"..) = 7 — T T C O V ( Z Z x" , I Z y" ) (nmp) . = 1 j = 1 i j . I = 1 j l = 1 1 ^ np np m m Z Z Z Z cov(x" ,,y" , ,,) (nmp) 2 1 " 1 r=l .; = 1 — I J " i ' j ' ^ np np m m T r r ^ 2 I Z n..,(p,a a + n..,P2o a ) (nmp) 2 . = 1 . , = 1 j = 1 j t = 1 i i ' a, a 2 J J ' E, E 2 25 ^ np np. m m Z r,,, , ( m 2 P l a a + Z Z n .., P 2 ° o ) (nmp) 2 . = 1 . l = 1 i i ' o, a 2 . = 1 j l = 1 j j ' x n P np , N , £ n..,(m 2p,cr a + m p 2 a a ) (nmp) 2 i = 1 i l = 1 11' a, a 2 e t E 2 / 1 s 2 (nmp) [mpjd a + p 2 a a ] (nmp) z 1 a 1 a 2 e, e 2 ^ (mpjO a + p 2a a ) nmp ' " ' r ' ' o i 1 ' a ! ' E I E 2 and i t can e a s i l y be seen from the c o r r e l a t i o n s t r u c t u r e assumed th a t c o v ( x ' . . , x " . . ) = 0 c o v ( 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. In a l l the d e r i v a t i o n s i t w i l l be assumed t h a t a 2 , a 2 , a 2 , a 2 , P i , and p, are known. al az e 2 e 2 The c u r r e n t mean y y on the 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 s t i m a t o r of the form y 2 £ = a 2 x'>« + b 2 x".. + c 2 y".. + d 2 y'.. where a 2 , b 2 , c 2 , and d 2 are c o n s t a n t s . 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 u n b i a s e d , t h a t i s , E ( y H ) = V S i n c e E(x'..) = E(x".•) = u x and E ( y ' . . ) = E ( y " . .) = V y then to be u n b i a s e d we r e q u i r e t h a t a 2 + b 2 = 0 and c 2 + d 2 = 1. C o n s e q u e n t l y , we o b t a i n t h a t y 2 n = a 2 ( x ' . . - x"..) + c 2 y".. + (1 - c 2 ) y ' . . (1) 26 The variance of th i s estimator i s Var(y 2 ) = a|[Var(x"..) + Var(x'..')] + c\ Var(y"..) JO + (1 - c 2 ) 2 Var(y'..) - 2a 2c 2cov(x"..,y"..) (2) 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 subs t i t u t i o n V a r ( y 2 o ) = a|{[(a 2 /np) + (o 2 /nmp)] + [ ( a 2 /nq) + ( a 2 /nmq)]} + c | [ ( a 2 /np) + ( a 2 /nmp)] + (1 - c 2 ) 2 {(a 2 /ns) + ( a 2 /nms)} 0 t 2 E 2 2 ^ 2 - 2a 2c 2(mp 1a a + p 2a a )/nmp. a x a 2 E j e 2L e t t i n g e,, = ( a 2 /n) + ( a 2 /nm) t»l E l 9 2 2 = ( a 2 /n) + ( a 2 /nm) a 2 E 2 6, = (mp.a a + p 2a a )/nm a : a 2 E i E 2 and i>2 = B 2 /(/e. 2 1 e 2 2 ) we obtain that Var(y 2 j i) = a|[(9 2 1/p) + ( e 2 1 / q ) ] + c 2 ( 6 2 2 / p ) + (1 - c 2 ) 2 ( e 2 2 / s ) - 2a 2c 2B 2/p = a 2 ( - + i ) 6 2 1 + + U ~ C z ) 2 ] e 2 2 - 2a 2c 2B 2/p (3) p q p ^ We now choose those values of the constants a 2 and c 2 i n (3) such that the variance of y 2 . i s minimized. We do this by d i f f e r e n t i a t i n g equation (3) with respect to (w.r.t.) each of a 2 and c 2 , set t i n g the re s u l t s equal to zero, and then simultaneously solving for a 2 and c 2 . We obtain, a f t e r s i m p l i f i c a t i o n , that a* 2 = '{(p/[s(l - q i p f . ) + p])B 2q}/9 2 i = c- 2B 2q/6 2 l . c * 2 = p / [ s ( l - q*l) + p] where a* 2 and c"'"2 are the 'optimal 1 values of a 2 and c 2 , respectively, that minimize Var(y 2 ), and i>2 2 = S 2 2 /( 9 2x 9 22 ). Substituting these 'optimal' values into equation (1) y i e l d s y 2 j / = (c-'- 2e 2q/9 2 1 )(x' .. - x"..) + c-2y"..+ (1 - c* 2 ) y ' . . = c* 2{y"-- + ( 6 2 q / 9 2 1 ) ( x ' . . - x"..)} + (1 - c* 2 ) y * . . B u t x.. = qx'.. + (1 - q)x".. = the grand mean on occasion 1 or x.. - x".. = q(x'.. - x"..) and @ 2/6 2 l = cov(x".. , y"..) /Var (x". .) = ^2YX' 3 r e S r e s s : L O n c o e f f i c i e n t . Then y i % = c» 2{y".. + f 3 2YX (^" " *"-- ) } + ( 1 " c " 2 ) y ' - . Let the quantity {y".. + 6. (x.. - x"..)} = y 2 , a regression estimator. Then y H = c * 2 ( y 2 r e ) + (1 - c* 2 ) y ' .. (4) or, s u b s t i t u t i n g the value of c* into equation (4), y H = U e 2 2 / s ) ( y 2 r e ) + [ ( e 2 2 ( i - * 2 2 ) ) / P + t 2 2 e 2 2 ] y - . . } / {( 9 2 2 / s ) + [ e 2 2 ( i - <i>22)/p + ^ 2 2 e 2 2 ]} = C P y 2 r e + s ( l - q ^ 2 2 ) y ' . . ] / [ p + s ( l - q * 2 2 ) ] . (5) We notice that the current mean estimator i s a weighted average of two uncorrelated estimates y and y'.., and i s unbiased. re The variance of t h i s estimator, which i s the minimum possible for such l i n e a r estimators, stated in equation (2) can be otherwise simply obtained as follows. We can write that, using equation (4) V a r ( y 2 o ) = c * 2 2 Var(y 2 ) + (1 - c * 2 ) 2 Var(y'..) since y, and y'.. are uncorrelated of each other (from the c o r r e l a t i o n re structure assumed). After s u b s t i t u t i o n and further s i m p l i f i c a t i o n we get 28-Var(y 2 j l) = 6 2 2 { [ l - q^22]/[p + s( l - q*|)]> (6) = 9 2 2{(1 - c- 2)/s}. 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 - U M 2 ] and Var(y 2 j i) = { 6 22 ( 1 - q* 2)}/ [ l - (q* 2) 2 ] 2. If, in addition to (1), 6 2 1 = 9 2 2 = 62, i.e., the variances within the stages are the same on both occasions, then y 2 j < = <p y 2 r e + q(i - q^y'..}/ [ i - (q * 2 ) 2 ] and VarCy^) = {92(1 - q*l)}/ [1 - ( q * 2 ) 2 ]• It can be seen that for q = 0 or q = 1, in this special case 2, Var(y 2 ) = 62. This indicates that whether the sample is completely Jo retained or completely replaced by a new sample, the variance of the e s t i -mator is the same. For a l l values of 0 < q < 1, Var(y 2 ) < 62, which Jo indicates that a replacement policy w i l l improve the estimate of current mean i f I|J 2 ^  0 (i.e., p1 ^ 0 and p 2 £ 0). Estimator of the change. The change in means between the two occa-sions of sampling, A = u - y , w i l l be estimated by g 2„, which is of a linear form g 2 = e 2y".. + f 2x".. + h2y'.. + t 2x'.. 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 y then to be u n b i a s e d we r e q u i r e t h a t e 2 + h 2 = 1 and f 2 + t 2 = - 1 . C o n s e q u e n t l y we o b t a i n t h a t g 2 = e 2 y " . . + (1 - e 2 ) y ' . . + f 2 x " . . - (1 + f 2 ) x ' . . (7) 36 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 V a r ( g 2 o ) = e 2 V a r ( y " . . ) + (1 - e 2 ) 2 V a r ( y ' . . ) + f 2 V a r ( x " . . ) 3C-+ (1 + f 2 ) 2 V a r ( x ' . . ) + 2 e 2 f 2 c o v ( x " . . , y " . . ) (8) 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 e a r l i e r 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 s u b s t i t u t i n g i n the v a r i a n c e s and c o v a r i a n c e s , we o b t a i n t h a t v a r ( g 2 o ) = e 2 ( 9 2 2 / p ) + (1 - e 2 ) 2 ( e 2 2 / s ) + f 2 ( 6 2 l / p ) + (1 + f 2 ) 2 ( 6 2 1 / q ) + 2 e 2 f 2 B 2 / p (9) where 6 2 l , 9 2 2 , 0 2 are as d e f i n e d e a r l i e r . We now choose 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 2 and f 2 i n (9) such t h a t the v a r i a n c e o f g 2 i s m i n i m i s e d . We do t h i s by d i f f e r e n t i a t i n g e q u a t i o n (9) w . r . t . each o f e 2 and f 2 , s e t t i n g t he r e s u l t s e q u a l t o z e r o , .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 f o r e 2 and f 2 . We o b t a i n t h a t f * 2 = C- p q B 2 Y X / K 2 ] - [ p ( s + p ) / K 2 ] e- 2 = [p/K 2] + [ p s / K 2 ] B 2 X Y where e' v 2 and f * 2 are the ' o p t i m a l ' v a l u e s of e 2 and f 2 , r e s p e c t i v e l y , t h a t m i n i m i s e the V a r ( g 2 ) 30 6 2 / 6 2 2 = c o v ( x " . . , y " . . ) / V a r ( y " . . ) = B 2 X Y , a r e g r e s s i o n c o e f f i c i e n t K 2 = p + s ( l - q*|) and the 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 . 30 Substituting in the values of e* 2 and f * 2 into equation (8) gives (after multiplying out) 8 2 a = (p/K 2 Vy".. + [ ( p s B 2 X Y ) / K 2 ] y ' \ . + [(s(1 - q* 2))/K 2]y'.. - [(psB 2 X Y)/K 2]y«.. - [ ( p q B 2 Y X ) / K 2 ] x " . . - [(p(s + p))/K 2]x". + [ ( p q 6 2 Y X ) / K 2 ] x ' . . - [q(s + p - s*|)/K 2]x'.. After further s i m p l i f i c a t i o n , we obtain that & 2 t = Up/K2)y2re + [ s ( l - qH»|)/K2]y'..} - {[p(s + p)/K 2]x 2 ]_ e + [q(p + s ( l - ^ 2))/K 2]x'..} (10) where x 2 r e = x".. + B 2 X Y [ s / ( s + p)][y'.. - y"..] This estimate of change i s seen to be a l i n e a r combination of the unbiased current and previous mean estimators y 2 and x 2 , respectively. (Note Ki J6 that the second p r i n c i p a l piece in equation (10) i s x 2 , the BLUE of the JO population mean on occasion 1, given the observations on occasion 2.) The variance of g 2., which i s the minimum possible for such l i n e a r estimators, i s e a s i l y derived by considering equation (8). V a r ( g H ) = e 2 2 [ e * 2 / p + (1 - e * 2 ) 2 / s ] + 6 2 1 [ l + ( f * 2 + p) 2/pq] + 2 e * 2 f * 2 \ p 2 / e 2 1 e 2 2 / p . Substituting i n the values of e* 2 and f * 2 and a f t e r some lengthy algebraic manipulation, we obtain that Var(g 2 j l) = {[p + s(l - *|)]e21 + (1 - q<l>2) 6 2 2 - 2pvp2/6 2 1 9 2 2 }/ [p + s ( l - qtpl)]. (10.1) We s h a l l now examine some sp e c i a l cases. 1. If q = s, g H = [ p / ( l - q 2 ^ I ) ] ( y 2 r e - x 2 r e ) + [ q ( l - qipl)/(l - qa*J)](y'.. - x'..) and Var(g 2 x <) = [(1 - q* 2 )( 6 2 1 + 6 2 2 ) - 2p^ 2/6 2 , 6 2 2 ] / (1 - q 2 ^ 2 ) . 31 2. I f , in addition to ( 1), 6 2 1 = 6 2 2 = 6 2, then 3 ^ = 6 2 y x and g 2 j l = [ p / ( l - q * 2 ) j ( y " - - - x"..) + [ q ( l - 1>2)/(1 - q * 2 ) ] ( y ' . . - x'..) and V a r ( g 2 £ ) = 26 2(1 - q*| - p * 2 ) / [ l - ( q * 2 ) 2 ] (11) It can be seen in equation (11) that for q = 0, Var(g 2^) = 29 2 ( 1 - i>2) (= fixed sampling variance of change) = 1, Var(g 2^) = 26 2 (= independent sampling variance of change) So that forvalues of 0 < q < 1, the variance of the.growth estimator w i l l vary between 26(1 - \|J2) and 26. This indicates that t r a d i t i o n a l CFI or fixed sampling gives improved estimates of growth over p a r t i a l replacement as long as ty2 ^ 0. The e f f i c i e n c y of the change estimator g 2 depends on i>z and q: i t increases with increases i n ty2 and q, that i s , ( i = 1,2) must be high and o , a (t = 1,2) be as low as possible since a e t t ip 2 = [mpjO a + p 2 a a ]/[/9 2 1 6 2 2 (nm) ] . a 1 a 2 E 1 E 2 Three-stage SPR Consider a population 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 of M ssu's, and each ssu containing T t e r t i a r y sample units ( t s u ' s ) . Further, suppose that the sample units (psu's, ssu's, or tsu's) are of equal s i z e . In p a r t i c u l a r , suppose n psu's are selected by srswor on the f i r s t occasion, m ssu's are selected by srswor from each sample psu, and r tsu's are selected by srswor from each of the sample ssu's. A random sample (selected by srswor) of size np (0 < p < 1) of the n psu's i s retained for the second occasion together with i t s respective ssu's and tsu's- drawn from the f i r s t occasion. In addition, a random sample of size ns (s > 0) of the N-n other psu's i s selected by srswor for 32 i n c l u s i o n i n the sample on the second o c c a s i o n . A g a i n , m ss u ' s a r e s e l e c t e d by srswor from each of the ns psu's and r t s u ' s are s e l e c t e d by s r s w o r from each of the nm s s u ' s . The numbers N, M, and T w i l l a g a i n be assumed t o be i n f i n i t e l y l a r g e . O b s e r v a t i o n s a r e t a k e n i n each o f the nmr t s u ' s . on the f i r s t o c c a s i o n and nmr (p + s) t s u ' s on t h e second o c c a s i o n . We 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 f o l l o w s : O c c a s i o n 1 No. of unmatched psu's No. o f matched psu's No. of unmatched ssu's No. o f matched s s u ' s No. of unmatched t s u ' s No. o f matched t s u ' s Matched o b s e r v a t i o n s Unmatched o b s e r v a t i o n s nq np nmq nmp nmrq nmrp x". i j k x' . .. i j k i = l , 2 , . , j = l , 2 , . , k = l , 2 , .. 1=1,2... j = l , 2 , . k = l , 2 , . ,np ,m ,nq ,m , r ns np nms nmp nmr s nmrp y " i j k y i j k i = l , 2 , . j = l , 2 , . k = l , 2 , . 1=1,2,. j = l , 2 , . k = l , 2 , .np ,m , ns ,m , r where q = 1 " P • R e c a l l t h a t X and Y do not r e f e r t o d i f f e r e n t v a r i a b l e s o f i n t e r e s t : t h e y r e f e r t o the same v a r i a b l e 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 1 and Y on o c c a s i o n 2. 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 and the change i n means A = u - u of the v a r i a b l e of i n t e r e s t . I t w i l l Y A be assumed t h a t on the f i r s t o c c a s i o n the o b s e r v a t i o n s (matched or 33 unmatched) are described by the lin e a r nested model i=l,2,...,N i j k l i K i ) j K i j ) k k=l 2 ... T where x.., = observation on the kth tsu within the i t h ssu within the I jk J i t h psu y = o v e r a l l mean of the observations A a, . = ef f e c t of the i t h psu l i r e w . , . = e f f e c t of the i t h ssu within the i t h psu ^ l ( i j ) k = e ^ ^ e c t °f t n e ^th t s u w i t n i n the j t h ssu within the i t h psu and a l l the Y i ( - L j ) k ' s a r e independent random variables each with expected value = 0 and variance = a2 , a l l the E,,.. ,'s are independent random v a r i a b l e s , Yi l ( i ) j independent of ^ ^ ( ^ j ^ ' i each with expected value = 0 and variance = a2 , and a l l the a ^ £ ' s a r e independent random v a r i a b l e s , independent of a 2 {E, , . . i} and iy,,. . x. }, each with expected value = 0 and variance K i ) l ( i j ) k ^ ct! Then cov(x. ,x. , . ,. = r\. (a1 +n...o2 +n...n. , . a 2 ) ijk l'j'k' u ' o, J J ' E, J J 1 kk' y j where 0 i f u 4 v n = i • r uv 1 i f u = v Note, in particular, that cov(x... , x . ) = cov(a..,a...) + cov(e,,...,£,,...)= a 2 + a 2 i f k ^ k ' ijk i jk' l i l i ' l ( i ) j l ( i ) j a t c, cov(x. ,, ,x. ... ,) = cov(a, . ,a, . ) = a 2 i f j £ j 1 and k 4 k1 Ijk I j k' h h otj J cov(x.. k,x.., k) = cov(a 1.,a 1.) = a 2^. i f j 4 j ' C 0 V < : x i j k ' X i ' j ' k " * = 0 i f 1 ^ 1 1 and j ^  j ' . cov(x. .. , x . , .. ) = 0 i f i 4 i ' . ijk l'jk 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; and observations on the tsu's within different psu's are uncorrelated. Note in particular that i f i = i ' , j = j ' , and k = k', then cov(x. ., ,x. ., ) = Var(x. ., ) = a 2 + a 2 + a 2 i j k ' i j k i j k a, E , Y i Similarly, 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 2 ( i j ) k llllly/.^l where y. ., = observation on the kth tsu within the ith ssu within the ith ijk psu u .^ = overall mean of the observations a„. = effect of the ith psu 2i E 2 ( i ) j = e ^ e c t °f J t n s s u W l t h i n the ith psu Y 2 ( i j ) k = e ^ e c t °^ t n e kth tsu within the jth ssu within the ith psu and a l l the ^ 2 ( i j ) k ' S a r e independent random variables each with expected value = 0 and variance = a 2 , a l l the E , , , . , ' s are independent random Y 2 2 ( i ) j variables, independent of ^ Y 2 ( i j ) k ^ ' e a c ^ w i t h expected value = 0 and variance = a 2 , and a l l the ot 's are independent random variables, independent of ^ 2 ( i j ) k ^ a n ^ ^ e 2 ( i ) j ^ ' e a c n w i t h expected value = 0 and A similar correlation structure w i l l be assumed for the observations on the second occasion as that on the f i r s t occasion. 35 In order to impose a correlation structure between occasions 1 and 2, i t w i l l be further assumed that c o v ( x i j k , y i I j I k l ) = n . - . L - p ^ o ^ + ^ . ( P . a ^ o ^ + \ k . > ] where ' * p 3 = the correlation between the effects due to the tsu's within the ssu's within the psu's. Note, in particular, that c o r r ( a l i , a 2 i , ) = n^, Pi /• c o r r ( e i ( i ) . , e 2 ( . , ) . l ) = n.^n.-.P, C O r r ( Y l ( i j ) k ' Y 2 ( i ' j ' ) k ' ) = T 1 i i ' n j j ' \ k ' P s C O r r ( a l i ' £ 2 ( i ) j ) = ° c o r r ( a 2 i ' £ l ( i ) j ) = ° c o r r ( a u , Y 2 ( i . ) k ) = 0 corr(a 2., Y 1 ( i j ) k> = 0 c o r r ( c 2 ( i ) . , Y = 0 and c o r r ( £ l ( i ) j ' Y 2 ( i j ) k } = ° Furthermore, i f i = i ' and j = j ' then C O v ( x i j k ' y f ' j ' k ' ) = C O v ( x i j k ' y i j k ) = P>°0l°«. + P , < V e , + P 3 ° Y ! a Y 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'. )/nmrq i=l j=l k=l 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 m r y"... = ( E £ E y". )/nmrp i = l j = l k=l 1 J where x 1... i s the mean of unmatched o b s e r v a t i o n s on o c c a s i o n 1 y'... i s the mean of unmatched o b s e r v a t i o n s on o c c a s i o n 2 x"... i s the mean of matched o b s e r v a t i o n s on o c c a s i o n 1 y"... i s the mean of matched o b s e r v a t i o n s on o c c a s i o n 2. The expected v a l u e s of the above p r e l i m i n a r y e s t i m a t o r s are as f o l l o w s : nq m r nq m r E(x'.. . ) = — — z £ E E(x'..,) = — — s i i E ( i r + a, . nmrq . . . , , , i j k nmrq X l i M i = l j = l k=l i = l j = l k=l + ^ ( D j ^ K i j ) ^ = y x S i m i l a r l y , E(x"....) = p x . and E ( y ' . . . ) = E ( y " . . . ) = y y , s i n c e a u , a ^ , c 2 ( i j ) J , Y 1 ( i J ) k , and Y 2 ( . . ) k each have expecta-t i o n e q u a l t o z e r o . 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 are as f o l l o w s : nq m r V a r ( x ' . . .) = Var( Z Z E x ' . . . ) n m r q i = l j = l k=l l j k nq ^ m r -. ^ E V a r ( — E E x ' ) ( n q ) 2 . = 1 mr j = 1 ^ i j k 1 . m r 1 V a r ( — E E x : . . ) (nq) mr k = 1 i j k { E E V a r ( x ' ) + r E E c o v ( x ' ,x' , ) (nq) ( m r ) 2 1 R = 1 i j k ^ , i j k i j ' k + E E E E cov(x ! .. ,x!.,,.,)+ m E E cov(x ! ,x! , )} W k^k' l j k 1 J k k^k' l j k l j k 38. {mr(a 2 + a 2 + a 2 ) + mr(m - 1) nq(mr) 2 04 E l Yi + mr(r - l)(m - l ) a 2 + m r ( r - l ) [ a 2 + a 2 ]} — { a 2 + ( a 2 . /m) + ( a 2 /mr)} nq E l Yi and s i m i l a r l y , 1 Var(x"...) = — [ np a , Var(y'...) = — [a ns a-+ ( a 2 /m) + ( a 2 /mr)] E l Yl + ( a 2 /m) + ( a 2 /mr)] e 2 Y 2 and Var(y"...) = — [ a 2 + ( a 2 /m) + ( a 2 /mr)] . np a Y 2 Further, cov(x"...,y"...) = np m r np m • r (nmrp) 2 C O v ( Z 1 1 x " i i k ' Z Z Z y" ) ^ , . = 1 j = 1 k = 1 i j k j l = 1 y i , j , k I ; np np m m r 7 \T 2 I; S • _ Z • Z Z cov(x". .. ,y". , . ,, ,) (nmrp) 2 , = 1 ., = 1 j = 1 . j , = 1 . k = 1 • k , = 1 " U k y I ' j ' k " ' (nmrp) : np np m m r r Z Z Z Z Z Z TI. i = l i'=l j=l j'=i k=l k'=l 3 (p! a a + n . . , p 2 a a + n . . . n. . . p 3 a a ) a , a 2 j j ' E , e 2 J J ' k k ' K 3 Y l Y 2 ^ np np m m (nmrp) 2 , * , ^ , ^ i i ' ( r * p ' ° a , V + r ^ - . P ' ° 0 i=l i'=l j=l j'=l r r + Z Z n k k , P 3 a a ) k=l k'=l K K Y l Y z np np m m 7 r r Z Z n. . , (m 2r 2 p, a a + r 2 Z Z (nmrp) 2 n ' 1 a, a. 1=1 i'=l j=l j'=l m m r i n..,p 2a a + Z Z Z Z n . . , n u u , p3o\. a.. ) J J 1 2 j=l j'=l k=l k'=l I np np Z Z n.. A , (m 2r 2PiO^ + mr 2p 2a a / \ 2 ^ , , . i l l 1 UiU u (nmrp) 2 . = 1 . l = 1 'n' M» C l a. + mrp,a a ) Yi Y 2 39 T - - i - T r - ( ™ r p ) [ m r p l « 7 a i a a i + r p . o ^ o ^ + P . a ^ a ^ ] [mrp,o a + r p 2 a a + p 3a a ] nmrp L"""~ " 1" a," a 2 ' ' ^ i e 2 ' * Yi Y 2 and i t can e a s i l y be seen that cov(x'...,x"...) = 0 cov(x'...,y'...)=0 and cov(y'...,y"... ) = 0. Estimator of the current mean. Again, in a l l the derivations i t w i l l be assumed that a2 , a 2 , a 2 , a 2 , a 2 , a 2 , p,, p 2, and ai a 2 £1 e 2 Y i ' Y 2 p 3 are known. The current mean ji o n t n e second occasion i s estimated by a l i n e a r estimator of the form y 3 = a3x'... + b 2x"... + c 2y"... + d 2y'... where a 3 , b 3, c 3 , and d 3 are constants. We require that this estimator be unbiased, that i s , Since E(x'...) = E(x"...) = y x and E(y'...) = E(y"...) = u y then to be unbiased we require that a 3 + b 3 = 0 and c 3 + d 3 = 1. Consequently we obtain that y 3 ^ = a 3(x'... - x"...) + c 3y"... + (1 - c 3 ) y ' . . . The variance of t h i s estimator i s Var(y 3 ) = a|[Var(x'...) + Var(x"...)] + c|Var(y"...) + (1 - c 3 ) 2 V a r ( y ' . . . ) - 2a 3c 3cov(x"...,y"...) since a l l other covariances are zero (given the correlation structure assumed). Then by substitution Var(y 3 n) = a 2 3 { [ ( a 2 /np) + (a 2 /nmp) + (a 2 /nmrp)] + [(a 2 /nq) + (a 2 /nmq) + (a 2 /nmrq)]} + c 2 3 [ ( o 2 /np) H- (a 2 /nmp) e i Y i a 2 e 2 + (a 2 /nmrp)] + (1 - c 3 ) 2 [ ( a 2 /ns) + (a 2 /runs) + (o 2 /nnirs)] Y 2 a 2 E 2 1 2 - 2a 3c 3(mrp 1o a + rp 2a a + p3cr a )/nmrp. Q j a 2 e i ^ - 2 Y 1 Y 2 Letting e,. = (a 2 /n) + (a 2 /nm) + (a 2 /nmr) O i E j Y l 8 , , = ( a 2 /n) + ( a 2 /nm) + ( a 2 /nmr) a 2 e 2 o Y 2 B3 = [mrpjO- a + rp 2cr a + P3a a ]/nmr ( X 1 0 l 2 £ l E 2 i ] 1 2 and * 3 = B 3 / (/6 3 ,9 J 2) we obtain that V a r ( y H ) = a 2 3 [ ( 6 3 1 / q ) +(e s l/p)] +c 2 3 ( 6 3 2 /p) + (1 - c 3 ) 2 ( 9 3 2 /s) - 2a 3c 3B 3/p. (13) We now choose those values of a 3 and c 3 such that the variance of y 3 is minimized. We obtain that a* 3 = {(p/[s(l - q^ 2 3) + p])B 3q } /6 3 1 = c* 3B 3q/6 3 1 c* s = p/[s(l - qi|j23) + p] where a* 3 and c* s are the 'optimal' values of a 3 and c 3, respectively that minimize Var(y 3 ). Substituting in the values of a* 3 and c* 3 into equa-tion (12) and modifying as shown in the two-stage SPR, we obtain that y,z = c*,(y, r e) + (1 - c*,)y'... (14) = [ p y 3 r e + s ^ - q^ 2 3)y'---l/tp + s(i - q* 2 3)] 41 • where y 3 r e = y".-. + B 3 y x ( x - x»...) and B 3/631 = cov(x"..,y"..)/Var(x"..) -• S^xy' 3 r e 8 r e s s i o n c o e f f i c i e n t . Using equation (14) we determine that (given the c o r r e l a t i o n structure assumed) Var(y ) = c* 2 V a r ( y 3 ) + (1 - c*,) 2Var (y' . . ..) JO LQ = 6, 2(1 - q+|)/[p + s ( l - qi>23)] = 9 3 2 {(1 - c*,)/s}. Again, we can obtain some special cases. 1. If q = s -y H = [ p y , r e + q ( i - q+!)y'.••]/[!- ( q * 3 ) 2 l and Var(y 3 j l) = 6 3 2 ( 1 - qi|»f)/[l - (q+,) 8]. 2. I f , in addition to (1), 63, = e 3 2 = 63, then Var(y 3 j l) = 9 3(1 - q * f ) / [ l - ( q t 3 ) 2 ] . Similar conclusions can be drawn about Var(y 3 ) i n th i s case, as in the JO s-two-stage SPR. Estimator of change. The change i n means between the two occasions of sampling, A = u - u , w i l l be estimated, as before, by I A S,3l = e 3y"... + f 3 x " . . . + h 3y'... + t 3x'... We require that t h i s estimator be unbiased, that i s , E ( g 3 j ; ) = V Y - Mx = A. Since E(x"...) = E(x'...) = n X and E(y"...) = E(y'...) = u then to be unbiased we require that e 3 + h 3 = 1 r and f s + t s = -1. 42 C o n s e q u e n t l y , we o b t a i n t h a t g 3 = e',y"... + (1 - e 3 ) y ' . . . + f 3 x " . . . - (1 + f 3 ) x ' . . . (15) 36 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 V a r ( g 3 j l ) = efVar(y.'-'. . .) + (1 - e s ) 2 V a r (y' . . .) + f f V a r ( x " . . . ) + (1 + f 3 ) 2 V a r ( x ' . . . ) + 2 e 3 f 3 c o v ( x " . . . , y " . . . ) (16) 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 e a r l i e r . By s u b s t i t u t i n g i n the v a r i a n c e s and the c o v a r i a n c e s , we get V a r ( g 3 j l ) = e 2 ( 9 3 2 /p) + (1 - e 3 ) 2 ( 9 3 2 / s ) + f 2 ( 9 3 , / p ) + (1 + f 3 ) 2 ( 6 3 1 / q ) + 2 e 3 f 3 6 3 / p (17) where 9 3 1 , 9 3 2 , and B 3 a r e as d e f i n e d e a r l i e r . We choose the v a l u e s of e 3 and f 3 i n (17) such t h a t the v a r i a n c e of g 3 36 i s m i n i m i s e d . We o b t a i n t h a t f * s = [- p q B 3 Y X / K 3 ] - [ p ( s + p ) / K 3 ] .. -e* s = [ p / K 3 ] + [ p s / K 3 ] B 3 X Y where K 3 = p + s ( l - qi|if) e * 3 and f * 3 are the ' o p t i m a l ' v a l u e s of e 3 and f 3 , r e s p e c t i v e l y , t h a t m i n i m i z e V a r ( g 3 ) 16 B 3 / 6 3 2 = c o v ( x " . • • ) / V a r ( y " . . ) = B 3 X Y , a r e g r e s s i o n ' c o e f f i c i e n t and o t h e r symbols a r e as p r e v i o u s l y d e f i n e d . S u b s t i t u t i n g i n the v a l u e s of e* 3 and f * 3 i n t o e q u a t i o n (15) and a f t e r a l e n g t h y s i m p l i f i c a t i o n , we o b t a i n t h a t g H = ( ( p / K 3 ) y 3 r e + [ s ( l - q * 2 ) / K 3 ] y ' . . . } - { [ p ( s + p ) / K 3 ] x 3 r e + [ q ( p + s ( l - ^ 2 ) ) / K 3 ] x ' . . . (18) where x 3 r e = x"... + 6 3 x y [ s / ( s + p ) ] [ y ' . . . - y"...] = x"... + ^ 3 X Y ^ * * * ~ y " , # , - l 43 Then V a r ( g H ) = {[p + s ( l - * 2 3 ) ' ] e 3 1 + (1 - q * i ) e , 2 - 2pip 3^ 9 3 1 9 3 2 " } / K , . The sp e c i a l cases obtained i n two-stage SPR and the conclusions made therein can s i m i l a r l y be obtained here too. Multistage SPR Following from the derivation of the two- and three-stage SPR, we s h a l l how generalize the derivation to h-stage (h > 1) SPR. Consider a population co n s i s t i n g of N psu's, each psu containing M ssu's, each ssu containing T tsu's, and so on, and each penultimate unit c o n s i s t i n g of W ultimate sample units (at the hth stage). Further suppose that the sample units at each stage of the multistage design are of equal s i z e . In p a r t i c u l a r , suppose n psu's are selected by srswor on the f i r s t occasion, m ssu's are selected by srswor from each of the sample psu's, r tsu's are selected by srswor from each of the sample ssu's, and so on u n t i l a random sample u of the ultimate units i s obtained by srswor from each of the nmr ...penultimate units. A random sample (selected by srswor) of size np (0 < p < 1) of the n psu's i s retained for the second occasion together with i t s respective sub-units drawn from the f i r s t occasion. In addition, a random sample of size ns (s > 0) of the N-n other psu's i s selected by srswor for in c l u s i o n i n the sample on the second occasion. Sub-units are selected from each of the ns psu's by srswor as on the f i r s t occasion. It w i l l be assumed that N, M, T, W are i n f i n i t e l y large numbers. Observations are made i n each of the mnr ... u ultimate units on occasion 1 and in mnr ... u(p + s) ultimate units on occasion 2. The observations w i l l be designated as follows: 44 Occasion No. unmatched psu's No. matched psu's No. unmatched ssu's No. matched ssu's nq np nmq nmp ns np nms nmp No. unmatched ultimate units No. matched ultimate units Unmatched observations nm uq up x' . .. ( i = l , 2 , . . . ,nq) ijk...w x". .. ( i = l , 2 , . . . ,np) 1jk.. .w Matched observations where, for matched or unmatched observations, j = 1,2,...,m k = l , 2 , . . . , r nm us up y' . ., ( i = l , 2 , . . . ,ns) 1 Ijk. . .w y". (i=l , 2 ,...,np) J I j k . . .w w = 1,2 ,... , u and q = 1 - p. Assuming a l i n e a r h - f o l d nested model for the observations at each occasion (matched or unmatched), and given s i m i l a r assumptions at each stage of the h-stage design and across the two occasions, as was done in the two- and three-stage designs, we can define the means of the obser-vations as nq m r u - ( Z Z Z ... Z x'... )/nmr ... uq i=l j=l k=l w=l np m r ( Z Z Z i=l j=l k=l u Z x". .. )/nmr ... up I l k . . .w w=l 45 a n d ns m r = ( E y. E - 1=1 j=l k=l np m r = ( E E E i=l j=l k=l u E y w=l ijk...w ) /nmr ... us E y". i j k . . .w ) /nmr up Var(x'. . Var(x".. Var(y' and Var(y".. Further, cov(x". . The variances of the above means are as follows: + (° 2_ /mr...u))/nq T i ) = ( a 2 + ( a 2 /m) + /mr) + ... «i £ I ) = {a 2 + ( a 2 /m) + ( a 2 /mr) + ... + E I Yi ) = {a 2 + ( a 2 /m) + ( a 2 /mr) + ... + a 2 e 2 Y 2 ) = {a 2 + ( a 2 /m) + ( a 2 /mr) + a 2 E 2 Y 2 + ( a2 /mr...u)}/np T, .....y" ) = B,/p = {(mr...up,a a ) + (r...up,a a ) h ' c ^ c x ^ ^ 1 ^ 2 + ... + (p,a a )} / (nmr. .. up ) . h T j T 2 cov(x' ....... ,x" ) = 0 cov(x' ,y ' ) = 0, and cov(y' ,y" ) = 0. Estimator of the current mean. Using s i m i l a r assumptions given in the two- and three-stage designs, the BLUE of the current mean u Y for the h-stage design i s given by ^h = ^ P ^h + s ( 1 " ^ h ^ ' ^ P + s ( 1 " q ^ h ^ re where 7 h l 3h2 ( o 2 /n) + ( a 2 /nm) + ... + [ a 2 /(nmr...u)] t*i £i T i ( a 2 /n) + ( a 2 /nm) + ... + [ a 2 /(nmr...u)] *h = B h / / 6 h l 9 h 2 = y". + B h y x (x - x' and B h Y X = B h / 6 h l . 46 The variance of t h i s estimator, which i s the minimum possible for such l i n e a r estimators, i s given as Var(y ) = 9^(1 - qip * } / [ p + s ( l - q * J ) ] -I If q = s, then yh = (P yh + q d - q^ )y' }/[P - ( q * h ) 2 ] il r e and V a r ( y h ) = 9^(1 - q i p * ) / [ l - ( q ^ ) 2 ] . I In addition, i f 9,, = 9, „ = 9, , then ni hi n V a r ( 9 h ) = 9 h ( l - q ^ ) / [ l - (q1> h) 2]. I Estimator of the change. The BLUE of the change in means on the two occasions i s given by §h = {(p/Vyh + [ s ( 1 " q ^ h ) / K h ] y ' } " { [ p ( s + p ) / K h ] x h H re re + Jlqtp + s ( l -* 2 h)]/K hIx'.......} where 3hXY = 6 h / 6 h 2 and = p + s ( l - q ^ ) -The variance of t h i s estimator, which i s the minimum possible for such l i n e a r estimators, i s given by V a r ( g h ) = {[p + s ( l - * 2 ) ] 9 h l + (1 - q * 2 ) e h 2 I ~ 2 P V e h l 6 h 2 } / K h -If q = s, then § h = { P / [ i - ( q * h ) 2 ] > ( y h - x h ) + { q ( l - q ^ h ) 2 ] } ( y ' - x' il re re and V a r ( g ) = {[1 - q* h«] ( 9 ^ + 8 ^ ) - 2 p V l ^ } / [ l - ( q * h ) « ] . 47 In a d d i t i o n , i f 6,, = 6, „ = 6,, then h i nz n 6hYX = BhXY and V a r ( g h ) = 29 (1 - q^ - p * h ) / [ l - ( q * . ) 2 ] . I Unequal S i z e Sampling U n i t s In the p r e v i o u s s e c t i o n s , i t was assumed t h a t the sampling u n i t s at each stage were of equal p h y s i c a l s i z e . However, i n sampling e x t e n s i v e p o p u l a t i o n s , such as f o r e s t s , sampling u n i t s t h a t v a r y i n s i z e are f r e q u e n t l y e n c o u n t e r e d . I f the s i z e s do not v a r y g r e a t l y , one method of a n a l y s i s would be to s t r a t i f y the u n i t s by s i z e , say, so t h a t the u n i t s w i t h i n a s t r a t u m become equal i n s i z e , or n e a r l y so. In t h i s c a s e, the formulae a l r e a d y d e v e l o p e d c o u l d be used w i t h i n each s t r a t u m . O f t e n , however, t h e r e e x i s t s u b s t a n t i a l d i f f e r e n c e s i n s i z e between the sampling u n i t s at each s t a g e . Separate e s t i m a t o r s must be developed to handle the case i n which the u n i t s v a r y i n s i z e . E s t i m a t o r s of the c u r r e n t mean and the change are now d e r i v e d f o r sampling u n i t s t h a t v a r y i n s i z e i n two-stage SPR on two o c c a s i o n s . (The cases f o r t h r e e - s t a g e and m u l t i s t a g e d e s i g n s w i l l not be c o n s i d e r e d here.) 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 psu's and the i t h psu c o n s i s t i n g of M\ s s u ' s . Suppose n psu's are s e l e c t e d by srswor on the f i r s t o c c a s i o n and nu ssu's are s e l e c t e d by srswor from the i t h sample psu. A random sample ( s e l e c t e d by srswor) of s i z e np (0 < p < 1) of the n psu's i 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 ssu's drawn from the f i r s t o c c a s i o n . ' 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 i s s e l e c t e d by srswor f o r i n c l u s i o n i n the sample on the second o c c a s i o n . A g a i n , m^ ssu's a r e s e l e c t e d by srswor from the i t h psu of the ns psu's. I t w i l l be assumed t h a t N and M. i = l , 2 , . . . , N a r e i n f i n i t e l y 48 l a r g e . L e t m ' t i = t n e number of unmatched s s u ' s on the i t h psu on the t t h o c c a s i o n ( t = 1 , 2 ) , and m" = t h e number of matched s s u ' s on the i t h psu on the t t h o c c a s i o n ( t = 1 , 2 ) . (Note t h a t m" . = m"„. by m a t c h i n g . ) l i 2 i . " n O b s e r v a t i o n s a r e t a k e n on the Z m, . ssu's on the f i r s t o c c a s i o n 1 = 1 U n(p+s) and on the Z m„. ssu's on the second o c c a s i o n . The o b s e r v a t i o n s i = l 2 1 w i l l b e . d e s i g n a t e d as f o l l o w s : O c c a s i o n Unmatched o b s e r v a t i o n s x'.. { ^  } ' 2 ' * * * 'n<? y ' . . {* } ' ? ' " " * ' n S i j j = l , 2 , . . . , m ' u - J i j j = l , 2 , . . . , m ' 2 i Matched o b s e r v a t i o n s x".. { 1 = J ' \ **" " ' n p, y" . . { i =J . 2 , ... ,np i j j = l , 2 , . . . .m"^ i j j = l , 2 , . . . ,m"2i I t w i l l be assumed t h a t on the f i r s t o c c a s i o n , the o b s e r v a t i o n s (matched or unmatched) a r e d e s c r i b e d by the l i n e a r n e s t e d model X i j = + °li + e l ( i ) j ^ = } , o , , , , , S J — i ,Z, •.•, where u v , a,., and e w . s . are as d e f i n e d e a r l i e r . X l i l ( i ) j The same c o r r e l a t i o n s t r u c t u r e assumed i n the e q u a l s i z e case o f the two-s t a g e SPR w i l l be assumed h e r e , f o r the o b s e r v a t i o n s on o c c a s i o n 1. S i m i l a r l y , the o b s e r v a t i o n s on the second o c c a s i o n w i l l be d e s c r i b e d by a l i n e a r n e s t e d model and the same c o r r e l a t i o n s t r u c t u r e assumed i n the e q u a l s i z e case o f the two-stage SPR w i l l a l s o be assumed h e r e , f o r the o b s e r v a t i o n s on o c c a s i o n 2. 49 Furthermore, the same c o r r e l a t i o n s t r u c t u r e , as t h a t f o r the equ a l s i z e c a s e, w i l l be assumed f o r the o b s e r v a t i o n s on the f i r s t and second o c c a s i o n s . From the sample o b s e r v a t i o n s we o b t a i n unweighted p r e l i m i n a r y e s t i -mators as f o l l o w s : nq m' u nq x' . . = ( Z Z x ' . . ) / Z m', . . . . , i j . , l i i = l j = l J i = l np m" l i np x".. = ( Z Z x". . ) / Z m". . . . . . . . I J . . . . . l i i = l j = l i = l ns m' . ns 2 i y'.. = ( £ Z y ' . . ) / Z m' . . . . , i j . , 2 i i = l j = l i = l np m"2. np y".. = ( Z Z y " . . ) / Z m" . i = l j = l i = l The expected v a l u e s of the above e s t i m a t o r s are as f o l l o w s : E ( x ' . . ) = E{( Z Z - ' . . ) / Z m' } i = l j = l 1 J i = l nq m^. nq = E{[ Z Z (v. + a + e. . * , ) ] / Z m' } i = l j = l * 1 U , J i = l 1 1 nq nq nq m1^ nq = E { u + ( Z m ' . a . ) / Z m ' . + ( Z Z e . ) / i m< } X 1 = 1 1 1 1 i = l 1 1 1=1 j = l i = l U nq nq nq m ' j , nq = p v + E [ ( Z m'..a.)/ Z m. + ( Z Z c . . . . ) / Z m' .] X i = l 1 1 1 1=1 1 i = l j = l i = l 1 1 = u x. / 50 S i m i l a r l y , i t can be shown t h a t E(x"..) = u} E(y'..) = and E(y"..) = We can d e f i n e the v a r i a n c e s o f the p r e l i m i n a r y e s t i m a t o r s as f o l l o w s : nq nq m' V c • ( i ) j V a r ( x ' . . ) = Var I m ' a . Z i = l 1 1 1 l i Z z, i = l j = l + J nq nq Z m" Z m' ...... i = l 1=1 nq nq nq m ' u nq = a 2 ( Z tn" . 2 ) / ( 2 m' . ) 2 + ( Z Z a 2 )/( Z m'..! a, . , i i . . l i . . . n e, . , l i 1 i = l i = l i = l j = l 1 i = l nq nq nq nq = a 2 ( Z m 2 ) / ( Z m' . ) 2 + a 2 ( Z m' . ) / ( Z m'..) : a , . , . ! ! .-. . l i • e, . , l i . . l i 1 1=1 1=1 1 i = l i = l nq nq nq = a 2 ( Z m' . 2 ) / ( Z m< . ) 2 + a 2 I Z m'. . a, . . 1 l . , 1 l E , . . 1 l 1 1=1 i = l 1 i = l = a2n ( n q m * 2 ) / ( n q m j ) 2 +a 2_ /nqm'i i 1 m » 1 { 0 2 + — a 2 } where nq nq nq ( m p 2 . a i e> {u[ a'' a 2 } mi m'j = ( Z m' l i )/nq i = l m , 2 and TT [ nq = (* Z m'2 . )/nq i = l _ nq nq - m T ] / ( m ' 1 ) 2 = nq( Z m'2 )/( Z m' ) : i = l i = l 1 1 51 S i m i l a r l y , V a r ( x " . . ) 1 •CTI V o 2 + _1_ a 2 np «i S " J _ {TT 2 a 2 + J _ a 2 ns ™ 2 _1_ {TT'2' a 2 + J _ a 2 np a 2 my where np np TT", = np( 2 m" . 2 ) / ( £ m" )• i = l 1=1 ns ns i r ' 2 = ns( Z m' . *.)/( z m'2i)2 a n d i = l . i = l np np „» 2 = np( 2 m " 2 . 2 ) / ( S m " 2 . ) 2 i = l i = l (Note t h a t ny = TTy = IT" say, and m" = my = m" say.) F u r t h e r , i t can be e a s i l y shown t h a t i n t h i s case o f unequal s i z e u n i t s c o v ( x " . . ,y". . ) = — [TT"P,O- 2 a 2 + — p 2 a a ] c o v ( x ' . . , x " . . ) = 0 c o v ( x ' . . , y ' . . ) = C and c o v ( y ' . . ,y"..) = 0. E s t i m a t o r o f the c u r r e n t mean. The c u r r e n t mean p y on the 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 c o m b i n a t i o n o f t h e p r e l i m i n a r y e s t i m a t o r s as b e f o r e [ s e e e q u a t i o n ( 1 ) ] . The u n b i a s e d n e s s r e q u i r e m e n t l e a d s t o y 2 j l = a 2 ( x ' - x") + c 2 y " + (1 - c 2 ) y ' (18.1) Then V a r ( y 2 . ) = a 2 [ V a r ( x " . . ) + V a r ( x ' . . ) ] + c 2 V a r ( y " . , ) + (1 - c 2 ) 2 V a r ( y ' . . ) - 2 a 2 c 2 c o v ( x " . . , y " . . ) . 52 If we l e t Var(x"..) = kh*\> o 2 + ± o 2 )] = i 6", p n * o, - E, mi Var(x-..) o* 4 a ' ( i ) ) = i e - , Var(y"..) = — [ — ( i r r t o 2 + — o 2 )] = - 9" P n 2 a 2 - E 2 p Var(y'..) = ^ ( r r ' o 2 + o 2 )] = i 6« s n 2 a - E . z m2 '(x"..,y"..) = -[-(ir"p,a a + — p 2a a ) ] = ~~ &1 P n a, a 2 -„ e t £ 2 p Then Var(y 2 j, :) = a 2[±9'\ + ± 6 \ ] + c 2 ( i 6 " 2 ) + (1 - c 2 ) 2 ( ^ 6'2) 3 2 C 2 p We now choose the values of a 2 and c 2 that minimize V a r ( y 2 ) as before. We obtain that a* 2 = c* 2qB'/(p9' I + qQ'\) e'2p(pe'!+ qe"!) and (pe' 2+ seyXpe'i-f qey) - pqsB' 2 These values of a* 2 and c* 2 can then be substituted into equation (18.1) to give the BLUE of the current mean for the case of unequal size sampling units. The variance of the estimator so obtained, which i s the minimum possible for such l i n e a r estimators i s given as 92[9'2'(p9| + q 8y) - qpB , 2](p9; + qey) Var(y ) = s[p9{ + q9'/][(p9 2 + se'2')(p0; + qe*,') - pqsB' 2] e'2p(Pe; +'qey) = (e 2/s) {l • } s[(p9 2 + seyHpe; + qey) - pqse 1 2 ] The sp e c i a l case in which the sampling units within stages are of equal s i z e , considered in the e a r l i e r section, can be obtained from t h i s 53 general r e s u l t by su b s t i t u t i n g the following equivalent forms: 9*2 = 0 2 2 9"2 = 6 2 2 6 ' 1 = 6 2 1 9"! = 9 2 i and 6' 2 = B 2. -Estimator of the change. S i m i l a r l y , to estimate change in t h i s case of unequal size sampling units, we combine the preliminary estimators as before [equation (7)]. In this case, however, the values of e 2 and f 2 that : minimize the V a r ( g 2 o ) , e* 2 and f * 2 > res p e c t i v e l y , are e*a =; {•He,j(q8"i + pe ' i)[(se" 2 + p9 ' 2 ) ( q e ' \ + pe\) - PqsB ' 2 ] 7 S 2 p 2 q(q9" 1 + p 9 \ ) l ] - [(B'e'^pS'j + s9" 2) - qB' 29' 2 ) / p 2qs]}/ {[ICpg1! + q 9 " 1 ) ( P 9 ' 2 + S 9 " 2 ) [ ( s 9 " 2 + P 9 J 2 ) ( q 9 " i + p 6 \ ) - pqsB' 2]]]/ s 2p 3 q(q9" 1 + p9' i)} f " 2 = - [ e ' j ( p e ' 2 + s9" 2) + qB ' e ^ C q e " , + pe\)/ q[(s0" 2 + P 9 ' 2 ) ( q 9 " 1 + p6' x - qspB' 2)]. These values can then be substituted into equation (7) to obtain the BLUE of change for the unequal size, sampling units case. The variance of the estimator so obtained, the minimum possible for such l i n e a r estimators, i s s i m i l a r l y obtained by s u b s t i t u t i n g e* 2 and f * 2 into equation (9). Other Estimators The discussion so far has been confined to best, l i n e a r and unbiased estimators. These were obtained by combining a regression estimate from the matched portion of the sample with a mean per unit estimate based on the current sample. We s h a l l now derive the theory of two-stage SPR (equal-sized sampling units at each stage) using another estimator, the r a t i o estimator (RE). Assume that sampling i s done as was done e a r l i e r 54 f o r the two-stage e q u a l - s i z e d s a m p l i n g u n i t s c a s e . C u r r e n t mean. U s i n g a dou b l e s a m p l i n g r a t i o e s t i m a t e , we can o b t a an improved e s t i m a t o r y"^ of an f o l l o w s : y " r = ( y " . • / x " . . ) x . . = R x.. where x.. = p x".. + q x'.. = o v e r a l l sample mean on the f i r s t o c c a s i o n . We can r e w r i t e y" as f o l l o w s : r y " r = ^ + - ^ = £ - 1 1 u + Z _ L L ( x . . - U ) x".. X x".. X We n o t i c e t h a t the p i e c e ( y " . . / x " . . ) p x i s t he u s u a l r a t i o e s t i m a t o r of and t h a t the' q u a n t i t y ( y " . . / x " . . ) ( x . . - p x ) i s e x p e c t e d t o be v e r y s m a l l ( n e g l i g i b l e ) . Then we can w r i t e y" = ( y " . . / x " . . ) x . . - ( y " . . / x " . . ) u v . r A C o n s e q u e n t l y , u s i n g t h e r e a s o n i n g o f C o c h r a n (1977:343) we o b t a i n t h a t V a r ( y " r ) = ^ ( 6 2 2 - q[ 2Ri|; 2 /e 2 1 9 2 2 - R'"6 2 1 ]} where R = u /u ( e s t i m a t e d by R = y"../x"..) Y X An e s t i m a t o r y2^_ of the p o p u l a t i o n mean u y on the second o c c a s i o n i s g i v e n by t a k i n g a w e i g h t e d average of y"^ and y'.. as f o l l o w s : y * r = w y " r + (1 - w)y'.. (19) where w and (1 - w) a r e w e i g h t s . 55 The minimum-variance e s t i m a t o r of y 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 of w w h i c h m i n i m i z e s the V a r ( y 2 ^ ) . S i n c e y"^ i s s t a t i s t i c a l l y u n c o r r e l a t e d w i t h y'.. ( g i v e n the assumed c o r r e l a t i o n s t r u c t u r e ) , t h e n V a r ( y 2 r ) = w 2 V a r ( y " r ) + (1 - w ) 2 V a r ( y ' . . ) (20) D i f f e r e n t i a t i n g (20) w . r . t . w, s e t t i n g the r e s u l t s e q u a l t o z e r o , and s o l v -i n g f o r w g i v e s ' ' w* = V a r ( y ' . . ) / [ V a r ( y " ) + V a r ( y ' . . ) ] (21) where w" i s the v a l u e o f w w h i c h m i n i m i z e s V a r ( y 2 ^ _ ) . By s u b s t i t u t i n g i n the v a r i a n c e s i n t o e q u a t i o n (21) we o b t a i n t h a t w* = ( 6 2 2 /s)/H {( 9 2 2 - q [ 2 R i p 2 ^ 9 2 l e 2 2 - R 2 9 2 I ] ) / p } + ( 6 2 2 / s ) l | = p/{p + s [ l - q ( 2 ^ 2 A - 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 o f the a v e r -ages o v e r the ssu's on o c c a s i o n 1 t o the c o e f f i c i e n t o f v a r i a t i o n of t h e a v e r a g e s o v e r the s s u ' s on o c c a s i o n 2. We t h e n o b t a i n t h a t y 2 r = <P y " r + s [ l - q(2+ 2A - A 2 ) ] y ' . . } / { p + s [ l - q ( 2 * 2 A - A 2 ) ] } _ and t h a t V a r ( y 2 r ) = 9 2 2 [ l - q( 2i|>2 A- . A 2 ) ] / {p + s [ l - q ( 2 * 2 A - A 2 ) ] } = 9 2 2 { ( 1 - w * ) / s } . We can o b t a i n some s p e c i a l c a s e s as f o l l o w s : 1. I f q = s ? 2 r - {p y " r + q [ l - q ( 2 * 2 A - A 2 ) ] y ' . . } / [ 1 - q 2 ( 2 * 2 A - A 2 ) ] and V a r ( y 2 r ) - 9 2 2 [ l - q ( 2 ^ 2 A - A 2 ) ] / { 1 - q 2(2i|; 2A - A 2 ) } . 56 2. I f q. = s and 6 2 1 = 92-2 = 8 2 y 2 r = <P y " r .+ q [ l - q R ( 2 ^ 2 - R ) ] y ' . . } / [ 1 - q 2 R ( 2 ^ 2 - R ) ] and V a r ( y 2 ] _ ) = 6 2 [ l - q R ( 2 * 2 - R ) ] / U - q 2R(2<f 2 - R)> A g a i n , h e r e t o o , i f q = 0 or q = 1, V a r ( y 2 ^ _ ) = 9 2 and f o r a l l o t h e r v a l u e s of q, 0 < q < 1, V a r ( y 2 ] _ ) < 6 2 i f i>2, R > 0. T h i s a l s o i n d i c a t e s t h a t a r e p l a c e m e n t p o l i c y w i l l improve the e s t i m a t e of y y , i f i>2 4 0. As was p o i n t e d out e a r l i e r , y 2 ^ i s b i a s e d . Now we s h a l l d e t e r m i n e the amount of t h i s b i a s . We can r e w r i t e y 2 ]_ i n (19) as y 2 j_ = w{p y".. + q ( y " . . / x " . . ) x ' .. - y'..} + y'.. T a k i n g the e x p e c t e d v a l u e of t h i s q u a n t i t y E ( y 2 r ) = w E[p y".. + q ( y " . . / x " . . ) x ' . . - y'..] + E ( y ' . . ) = w[p y y + q E ( y " . . / x " . . ) y x - y y ] + y y b u t , a c c o r d i n g t o Murphy (1967:304) E ( y " . . / x " . . ) = ( y y / y x ) [ l + ( 9 2 1 / p y 2 x ) - ( i | i 2 / 9 2 x 6 2 2 ) / y x y y p] t o terms of o r d e r 1/n. T h i s means t h a t E ( y 2 r ) * w q[R9 2 1/p y x - R ^ 2 / 9 2 1 9 2 2 / y y p] + y y . Then b i a s of y2^_ w i l l be g i v e n by B ( y 2 ] . ) = E ( y 2 r - y y ) = ( w q [ R 6 2 1 / y x ) - R ^ 2 / 9 , 9 2 / u y ] / p + y y } - y y = wq R ( 9 2 1 / y y - * 2 / 6 2 1 9 2 2 / y y ) / p o r s u b s t i t u t i n g w* f o r w B ( y 2 ) » q[R 9 2 l - * 2 / 9 , 9 2 ] / { (s + p - q s [ 2 ^ 2 A - A 2 ] ) y }. r A We can see t h a t f o r l a r g e n, the b i a s becomes n e g l i g i b l e and, i n p r a c -t i c e , t h e b i a s i s i n s i g n i f i c a n t even f o r moderate samples. ( F o r example, i f A = 1, q = 0.7, s = 0.4, R = 1.1, i>z = . 9 , 6 2 , = 9 2 2 = 1 , n = 30, y v = 530 m 3/ha, and y'.. = 550 m 3/ha, B ( y 2 ) = 0.00277 f o r y 2 = 552.50302.) 57 Comparison of the c u r r e n t mean e s t i m a t o r s . . I t would be u s e f u l t o compare the r e l a t i v e e f f i c i e n c i e s of the BLUE and the RE i n e s t i m a t i n g the c u r r e n t mean on the second o c c a s i o n i n a two-stage d e s i g n . 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 2 ^ o v e r y 2 j_ i s g i v e n by Q = [ V a r ( y 2 r ) - V a r ( y 2 j l ) ] / V a r ( y 2 j t ) . = ( [ p + s ( l - g ^ 2 ) ] H i - ( q / e 2 2 ) [ 2 R » 2 / ' e 2 1 e 2 2 - R 2 e 2 1 ] } _ l (LTP + s [ l - ( q / 6 2 2 )( 2 R ^ 2 / 6 2 1 e 2 2 - R 2 9 2 1 ) T J j ( l - q* l ) = H{(n - q s * | ) [ l - q ( 2 ^ 2 A - A 2 ) ] } / { [ f i - q s ( 2 * 2 A - A 2 ) ] ( l - qi>l)}J\-l where .fi =' s + p = r a t i o o f sample s i z e on o c c a s i o n 2 to t h a t on o c c a s i o n 1 We can 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 -C - qs(p,4>1 + P 2 * 2 ) 2 ] [ l - q(2A(p I«D L + P2<t>2) - A 2 ) ] } / {[fi - qs(2A(p1<t>1 + p24>2) - A 2 ) ] ( l - q(Pi«t>,„.+ P 2<t> 2)f[] - 1 where 4 > 1 = a a / ( n / 9 2 1 9 2 2 ) a j a 2 and a2 = a a / ( n m / 9 2 , 9 2 2 ) e 1 G 2 We can now t h e n t a b u l a t e the v a l u e s o f Q f o r each v a l u e of fi, A, p,, p 2, <)>!, and <t>2. T h i s has been done f o r some s e l e c t e d v a l u e s o f fi, A, P i , P 2, <t>i, and <t>2, and are g i v e n i n T a b l e I . Note t h a t i n T a b l e I , Oj = <(>2 = 0.5. From T a b l e I we can make the 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, the e f f i c i e n c y g a i n i n c r e a s e s as A i n c r e a s e s , 2. f o r f i x e d A, the e f f i c i e n c y g a i n d e c r e a s e s as fi i n c r e a s e s , 3. as A tends t o p l 5 e f f i c i e n c y d e c l i n e s , 4. e f f i c i e n c y i n c r e a s e s as px i n c r e a s e s , 5. f o r v a l u e s of pt l e s s than 0.9, e f f i c i e n c y i s h i g h e s t f o r p = .4 or RH1 RH2 0 .6 0 .6 0 .6 0 . 7 0 .6 0 . 8 0 .6 0 . 9 RH1 RH2 RH1 RH2 RH1 RH2 RH1 RH2 0 .4 0 . 9 1 .9 3.4 TABLE I: PERCENT GAIN IN EFFICIENCY (0%) OF J2-^ OVER y 2 ^ DELTA= 0 .50 OMEGA=1.00 P 0 .2 0 .3 0 .4 0 .5 0 .6 0MEGA=O.75 P 0 .2 0 . 3 0 .4 0 .5 0 .6 OMEGA= 1 .25 P 0 .2 0.3 0.4 0 .5 0 .6 0 .4 0 .5 1.1 1.1 2.1 2.1 3.7 3.5 0 .4 1 .0 1 .9 3.2 0 . 4 0 . 9 1.7 2.7 0 .3 0 .7 1 .5 2.7 0 .3 0 .4 0 .3 0 .3 0 .2 0 .3 0 .3 0 .3 0.2 0 .8 0 .8 0 .8 0 .7 0 .6 0 .7 0 .7 0 .6 0 .6 1.6 1.6 1.5 1.3 1.2 1 . 3 . 1 . 3 1.2 1.1 2 .9 2.8 2.5 2.1 2.2 2.4 2.3 2 .0 1.7 DELTA= 0.75 OMEGA=0.75 P OMEGA=1.00 P OMEGAM.25 P 0 . 2 0 . 3 0 .4 0 . 5 0 . 6 0 . 2 0 . 3 0 .4 0 .5 0 .6 0. 2 0. 3 0 .4 0 . 8 1 . 0 1.0 1.0 0 . 9 0 . 6 0. 8 0 .8 0 .7 0.7. 0. 5 0. 6 0 .6 0 . 4 0. 5 0 . 5 0 .5 0 . 4 O. 3 0. 4 0 .4 0 .4 0 .3 0. 3 0. 3 0 .3 0 . 1 0. 1 O.1 0.1 0 . 1 0 . 1 0. 1 0.1 0 . 1 0 . 1 0. 1 0. 1 0 . 1 0 . 0 O. 0 -o.o -O.O -0 . 0 -O. 0 -0 . 0 - 0 . 0 -O.O - 0 . 0 0. O -0. 0 0 . 0 DELTA 1 . 00 OMEGA=0.75 OMEGA=1 .00 OMEGA= P P I 0. 2 0. ,3 0 .4 0 . 5 0. 6 O. 2 0. 3 0 .4 0 .5 0 .6 0. 2 0. .3 0 .4 5. ,4 6. .5 6 .9 6 .7 6. 1 4. 1 5 , .0 5.3 5.1 4 .6 3, ,3 4 , .0 4.3 4 . 8 5. .6 5.7 5.4 4 . 8 3. 7 4 . 3 4 .4 4.2 3.7 3. .0 3. ,5 3.6 4 . 1 4 .6 4 .6 4 .3 3. . 7 3. .2 3, .6 3.6 3.3 2.9 2 .6 2 .9 2.9 3. .4 3 .7 3.5 3.2 2 .7 2. .7 2 .9 2 .8 2.5 2. 1 2 .2 2 .4 2.3 DELTA = 1 . ,25 OMEGA=0.75 OMEGA=1 p .00 OMEGA= i 0 .2 0 P .3 0 .4 0 .5 0 .6 0 .2 0 .3 0 .4 0 .5 0 .6 0 .2 0 .3 0.4 12 .2 15 .4 16.9 17.0 15 .7 9 . 1 11 .5 12.6 12.7 11.7 7 .3 9 .2 10. 1 12 .0 14 .7 15.7 15.4 13 .9 9 . 1 1 1 . 1 11.8 11.6 10.5 7 .3 8 .9 9.5 1 1 .9 14 .0 14.5 13.8 12 . 2 9 .0 10 .6 11 .0 10.5 9.3 7 .3 8 .6 8.9 1 1 .9 13 .3 13.2 12.2 10 .6 9 . 1 10 .2 10.2 9.4 8. 1 7 .4 8 .3 8.3 DELTA = 1 .50 OMEGA=0.75 OMEGA=1 ,00 OMEGA= P P OA 0 .2 0 .3 0 .4 0 . 5 0 .6 0 .2 0 .3 0 .4 0 .6 0 .2 0 .3 0 .4 0 .6 0 .3 0. 1 0 . 0 0 .5 0 .3 0. 1 -0.0 4.2 3.4 2.7 2 .0 3.7 3.0 2.3 1 .7 10. 1 9.3 9.4 8.4 7.5 6 .6 19.2 2 5 . 5 29 .4 3 0 . 6 29 .2 19.8 2 5 . 5 28 .6 29.1 27 .2 2 0 . 6 2 5 . 7 27 .8 27 .6 25 .2 2 1 . 8 2 6 . 0 27 .2 26.1 23 .2 14.1 18.6 21 .3 22.2 21 .2 14.6 18.8 2 1 . 0 21 .3 19.9 15.4 19.1 20 .6 20.4 18.7 16.4 19.5 20 .4 19.6 17.4 11.2 14.7 16.8 17.4 16.6 11.6 14.9 16.6 16.8 15.7 12.2 15.2 16.4 16.2 14.8 13.1 15.6 16.3 15.7 13.9 0MEGAM.5O P 0 .2 0.3 0 .4 0 .5 0 .6 0.2 0.2 0 .2 0 .2 0.2 0 .5 0.6 0 .6 0 .5 0 .5 1.0 1 1 1.1 1 0 0 .9 1.9 2.0 1.9 1.7 1.5 OMEGA=1.50 P 0 .2 0.3 0 .4 0 . 5 0 .6 0 .4 0.5 0 .5 0 .5 0 .5 0 .2 0 .3 0 .3 0 .2 0.2 0.1 0.1 0.1 0.1 0.1 O.O 0 .0 - 0 . 0 0 . 0 0 . 0 OMEGA=1.50 P 0 .2 0.3 0 .4 0 .5 0 .6 2.8 3.4 3.6 3.5 3.1 2.5 2.9 3 .0 2 .9 2.5 2.2 2.5 2.5 2.3 2 .0 1.9 2.0 1.9 1.7 1.5 OMEGA=1.50 P 0 .2 0.3 0 .4 0 .5 0 .6 6.1 7.6 8.4 8.4 7.8 6.1 7.4 7.9 7.8 7.0 6.1 7.2 7.5 7.1 6 .3 6.2 7.0 7 .0 6 .5 5.6 0MEGAM.5O P 0 .2 0.3 0 .4 0 . 5 0 .6 9.2 12.1 13.8 14.3 13.7 9.6 12.3 13.7 13.9 13.0 10.2 12.6 13.6 13.5 12.3 10.9 13.0 13.6 13 .0 11.6 00 NOTE: DELTA=Rje„/e i,;.OMEGA = S+P. RH1= ( . RH2= £ TABLE I: PERCENT GAIN IN EFFICIENCY (0%) OF OVER DELTA= 0.50 OMEGA=0 .75 OMEGA*1 .00 OMEGA=1 D .25 OMEGA =1 P .50 RH1 RH2 O. 2 0. P 3 0.4 0.5 0. 6 0'. 2 0. 3 r 0.4 0.5 0.6 0. 2 O. 3 0.4 0.5 0. 6 0. 2 0. 3 0.4 0.5 0. 6 0.7 O. 6 0. 9 1 . 1 1.1 1 .0 0. 9 0. 7 0. 8 0.8 0.8 0.7 0. 6 0. 7 0.7 0.6 0. 6 0. 5 0. 6 0.6 0.5 0. 5 0.7 0.7 1 . 9 2. 1 2.1 1 .9 1 . 7 1 . 5 1 . 6 1 .6 1 .5 1 .3 1 . 2 1 . 3 1 . 3 1 .2 1 . 1 1 . 0 1 . 1 1 . 1 1 .O 0. 9 0.7 0.8 3. 4 3 . 7 3.5 3.2 2. 7 2. 7 2. 9 2.8 2.5 2. 1 2. 2 2. 4 2.3 2.0 1 . 7 1 . 9 2. 0 1 .9 1 . 7 1 . 50.7 0.9 • 5 . 8 5. 9 5.5 4.9 4. 0 4. 6 4 . 7 4.4 3.8 3.2 3. 7 3. 9 3.6 3.2 2 . 6 3. 2 3. 3 3. 1 2 . 7 2 . 2 DELTA 0. 75 OMEGA=0 . 75 OMEGA=1 .00 OMEGA=1 D • 25 . OMEGA=1 P .50 RH1 RH2 0. 2 0. P 3 0.4 0.5 0. .6 0. 2 0. 3 r 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 0.7 0.6 0. 4 0. 5 0.5 0.5 0. .4 0. 3 0. 4 0.4 0.4 0.3. 0. 3 0. 3 0.3 0.3 0. 3 0. 2 0. 3 0.3 0.2 0. 2 0.7 O. 7 0. 1 0. 1 O. 1 0. 1 0, , 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 0.7 0.8 0. 0 0. 0 -0.0 -0.0 -o. 0 -0. 0 -0. C 0.0 -0.0 -0.0 0. 0 -o. 0 0.0 0.0 -0. 0 0. 0 0. 0 -O.O 0.0 0. 0 0.7 0.9 O. 2 0. 2 0.2 0. 1 0, . 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. . 1 0. . 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 0. 1 DELTA = 1 . 00 OMEGA=0 .75 OMEGA=1 .00 OMEGA= 1 D .25 0MEGA=1 p .50 RH 1 RH2 0. . 2 0. P .3 0.4 0.5 0 .6 0. 2 0. 3 f 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 0.7 0.6 4 . 8 5. .6 5.7 5.4 4 .8 3. .7 4 . , 3 4.4 4.2 3.7 3. .0 3, .5 3.6 3.4 3. .0 2 . 5 2 .9 3.0 2.9 2 . 5 0.7 0.7 4 . 1 4 .6 4.6 4.3 3 .7 3. 2 3. .6 3.6 3.3 2.9 2 .6 2. .9 2.9 2.7 2 , .3 2 , .2 2, ,5 2.5 2 .3 2. 0 0.7 0.8 3 .4 3 .7 3.5 3.2 2 .7 2. ,7 2 .9 2.8 2.5 2. 1 2 .2 2 .4 2 . 3 2.0 1 .7 1 , ,9 2 .0 1 .9 1 . 7 1 . . 5 0.7 0.9 2 .7 2 .7 2.5 2 . 2 1 .8 2. . 1 2 .2 2.0 1 .7 1 .4 1 .8 1 . .8 1 .7 1 .4 1 .2 1 . .5 1 .5 1 .4 1 .2 1 . 0DELTA = 1 . . 25 OMEGA=0 . 75 OMEGA=1 .00 OMEGA=1 o . 25 OMEGA=1 p .50 RH'I RH2 0 . 2 0 P .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 .3 0.4 0.5 0 .6 0.7 0.6 12 .0 14 .7 15.7 15.4 13 .9 9 . 1 1 1 . 1 11.8 11.6 10.5 7 . 3 8 .9 9.5 9.3 8 .4 6 . 1 7 .4 7.9 7.8 7 .0 0.7 0.7 1 1 .9 14 .0 14.5 13.8 12 .2 9 .0 10 .6 1 1 .0 10.5 9.3 7 .3 8 .6 8.9 8.5 7 .5 6 . 1 7 .2 7.5 7 . 1 6 . 3 0.7 0.8 1 1 .9 13 .3 13.2 12.2 10 .6 9 . 1 10 . 2 10.2 9.4 8. 1 7 .4 8 .3 8.3 7.7 6 .6 6 . 2 7 .0 7.0 6 .5 5 . 6 0.7 0.9 1 1 .9 12 .6 12.0 10.7 9 .0 9 .2 9 .8 9.4 8.4 7.0 7 .5 8 .0 7.7 6.9 5 .7 6 .4 6 .8 6.5 5.8 4 . 9 DELTA = 1 .50 OMEGA=0 .75 OMEGA= 1 .00 OMEGA=1 D . 25 OMEGA=1 P .50 RH1 RH2 0 .2 0 P .3 0.4 0.5 0 .6 0 .2 0 .3 r 0.4 0.5°i 0.6 0 .2 0 .3 0.4 0.5 0 .6 0 .2 0 .3 0.4 0.5 0 .6 0.7 0.6 19 . 8 25 .5 28.6 29 . 1 27 .2 14 .6 18 .8 21.0 21.3 19.9 11 .6 14 .9 16.6 16.8 15 .7 9 .6 12 .3 13.7 13.9 13 .0 0.7 0.7 20 .6 25 .7 27.8 27 .6 25 .2 15 .4 19 . 1 20.6 20.4 18.7 12 .2 15 .2 16.4 16.2 14 .8 10 .2 12 .6 13.6 13.5 12 . 3 0.7 0.8 2 1 .8 26 .0 27.2 26. 1 23 . 2 16 .4 19 .5 20.4 19.6 17.4 13 . 1 15 .6 16.3 15.7 13 .9 10 .9 13 .0 13.6 13.0 1 1 . 6 0.7 0.9 23 .5 26 .5 26.6 24.6 21 .3 17 .8 20 . 1 20.2 18.7 16.2 14 .3 16 .2 16.3 15. 1 13 . 1 12 .0 13 .6 13.6 12.6 10 .9 NOTE: D.ELTA = Rje4|/0^.OMEGA = S+P, RH1= ^ U l VO RH2 = T A B L E I : P E R C E N T G A I N I N E F F I C I E N C Y ( 0 % ) OF Y2± O V E R D E L T A = 0 . 5 0 0 M E G A = 0 . 7 5 P 0 M E G A = 1 . 0 0 P O M E G A = 1 . 2 5 P R H 1 R H 2 0 . 2 0 . 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 . 8 0 . 6 1 . 9 2 . 1 2 . 1 1 . 9 1 . 7 1 . 5 1 . 6 . 1 . 6 1 . 5 1 . 3 1 . 2 1 . 3 1 . 3 1 . 2 1 . 1 0 . 8 0 . 7 3 . 4 3 . 7 3 . 5 3 . 2 2 . 7 2 . 7 2 . 9 2 . 8 2 . 5 2 . 1 2 . 2 2 . 4 2 . 3 2 . 0 1 . 7 0 . 8 0 . 8 5 . 8 5 . 9 5 . 5 4 . 9 4 . 0 4 . 6 4 . 7 4 . 4 3 . 8 3 . 2 3 . 7 3 . 9 3 . 6 3 . 2 2 . 6 0 . 8 0 . 9 9 . 6 9 . 3 8 . 3 7 . 1 5 . 8 7 . 5 7 . 4 6 . 6 5 . 6 4 . 6 6 . 2 6 . 1 5 . 5 4 . 7 3 . 8 D E L T A 0 . 7 5 0 M E G A = 0 . 7 5 0 M E G A = 1 . 0 0 0 M E G A = 1 . 2 5 P P P R H 1 R H 2 0 . 2 0 . 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 . 8 0 . 6 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 8 0 . 7 0 . 0 0 . 0 - 0 . 0 - 0 . 0 - 0 . 0 - 0 . 0 - 0 . 0 - 0 . 0 - 0 . 0 - 0 . 0 0 . 0 - 0 . 0 0 . 0 0 . 0 - 0 . 0 0 . 8 0 . 8 0 . 2 0 . 2 0 . 2 O . 1 0 . 1 0 . 1 0 . 1 O . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 0 . 8 0 . 9 O . 9 0 . 8 0 . 7 0 . 6 O . 5 0 . . 7 0 . 7 0 . 6 0 . 5 0 . 4 0 . 6 0 . 5 0 . 5 0 . 4 0 . 3 D E L T A 1 . 0 0 0 M E G A = O . 7 5 0 M E G A = 1 . 0 0 0 M E G A = 1 . 2 5 P P P R H 1 R H 2 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 . 8 0 . 6 4 , 1 4 . . 6 4 . 6 4 . 3 3 . . 7 3 . . 2 3 . . 6 3 . 6 3 . 3 2 . 9 2 . 6 2 . . 9 2 . 9 2 . 7 2 . 3 0 . 8 0 . 7 3 . 4 3 . . 7 3 . 5 3 . 2 2 . 7 2 . . 7 2 . 9 2 . 8 2 . 5 2 . 1 2 . 2 2 . . 4 2 . 3 2 . 0 1 . . 7 0 . 8 0 . 8 2 . . 7 2 . . 7 2 . 5 2 . 2 1 . 8 2 . . 1 2 , . 2 2 . 0 1 . 7 1 . 4 1 . 8 1 . 8 1 . 7 1 . 4 1 2 0 . 8 0 . 9 1 9 1 . 8 1 . 6 1 . 3 1 . 1 1 . 5 1 . 5 1 . 3 1 . 1 0 . 9 1 . 3 1 . 2 1 . 1 0 . 9 0 . 7 D E L T A = 1 . 2 5 0 M E G A = O . 7 5 O M E G A = 1 n . 0 0 O M E G A = 1 p . 2 5 R H 1 R H 2 0 . 2 0 . 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 . 8 0 . 6 1 1 . 9 14 . 0 1 4 . 5 1 3 . 8 1 2 . 2 9 . 0 1 0 . 6 1 1 . 0 1 0 . 5 9 . 3 7 . 3 8 . 6 8 . 9 8 . 5 7 . 5 0 . 8 0 . 7 1 1 . 9 1 3 . 3 1 3 . 2 1 2 . 2 1 0 . 6 9 . 1 1 0 . 2 1 0 . 2 9 . 4 8 . 1 7 . 4 8 . 3 8 . 3 7 . 7 6 . 6 0 . 8 0 . 8 1 1 . 9 1 2 . 6 1 2 . 0 1 0 . 7 9 . 0 9 . 2 9 . 8 9 . 4 8 . 4 7 . 0 7 . 5 8 . 0 7 . 7 6 . 9 5 '. 7 0 . 8 0 . 9 1 2 . 1 1 1 . 9 1 0 . 7 9 . 2 7 . 5 9 . 5 9 . 4 8 . 5 7 . 3 5 . 9 7 . 8 7 . 8 7 . 0 6 . 0 4 . 9 D E L T A = 1 . 5 0 O M E G A = 0 . 7 5 O M E G A = 1 . 0 0 0 M E G A = 1 . 2 5 P P P R H 1 R H 2 0 . 2 0 . 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 . 8 0 . 6 2 0 . 6 2 5 . 7 2 7 . 8 2 7 . 6 2 5 . 2 1 5 . 4 1 9 . 1 2 0 . 6 2 0 . 4 1 8 . 7 1 2 . 2 1 5 . 2 1 6 . 4 1 6 . 2 14 . 8 0 . 8 0 . 7 2 1 . 8 2 6 . 0 2 7 . 2 2 6 . 1 2 3 . 2 1 6 . 4 1 9 . 5 2 0 . 4 1 9 . 6 1 7 . 4 1 3 . 1 1 5 . 6 1 6 . 3 1 5 . 7 1 3 . 9 0 . 8 0 . 8 2 3 . 5 2 6 . 5 2 6 . 6 2 4 . 6 2 1 . 3 1 7 . 8 2 0 . 1 2 0 . 2 1 8 . 7 1 6 . 2 14 . 3 16 . 2 1 6 . 3 1 5 . 1 1 3 . 1 0 . 8 0 . 9 2 5 . 9 2 7 . 4 2 6 . 1 2 3 . 2 1 9 . 5 1 9 . 8 2 1 . O 2 0 . 1 1 7 . 9 1 5 . 0 16 . 1 17 . 1 1 6 . 3 1 4 . 5 1 2 . 2 N O T E : D E L T A = R ^ / e j i \ O M E G A = S + P , R H 1 = 0 M E G A = 1 . 5 O P 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 1 . 0 1 . 1 1 . 1 1 . 0 0 . 9 1 . 9 2 . 0 1 . 9 1 : 7 1 . 5 3 . 2 3 . 3 3 . 1 2 . 7 2 . 2 5 . 3 5 . 2 4 . 7 4 . 0 3 . 2 0 . 2 0 . 3 0 . 1 0 . 0 O . 1 0 . 5 1 2 . 0 1 3 . 5 O M E G A = 1 . 5 0 P 0 . 4 0 . 5 0 . 6 0 . 1 0 . 1 0 . 0 - 0 . 0 0 . 1 0 . 5 O . 1 0 . 4 0 . 1 0 . 0 O . 1 0 . 3 1 4 . 4 0 . 1 0 . 0 0 . 1 0 . 3 0 M E G A = 1 . 5 O P 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 2 . 2 2 . 5 2 . 5 2 . 3 2 . 0 1 . 9 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 O M E G A = 1 . 5 0 P 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 6 . 1 7 . 2 7 . 5 7 . 1 6 . 3 6 . 2 7 . 0 7 . 0 6 . 5 5 . 6 6 . 4 6 . 8 6 . 5 5 . 8 4 . 9 6 . 6 6 . 6 6 . 0 5 . 1 4 . 2 O M E G A = 1 . 5 0 P 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 1 0 . 2 1 2 . 6 1 3 . 6 1 3 . 5 1 2 . 3 1 0 . 9 1 3 . 0 1 3 . 6 1 3 . O 1 1 . 6 1 3 . 6 1 3 . 6 1 2 . 6 1 0 . 9 1 3 . 7 1 2 . 2 1 0 . 2 ON o R H 2 = TABLE I: PERCENT GAIN IN EFFICIENCY (0%) OF y 2 ^ OVER y 2 . I DELTA= 0. 50 0MEGA=0.75 OMEGA=1 .00 OMEGA^1 . 25 OMEGA=1 .50 P P P P RH 1 RH2 O. 2 0. 3 0 .4 0 . 5 0 . 6 o: 2 0 . 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 O. 6 0 . 9 0 .6 3. 4 3. 7 3.5 3.2 2 . 7 2. 7 2. 9 2.8 2.5 2.1 2. 2 2 . 4 2.3 2 .0 1 . 7 1 . 9 2. 0 1.9 1 .7 1 . 5 0 . 9 O. 7 5. 8 5. 9 5.5 4 . 9 4 . 0 4 . 6 4 . 7 4 .4 3.8 3.2 3. 7 3 . 9 3.6 3.2 2 . 6 3. 2 3. 3 3.1 2 . 7 2 . 2 0 . 9 0 .8 9 . 6 9. 3 8 .3 7.1 5. 8 7 . 5 7 . 4 6 .6 5.6 4 .6 6. 2 6 . 1 5.5 4.7 3. 8 5. 3 5. 2 4 . 7 4 . 0 3 . 2 0 . 9 0 . 9 . 15 . 7 14. 2 12.2 10.1 8 . 0 12. 5 1 1 . 4 9 .8 8 .0 6.3 10. 3 9. 5 8.1 6 .7 5 . 3 8 . 8 8. 1 7 .0 5 . 7 4 . 5 DELTA= 0. 75 0MEGA=0.75 OMEGA=1 .OO OMEGA=1 .25 OMEGA31 .50 P P P P RH1 RH2 0. 2 0. 3 0 .4 0 . 5 O. 6 0. 2 0. 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 . 9 0 .6 0. 0 0. .0 - 0 . 0 - 0 . 0 -0 . 0 -0 . 0 -0 . 0 -o.o - 0 . 0 - 0 . 0 0. 0 -0 . 0 0 . 0 0 . 0 -0. 0 0. 0 0. 0 -o.o 0 . 0 0. 0 0 . 9 0 . 7 0. 2 0. 2 0 .2 0.1 0. 1 0. 1 0. , 1 0.1 0. 1 0.1 0. 1 0. 1 0. 1 0. 1 0. . 1 0. 1 0. 1 0.1 0 . 1 0. 1 0 . 9 0 .8 0. .9 0. 8 0 .7 0 . 6 0. ,5 0. 7 0. .7 0 . 6 0 .5 0 .4 0. 6 0. 5 0 .5 0 .4 0. .3 0. 5 0. 5 0 .4 0 .3 0. 3 0 . 9 0 . 9 2. ,5 2. 2 1.8 1.5 1 . . 1 2 . 1 1 . 8 1.5 1 .2 0 . 9 1 . 7 1 . 5 1.2 1 .0 0. 8 1 . 5 1 . 3 1.1 0 . 9 0. 7 DELTA= 1 . 00 OMEGA=0.75 OMEGA=1 .00 OMEGA=1 .25 OMEGA=1 .50 P P P P RH 1 RH2- ' 0, .2 0. .3 0 .4 0 .5 0 .6 0. 2 0. .3 0 .4 0 .5 0 .6 0. 2 o. 3 0.4 0 .5 0. 6 0. .2 0. 3 0 .4 0 .5 0. 6 0 . 9 0 . 6 . 3 .4 3, .7 3.5 3.2 2 .7 2. 7 2 .9 2 .8 2.5 2.1 2 .2 2 . 4 2.3 2.0 1 .7 1 . .9 2 , .0 1.9 1 .7 1 .5 0 . 9 0 . 7 2 .7 2. .7 2.5 2.2 1 . .8 2. 1 2 .2 2 .0 1 .7 1.4 1 . .8 1 . 8 1.7 1 .4 1 . . 2 1 . , 5 1; .5 1.4 1 .2 1 . .0 0 . 9 0 .8 1 !9 1 .8 1.6 1.3 1 . 1 1 . 5 1 .5 1.3 1 . 1 0 .9 1 . .3 1 . 2 1.1 0 .9 0 . 7 1 . 1 1 .0 0 . 9 0 .8 0. .6 0 . 9 0 . 9 1 . 1 1 .0 0 .8 0 .6 0 .5 O. ,9 0 .8 0 .7 0 .5 0.4 0. .8 0. 7 0 .6 0 .4 0 .3 0. , 7 0. ,6 0 .5 0 .4 0. . 3 DELTA= 1 .25 OMEGA=0.75 OMEGA=1 .00 OMEGA=1 .25 OMEGAM .50 P P P P RH1 RH2 0 .2 0 .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 .3 0 .4 0 .5 0. .6 0 . 9 0 .6 1 1 .9 13 .3 13.2 12.2 10 .6 9 . 1 10 .2 10.2 9.4 8.1 7 . 4 8 3 8.3 7.7 6 .6 6 . 2 7 .0 7.0 6 . 5 5 .6 0 . 9 0 . 7 1 1 .9 12 .6 12 .0 10.7 9 .0 9 . 2 9 .8 9.4 8.4 7.0 7 .5 8 .0 7.7 6 .9 5 .7 6 .4 6 .8 6 .5 5.8 4 . 9 0 . 9 0 .8 12 . 1 1 1 .9 10.7 9.2 7 .5 9 .5 9 .4 8 .5 7.3 5.9 7 .8 7 .8 7.0 6 .0 4 .9 6 .6 6 .6 6 . 0 5 . 1 4 . 2 0 . 9 0.9- 12 . 4 1 1 .1 9.4 7.8 6 . 1 9 .9 9 .O 7.6 6.2 4 .9 8 . 3 7 . .5 6.4 5.2 4 . 1 7 . 1 6 .4 5.5 4 . 5 3 . 5 DELTA= 1 .50 OMEGA=0.75 OMEGA=1 .OO 0MEGA=1 .25 OMEGA=1 .50 P P P P RH1 RH2 O .2 0 .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 .3 0 .4 0 .5 0 .6 0 . 9 0 . 6 21 .8 26 .0 27 .2 26.1 23 . 2 16 .4 19 .5 20 .4 19.6 17.4 13 . 1 15 .6 16.3 15.7 13 .9 10 .9 13 .0 13.6 13 .0 1 1 .6 0 . 9 0 .7 23 .5 26 .5 26 .6 24 .6 21 . 3 17 .8 20 .1 20 .2 18.7 16.2 14 .3 16 .2 16.3 15. 1 13 . 1 12 .0 13 .6 13.6 12.6 10 . 9 0 . 9 O. 8 25 .9 27 .4 26.1 23 .2 19 .5 19 .8 21 .0 20.1 17.9 15.0 16 . 1 17 . 1 16.3 14.5 12 .2 13 .5 14 .4 13.7 12.2 10 . 2 0 . 9 0 . 9 29 .6 28 .7 25 .6 2 1 . 8 17 .7 23 .0 22 .4 20.1 17.0 13.8 18 .8 18 .4 16.5 14.0 1 1 .3 15 .9 15 .6 14 .0 11.9 9 .6 NOTE: DELTA=R^J7g^ L.0MEGA = S+P, RH1= , RH2= . ^ 62 .5, but f o r P! = .9, t h e r e i s a s t e a d y d e c l i n e i n e f f i c i e n c y , 6. at fi = 1, and A > 0.5, e f f i c i e n c y d e c r e a s e s w i t h i n c r e a s e i n P 2, but f o r A = 0.5, e f f i c i e n c y i n c r e a s e s w i t h i n c r e a s e i n p 2 . From t h i s l i m i t e d n u m e r i c a l s t u d y , i t i s n o t e d t h a t y 2 ^ has a s l i g h t edge ov e r y 2 r « For most p r a c t i c a l p u r p o s e s and e s p e c i a l l y when Pi and P 2 a r e h i g h and fi = A = 1, b o t h e s t i m a t e s seem t o be e q u a l l y d e s i r a b l e . T h i s n u m e r i c a l s t u d y was done on t h e a s s u m p t i o n t h a t b o t h e s t i m a t o r s were e q u a l l y e x p e n s i v e t o compute. E s t i m a t o r of the change. A r a t i o e s t i m a t o r of the change i n means between two s u c c e s s i v e o c c a s i o n s i s o b t a i n e d as a w e i g h t e d average of the means on o c c a s i o n s 1 and 2, e s t i m a t e d by t h e r a t i o method. Thus g 2 r = U y " r + (1 - a )y'..} - {b x " r + (1 - b)x'..} (22) where 0 x " r = ( x " . . / y " . . ) y . . — p — s — v.. = 7— c —c- y".. + T r y'.. = o v e r a l l sample mean on o c c a s i o n 2 J (s+p) y (s+p) J v a,b a r e w e i g h t s and o t h e r symbols a r e as d e f i n e d e a r l i e r . The v a r i a n c e of g 2 ^ i s g i v e n by V a r ( g 2 r ) = a 2 V a r ( y " r ) + (1 - a ) 2 V a r ( y ' . . ) + b 2 V a r ( x " r ) + (1 - b ) 2 V a r ( x ' . . ) - 2 ab c o v ( y " r , x " r ) + 2 a ( l - b)cov(y" ,x'..) - 2 b ( l - a)cov(x" ,y 1..) (22.1) s i n c e y" 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'.. . J r r Now, u s i n g the r e a s o n i n g of Cochran (1977:343) (and as i n d i c a t e d i n the e s t i m a t i o n of t h e c u r r e n t mean i n t h i s s e c t i o n ) c o y ( x " ,x" > = cov{[y"..'+ R ( x . . - x " . . ) ] , [ x " . . + ( 1 / R ) ( y . . - y". . ) ] } = c o v ( y " . . , x " . . ) + c o v [ y " . . , ( 1 / R ) y . . ] + c o v [ y " . . , ( - 1 / R ) y " ] + c o v ( R x . . , x " . . ) + c o v [ R x . . , ( 1 / R ) y . . ] + c o v [ R x . . , ( - 1 / R ) y " . . ] + c o v ( - R x " . . , x " . . ) . + c o v [ - R x " . . , ( 1 / R ) y . . ] + c o v [ - R x " . . , ( - 1 / R ) y " . . ] = R ( s ! p ) p { ( S + P ) f " P ^ A * ^ " s } c o v ( x " r , y ' . . ) = c o v { [ x " . . + ( 1 / R ) ( y . . - y " . . ) ] , y'..} = ( l / R ) c o v ( y . . , y ' . . ) = e 2 2 / R ( s + p) c o v ( y " r , x ' . . ) = c o v { [ y " . . + R ( x . . - x " . . ) ] , x'..} = R c o v ( x . . , x ' . . ) = R 6 2 j and V a r ( x " r ) = V a r ( x " . . ) + ( 1 / R ) 2 V a r ( y . . - y"..) + ( 2 / R ) c o v [ x " . . , ( y . . - y " . . ) ] = 6 2 1 [ s ( l - A 2 ) + p ( l - A ) ] / A 2 p ( s + p ) . Then, by s u b s t i t u t i n g i n the v a r i a n c e s and c o v a r i a n c e s above i n t o e q u a t i o n ( 2 2 . 1 ) , we o b t a i n t h a t V a r ( g 2 ] . ) = a 2 { 6 2 2 [ l - q ( 2 * 2 A - A 2 ) ] / p } + (1 - a) 2 ( 6 2 2 /s) + b 2 { 6 2 I [ s ( l - A 2 ) + p ( l - A ) ] / A 2 p ( s + p)} + (1 - b ) 2 ( 9 2 1 / q ) - 2ab{[ 6 2 2 / R ( s + p ) p ] [ [ ( s + p ) [ - p i | < 2 A - q A 2 ] - sjj } + 2 a ( l - b ) R 6 2 1 - 2 b ( l - a) [ e 2 2 / R ( s + p ) ] (22.2) D i f f e r e n t i a t i n g e q u a t i o n (22.2) w . r . t . a and b, s e t t i n g t he r e s u l t s e q u a l t o z e r o and the n s i m u l t a n e o u s l y s o l v i n g f o r a and b g i v e s a* = (AT - Z<)>)/(AT - <t>2) b* = (AZ - <(>A)/(AT - <t>2) where A = ( 9 2 2 / s ) - R 6 2 1 T = [ 6 2 2 / A 2 p q ( s + p ) ] { q [ s ( l - A 2 ) + p ( l - A ) ] + A 2 p ( s + p)} Z = ( 6 2 l / q ) + [ 6 2 2 / R ( s + p ) ] A = ( e 2 2 / s p ) { p + s [ l - q ( 2 ^ 2 A - A 2 ) ] } $ = 6 2 2 / R p [ p ^ 2 A - qA 2 + 1] - R 6 2 1 and a* and b* are the ' o p t i m a l ' v a l u e s of a and b, r e s p e c t i v e l y , t h a t m i n i m i z e V a r ( g 2 r ) . The minimum-variance e s t i m a t o r o f g 2 ^ i s o b t a i n e d by s u b s t i t u t i n g a and b* i n t o e q u a t i o n ( 2 2 ) , and 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 i s g i v e n by s u b s t i t u t i n g a* and b* i n t o e q u a t i o n ( 2 2 . 1 ) . L i k e y 2 r > i t i s e x p e c t e d t h a t g 2 ^ i s b i a s e d . The amount of t h i s b i a s w i l l now be d e t e r m i n e d as f o l l o w s . B(g2].) - E [ g 2 r - (u Y - M X ) ] = E(y2].) - E ( x 2 r ) - i i y + V X where x 2 i s t h e second p r i n c i p a l p i e c e i n e q u a t i o n ( 2 2 ) . Now, u s i n g t h e arguments g i v e n i n Murphy (1967:364) X E ( y 2 r ) = a q [ ( R 6 2 1 / p u x ) - ( R ^ 2 / 6 2 , 9 2 2 / p u y ] - u y Then by s u b s t i t u t i o n B ( g 2 r ) - { a q [ ( R 6 2 1 / P M x ) + ( s 6 2 2 / p R u y ) (R\|) 2/6 2 1e 2 2)/pu y] + u y} ( b [ y x ( s - q) ( s ^ 2 / e 2 ! e 2 2 / p R u x ) ] + u x> Y + y X 65 = a q [ ( R 9 2 1 / p u x ) -• ( 2 / 6 2 , 6 2 2 /pu y) ] - b [ u x ( s - q) + ( s 6 2 2 /pRM y) - ( s i p 2 / e 2 1 e 2 2 / P R M x ) ] . T h i s b i a s becomes n e g l i g i b l e f o r l a r g e v a l u e s of n, m. Comparison of the BLUE and r a t i o e s t i m a t o r . A g a i n , i t would be u s e f u l t o compare the e f f i c i e n c y of t h e BLUE and the RE i n e s t i m a t i n g change. Because of the c o m p l e x i t y of the v a r i a n c e f u n c t i o n s of change, o n l y a n u m e r i c a l c o m p a r i s o n w i l l be done. F o l l o w i n g t h e same p r o c e d u r e adopted i n c omparing the e f f i c i e n c y w i t h r e s p e c t t o e s t i m a t i n g the c u r r e n t mean,we p r o c e e d as f o l l o w s . The g a i n i n p r e c i s i o n of g 2 over g 2 i s X/ r g i v e n by V a r ( g 2 ) - V a r ( g 2 ) Qi = ~ — V a r ( g H ) U s i n g e q u a t i o n s (10.1) and ( 2 2 . 2 ) , the v a l u e s of Ch f o r some v a l u e s of Q, A, p j , p 2 , <t>i, and <t>2 were t a b u l a t e d . The r e s u l t s a r e g i v e n i n T a b l e I I . I t seems o b v i o u s from T a b l e I I t h a t the r a t i o e s t i m a t o r of change i s o v e r a l l v e r y i n e f f i c i e n t as compared to the BLUE. TABLE I I : PERCENT GAIN IN EFFICIENCY (0,%) OF g • OVER g 1 I r DELTA= 0.50 OMEGA-O.75 OMEGA=1.00 OMEGA-1.25 OMEGA-1.50 P P P P RH 1 RH2 0. 2 0.4 0. 6 0. 2 0.4 0.6 0.2 0.4 0. 6 0. 2 0.4 0. 6 0.6 0.6 -47 . 2 2063.7 549 . 3 4 . 0 -361.8 498.8 25 .0 -14.5 4 10. 5 36 . 3 76 . 9 209 . 1 8 0.6 0.7 -33 . 3 4257 . 4 637 . 1 14 . 1 -262.3 590.4 33.4 14.3 505 . 4 43 . 6 95 . 8 239 . 0.6 0.8 - 18 . 2 29485.5 754 . 3 26. 2 -192.9 716.0 44 .0 44 . 1 650. 8 53 . 2 119.2 276 . 8 0.6 0.9 -0. 3 -193.0 918 . 7 4 1 : 7 - 136 .8 897 . 4 58 .0 78 . 9 898 . 4 66 . 2 150. 5 4 16. 5 DELTA- 0.75 OMEGA-O.75 OMEGA =1.00 OMEGA =1.25 OMEGA-1.50 P P P P RH 1 RH2 0. 2 0.4 O. 6 0. 2 0.4 0.6 0..2 0.4 0. 6 0. 2 0.4 0. 6 0.6 0.6 12 . 7 - 1104.6 525. 3 38 . 5 1 .6 316.4 50. 2 137 . 2 40. 7 56. 8 188 . 3 -339 . 9 0.6 0.7 22 . 9 -834 .4 611. 5 47 . 0 26.7 375.9 57 . 7 153 . 1 35. 5 63. 6 201 .6 -513 . 0 O.G 0.8 35 . 7 -679.5 726. 7 58 . 2 54 .5 456.6 67 . 7 175.6 21 . 1 72 . 8 222 . 5 -857 . 2 0.6 0.9 52 . 7 -582 . 7 888 . 1 73. 5 88.8 571 .9 81.8 208 . 3 -13. 7 . 86. 0 254.5 - 1706 . 3 D E L T A = 1 .00 OMEGA =0.75 OMEGA - 1.00 OMEGA- 1 . 25 D OMEGA-1.50 P RH 1 RH2 0. 2 y 0.4 O. 6 0. 2 r 0.4 0.6 0.2 0.4 0. .6 0. 2 0.4 0. .6 0.6 0.6 44 . 1 - 169.2 397 . 1 58 . 8 166.3 96.5 65.9 245.7 -189. . 7 70. .0 280. 1 -467 . 6 0.6 0.7 53 7 - 137 .0 466 . 3 67 . 1 182 . 2 118.9 73.4 259 . 3 -232 .0 76. .9 292.5 -596 8 0.6 0.8 66 . 3 - 106.7 558 . ,6 78 . 3 205.8 148.5 83.6 281 .8 -295 .4 86. .4 314.0 -798 . 8 0.6 0.9 83 .8 -75 . 2 687 .8 94 . 1 240.9 189 . 5 98. 1 317.4 -395 . 7 100 . 1 349.1 - 1 140 . 5 DELTA- 1 .25 OMEGA =0.75 OMEGA-1.00 OMEGA-1.25 D OMEGA-1.50 P RH 1 RH2 0 .2 Y 0.4 0 .6 0 . 2 f 0.4 0.6 0.2 0.4 0 .6 0 . 2 o:4 0 .6 0.6 0.6 64 . 1 93 .0 265 .0 72 . 3 282 .6 -33.3 76.5 336.7 -268 .3 78 .9 361 . 3 -463 . 1 0.6 0.7 73 . 9 110.4 315 .8 81 .0 298 . 3 -29. 1 84 . 3 35 1 . 3 -3 11 .6 86 . 2 374 .8 -554 . 6 0.6 0.8 87 .0 134 .0 383 .4 92 .6 324 .0 -24 .0 94 . 9 376.3 -372 .9 96 .2 398.8 -686 . 2 0.6 0.9 105 . 4 167 .6 478 .O 109 . 1 364 .4 -17.6 1 10. 2 416.8 -464 .0 1 10 .5 438.4 -886 . 3 DELTA- 1 .50 OMEGA-O.75 OMEGA-1.00 OMEGA =1.25 D OMEGA- 1 .50 P RH1 RH2 0 .2 r 0.4 0 .6 0 .2 f 0.4 0.6 0.2 0.4 0 .6 0 . 2 0.4 0 .6 0.6 0.6 77 .9 254 .4 186 .9 81 . 7 380.7 -88.2 83.6 417.7 -284 .6 84 .9 434 . 1 -435 . 3 O. 6 0.7 88 .2 272 . 1 226 .8 90 .7 398 .9 -90. 1 91.9 434 .9 -322 .6 92 .6 450. 3 -504 . 8 0.6 0.8 102 .0 299 .6 279 .8 103 .0 428.8 -93.0 103 . 1 463 .9 -375 .0 103 . 1 478 . 1 -601 . 4 0.6 0.9 121 . 5 342 .0 353 .9 120 .3 476 . 2 -97 .6 119.1 510. 7 -451 .0 1 18 . 1 523.5 -742 . 7 NOTE: DELTA = R/e 2 j/8 2z OMEGA-S+P , RH 1 = p x , RH2- p 2 RH1 RH2 8 0.6 8 0.7 8 0.8 8 0.9 0.2 TABLE I I : PERCENT GAIN IN EFFICIENCY (Q,%) OF g 2 ^ OVER g2^_ DELTA= 0.50 0MEGA=0.75 OMEGA=1.00 P . p 0.2 0.4 0.4 0.6 -18.2 29485.5 754.3 -0.3 -193.0 918.7 23.1 - 1956.1 1 165.8 58.2 -1706.7 1578.3 0.6 26.2 -192.9 716.0 41.7 -136.8 897.4 63.5 -83.5 1178.9 98. 1 -21.6 1665.2 0.2 OMEGA - 1 .25 P 0.4 0 . 6 4 4 . 0 44.1 650.8 58.0 78.9 898.4 78.4 124.9 1401.0 111.6 195.6 2852.3 OMEGA=1.50 P 0.2 0.4 0.6 53.2 119.2 276.8 66.2 150.5 4 16.5 85.6 196.1 449.0 117.4 270.7 596.6 RH1 RH2 8 0.6 8 0.7 8 0.8 8 0.9 OMEGA=0.75 P 0.2 0.4 0 . 6 35.7 -679.5 726.7 52.7 -582.7 888.1 77.1 -521.8 1130.5 116.7 -489.7 1535.0 0MEGA=1.00 P 0.2 0.4 DELTA= 0.6 58.2 54.5 456.6 73.5 88.8 571.9 96.2 135.7 748.7 133.6 209.5 1050.4 0. 75 OMEGA=1.25 P 0.2 0.4 0.6 67.7 175.6 21.1 8 1.8 208.3 -13.7 102.9 258.5 -100.4 138.2 343.2 -347.7 OMEGA=1.50 P 0.2 0.4 0.6 72.8 222.5 -857.2 86.0 254.5 -1706.3 106.0 305.3 -3390.1 139.7 393. 1 1 1 177. 1 RH1 RH2 8 0.6 8 0.7 8 0.8 8 0.9 0MEGA=0.75 P 0.2 0.4 0.6 66.3 -106.7 558.6 83.8 -75.2 687.8 109.8 -38.0 881.6 153.0 1 3 5 1204.7 0MEGA=1.00 P 0.2 0.4 DELTA= 0.6 78.3 205.8 148.5 94.1 240.9 189.5 117.9 295.7 250.4 157.7 389.3 351.O 1 .00 OMEGA=1.25 P 0.2 0.4 0.6 83.6 281.8 -295.4 98. 1 3 17.4 -395 . 7 120.2 375.1 -567.9 157.4 476.0 -899.8 OMEGA =1.50 P 0.2 0.4 0.6 86.4 314.0 -798.8 100. 1 349 .1 - 1140.5 120.9 406.8 -1787.9 156.3 508.8 -3255.5 RH1 RH2 OMEGA=0.75 P 0.2 0.4 0.6 87.0 134.0 383.4 105.4 167.6 478.0 133.2 218.6 619.6 179.7 303.9 855.6 OMEGA=1.00 P 0.2 0.4 DELTA= 0.6 92.6 324.0 -24.0 109.1 364.4 -17.6 134.2 429.9 -9.0 176.3 544.4 3.7 1 . 25 0MEGA=1.25 P 0.2 0.4 0.6 94.9 376.3 -372.9 110.2 416.8 -464.0 133.4 483.5 -609.1 172.5 601.6 -866.2 OMEGA= 1 .50 P 0.2 0.4 0.6 96.2 398.8 -686.2 110.5 438.4 -886.3 132.4 504.3 -1214.8 169.5 621.6 -1818.8 RH1 RH2 0.8 0.8 0.6 0.7 0.8 0.8 0.8 0.9 OMEGA=0.75 P 0.2 0.4 0.6 102.0 299.6 279.8 121.5 342.0 353.9 151.0 409.9 464.7 200.4 527.8 649.1 DELTA= OMEGA=1.00 P 0.2 0.4 * 0.6 103.0 428.8 -93.0 120.3 476.2 -97.6 146.6 553.7 -105.3 190.8 690.0 -119.2 1 .50 OMEGA=1.25 P 0.2 0.4 0.6 103.1 463.9 -375.0 119.1 510.7 -451.0 143.3 587.6 -568.7 184.2 723.9 -771.1 OMEGA=1.50 P 0.2 0.4 0.6 103.1 478.1 -601.4 118.1 523.5 -742.7 14 1.0 598.7 -964.0 179.6 732.5 -1349.4 NOTE : DELTA = R/6 2 ^ 2 2OMEGA = S+P, RH1=pj . RH2=p 2 CHAPTER 4 OPTIMUM ALLOCATION AND REPLACEMENT In most applications of sampling, cost is an important factor since the funds available for sampling are usually limited. It is therefore necessary to include methods of optimal sample allocation in the overall design of a sampling scheme. In this chapter we discuss the use of dynamic, programming (DP) in the determination of the optimum replacement policy for multistage sampling on successive occasions for several v a r i -ables of interest. For sampling on successive occasions, the sampling design is expected to be s t a t i s t i c a l l y efficient over the whole series of succes-sive occasions considered in i t s entirety. In particular, the following questions must be answered. 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 optimal with respect to each of the several variables of interest estimated at the current occasion, and have the side conditions imposed on the estimation proce-dure been met? These and other questions constitute an optimal SPR design problem. For example, the objective of a two-occasion timber inventory design may 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 is 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). Typically, the objective and constraint functions are non-linear, and the problem of determining the optimal SPR design is a non-linear decision problem. Several authors, for example, Rana (1976), Singh and Kathuria (1969), Kulldorff (1963), Tikkiwal (1953), Patterson (1950), Yates (1949), and Jessen (1942) , have considered the problem of deter-mining the optimum replacement policy in SPR. They were interested mainly in the situation where there was only one variable of interest at a time. Ware and Cunia (1962) and Hazard and Promnitz (1974) determined optimal: SPR designs in 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 succes-sive occasions. Ware and Cunia used a graphical technique to solve the non-linear decision problem. Graphical methods are suitable for the case of two-occasion SPR, where there are no more than two decision variables and the number of side conditions is relatively small. Hazard and Promnitz solved their optimal SPR problem with an algo-rithm that required that the cost and constraint equations be differen-tiable convex functions. (The assumption of convexity w i l l usually be met in optimal SPR problems in forestry.) Although a number of iterative solution techniques have been developed for such non-linear programming problems, there is no assurance that a solution w i l l always be reached in a reasonable number of iterations. Furthermore, a subsequent sensitiv-ity analysis on the derived decisions is recommended in order to firmly establish the optimality of the decision variables. Optimal sample design for SPR involves a sequence of interrelated 70 decisions over time. Previous workers i n t h i s f i e l d apparently did not take advantage of t h i s underlying process i n determining t h e i r optimal replacement p o l i c i e s . We s h a l l e x p l o i t the sequential nature of the problem to cast the optimal sample design problem as a multistage model which can be optimized through dynamic programming. F i r s t , dynamic programming i s discussed i n general. Next, the dynamic nature of the optimal SPR design problem i s examined. Then the s o l u t i o n procedure of the optimal two-stage SPR problem i s presented. Dynamic programming (DP) i s an optimization method for multistage d e c i s i o n processes. DP involves separating the multivariable optimization problem into a ser i e s of one-variable optimization problems. The re s u l t a n t one-variable problems may then be solved r e a d i l y using standard methods of d i f f e r e n t i a l calculus or simple search procedures. The theory of DP i s covered extensively elsewhere, f o r example, Dano ( 1 9 7 5 ) , . Wilde and Beightler (1967), and Nemhauser (1966). B r i e f l y , the c h a r a c t e r i s t i c s of a DP problem are reviewed and these are as follows: (i) the problem can be divided into stages, with a d e c i s i o n at each stage; ( i i ) each stage has a number of states associated with i t , and the e f f e c t of the decision at each stage i s to transform the current state into a state associated with the next stage; ( i i i ) the p r i n c i p l e of optimality as stated by Bellman (1957) holds at each stage: An optimal p o l i c y has the property that whatever the i n i t i a l s tate and i n i t i a l decisions are, the remaining decisions must co n s t i t u t e an optimal p o l i c y with regard 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 recursive 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 for each state with r stages remaining, given the optimal decisions for each state with (r - 1) stages remaining. This relationship is of the general form f r ( X r ) = optimize {c r(X r,d r) + f ^ O ^ ) } d eD r r where, D is the constraint set for the decision variable d , r r' f (X ) is the optimal value when starting in state X with r stages remaining, X , = t(X ,d ) is the transformation of state when decision d r-1 r r r is used when in state X , r' c^CX^jd^) is a value function (profit, cost, etc.) for using d^ when in state X r = 1, 2 , . . . ; (v) the problem is solved using the recursive relationship. At each stage, an optimal solution from a l l previous stages, under any conditions, is found and carried into the next stage, until the last stage when the optimal decisions are found for the whole problem. Unlike linear or other non-linear programming problems, there is no standard mathematical formulation of dynamic programming problems; specific formulations must be developed to f i t individual problems. Now 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 72 i , j ,... , and w where r i s a measurement occasion r = l , 2 , . . . , t i,j,...,v> are indic a t o r variables and t h e i r p o s i t i o n s correspond to occasion of measurement such that i represents cur-rent occasion r, j represents (r - l ) * " ^ occasion, and so on and w represents 1st occasion n. . i s the t o t a l number of sample units observed on occasions 1 j .. . w i , j , . . . , and w, and > n i s the t o t a l sample size on occasion 1. The i n d i c a t o r variables take on the value 1 i f the sample unit was measured on corresponding occasion and 0 (zero) otherwise. For example, p 1 0 i o i s the proportion of sample units measured on both current and second occa-sions i n four-occasion sampling. S i m i l a r l y , p 1 0 i s the proportion of units measured on current occasion only i n two-occasion sampling. We note that the number of groups of sample units measured at u occasions at occasion I i s m r u = (r ~ = r ! / t ( r - u ) ! u ! 1 For example, in two-occasion sampling r = 2 m 2 l = 2!/[(2 - 1)!1!] = 1 m 2 2 = 2!/[(2 - 2)!2!] = 1 That i s , there i s only one group of sample units measured only once and only one group measured on both occasions (see Figure 1.). Note also that r i n p i s not an exponent, and that at occasion r, there are r - 1 occasions remaining. In SPR, the t o t a l sample at each successive measurement occasion r w i l l consist of several groups of remeasured and new sampling units. For 73-£ ' 9 u r e 1- Groups of sampling units in SPR on two occasion*: sampling o c c a s i o n 1 P = P. 11 s = p; 10 q = p, 01= proportion of units measured on both occas ions = proportion of units measured on the second occasion o n l y = proportion of units measured on the f i r s t o c c a s i o n on l y 74 example, i n t w o - o c c a s i o n s a m p l i n g r = 2, t h e r e would be the f o l l o w i n g groups of s a m p l i n g u n i t s as shown i n F i g u r e 1. The t o t a l sample s i z e a t measurement t i m e r i s 1 1 1 1 n = n ( E E E ... E p . . ) r = l , 2, . . . , t r i=0 j=0 k=0 w=0 X J - - ' W F o r the example of t w o - o c c a s i o n s a m p l i n g , the sample s i z e on t h e c u r r e n t o c c a s i o n 2 w o u l d be 2 2 n 2 = n ( p 1 0 + P u ) [ o r as i n e a r l i e r n o t a t i o n = n ( s + p ) ] . DP i s used 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 ) I t i s r e c o g n i s e d 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 by " t i m e " s t a g e s , a s t a g e b e i n g d e f i n e d as a measurement t i m e r ( r = 1, 2 t ) . A t each s t a g e a d e c i s i o n n i s r e q u i r e d . I f we . . . r-1 assume t h a t n i s known, t h e d e c i s i o n t h e n c o n s i s t s o f 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 and remeasured sample u n i t s . We c a n c r e a t e r-1 " s u b - p r o b l e m " s t a g e s 2 w i t h i n each t i m e s t a g e such t h a t t h e sub-p r o b l e m s t a g e s a r e n e s t e d w i t h i n the time s t a g e s . An example o f t h i s n e s t i n g i s g i v e n f o r t w o - o c c a s i o n s a m p l i n g i n F i g u r e 2. ( i i ) A t each 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 each 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 as t h e amount of v a r i a n c e o f t h e v a r i a b l e o f i n t e r e s t ( i = 1, 2, z) c o r r e s p o n d i n g 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 t o be a c c o u n t e d f o r by t h e sample n^ _ t a k e n . ( i i i ) 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 the SPR o p t i m a l d e s i g n model. That i s , a t a p a r t i c u l a r measurement o c c a s i o n r , an o p t i m a l sample s i z e f o r t h e r e m a i n i n g measurement o c c a s i o n s i s in d e p e n d e n t of t h e sample 75 s i z e s t a k e n on p r e v i o u s measurement o c c a s i o n s . ( i v ) A r e c u r s i v e r e l a t i o n s h i p used t o s o l v e t h e SPR p r o b l e m can be d e v e l o p e d w h i c h i d e n t i f i e s the optimum r e p l a c e m e n t p o l i c y (p ^ ^' s ) f o r the s p e c i f i e d v a r i a n c e l e v e l s of v a r i a b l e s of i n t e r e s t w i t h r more o c c a s i o n s r e m a i n i n g , g i v e n the o p t i m a l p o l i c y w i t h r-1 o c c a s i o n s r e m a i n i n g . I n o r d e r t o d e v e l o p the r e c u r s i v e r e l a t i o n s h i p we r e q u i r e (a) a c o s t  ( o b j e c t i v e ) f u n c t i o n r e l a t i n g the d e c i s i o n v a r i a b l e s t o the c o s t of s a m p l i n g , and (b) v a r i a n c e ( c o n s t r a i n t ) f u n c t i o n s r e l a t i n g s p e c i f i e d v a r i a n c e ( p r e c i -s i o n ) l e v e l s of the v a r i a b l e s of i n t e r e s t t o t h e d e c i s i o n v a r i a b l e s . The o p t i m a l d e s i g n p r o b l e m can t h e n be e x p r e s s e d as a m u l t i - s t a g e p r o c e s s by (1) s e p a r a t i n g the c o s t and v a r i a n c e f u n c t i o n s i n t o s t a g e components, and (2) decomposing t h e d e c i s i o n p r o b l e m , t h a t i s , r e p l a c i n g t h e r—1 r — 1 r—1 Z 2 - d e c i s i o n p r o b l e m w i t h Z 2 o n e - d e c i s i o n problems. The Z 2 r = l r = l r = l o n e - d e c i s i o n problems a r e t h e n s o l v e d r e c u r s i v e l y . We s h a l l now r e t u r n t o the s p e c i f i c p r o b l e m of d e t e r m i n i n g the o p t i m a l r e p l a c e m e n t p o l i c y f o r the s a m p l i n g p l a n s d e s c r i b e d i n c h a p t e r 3. We s h a l l r e s t r i c t o u r s e l v e s t o t w o-stage SPR w i t h e q u a l - s i z e d s a m p l i n g u n i t s at each s t a g e on two s u c c e s s i v e o c c a s i o n s . E x t e n s i o n t o t h e o t h e r s a m p l i n g d e s i g n s i s s t r a i g h t f o r w a r d . I t w i l l be f u r t h e r assumed t h a t n and m are a l r e a d y known, t h a t i s , we are somewhere i n between the f i r s t and second o c c a s i o n s . We now want t o p l a n t h e i n v e n t o r y on the second o c c a s i o n , t h a t i s , t o choose the sample s i z e s n" psu's (and mn" s s u ' s ) and n' p s u ' s (and mn' s s u ' s ) i n the most e x p e d i e n t way as r e g a r d s the e s t i m a t i o n o f the c u r r e n t p o p u l a t i o n mean and change between the two s u c c e s s i v e o c c a s i o n s . I n c h o o s i n g n" and n', we s h a l l pay a t t e n t i o n t o t h e aim of k e e p i n g t h e c o s t of the 76 i n v e n t o r y as low as p o s s i b l e . ( n " i s number of remeasured p s u ' s and n' i s the number of new ones and assume n" = np and n' = ns.) To f o r m u l a t e the problem we s h a l l assume t h a t the t o t a l c o s t C p e r -t a i n i n g t o the second o c c a s i o n i s g i v e n by the simple c o s t f u n c t i o n C = k 2 p + k i S (23) where k t = c',n + c ' 2 m n k 2 = c",n + c" 2mn c 1 = c o s t of a new psu ( i = 1) and a s s u ( i = 2 ) , assumed t o be known, and c"^ = c o s t of a remeasured psu ( i = 1) and a s s u ( i = 2) assumed t o be known. F u r t h e r , we s h a l l assume the v a r i a n c e f u n c t i o n s d e v e l o p e d e a r l i e r ( e q u a -t i o n s 6 and 10.1) t h a t - r e l a t e the v a r i a n c e of c u r r e n t mean and change, r e s p e c t i v e l y , t o p and s. The tw o - s t a g e SPR o p t i m a l d e s i g n d e c i s i o n p r o b l e m i s t h e n s t a t e d as f o l l o w s . F i n d p and s such t h a t the c o s t o f s a m p l i n g C i s m i n i m i z e d and such t h a t t h e s p e c i f i e d v a r i a n c e l e v e l s V! and V 2 o f c u r r e n t mean and g r o w t h , r e s p e c t i v e l y , a r e met. E x p r e s s e d i n a n o t h e r way, f i n d p, s such t h a t C = minimum { k j S + k 2 p } p,s and v a r ( y 2 j l ) < V! v a r ( g 2 j ^ ) < V 2 p,s > 0. (We a r e g o i n g back t o our o l d n o t a t i o n where p = p 2 j , and s = p 2 l 0 , i n t h i s c a s e . ) I f we s e t v a r ( y 2 ) < V1 and v a r ( g 2 ) < V 2 , then e q u a t i o n (6) y i e l d s . e 2 2 ( [ l - q * | ] / [ s + p - qs * i ] < V, (24) and e q u a t i o n (10.1) y i e l d s [ ( s + p - s i p | ) e 2 I + ( l - q\p|)e 2 2 - 2 p i j j 2 / e 2 1 e 2 2 ] / (s + p - qs*!) < V 2 (25) where q = (1 - p ) . F i r s t , we s e p a r a t e the c o s t and v a r i a n c e f u n c t i o n s i n t o s t a g e com-p o n e n t s . S e p a r a t i o n of the c o s t f u n c t i o n C i n t o s t a g e c o s t components C i s a l r e a d y a c c o m p l i s h e d by v i r t u e of i t s l i n e a r and a d d i t i v e n a t u r e . S e p a r a t i o n of the v a r i a n c e f u n c t i o n s i s somewhat more d i f f i c u l t . How-e v e r , a f t e r some l e n g t h y a l g e b r a i c m a n i p u l a t i o n , i n e q u a t i o n (24) becomes [p/(l - +| +.p* 2 2)] + s V [V, - ( e 2 2 / V j - ) ] < V, (26) and (25) becomes { [ p ( e 2 1 - v 2 - 2 * 2 / e 2 l e 2 2 ) + ( l - ip* + p * | ) e 2 2 ] / [ v 2 ( l - ip* + pi>22) -e 2 l ( i - *|)]} - s + v 2 < V 2 (27) The s e p a r a t i o n of the c o s t and v a r i a n c e f u n c t i o n s i s c o m p l e t e ; and we can now use (26) and (27) t o c r e a t e s t a g e t r a n s i t i o n f u n c t i o n s of the form X , = t ( X ,d ) which compute the amount of v a r i a n c e l e f t t o be r-1 r r a l l o c a t e d subsequent to an i n v e n t o r y on o c c a s i o n 2, as a f u n c t i o n of the v a r i a n c e l e f t t o be a l l o c a t e d p r i o r t o o c c a s i o n 2, and n(p + s) psu's and mn(p + s) ssu's u n d e r t a k e n a t o c c a s i o n 2. These a r e shown i n a s c h e m a t i c d i a g r a m ( F i g u r e 2). (Note t h a t n(p + s) = n 2 and mn(p + s) = mn 2.) The e x p r e s s i o n s on the l e f t hand s i d e of i n e q u a t i o n s - (26) and (27) may now be r e g a r d e d as the " s t a t e s " of the model w i t h Xj = ( X 1 1 5 X 1 2 ) 78 F i gu re 2. The stage d iagram f o r t h e o p t i m a l S P R d e s i g n p r o b l e m . X l t = X, 0 =0 X a 1 = X M = 0 T r a n s i t i o n f u n c t i o n s : . V -X v : x** = x „ - J {[p(ea,- v* - 2%7eI7en) • 0 - ^ * • »£ p)e«] / Lv.(i -'Hf • P*;*) - e„ o - O]} • vj| X|f = X 1 t - s X u = X,* - {[p/(1 - • p*l)] • V, - (9*/V,)} X*i = X** .• s 7 9 and X 2 = ( X 2 1 , X 2 2 ) s t a t e s p e r t a i n i n g t o (26) and (27), r e s p e c t i v e l y . C o m p u t a t i o n o f t h e s t a t e s o f t h e model a t t h e i n v e n t o r y on o c c a s i o n 2 i s a t w o - s t a g e p r o c e s s , as i n d i c a t e d i n F i g u r e 2. The v a l u e o f t h e f i r s t 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 X 1 2 = X 1 2 - { [ p / ( l - i>\ + p**)] + V x - (e 2/Vi)} (28) and the v a l u e o f t h e f i r s t s t a t e X x subsequent to measurement o f n' psu's and mn' ss u ' s i s X n = X 1 2 - s (29) (The t i l d e [~] de n o t e s t h e i n t e r m e d i a t e s t a t e o r o u t p u t a t s t a g e 1 o r 2; a s t a t e w i t h o u t [~] denotes an i n p u t t o s t a g e 1 o r 2.) S i m i l a r l y , t h e v a l u e o f the second s t a t e X 2 subsequent t o t h e remeasurement o f n" psu's and mn" ss u ' s i s . 2 2 x 2 2 = x 2 2 - [ [ { [ p ( e 2 1 - v 2 - 2^ 2/e 2 1e 2 2) + (1 - ^ 2 + i|; 2p)e 2 2]/ t v 2 ( i - ^  + P^) - e 2 i ( i - * 2 > ] } + v 2 J (30) and a f t e r n ' psu's and mn' s s u ' s i s X 2 1 = X 2 2 + s (31) (Note X.. = s t a t e o f model a t j * " * 1 s t a g e f o r s t a t e v a r i a b l e , and r e c a l l t h a t n' = ns and n" = np.) The i n i t i a l and f i n a l s t a t e s o f t h e s t a t e v a r i a b l e s X x and X 2 a r e , r e s p e c t i v e l y : x i : X12 £ v i a n d x i o = 0 X 2 : X 2 2 <_ V 2 and X 2 0 = 0 Note t h a t t h e i n c i d e n c e i d e n t i t y X. = X.., i , j = 1,2 e x i s t s i n t h e x ( j + l ) i j model 'Wilde & B e i g h t l e r [ 1 9 6 7 ] ) . S e p a r a t i o n o f the c o s t and v a r i a n c e f u n c t i o n s has produced a m u l t i -s t a g e model ( F i g u r e 2) of s a m p l i n g w i t h p a r t i a l r e p l a c e m e n t on two s u c -c e s s i v e o c c a s i o n s . The 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 had two d e c i -80 s i o n s , p and s, can now be decomposed i n t o two problems c o r r e s p o n d i n g t o the two s t a g e s o f the model, each 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 or s. The two s i n g l e - d e c i s i o n problems can then be s o l v e d r e c u r s i v e l y by c a l c u l u s o r s i m p l e s e a r c h p r o c e d u r e s . The d e c o m p o s i t i o n i s done n e x t as f o l l o w s . At s t a g e 1, the d e c i s i o n p r o b l e m i s t o f i n d t he optimum p r o p o r t i o n of new p r i m a r y s a m p l i n g u n i t s s* such t h a t f i ( X 1 1 5 X 2 1 ) = min {k,s + f 0(X,„,X 2„)} (32) s>0 f o r a l l p o s s i b l e v a l u e s of [ X 1 1 , X 2 1 ] , w i t h f 0 ( X i 0 , X 2 0 ) p r e d i c t e d from t r a n s i t i o n e q u a t i o n s (29) and ( 3 1 ) ; i n t h i s case f ( X l 0 , X 2 0 ) = 0. At s t a g e 2, the d e c i s i o n p r o b l e m i s t o f i n d t h e optimum p r o p o r t i o n of p r i m a r y sample u n i t s t o be remeasured p* on o c c a s i o n 2 such t h a t f 2 ( X l 2 , X 2 2 ) = min { k 2 p + f , ( X u , X n ) } (33) 0<p<l f o r a l l p o s s i b l e v a l u e s of [ X 1 2 , X 2 2 ] , w i t h ( X 1 1 , X 2 1 ) p r e d i c t e d from the t r a n s i t i o n e q u a t i o n s (28) and ( 3 0 ) . S o l u t i o n P r o c e d u r e The e q u a t i o n s (32) and (33) can then be s o l v e d r e c u r s i v e l y . The s o l u t i o n p r o c e d u r e i s as f o l l o w s . C o n s i d e r the i n i t i a l and f i n a l s t a g e s of the decomposed d e c i s i o n p r oblem. At s t a g e 1, X j 0 = 0. S u b s t i t u t i n g the f i n a l c o n d i t i o n i n t o the s t a g e 1 s t a t e t r a n s i t i o n f u n c t i o n (29) one o b t a i n s X,, B X10 = 0 = X 1 2 - s, t h a t i s , s* = s i n c e X 1 2 E X,,, t h a t i s , the o p t i m a l d e c i s i o n i s e q u a l t o the i n p u t s t a t e . S i m i l a r l y , X 2 0 = 0, and s u b s t i t u t i n g i n t o the second t r a n s i t i o n f u n c t i o n (31) p r o v i d e s X 2 x 5 X 2 0 = 0 = X 2 2 + s t h a t i s , S" = - X 2 1 s i n c e X 2 2 = X 2 s . Hence, at s t a g e 1, w = raax(X,i, _ X 2 i ) 81 and the d e c i s i o n problem i s s i m p l i f i e d t o f i n d i n g s* = w such t h a t f,(w) = min ( k,s) (31.1) s=w f o r a l l f e a s i b l e v a l u e s of w. P r o c e e d i n g w i t h the r e c u r s i v e s o l u t i o n , the s t a g e 2 pr o b l e m i s t h e n t o f i n d p* such t h a t f 2 ( X 1 2 ) = min { k 2 p + f ^ X j } (32) p>0 but we know t h a t from e q u a t i o n (28) f 2 ( x , ) = k ! ( x 1 2 - { [ p / ( i - V2 + P*|)].+ v t - ( e 2 / v x ) } ) (33) Now i f we s e t the i n i t i a l c o n d i t i o n X 1 2 = V t i n (33) and s u b s t i t u t e (33) i n t o (32) we o b t a i n t h a t f 2 ( X 1 2 ) = min { k 2 p + k . C f l - e2 /Vt ) + [ p / ( l - 4<22 + p * 2 ) H } . (34) p>0 2 P j " i s e q u a l t o t h a t v a l u e o f p t h a t m i n i m i s e s ( 3 4 ) . S i m i l a r l y , i f we s e t X 2 2 = V 2 i n (30) and s u b s t i t u t e i n t o (32) we o b t a i n t h a t f 2 ( X 2 2 ) = min { k 2 p + k, p>0 p ( e 2 2 - v2 - 2 * 2 / e 2 2 e 2 1 ) + ( l - ipj + p\p|)i } (35) v 2 ( i - ^\ + p^l) - e 2 1 ( i - i>22) A g a i n p 2 " i s e q u a l t o t h a t v a l u e of p t h a t m i n i m i s e s ( 3 5 ) . Hence a t the second s t a g e 2 z = ma x ( p 1 " , p 2 " ) and the d e c i s i o n problem i s s i m p l i f i e d t o f i n d i n g p* = z such t h a t f 2 ( z ) = min { ( k 2 + k j ) p } p = z f o r a l l f e a s i b l e v a l u e s of z. Once we have found p-', we now t r a c e back t o s t a g e 1 t o o b t a i n s*. U s i n g t r a n s i t i o n f u n c t i o n s (28) and ( 3 0 ) , we o b t a i n X1j and X 2 1 by sub-s t i t u t i n g p = p". Then s* w i l l be g i v e n by 82 s* = m a x ( X l j , - X 2 1 ) . T h i s c o m p l e t e s t h e s o l u t i o n p r o c e d u r e . A l l a l o n g i t has been assumed t h a t n, m and a l l v a r i a n c e s and c o -v a r i a n c e s a r e known. However, i f t h e s e v a l u e s a r e not known, t h a t i s , we are p l a n n i n g a f u t u r e i n v e n t o r y r i g h t from o c c a s i o n 1, t h e n n and m can be o b t a i n e d by an i t e r a t i v e p r o c e d u r e : r e p e a t i n g t h e p r o c e d u r e d e s c r i b e d above f o r a l l f e a s i b l e v a l u e s of ( n , m) and i d e n t i f y i n g t h a t p a i r w h i c h m i n i m i z e s the t o t a l c o s t . The v a r i a n c e s and the c o v a r i a n c e s w i l l have to be e s t i m a t e d . The use of DP i n o p t i m a l SPR d e s i g n problems w i l l be b e t t e r u n d e r s t o o d i n c h a p t e r 5 where a sample problem i s s o l v e d . CHAPTER 5 SAMPLE PROBLEM I n t h i s c h a p t e r we i n v e s t i g a t e the a p p l i c a t i o n of the g e n e r a l t h e o r y d e v e l o p e d i n the p r e c e d i n g c h a p t e r s t o a s p e c i f i c f o r e s t i n v e n t o r y p r o b l e m . A t t e n t i o n w i l l be r e s t r i c t e d t o the use of two-stage SPR on two s u c c e s s i v e o c c a s i o n s . The p r o b l e m i s t o d e t e r m i n e t h e c u r r e n t mean volume per ha and the p e r i o d i c change i n volume per ha ( s a y , over a 15-year p e r i o d ) i n a f o r e s t a r e a . I n v e n t o r y d a t a c o l l e c t e d i n r e c e n t y e a r s from B r i t i s h C olumbia's Cranbrook P u b l i c S u s t a i n e d Y i e l d U n i t (PSYU) w i l l be used t o demon s t r a t e the s o l u t i o n of the sample problem. The s o u r c e and n a t u r e of t he d a t a and the d e t e r m i n a t i o n of t h e optimum r e p l a c e m e n t p o l i c y t h r o u g h dynamic programming a r e d e s c r i b e d , and the n sample c a l c u l a t i o n s a r e p e r -formed based on the e x i s t i n g d a t a base. C ranbrook i s one of the 81 PSYU's^ i n B r i t i s h C o l u m b i a . I t c o n t a i n s 2 a p p r o x i m a t e l y 506,006 ha of crown f o r e s t l a n d and 233,032 ha of n o n - f o r e s t l a n d ( F o r e s t Survey and I n v e n t o r y D i v i s i o n , 1965). The p r i n c i p a l f o r e s t t r e e s p e c i e s i n c l u d e : spruce ( P i c e a e n g e l m a n n i i P a r r y ) , w e s t e r n hemlock (Tsuga 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 l a s i o c a r p a [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 o f l o d g e p o l e p i n e ( P i n u s  c o n t o r t a D o u g l . ) . The u n i t i s d i v i d e d i n t o 40 compartments of v a r y i n g a r e a s . The number of samples e s t a b l i s h e d v a r i e d from compartment t o com-partment . ^Timber R i g h t s and F o r e s t P o l i c y i n B r i t i s h C o l u m b i a , V o l . I , 1976. R o y a l Commission on F o r e s t R e s o u r c e s , V i c t o r i a , B.C. 2 Crown f o r e s t l a n d i s l a n d b e l o n g i n g t o the s t a t e o r government. 83 84 S e v e r a l i n v e n t o r i e s have been c o n d u c t e d by the B r i t i s h C olumbia F o r e s t S e r v i c e (BCFS) i n t h i s u n i t s i n c e 1952. D u r i n g the p e r i o d 1953 to 1964 i n c l u s i v e , 462 samples ( f i x e d a r e a , h a l f - a c r e and t w o - f i f t h a c r e ) were e s t a b l i s h e d i n a l l t i m b e r t y p e s . D u r i n g the 1979 i n v e n t o r y , 176 p o i n t samples were e s t a b l i s h e d i n a l l the t i m b e r t y p e s . (Permanent sample p l o t s were e s t a b l i s h e d i n 1968 and the f i r s t remeasurement was i n 1978.) The b a s i c i n v e n t o r y t e c h n i q u e used o v e r a l l t h e s e y e a r s was s t r a t i f i e d random s a m p l i n g , w i t h mature t i m b e r t y p e s b e i n g sampled more i n t e n s i v e l y t h a n the immature or o t h e r t y p e s . Data summaries based on the 1964 and on the 1979 s u r v e y s are a v a i l a b l e by sample number and the a t t r i b u t e s measured i n c l u d e d volume p e r ha ( t o v a r i o u s l e v e l s o f u t i l i z a t i o n ) and the number of stems per ha. For our p u r p o s e s , the compartments w i l l c o n s t i t u t e the psu's and the samples w i t h i n the compartments, the s s u ' s . I n o t h e r words, we have a two-stage 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 , but e q u a l - s i z e d s s u ' s . The 1964 sample d a t a w i l l be assumed t o be the f i r s t o c c a s i o n measurements and t he 1979 sample d a t a as t h e second o c c a s i o n measurements. The o b j e c -t i v e w i l l , t h e r e f o r e , be t o d e t e r m i n e the c u r r e n t (1979) mean volume p e r ha and the change i n mean volume p e r ha between 1964 and 1979 ( t h a t i s , over a 15-year p e r i o d ) . Twenty-seven out of the 40 psu's were sampled i n 1964 and the number of samples per compartment ranged from 1 to 25 (mean = 1 1 ) . I n 1979, 35 compartments were sampled w i t h an average of 9 s s u ' s per p s u . (Of the 35 p s u ' s , 16 had not been sampled i n 1964.) None of the 1964 samples were a c t u a l l y remeasured i n 1979. F o r the purposes of d e t e r m i n i n g t h e a p p r o x i m a t e number of psu's t o remeasure, we s h a l l t a k e the i n i t i a l sample s i z e t o be as f o l l o w s : psu's n = 27 and ssu's per psu m = 11. I t w i l l 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 is required: (a) As always for planning an inventory, a knowledge of the estimates of the population parameters for the forest area to be sampled is required. We shall assume the following estimates of the population parameters in the Cranbrook PSYU: .(i) average volume per ha u y = 475.41 m3 ( i i ) periodic change (over 15 years) of volume per ha + 321.81 m3/ha ( i i i ) variance of volume per ha between psu's a 2 = 189876.06, (i=l,2) i (iv) variance of volume per ha between ssu's within the psu's o2 = i 1189.56, (i=l,2) (v) correlation between the effects due to the psu's in 1964 and 1979 P i = 0.95 (vi) correlation 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 • 62-, = 6 2 2 = [(189876.06/27) + (1189.56)/(27 x 11)] = 7420.65 and \p2 = [(11 x 0.95 x 189876.062) + (0.85 x 1189.562)]/ [27 x 11 x /6 2 16 2 2] = 0.88 (b) 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 is ± l07o at the 957» confidence level per 3 Guidelines for Forest and Range Inventory in British Columbia, 1980. Inventory -Branch, Ministry of Forests, B.C. ) 86 u n i t . I n o r d e r t o be w i t h i n the range of the d a t a a v a i l a b l e , however, we s h a l l assume an a l l o w a b l e e r r o r o f + 30% a t the 95% c o n f i d e n c e l e v e l p er mean volume per ha or p e r i o d i c change p e r ha. T h i s i m p l i e s t h a t the a l l o w a b l e v a r i a n c e l e v e l s f o r e s t i m a t i n g c u r r e n t mean volume p e r ha i s V- = {[0.30 x 4 7 5 . 4 l ] / 2 } 2 = 5085.116 and f o r p e r i o d i c change ( o v e r 15 y e a r s ) i s V = {[0.30 x 3 2 1 . 8 l ] / 2 } 2 =2330.137. 'g ( c ) I t w i l l a l s o be assumed t h a t t h e t o t a l c o s t p e r t a i n i n g t o the second o c c a s i o n ( i n v e n t o r y ) i s g i v e n by the s i m p l e c o s t f u n c t i o n ( e q u a t i o n [ 2 3 ] ) C = k j p + k 2 s " where k^ i = 1,2 a r e as d e f i n e d e a r l i e r i n e q u a t i o n ( 2 3 ) . We i n t r o d u c e some new n o t a t i o n . L e t . X = k 1./k 2 and C = C/k 2. Then, t h e c o s t r e l a t i o n above can be w r i t t e n as C = Xp' + s the e x p r e s s i o n t o be m i n i m i z e d . T h i s form of the c o s t r e l a t i o n i s more u s e f u l , e s p e c i a l l y when the a b s o l u t e c o s t v a l u e s a r e not a v a i l a b l e ; Now we s t a t e the i n v e n t o r y p l a n n i n g p r o b l e m . The p r o b l e m i s t h a t of d e t e r m i n i n g the optimum number of psu's t o remeasure and new ones t o t a k e on t h e second o c c a s i o n , and 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 : F i n d p, s > 0 such t h a t C i s m i n i m i z e d and such t h a t t h e a l l o w a b l e e r r o r s o f c u r r e n t mean volume p e r ha and p e r i o d i c change i n volume p e r ha a r e met. F o l l o w i n g t he o p t i m i z a t i o n p r o c e d u r e d e v e l o p e d i n c h a p t e r 4, the s t a g e diagram f o r t h i s sample p r o b l e m i s s i m i l a r t o t h a t shown i n F i g u r e 2. The s t a t e t r a n s i t i o n f u n c t i o n s a r e : f o r c u r r e n t mean volume p e r ha Xj x,, = x l 2 - [p/(l - *| + p*|)] X l 0 = X l i - s and f o r p e r i o d i c change i n volume p er ha X 2 P ( e 2 l - v - 2^ 2/e 2,e 2 2) + ( l - \p| + p ^ 2 ) e 2 2 X 2 i = X 2 2 - C ] v ( l - *f + PV2) - e 2 I ( l - + |) X 2 0 = X 2 1 + S . We now p r o c e e d w i t h the dynamic programming s o l u t i o n as f o l l o w s . At s t a g e 1, we w i s h t o f i n d s * ( X 1 1 9 X 2 1 ) such t h a t C ' 1 ( X U ) X 2 1 ) = min {s + C 1 „ (X, 0 ,X 2 0 ) } s>0 f o r a l l f e a s i b l e v a l u e s of ( X l l , X 2 X ) , where C . i = 0, 1 are- the c o s t s a s s o c i a t e d w i t h t he i t h s t a g e s . U s i n g the t r a n s i t i o n f u n c t i o n s we see t h a t X 1 0 = X „ - s = 0 X 2Q = X 2 X + s = 0 i m p l y i n g t h a t s w = ( X j j , — X 2 j ) S i n c e C ' 0 ( X 1 0 , X 2 0 ) = 0, the s o l u t i o n f o r the s t a g e 1 problem i s g i v e n by the f u n c t i o n w = raax[X,j, - X 2 ! ] and the a s s o c i a t e d c o s t i s C j(w) = min {s} s=w We n o t i c e t h a t s* has been d e t e r m i n e d as a f u n c t i o n of p. Hence we 88 p r o c e e d t o s t a g e 2, t o d e t e r m i n e the v a l u e o f p * and t h e n t r a c e back t o s t a g e 1 t o f i n d s*. At s t a g e 2, we w i s h t o f i n d p * ( X j 2 , X 2 2 ) such t h a t C ' 2 ( X 1 2 , X 2 2 ) = min U p + C ' jCw)} p>0 f o r a l l f e a s i b l e v a l u e s o f ( X 1 2 , X 2 2 ) . We s e t X 1 2 = e 2 2/v- = 1.459 x 2 2 = 0 and compute C'^w) and C ' 2 ( X l 2 , X 2 2 ) f o r a l l f e a s i b l e v a l u e s of 0 < p < 1. ( I n t h i s example, X = 1, k t = k 2 = $500.) The e n u m e r a t i o n r e s u l t s are summarized i n T a b l e I I I . I t can be seen from T a b l e I I I t h a t w i t h X 1 2 = 1.459 and X 2 2 = 0, p* = 0.67, and t r a c i n g back t o s t a g e 1 u s i n g p* = 0.67, we see t h a t s* = 0.55. T h i s c o m p l e t e s t h e s o l u t i o n . The s o l u t i o n s o b t a i n e d i m p l y t h a t g i v e n i n i t i a l l y n = 27 p s u ' s , on the second o c c a s i o n we must remeasure ( 0 . 6 7 ) ( 2 7 ) = 18 psu's and t a k e ( 0 . 5 5 ) ( 2 7 ) = 15 new p s u ' s , g i v i n g a t o t a l o f 33 p s u ' s , w i t h an average of 11 ssu's per p s u . The t o t a l c o s t of t h e i n v e n t o r y a f t e r the second o c c a s i o n i s a p p r o x i m a t e l y $181,663. A random sample of 18 psu's was t a k e n from t h e i n i t i a l 27 p s u ' s , t o g e t h e r w i t h t h e i r s s u ' s . I n a d d i t i o n , 15 psu's measured i n 1979 but not i n 1964 were 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 s s u ' s . The volumes p e r ha at each of the sample ssu's i n each of Che s e l e c t e d psu's are sum-m a r i z e d i n Appendix I . I t s h o u l d be n o t i c e d t h a t t h e remeasurement d a t a on the 18 psu's were s i m u l a t e d u s i n g the e x i s t i n g volume-age c u r v e s f o r the a r e a (see A ppendix I I ) . F i r s t o c c a s i o n measurements a r e l a b e l l e d X and second o c c a s i o n Y. From the sample d a t a g i v e n i n A p p e n d i x I we o b t a i n the f o l l o w i n g sample s t a t i s t i c s . T a b l e I I I . E n u m e r a t i o n R e s u l t s •59 .60 .61 .62 .63 .64 .65 .66 .67* .68 .69 .58 19.613 ,58 ,57 ,57 .56 ,56 .56 .55 7.176 4.038 2.608 1.790 1.260 0.889 0.615 ,55* 0.404 ,55 0.236 54 0.100 2912567.00 1065673.00 599707.00 387341.06 265831.18 187150.50 132045.31 91296.18 82167.50 81601.43 81047.12 3000181.00 1154773.00 690291.81 479410.87 359386.06 282190.31 228570.18 189306.06 181662.37* 182581.25 183512.00 Note: (1) Those v a l u e s marked w i t h an a s t e r i s k (*) are the optimum s o l u t i o n s . (2) C j i s the c o s t of m e a s u r i n g the new (ns) psu's and t h e i r a s s o c i a t e d s su's C' 2 i s the t o t a l c o s t of m e a s u r i n g the n(p + s) psu's and t h e i r s s u ' s . 90 x".. = 414.64 x'.. = 406.90 y".. = 420.89 y'.. = 124.96 a 2 = [ Z (x. - x . . ) 2 ] / ( n p - 1) =17318.0336 a i i = l 1 np a 2 = [ 2 (y. - y . . ) 2 ] / ( n p - 1) = 17814.7747 1 = 1 1 np m. i np 6 2 = [ Z Z ( x . . - x . ) 2 ] / [ Z m.) - np] = 28859.2144 e, . . . . l i i . , i 1 i = l j = l i = l np m. , . l n(p+s) 6 2 = [ Z Z ( y . . - y . ) 2 ] / [ ( Z m.) - np] = 31673.3209 C z i = l j = l 1 J 1 i = l 1 n p P, = [ Z (x. - x . . ) ( y . - y . . ) ] / [ ( n p - 1)8 a ] = 0.99 1 = 1 1 i m. np l np P 2 = [ Z Z ( x . . - x . ) ( y . . . - y . ) ] / [ ( z m. - np)8 a ] = 0 . 9 6 i = l j = l X J 1 ^ 1 i = l 1 And u s i n g the above sample s t a t i s t i c s we o b t a i n t h a t = 1079.65034 e' 1 .= 1159.73642 §" 2 = 1116.92573 §' 2 = 1091.27952 I' = 1084.22304 We now have a l l the n e c e s s a r y s t a t i s t i c s to pr o c e e d w i t h the computa-t i o n of the e s t i m a t e s of ( i ) c u r r e n t mean g r o s s volume per ha (17.5 cm +) u y and ( i i ) p e r i o d i c change i n mean volume over the 15 y e a r i n -t e r v a l , A. ( i ) C u r r e n t mean volume R e c a l l t h a t the minimum-variance l i n e a r u n b i a s e d e s t i m a t o r o f f o r u n i t s ' of unequal s i z e i s y 2 = a*(x'.. - x"..) + c*y".. + (1 - C")y'.. 91 where a* and c* are as d e f i n e d e a r l i e r . U s i n g the sample e s t i m a t e s of 9'^, 9"^ i =1,2 and B', we o b t a i n t h a t c* = 0.63183 a* = 0.19947 and hence y 2 = [0.19947 x (406.90 - 414.64)] + (0.63183 x 420.89) + (0.36817 x 124.96) 36 = 310.39 m 3/ha. The v a r i a n c e of t h i s e s t i m a t e i s g i v e n by V a r ( y 2 j l ) = a * 2 ( e ' , / q + 9" 2/p) + c * 2 9 " 2 / p + (1 - c * ) 2 6 ' 2 / s - 2a*c*B'/p = 730.499. By t a k i n g / V a r ( y 2 ) = 27.0277 and u s i n g the t - v a l u e a t t h e 957» c o n f i d e n c e l e v e l o f a p p r o x i m a t e l y 2, we o b t a i n t h a t c u r r e n t volume p e r ha = 310.39 ± 54.055 m3. ( i i ) P e r i o d i c change i n mean volume p e r ha A g a i n , r e c a l l t h a t the minimum-variance l i n e a r u n b i a s e d e s t i m a t o r of A f o r u n i t s o f u n e q u a l s i z e i s g 2 0 = e*y".. + (1 - e * ) y ' . . + f * x " . . - (1 + f * ) ? . . 36 where e* and f * a r e as d e f i n e d e a r l i e r . U s i n g t h e sample e s t i m a t e s of 0' and 9"^ i = l , 2 and B', we see t h a t ?* = -0.99664 e* = 0.98514 92 and hence g 2 ••= (0.98514 x 420.89) + (0.01486 x 124.96) - (0.99664 x 414.64) - (0.00336 x 406.90) = 1.878 m 3/ha The v a r i a n c e of t h i s e s t i m a t e i s g i v e n by: V a r t g ^ ) = e * 2 9 " 2 / P + ( l - e * ) 2 9 ' 2 / s + b- 2Q"Jp + (1 + f *) 2 6' ,/q + 2e*f*S'/p = 41.2828. By t a k i n g / V a r ( g 2 ) = 6.4251 and u s i n g the t - v a l u e at t h e 957- c o n f i d e n c e l e v e l o f a p p r o x i m a t e l y 2, we o b t a i n t h a t change i n volume i n 15 y e a r s = 1.878 ± 12.8502 m3/ ha. The r e s u l t s o f the c a l c u l a t i o n s i n d i c a t e t h a t the t o t a l whole-stem volume ( l i v i n g t r e e s o n l y , dbh 17.5 cm +) i n the f o r e s t l a n d a r e a (506,006 ha) of t h e Cranbrook PSYU i n 1979 was (506,006) x (310.39 ± 54.055) = 157,059,202.3 ± 27,352,402.29 m3 and the change i n the volume between 1964 and 1979 was 506,006 x (1.878 ± 12.8502) = 950,279.3 ± 6,502,348.39 m3. The change i n volume i s a r e s u l t o f s u r v i v o r growth ( i n c r e m e n t on t r e e s p r e s e n t a t b o t h t h e 1964 and 1979 i n v e n t o r i e s ) , m o r t a l i t y (volume o f t r e e s r e n d e r e d u s e l e s s t h r o u g h n a t u r a l c a u s e s such as o l d age, i n s e c t s , w i n d f a l l , f i r e , e t c . ) , t u t ( l o g g i n g ) , and i n g r o w t h (volume o f t r e e s g r o w i n g i n t o m e a s u r a b l e s i z e , i n t h i s case 17.5 cm). For a complete d e f i n i t i o n of the growth components see Beers ( 1 9 6 2 ) . A c c o r d i n g t o the BCFS r e c o r d s , the t o t a l volume o f t i m b e r l o g g e d and t h a t l o s t t h r o u g h m o r t a l i t y between 1964 and 1979 i n the f o r e s t a r e a of the Cranbrook was e s t i m a t e d a t 24,023,584 m3. An e s t i m a t e o f the i n g r o w t h and s u r v i v o r growth volume o b t a i n e d from the permanent sample p l o t s i n the Cranbrook was 4.39 m /h a / y e a r , g i v i n g a t o t a l o f 93 (4.39 x 506,006 x 15) = 33,320,495 m3 between 1964 and 1979. Thus the net change in volume between 1964 and 1979 is 33,320,495 +(-24,023,584) = 9,296,911 m3. This result is slightly higher than the upper 95% con-fidence limit of the estimate obtained through the sample problem calcu-lation. 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 limits on estimates of current timber mean volume and change were constructed based on the central limit theorem that the probability distribution of the SPR estimators was sufficiently close to the normal distribution and for practical 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 relatively high and hence inflated the variance estimates. A further discussion of the various aspects of the sample problem and of the theory derived in chapters 3 and 4 is given in the next chapter. T. Cunia, Lecture notes, Workshop on sampling on successive occa-sions. Colorado State University, July 1979. CHAPTER 6 DISCUSSION AND CONCLUSION The theory of successive forest inventories with partial replacement of units presented by Ware and Cunia (1962) has been extended to use multi-stage sampling designs (with partial replacement of the primary sample units). Multistage designs have many desirable features particularly for an inventory of large forest areas. These designs (i) provide u l t i -mate sample units that can be cost efficiently measured, especially when construction of the sampling frame is 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 in incorporating data from high- and low-altitude and ground level sources simultaneously for efficiency. This is particularly more so i f variable probabilities of selection are used at the various stages of the multistage design. Simple random sampling was assumed at each stage in this thesis for simplicity of presentation. A logical extension of the theory developed here is to use variable probability sampling at the vari-ous stages. This would, for example, involve extending the work of Langley (1975, 1976) for one-occasion sampling to successive occasions. Multistage sampling is often applied to large regional and national inventories in order to reduce cost. The major potential disadvantage, however, is 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. T h i s makes the p r o v i s i o n of i n f o r m a t i o n on sub-d i v i s i o n s ( 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 . I f the s u b d i v i s i o n s are the same as the p s u ' s , making i n f e r e n c e s f o r the a r e a s not sampled i s u s u a l l y i m p o s s i b l e . I n such c a s e s o t h e r d e s i g n s may be employed. As was p o i n t e d out i n c h a p t e r 1, i n the c a s e of m u l t i s t a g e d e s i g n s , the t e c h n i q u e o f SPR g i v e s r i s e t o a number of s a m p l i n g a l t e r n a t i v e s , w i t h d i f f e r e n t c o m b i n a t i o n s o f r e p l a c e m e n t o f p r i m a r y , s e c o n d a r y , t e r t i a r y , e t c . , u n i t s o ver t i m e . For p r a c t i c a l r e a s o n s , i t was d e c i d e d t o c o n s i d e r o n l y the case i n w h i c h o n l y the 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 s e c o n d a r y , t e r t i a r y , e t c . u n i t s c o r r e s p o n d i n g t o the p r i m a r i e s . P a r t i a l r e p l a c e m e n t a t a l l s t a g e s of a m u l t i s t a g e d e s i g n may p r o v e t o be too complex i n t h e o r y and p r o h i b i t i v e l y too e x p e n s i v e t o a p p l y ( P r o f e s s o r T. C u n i a — p e r s o n a l c o m m u n i c a t i o n ) . I n a d d i t i o n , S i n g h and K a t h u r i a ( 1 9 6 9 ) , assuming e q u a l sample s i z e and e q u a l v a r i a n c e on b o t h o c c a s i o n s i n a two-stage d e s i g n , c o n c l u d e d t h a t 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 c o r r e l a t i o n were l a r g e r i n r e l a t i o n t o between-psu v a r i a n c e and c o r r e l a t i o n , the e s t i m a t e o f the c u r r e n t mean was, i n g e n e r a l , more e f f i c i e n t i n the c a s e of p a r t i a l r e p l a c e m e n t of psu's o n l y t h a n i n the case of p a r t i a l r e p l a c e m e n t of ssu's o n l y . ( T h i s c o n c l u s i o n seems l o g i c a l 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 between-psu and w i t h i n - p s u v a r i a n c e due t o p a r t i a l r e p l a c e m e n t of the p s u ' s , whereas o n l y the w i t h i n - p s u v a r i -ance component i s a f f e c t e d due t o p a r t i a l r e p l a c e m e n t of the s s u ' s . ) F u r t h e r , Rana and C h a k r a b a r t y (1976) c o n c l u d e d from a n u m e r i c a l s t u d y of the r e l a t i v e e f f i c i e n c i e s o f v a r i o u s s a m p l i n g p l a n s , g i v e n the a s s u m p t i o n s of S i n g h and K a t h u r i a ( 1 9 6 9 ) , t h a t i f s a m p l i n g was i n e x p e n s i v e and the 96 p r e c i s i o n o f the e s t i m a t e s o f the c u r r e n t mean and change was o f major i n t e r e s t , p a r t i a l r e p l a c e m e n t of o n l y the psu's was more e f f i c i e n t t h a n o t h e r p r o c e d u r e s t h e y c o n s i d e r e d i n most c a s e s . I t would be u s e f u l t o s t u d y the r e l a t i v e e f f i c i e n c i e s of the v a r i o u s s a m p l i n g a l t e r n a t i v e s a r i s -i n g from the d i f f e r e n t c o m b i n a t i o n s o f p a r t i a l r e p l a c e m e n t of the d i f f e r e n t i s t a g e u n i t s , f o r e s t i m a t i n g b o t h c u r r e n t v a l u e s and change, under the as s u m p t i o n s of unequal s i z e and u n e q u a l v a r i a n c e on s u c c e s s i v e o c c a s i o n s . The s p e c i a l case of one-stage SPR as p r e s e n t e d , f o r example, by Ware and C u n i a (1962) can be o b t a i n e d from the g e n e r a l r e s u l t p r e s e n t e d h e r e . I n p a r t i c u l a r , the two-stage SPR d e s i g n becomes one-stage SPR when a2 ( i =1,2) are s e t t o z e r o and hence * 2 becomes p. T h i s i s not a. I s u r p r i s i n g s i n c e s i m p l e random s a m p l i n g was used w i t h i n each s t a g e o f the m u l t i s t a g e d e s i g n . A l t h o u g h the o b j e c t i v e o f the s t u d y was t o p r e s e n t minimum-variance ( b e s t ) l i n e a r u n b i a s e d e s t i m a t o r s 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 e s t i m a t o r s , b i a s e d o r u n b i a s e d , were c o n s i d e r e d . S p e c i f i c a l l y , the use of the r a t i o e s t i m a t o r was i n v e s t i g a t e d . I f i t i s r e a l i z e d t h a t the BLUE f o r c u r r e n t mean i s a w e i g h t e d average of a r e g r e s s i o n double s a m p l -i n g e s t i m a t e and a mean based on c u r r e n t o b s e r v a t i o n s o n l y , t h e n i t seems l o g i c a l t o p o s t u l a t e an e s t i m a t o r , one wh i c h i s a w e i g h t e d average o f a doub l e s a m p l i n g r a t i o e s t i m a t e and a mean of c u r r e n t o b s e r v a t i o n s only. The BLUE was n e g l i g i b l y more e f f i c i e n t ( p r e c i s e ) under c e r t a i n c o n d i t i o n s t h a n the e s t i m a t o r based on the r a t i o e s t i m a t e f o r e s t i m a t i n g e i t h e r the c u r r e n t v a l u e s or change. F u r t h e r , the r a t i o e s t i m a t o r was b i a s e d ; the amount of the b i a s was e x p e c t e d t o be n e g l i g i b l e when the sample s i z e i n c r e a s e d . However, the w e i g h t s o f the e s t i m a t o r based on t h e r a t i o e s t i m a t e were d e r i v e d by m i n i m i z i n g the v a r i a n c e f u n c t i o n o f the 97 e s t i m a t o r of e i t h e r c u r r e n t mean or change. I t was i m p l i c i t l y assumed t h a t b i a s was z e r o . More a p p r o p r i a t e v a l u e s of the w e i g h t s would have been o b t a i n e d i f the mean square e r r o r (MSE) f u n c t i o n of c u r r e n t mean ( o r change) was m i n i m i z e d i n s t e a d . F o r example, i n e s t i m a t i n g the c u r -r e n t mean, the f u n c t i o n t o be m i n i m i z e d would have been ( s t a t e d here w i t h -out d e r i v a t i o n ) f o r a two-stage SPR d e s i g n M S E ( y 2 r ) = V a r ( y 2 r ) + [ E ( y 2 r ) - u y ] 2 However, i n p r a c t i c e and t o the o r d e r of a p p r o x i m a t i o n used, the v a l u e s of the w e i g h t s 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 s u f -f i c i e n t . A l t h o u g h the e s t i m a t o r of the c u r r e n t mean based on the r a t i o e s t i m a t e was s l i g h t l y l e s s e f f i c i e n t t h a n the BLUE, i t i s s u g g e s t e d t h a t where c o m p u t a t i o n of £3 and 3 i s c o s t l y and when c o r r e l a t i o n s p. ( j = 1,2, YA A Y J ...,h) are h i g h and the v a r i a n c e s 6 ^ ( i = 1,2) from o c c a s i o n t o o c c a s i o n are r o u g h l y the same, the e s t i m a t o r based on the r a t i o e s t i m a t e may be used. Both Sen e t a l . (1975) and Woodruff (1959) share the same v i e w . F u r t h e r , A r v a n i t i s and F o w l e r (1979: 307) s t a t e : B i a s e d s a m p l i n g e s t i m a t o r s are u s u a l l y s u r r o u n d e d by a vague, c o n t r o v e r s i a l meaning w h i c h works a g a i n s t t h e i r a c c e p t a n c e as more e f f i c i e n t t h a n u n b i a s e d ones 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 o f the s a m p l e r s i s t o employ minimum-variance u n b i a s e d 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 t h a t 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 or 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 exceed by f a r the s a m p l i n g e r r o r . The method adopted f o r the d e t e r m i n a t i o n of t h e optimum r e p l a c e m e n t p o l i c y o ver time used dynamic programming. Dynamic programming has been used i n o t h e r f o r e s t r y p r o b l e m s , f o r example, i n the d e t e r m i n a t i o n of the optimum t r e e b u c k i n g p o l i c y ( P n e v m a t i c o s & Mann, 1972), and i n the 98 d e t e r m i n a t i o n o f optimum l e v e l s of g r o w i n g s t o c k (Amidon & A k i n , 1968). U s i n g dynamic programming, the optimum rep l a c e m e n t p o l i c y can be d e t e r -mined f o r s i m u l t a n e o u s l y e s t i m a t i n g more t h a n two v a r i a b l e s of i n t e r e s t . T h i s w i l l n o t , i n g e n e r a l , a f f e c t the s e p a r a b i l i t y o f the o b j e c t i v e ( c o s t ) and c o n s t r a i n t f u n c t i o n s ; the c o s t o f c o m p u t a t i o n w i l l , however, i n c r e a s e . There i s no need f o r a subsequent s e n s i t i v i t y a n a l y s i s on the d e r i v e d o p t i m a l p o l i c y ; t h i s i s a u t o m a t i c a l l y b u i l t i n t o the dynamic programming f o r m u l a t i o n . The o p t i m a l p o l i c y and the a s s o c i a t e d c o s t are known f o r a l l f e a s i b l e v a l u e s of the s t a t e v a r i a b l e s , s i n c e the s o l u t i o n s a r e d e t e r -mined as f u n c t i o n s of the s t a t e v a r i a b l e s . F o r example, i f the s t a t e v a r i a b l e s X 1 2 and X 2 2 t a k e on v a l u e s o t h e r t h a n V t and V 2 , r e s p e c t i v e l y (see e q u a t i o n s [34] and [35]), a s o l u t i o n c o r r e s p o n d i n g t o the new v a l u e s of the s t a t e v a r i - a b l e s would e a s i l y be o b t a i n e d w i t h o u t n e c e s s a r i l y h a v i n g to r e - s o l v e the e n t i r e p r oblem. I n the sample problem ( c h a p t e r 5) optimum r e p l a c e m e n t p o l i c y was o b t a i n e d u s i n g complete e n u m e r a t i o n . However, n o n l i n e a r s e a r c h methods, such as the G olden s e c t i o n s e a r c h , c o u l d have been used i n s t e a d . F u r t h e r , i t w o u l d have been e a s i e r t o s o l v e the t w o - d e c i s i o n p r o b l e m by the c l a s s i c a l c a l c u l u s methods. However, when t h e r e a r e more t h a n two d e c i s i o n v a r i a b l e s , t h e s e c l a s s i c a l methods become d i f f i c u l t t o use and the s i m p l e s e a r c h p r o c e d u r e s ( s u c h as complete e n u m e r a t i o n ) may be the o n l y a l t e r n a t i v e . For problems w i t h s e v e r a l d e c i s i o n v a r i a b l e s , however, e n u m e r a t i o n i s o n l y p o s s i b l e a f t e r a dynamic programming d e c o m p o s i t i o n , Nemhauser (1966). The major drawbacks of dynamic programming as p o i n t e d out by Nemhauser (1966) are the s e p a r a b i l i t y and m o n o t o n i c i t y c o n d i t i o n s n e c e s -s a r y f o r the d e c o m p o s i t i o n of an N-stage p r o b l e m i n t o N p r o b l e m s . I f 99 .... the number of occasions involved i s not large, the variance functions should in general s a t i s f y these conditions. The problem of determining the optimum sample unit size at each stage of the multistage design has not been treated here. The subject has been discussed to some extent in some basic sampling texts. It suf-f i c e s to mention here that the optimum unit sizes depend on several fac-tors, such as sample c o e f f i c i e n t of v a r i a t i o n , cost of measurement at each l e v e l , and other p r a c t i c a l considerations. The optimum unit size should be determined from the experience of the inventory manager and af t e r considering the factors indicated above. The derived theory was i l l u s t r a t e d , for a two-stage SPR design, by working through a sample forest inventory problem. The nature of successive inventories did not permit an ideal planning and implementa-tion of the derived theory. The sample problem was designed to f i t an e x i s t i n g data set, so that several assumptions had to be made. For example, in order to determine the optimum replacement p o l i c y , i t was assumed that the number of ssu's per psu was equal, and the i n i t i a l population estimates were such that the r e s u l t i n g optimal p o l i c y was within the range of the e x i s t i n g data. However, the r e s u l t s obtained were within the range of the values expected. In the same problem, interest centered on estimating current timber volume and the change i n volume between occasions. Other variables of i n t e r e s t could have been estimated, for example, number of stems per ha, basal area per ha, number of deers, etc. The term "change" as used here means a type of growth which i s the difference between standing timber volume on occasion two and occasion one, termed "net increase" by Beers (1962). The term could be appropriately redefined in order to estimate other components of forest growth (for example, ingrowth) 100 as d e f i n e d by Beers ( 1 9 6 2 ) . I n the d e r i v a t i o n of the t h e o r y , i t was assumed t h a t the p o p u l a t i o n p a r a m e t e r s a 2 , a 2 , and p. ( i , j = 1,2) were known w i t h o u t e r r o r o r Q i E i -1 i n d e p e n d e n t of s a m p l i n g . I n r e a l i t y however, i t i s r a r e l y t r u e t h a t t h e s e v a l u e s a re known; they have t o be e s t i m a t e d . I n the sample p r o b -lem a 2 , a 2 and p. ( i , j = 1,2) were e s t i m a t e d from the matched sample a i E i d a t a . 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 e s t i m a t e s o f c u r r e n t mean and change a r e not u n b i a s e d . The b i a s , however, i s s m a l l i f n and m are r e l a t i v e l y l a r g e . F u r t h e r m o r e , the c a l c u l a t e d optimum r e p l a c e m e n t p o l i c y d e p a r t s from the t r u e optimum 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 of the p a r a m e t e r s d e p a r t from t h e i r t r u e v a l u e s . V e r y e f f e c t i v e s a m p l i n g methods f o r r e s o u r c e i n v e n t o r i e s i n c l u d e v e r s i o n s of m u l t i s t a g e s a m p l i n g . E x t e n s i o n of the t h e o r y from o n e - s t a g e SPR t o m u l t i s t a g e SPR was t h e r e f o r e of p r a c t i c a l i n t e r e s t , p a r t i c u l a r l y f o r the i n v e n t o r y of l a r g e f o r e s t a r e a s . I t would be u s e f u l t o e x t e n d f u r t h e r the t h e o r y t o use v a r i a b l e p r o b a b i l i t i e s o f s e l e c t i o n at t h e v a r i -ous s t a g e s of the m u l t i s t a g e d e s i g n ; and t o examine the c a s e s i n w h i c h p a r t i a l r e p l a c e m e n t o c c u r s a t o t h e r t h a n the p r i m a r y s t a g e . 101 REFERENCES Amidon, E. L., and A k i n , G. S. 1968. Dynamic programming t o d e t e r m i n e optimum l e v e l s o f gr o w i n g 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 of b i a s e d s a m p l i n g e s t i m a t o r s . 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 Sukhatme, B. V. 1970. A c o m p a r i s o n of two s a m p l i n g 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 t o s u c c e s s i v e s a m p l i n g . J . R o y a l S t a t . Soc. ( C ) , A p p l i e d S t a t . , 19: 251-259. A v a d h a n i , M. S., and Sukhatme, B. V. 1972. S a m p l i n g on s u c c e s s i v e o c c a s i o n s w i t h e q u a l and unequal p r o b a b i l i t i e s and w i t h o u t r e p l a c e m e n t . A u s t r a l i a n J . S t a t i s t i c s , 1 4 ( 2 ) : 109-119. B a r n a r d , J . E. 1974. Sampling w i t h p a r t i a l r e p l a c e m e n t c o n t r a s t e d w i t h complete remeasurement i n v e n t o r y d e s i g n s . An e m p i r i c a l examina-t i o n . P r o c e e d i n g s , M o n i t o r i n g F o r e s t Environment t h r o u g h S u c c e s -s i v e S a m p l i n g . T. C u n i a ( E d . ) , S t a t e U n i v e r s i t y o f New Y o r k , S y r a c u s e . B e e r s , T. W. Components of f o r e s t g r o w t h . J . F o r . , 6 0 ( 4 ) : 245-248. B e l l m a n , R. 1957. Dynamic programming. P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , New J e r s e y . B i c k f o r d , C. A. 1956. Proposed d e s i g n f o r c o n t i n u o u s i n v e n t o r y : a s y s -tem of p e r p e t u a l f o r e s t s u r v e y f o r the N o r t h w e s t . U.S. F o r e s t S e r v i c e , E a s t e r n Techniques M e e t i n g , F o r e s t S u r v e y , C u m b e r l a n d - T a i l s , K e n t u c k y . B i c k f o r d , C. A. 1959. A t e s t of c o n t i n u o u s i n v e n t o r y f o r n a t i o n a l f o r e s t management based upon a e r i a l p h o t o g r a p h s , double s a m p l i n g and r e -measured p l o t s . P r o c . Soc. Amer. F o r . : 143-148. B i c k f o r d , C. A. 1963. On s u c c e s s i v e f o r e s t i n v e n t o r i e s . P r o c . Soc. Amer. F o r . , 25-30. B i c k f o r d , C. A., Mayer, C. E., and Ware, K. D. 1963. An e f f i c i e n t sampl-i n g d e s i g n f o r f o r e s t i n v e n t o r y : the n o r t h w e s t f o r e s t s u r v e y . J . F o r . , 61: 826-833. B l i g h t , B. J . N., and S c o t t , A. J . 1973. A s t o c h a s t i c model f o r r e p e a t e d s u r v e y s . J . R o y a l S t a t . S o c , Ser B, 35: 61-66. C h a k r a b a r t y , R. P., and Rana, D. S. 1974. M u l t i - s t a g e s a m p l i n g w i t h p a r t i a l r e p l a c e m e n t of the sample on s u c c e s s i v e o c c a s i o n s . Amer. S t a t . A s s o c . , P r o c . S o c i a l S t a t . S e c : 289-291. Coc h r a n , W. G. 1977. Sampling t e c h n i q u e s . John W i l e y and Sons, I n c . , New York ( T h i r d E d i t i o n ) . 102 C u n i a , T. 1964. What i s s a m p l i n g w i t h p a r t i a l r e p l a c e m e n t and why use i t i n c o n t i n u o u s f o r e s t i n v e n t o r y . P r o c . Soc. Amer. F o r . : 207-211. C u n i a , T. 1965. C o n t i n u o u s f o r e s t i n v e n t o r y , p a r t i a l r e p l a c e m e n t of samples and m u l t i p l e r e g r e s s i o n . F o r e s t S c i . 1 1 ( 4 ) : 480-502. C u n i a , T. , and Chevrou, R. B. 1969. S a m p l i n g w i t h p a r t i a l r e p l a c e m e n t t h r e e or more o c c a s i o n s . F o r e s t S c i . 1 5 ( 2 ) : 204-224. on Dano, S. N o n l i n e a r and dynamic programming. S p r i n g e r - V e r l a g , Wien, Aus-t r i a . D i x o n , B. L., and H o w i t t , R. E. 1979. C o n t i n u o u s f o r e s t i n v e n t o r y u s i n g a l i n e a r f i l t e r . F o r e s t S c i . , 2 5 ( 4 ) : 675-689. E c k l e r , A. R. 1955. R o t a t i o n s a m p l i n g . Ann. Math. S t a t . 26:664-685. F o r e s t S urvey and I n v e n t o r y D i v i s i o n . 1965. Report on the 1964 U n i t Survey of the Cranbrook PSYU. B r i t i s h Columbia F o r e s t S e r v i c e . F r a y e r , W. E. 1966. Weighted r e g r e s s i o n i n s u c c e s s i v e f o r e s t i n v e n t o r -i e s . F o r e s t Sc. 12: 464-472. F r a y e r , W. E., van Aken, C , and S u l l i v a n , R. D. 1971. A p p l i c a t i o n of s a m p l i n g w i t h p a r t i a l r e p l a c e m e n t t o t i m b e r i n v e n t o r i e s , C e n t r a l Rocky M o u n t a i n s . F o r e s t S c i . , 1 7 ( 2 ) : 160-162. F r a y e r , W. E., and F u r n i v a l , G. M. 1967. A r e a change e s t i m a t e s from s a m p l i n g w i t h p a r t i a l r e p l a c e m e n t . F o r e s t S c i . 1 3 ( 1 ) : 72-77. Ghangurde, P. D., and Rao, J . N. K. 1969. Some r e s u l t s on s a m p l i n g o v e r two o c c a s i o n s . Sankya ( S e r . A ) , 31:463-472. Graham, J . E. 1973. Composite e s t i m a t i o n i n two c y c l e r o t a t i o n sample d e s i g n s . Comm. i n S t a t . , 1 ( 5 ) : 419-431. H a z a r d , J . W. 1969. O p t i m a l Replacement s t r a t e g y f o r s u c c e s s i v e f o r e s t s u r v e y s w i t h m u l t i p l e o b j e c t i v e s . Ph.D. t h e s i s , Iowa S t a t e U n i v e r s i t y . H a z a r d , J . W. 1977. E s t i m a t i n g a r e a i n s a m p l i n g f o r e s t p o p u l a t i o n s on two s u c c e s s i v e o c c a s i o n s . F o r e s t S c i . 2 3 ( 2 ) : 253-267. H a z a r d , J . W., and P r o m i t z , L. C. 1974. D e s i g n of s u c c e s s i v e f o r e s t i n v e n t o r i e s : o p t i m i z a t i o n by convex m a t h e m a t i c a l programming. F o r e s t S c i . 2 0 ( 2 ) : 117-127. J e s s e n , R. J . 1942. S t a t i s t i c a l i n v e s t i g a t i o n s o f a sample s u r v e y f o r o b t a i n i n g farm f a c t s . Iowa A g r i c . Exp. S t a . Res. B u l l 304. 104 pp. J o n e s , R. G. 1979. The e f f i c i e n c y of time s e r i e s e s t i m a t o r s f o r r e p e a t e d s u r v e y s . A u s t r a l . J . S t a t . , 2 1 ( 1 ) : 45-56. 103 K i s h , L. 1965. Survey s a m p l i n g . John W i l e y and Sons, I n c . , New York. K u l l d o r f f , G. 1963. Some problems o f optimum a l l o c a t i o n f o r s a m p l i n g on two o c c a s i o n s . Rev. I n t . S t a t . I n s t . , 31:24-57. L a n g l e y , P. G. 1975. M u l t i s t a g e v a r i a b l e p r o b a b i l i t y s a m p l i n g : t h e o r y and use i n e s t i m a t i n g t i m b e r r e s o u r c e s from space a i r c r a f t p h o t o -graphy. Ph.D. t h e s i s , U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y . L a n g l e y , P. G. 1976. S ampling methods u s e f u l t o f o r e s t i n v e n t o r y when u s i n g - d a t a from remote s e n s o r s . Paper p r e s e n t e d a t IUFRO XVI W o r ld C o n g r e s s , O s l o , Norway. L o e t s c h , F., and H a l l e r , K. E. 1964. F o r e s t I n v e n t o r y , V o l . I . BLV V e r l a g s g e s e l l s c h a f t , Munchen, B a s e l Wien. M a n o u s s a k i s , E. 1977. Repeated s a m p l i n g w i t h p a r t i a l r e p l a c e m e n t of u n i t s . A n n a l s of S t a t i s t i c s , 5 ( 4 ) : 795-802. Murthy , M. N. 1967. Sampling t h e o r y and methods. C a l c u t t a S t a t i s -t i c a l P u b l i s h i n g House. N a r a i n , R. D. 1953. On the r e c u r r e n c e f o r m u l a i n s a m p l i n g on s u c c e s -s i v e o c c a s i o n s . I n d i a n Soc. A g r i c . S t a t . J . 5: 66-69. N a r a i n , R. D. 1954. The g e n e r a l t h e o r y o f s a m p l i n g on s u c c e s s i v e o c c a s i o n s . B u l l . I n t . S t a t . I n s t . , 3 4 ( 3 ) : 87-89. Nemhauser, G. L. 1966. I n t r o d u c t i o n to dynamic programming. John W i l e y and Sons, I n c . , New Y o r k . Newton, C. M., C u n i a , T., and B i c k f o r d , C. A. 1974. M u l t i v a r i a t e e s t i m a t o r s f o r s a m p l i n g w i t h p a r t i a l r e p l a c e m e n t on two s u c c e s -s i v e o c c a s i o n s . F o r e s t S c i . 2 0 ( 2 ) : 106-116. Onate, B. T. 1960. Development of m u l t i s t a g e d e s i g n s f o r s t a t i s t i c a l s u r v e y s i n the P h i l i p p i n e s . Iowa S t a t e U n i v . S t a t i s t i c a l Lab. M i m e o - m u l t i l i t h S e r i e s NO. 3. P a t h a k , P. K., and Rao, T. J . 1967. I n a d m i s s i b i l i t y o f customary e s t i m a t o r s i n s a m p l i n g over two o c c a s i o n s . Sankya ( S e r . A ) , 29: 49-54. P a t t e r s o n , H. D. 1950. S ampling on s u c c e s s i v e o c c a s i o n s w i t h p a r t i a l r e p l a c e m e n t of u n i t s . J . R o y a l S t a t . S o c , Ser B, 12:241-255. P n e v m a t i c o s , S. M., and Mann, S. H. 1972. Dynamic programming i n t r e e b u c k i n g . F o r e s t P r o d u c t s J . , 2 2 ( 2 ) : 26-29. R a j , Des. 1965. On s a m p l i n g o v e r two o c c a s i o n s 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 t o s i z e . Ann. Math. S t a t i s t i c s , 36: 327-330. 104 Rana, D. S. 1978. R a t i o method of e s t i m a t i o n i n m u l t i - s t a g e s u c c e s -s i v e s a m p l i n g on two o c c a s i o n s . Amer. S t a t . A s s o c . , P r o c S o c i a l S t a t . Sec.: 289-291. Rana, D. S., and C h a k r a b a r t y , R. P. 1976. T h r e e - s t a g e s a m p l i n g on s u c c e s s i v e o c c a s i o n s . Amer. S t a t . A s s o c . , P r o c . S o c i a l S t a t . S e c : 700-704. Rao, J . N. K., and Graham, J . E. 1964. R o t a t i o n d e s i g n s f o r s a m p l i n g on r e p e a t e d o c c a s i o n s . A m e r i c a n S t a t . A s s o c . J . 59: 492-509. Rao, J . N. K., H a r t l e y , H. 0., and Coc h r a n , W. G. 1962. On a s i m p l e p r o c e d u r e of unequal p r o b a b i l i t y s a m p l i n g w i t h o u t r e p l a c e m e n t . J . R o y a l S t a t . S o c B, 24: 482-491. S c o t t , A. J . , S m i t h , T. M. F., and J o n e s , R. G. 1977. The a p p l i c a t i o n o f time s e r i e s methods t o the a n a l y s i s o f r e p e a t e d s u r v e y s . I n t . S t a t . Rev., 45: 13-28. S c o t t , A. J . , and S m i t h , T. M. F. 1974. A n a l y s i s o f r e p e a t e d s u r v e y s u s i n g time s e r i e s methods. J . Amer. S t a t . A s s o c , 6 9 ( 3 4 7 ) : 674-678. See, T. E. 1974. F o r e s t s a m p l i n g on two o c c a s i o n s w i t h p a r t i a l r e p l a c e -ment of sample u n i t s . M .Sc t h e s i s , F a c u l t y o f F o r e s t r y , U n i v e r -s i t y of B r i t i s h C o l u m b i a , Vancouver, Canada. Sen, A. R. 1971a. I n c r e a s e d p r e c i s i o n i n Canadian w a t e r f o w l h a r v e s t s u r v e y t h r o u g h s u c c e s s i v e s a m p l i n g . J . W i l d l i f e Managt. 35: 664-668. Sen. A. R. 1971b. S u c c e s s i v e s a m p l i n g w i t h two a u x i l i a r y v a r i a b l e s . Sankya B, 33: 371-378. Sen, A. R. 1972. S u c c e s s i v e s a m p l i n g w i t h p ( p > l ) a u x i l i a r y v a r i a b l e s . A n n a l s Math. S t a t . , 4 3 ( 6 ) : 2031-2034. Sen, A. R. 1973a. Some t h e o r y o f s a m p l i n g on s u c c e s s i v e o c c a s i o n s . A u s t r a l . J . S t a t i s t i c s , 1 5 ( 2 ) : 105-110. Sen, A. R. 1973b. Theory and a p p l i c a t i o n of s a m p l i n g on r e p e a t e d o c c a -s i o n s w i t h s e v e r a l a u x i l i a r y v a r i a b l e s . B i o m e t r i c s , 2 9 ( 2 ) : 381-385. Sen, A. R., S e l l e r s , S., and S m i t h , G. E. J . 1975. The use of a r a t i o e s t i m a t e i n s u c c e s s i v e s a m p l i n g . B i o m e t r i c s , 31: 673-683. S i n g h , D. 1968. E s t i m a t e s i n s u c c e s s i v e s a m p l i n g u s i n g a m u l t i - s t a g e d e s i g n . A merican S t a t . A s s o c . J . 6 3 ( 3 2 1 ) : 99-112. S i n g h , D., and S i n g h , B. D. 1965. Double s a m p l i n g f o r s t r a t i f i c a t i o n on s u c c e s s i v e o c c a s i o n s . American S t a t . A s s o c . J . 6 0 ( 3 1 1 ) : 784-792. 105 S i n g h , D. , and K a t h u r i a , 0. P. 1969. On two-stage s u c c e s s i v e s a m p l i n g . A u s t r a l . J . S t a t . , 1 1 ( 2 ) : 59-66. S i n g h , R. 1972. A n o t e on s a m p l i n g over two o c c a s i o n s . A u s t r a l i a n J . S t a t i s t i c s , 1 4 ( 2 ) : 120-122. Sukhatme, P. V., and Sukhatme, B. V. 1970. S a m p l i n g t h e o r y o f s u r v e y s w i t h a p p l i c a t i o n s . A s i a P u b l i s h i n g House, New D e l h i . Second ed. T i k k i w a l , B. D. 1951. Theory of s u c c e s s i v e s a m p l i n g . U n p u b l i s h e d manu-s c r i p t , I.C.A.R., New D e l h i . T i k k i w a l , B. D. 1953. Optimum a l l o c a t i o n i n s u c c e s s i v e s a m p l i n g . I n d i a n Soc. Agr. S t a t . J . 5: 100-102. T i k k i w a l , B. D. 1955. M u l t i - p h a s e s a m p l i n g on s u c c e s s i v e o c c a s i o n s . Ph.D. t h e s i s , N o r t h C a r o l i n a S t a t e C o l l e g e , R a l e i g h , N.C. T i k k i w a l , B. D. 1956a. A f u r t h e r c o n t r i b u t i o n t o the t h e o r y of u n i v a r -i a t e s a m p l i n g on s u c c e s s i v e o c c a s i o n s . I n d i a n Soc. A g r . S t a t . J . 8: 84-90. T i k k i w a l , B. D. 1956b. An a p p l i c a t i o n of the t h e o r y o f m u l t i p h a s e sampl-i n g on s u c c e s s i v e o c c a s i o n s t o s u r v e y s of l i v e s t o c k m a r k e t i n g . J . K a r n a t a k U n i v . 1: 120-130. T i k k i w a l , B. D. 1958a. An e x a m i n a t i o n of the e f f e c t of matched s a m p l i n g on t h e e f f i c i e n c y of e s t i m a t o r s i n the t h e o r y of s u c c e s s i v e sampl-i n g . I n d i a n Soc. Agr. S t a t . 10: 16-22. T i k k i w a l , B. D. 1958b. Theory of s u c c e s s i v e two-stage s a m p l i n g . A b s t . i n Ann. Math. S t a t . 29: 1291. T i k k i w a l , B. D. 1967. Theory of m u l t i p h a s e s a m p l i n g from a f i n i t e o r an i n f i n i t e p o p u l a t i o n on s u c c e s s i v e o c c a s i o n s . Review I n t . S t a t . I n s t . , 3 5 ( 3 ) : 247-263. Ware, K. D. I960. Optimum r e g r e s s i o n s a m p l i n g d e s i g n f o r s a m p l i n g of f o r e s t p o p u l a t i o n s on s u c c e s s i v e o c c a s i o n s . Ph.D. t h e s i s , Y a l e U n i v e r s i t y . Ware, K. D., and C u n i a , T. 1962. C o n t i n u o u s f o r e s t i n v e n t o r y w i t h p a r t i a l r e p l a c e m e n t of samples. F o r e s t S c i . Monograph No. 3. W i l d e , D. J . , and B e i g h t l e r , C. S. 1967. F o u n d a t i o n s of o p t i m i z a t i o n . P r e n t i c e - H a l l , I n c . , New J e r s e y . W o odruff, R. S. 1959. The use of r o t a t i n g samples i n the census bureau's monthly s u r v e y s . Amer. S t a t . A s s o c . P r o c . Soc. S t a t . S e c , 130-138. Y a t e s , F. 1949. S ampling methods f o r c e n s u s e s and s u r v e y s . Chas. G r i f f i n and Co., London. APPENDIX I 106 S A M P L E P R O B L E M D A T A P S U I D S S U I D W H O L E - S T E M V O L U M E , M 3 / H A X ( 1 9 6 4 ) Y ( 1 9 7 9 ) 1 8 - 8 1 3 2 7 5 . 8 — 1 8 - 8 1 4 8 1 . 4 — 1 8 - 8 1 5 1 9 9 . 0 — 1 8 - 8 1 6 1 4 6 . 0 — 1 8 - 8 1 7 1 5 5 . 8 — 1 8 - 8 1 8 1 4 3 . 3 — 1 8 - 8 1 9 1 7 7 . 1 — 1 8 - 1 0 5 1 1 2 . 3 — 1 8 - 1 9 5 8 2 . 9 — 1 8 - 2 1 3 2 4 0 . 1 — 1 8 - 2 1 4 2 0 6 . 9 — 1 8 - 2 5 4 1 3 0 3 . 1 • — 1 8 - 2 5 4 2 3 4 3 . 5 — 1 8 - 2 5 4 3 4 3 2 . 5 — 1 8 - 2 5 4 4 2 9 7 . 7 - -1 8 - 2 5 4 5 2 6 7 . 2 — 1 8 - 2 5 4 6 3 2 2 . 8 — 1 8 - 2 5 4 7 3 7 4 . 5 — 1 8 - 2 5 4 8 3 3 0 . 6 — 1 8 - 2 5 4 9 3 2 1 . 2 — 1 8 - 2 5 5 0 3 4 4 . 7 — -1 8 - 2 5 5 1 4 6 6 . 4 — 1 8 - 2 5 5 2 3 9 5 . 0 — 1 8 - 2 5 5 3 3 4 4 . 4 — 1 8 - 2 5 5 4 4 7 1 . 1 1 8 - 2 5 5 5 3 5 5 . 4 — 1 8 - 2 5 5 6 3 4 3 . 8 — 1 8 - 2 5 5 7 3 5 5 . 1 — 1 8 - 2 5 5 8 4 1 1 . 2 — 1 8 - 2 5 5 9 3 9 1 . 7 — 1 8 - 2 5 6 0 5 4 9 . 8 — 1 8 - 2 5 6 1 5 2 7 . 2 — 1 8 - 2 5 6 2 4 2 0 . 5 — 1 8 - 2 5 6 3 5 2 6 . 4 — 1 8 - 2 5 6 4 2 6 7 . 4 — 1 9 - 7 2 3 2 5 . 1 — 1 9 - 7 3 1 4 2 . 0 — 1 9 - 7 4 3 9 8 . 1 — 1 9 - 7 5 5 0 3 . 3 — 1 9 - 7 6 7 0 4 . 1 — 1 9 - 7 7 2 5 4 . 9 — 1 9 - 7 8 7 4 8 . 3 — 1 9 - 7 9 7 1 5 . 5 — 1 9 - 1 5 1 4 9 0 . 4 — 1 9 - 1 5 2 7 1 4 . 9 — 1 9 - 1 5 4 6 3 2 . 9 — 1 9 - 1 5 5 4 5 4 . 7 — 1 9 - 1 5 6 1 9 4 . 1 — N O T E : T H E V O L U M E S A R E F O R L I V I N G T R E E S O N L Y , D B H 1 7 . 5 CM+ 107 S A M P L E P R O B L E M D A T A P S U I D S S U I D W H O L E - S T E M V O L U M E , M X ( 1 9 6 4 ) Y (197S 19 - 15 7 4 9 2 . 8 — 19 - 15 8 3 7 8 . 3 — 19 - 15 9 5 7 4 . 6 — 19 - 15 10 4 0 7 . 5 — 19 - 15 11 7 6 4 . 5 --19 - 15 12 4 3 6 . 3 — 19 - 15 13 5 8 3 . 6 — 19 - 15 14 6 4 0 . 3 19 - 15 15 6 4 4 . 5 21 - 5 43 5 2 2 . 9 — 21 - 5 44 5 0 4 . 2 — 21 - 5 45 5 1 0 . 1 21 - •5 46 5 6 5 . 8 — 21 - 5 47 3 1 6 . 0 --21 - 5 48 4 8 4 . 2 --21 - 5 49 4 8 1 . 5 — 21 - 5 50 2 8 6 . 2 — 21 - 5 51 3 1 3 . 0 — 21 - 5 52 5 4 0 . 2 — 21 - 7 64 4 4 8 . 0 --21 - 7 65 1 3 1 . 3 — 21 - 7 66 7 4 2 . 7 --21 - 7 67 4 7 3 . 8 • 21 - 7 68 4 3 7 . 8 — • 21 - 7 69 -475.1 21 - 7 70 2 5 3 . 0 — 21 - 7 72 5 2 5 . 9 — 21 - 7 73 3 6 5 . 3 — 21 - 7. 74 6 2 5 . 8 — 21 - 7 75 3 2 8 . 6 — 21 - 7 76 4 8 0 . 4 — 21 - 7 77 3 5 2 . 3 --21 - 7 78 3 5 9 . 8 21 - 7 79 3 3 0 . 5 — 21 - 7 80 4 6 5 . 1 — 21 - 7 81 4 0 1 . 8 21 - 7 • 82 4 4 9 . 4 — 21 - 7 83 5 5 1 . 1 — 21 - 7 84 4 9 4 . 0 — 18 - 6 58 2 9 1 . 4 2 7 8 . 8 18 - 6 59 4 7 9 . 2 4 0 1 . 2 18 - 6 60 4 0 9 . 7 4 6 8 . 3 18 - 6 61 4 2 8 . 3 3 9 4 . 3 .18 - 6 62 4 3 1 . 4 4 4 2 . 7 18 - 6 63 3 4 0 . 5 3 1 4 . 4 18 - 6 64 7 7 3 . 2 7 3 2 . 5 18 - 6 65 5 7 2 . 7 5 3 7 . 6 18 - 6 66 5 6 2 . 0 5 7 9 . 2 N O T E : T H E V O L U M E S A R E F O R L I V I N G T R E E S O N L Y , D B H 1 7 . 5 CM+ 108 SAMPLE PROBLEM DATA PSU ID SSU ID WHOLE-STEM VOLUME, M 3 / ' X ( 1 9 6 4 ) Y ( 1 9 7 9 ) 18 - 7 32 ' 2 7 3 . 7 3 2 9 . 5 18 - 7 33 6 1 4 . 0 5 9 8 . 7 18 - 7 34 1 9 5 . 2 1 3 3 . 4 18 - 7 35 3 6 8 . 7 3 3 3 . 2 18 - 7 36 1 2 0 . 1 7 4 . 7 18 - 7 37 4 2 4 . 3 4 2 4 . 8 18 - 7 38 3 1 4 . 6 3 6 6 . 6 18 - 7 39 3 1 7 . 0 3 2 6 . 1 18 - 7 40 1 7 5 . 0 1 9 8 . 4 18 - 7 41 4 3 0 . 6 4 1 4 . 1 18 - 15 31 6 5 0 . 6 6 6 2 . 4 18 - 17 45 6 1 0 . 2 6 7 5 . 5 18 - 17 46 1 .0 1 6 . 0 18 - 17 47 5 2 7 . 9 5 4 6 . 6 18 - 17 48 4 2 9 . 1 3 4 5 . 1 18 - 17 49 4 3 9 . 8 4 7 4 . 3 18 - 17 50 3 9 2 . 1 2 4 2 . 1 18 - 17 51 7 4 0 . 5 7 4 1 . 2 18 - 17 52 6 2 1 . 8 5 6 7 . 2 18 - 17 53 3 7 4 . 6 4 0 1 . 4 18 - 17 54 2 4 8 . 3 2 7 8 . 9 18 - 17 55 2 1 3 . 9 1 9 7 . 8 18 - 17 56 4 9 8 . 5 ' 5 1 7 . 1 18 - 17 57 5 2 7 . 2 5 5 2 . 8 18 - 17 58 9 5 5 . 5 4 3 . 9 18 - 20 6 1 0 7 . 2 7 4 . 4 18 - 20 7 1 1 4 . 2 1 0 8 . 2 18 - 20 8 1 2 9 . 0 1 0 5 . 5 18 - 22 2 2 0 1 . 6 2 4 8 . 4 18 - 22 3 3 7 0 . 8 3 7 9 . 2 18 - 22 4 5 5 4 . 5 6 2 9 . 7 18 - 22 5 1 8 3 . 8 2 6 1 . 8 18 - 22 6 4 7 3 . 6 4 6 8 . 9 18 - 23 70 3 3 1 . 5 4 0 4 . 8 18 - 23 •71 4 0 6 . 1 4 9 5 . 3 18 - 23 72 3 8 6 . 2 3 6 4 . 6 18 - 23 73 4 2 3 . 8 4 5 1 . 5 18 - 23 74 6 1 9 . 1 6 6 3 . 4 18 - 23 75 4 4 2 . 0 4 5 8 . 7 18 - 23 76 4 5 3 . 7 4 9 7 . 0 18 - 23 77 3 2 8 . 0 3 7 6 . 3 18 - 23 78 5 3 6 . 0 5 7 8 . 1 18 - 24 15 4 8 6 . 9 4 3 3 . 1 18 - 24 16 4 0 3 . 4 3 5 0 . 2 18 - 24 17 2 1 4 . 3 2 3 6 . 4 18 - 24 18 3 6 2 . 2 3 4 3 . 2 18 - 24 19 3 5 4 . 2 3 2 4 . 5 18 - 24 20 3 1 8 . 9 3 5 0 . 5 NOTE : THE VOLUMES ARE FOR L I V I N G TREES ONLY , DBH 1 7 . 5 CM+ 109 SAMPLE PROBLEM DATA PSU ID SSU ID WHOLE-STEM VOLUME, M 3/HA X ( 1 9 6 4 ) Y ( 1 9 7 9 ) 18 - 24 21 4 6 1 . 9 4 6 6 . 7 18 - 24 22 4 2 6 . 0 4 1 5 . 3 18 - 24 23 3 7 8 . 7 3 2 7 . 9 18 - 24 24 2 9 3 . 8 2 6 6 . 5 18 - 24 25 3 8 7 . 3 4 6 4 . 5 18 - 24 26 3 6 7 . 7 3 6 9 . 8 18 - 24 27 3 1 2 . 7 3 1 1 . 3 18 - 24 32 1 5 0 . 2 1 5 5 . 7 18 - 24 33 4 7 1 . 9 5 3 1 . 1 18 - 26 4 1 5 3 . 3 1 5 3 . 2 18 - 31 10 5 0 5 . 0 4 4 4 . 8 18 - 31 11 3 8 1 . 1 3 4 0 . 2 18 - 31 12 3 1 2 . 6 3 1 2 . 7 18 - .31 13 3 7 8 . 6 3 6 9 . 7 18 - 31 16 3 5 4 . 5 3 8 9 . 6 18 - 31 17 5 4 4 . 0 5 7 1 . 7 18 - 31 18 1 3 9 . 2 1 5 7 . 2 18 - 31 19 5 7 8 . 5 5 7 9 . 7 18 - 31 20 5 1 5 . 3 5 8 2 . 6 18 - 31 21 5 9 3 . 4 6 0 0 . 8 18 - 31 22 6 3 8 . 8 6 4 6 . 8 18 - 31 23 5 1 4 . 1 5 5 9 . 6 18 - 31 24 . 4 8 2 . 7 4 8 8 . 9 18 - 31 25 3 0 1 . 2 2 8 5 . 1 •18 - 31 26 4 1 4 . 7 34 '4 .3 18 - 31 27 3 7 6 . 9 3 6 8 . 7 18 - 31 28 3 0 4 . 7 3 1 0 . 9 18 - 31 29 3 9 1 . 0 3 3 2 . 0 18 - 31 30 3 3 9 . 8 4 4 9 . 3 18 - 31 31 3 0 7 . 6 3 3 3 . 0 18 - 33 24 3 1 5 . 2 3 8 0 . 5 18 - 33 25 2 8 2 . 5 • 3 3 0 . 6 18 - 33 26 4 2 8 . 7 5 1 8 . 4 18 - 33 27 5 6 . 0 5 4 . 0 18 - 33 28 2 2 3 . 6 1 9 9 . 3 18 - 34 20 5 0 9 . 0 4 4 6 . 6 18 - 34 21 4 8 4 . 9 5 9 7 . 1 18 - 34 22 5 3 9 . 7 6 4 0 . 7 18 - 34 23 3 8 5 . 3 3 6 4 . 7 18 - 34 26 3 8 7 . 2 3 8 4 . 0 18 - 34 27 5 8 7 . 3 5 8 4 . 2 18 - 34 28 5 1 0 . 0 4 9 1 . 8 18 - 34 29 5 5 4 . 5 5 2 6 . 2 18 - 34 32 5 0 8 . 9 4 9 9 . 3 18 - 34- 33 2 4 9 . 3 2 0 4 . 5 18 - 34 34 5 6 2 . 8 6 2 2 . 2 18 - 34 35 3 8 3 . 1 3 8 4 . 2 18 - 34 36 3 2 5 . 0 3 7 5 . 9 NOTE : THE VOLUMES ARE FOR L I V I N G TREES ONLY , DBH 1 7 . 5 CM+ 110 SAMPLE PROBLEM DATA . PSU ID SSU ID WHOLE-STEM VOLUME, M 3 / H A X ( 1 9 6 4 ) Y ( 1 9 7 9 ) 18 - 34 37 5 7 1 . 2 6 7 7 . 7 18 - 34 38 6 3 0 . 2 6 3 4 . 3 18 - 34 39 2 6 8 . 6 2 5 0 . 3 18 - 34 40 444 . 5 4 8 7 . 2 18 - 34 41 8 7 5 . 8 8 8 1 . 4 18 - 34 42 3 8 5 . 5 4 1 8 . 0 18 - 34 43 4 4 9 . 0 4 5 4 . 2 18 - 34 44 6 0 0 . 3 5 9 9 . 8 18 - 34 45 5 4 6 . 3 5 3 5 . 2 18 - 34 46 4 4 9 . 4 4 5 3 . 5 18 - 34 47 5 8 4 . 3 5 9 8 . 8 18 - 34 48 3 3 1 . 3 3 1 4 . 7 18 - 34 49 5 0 7 . 7 4 8 8 . 7 18 - 34 50 1 8 7 . 4 1 5 3 . 4 18 - 34 51 4 4 9 . 9 5 7 0 . 0 18 - 34 52 7 1 1 . 9 7 3 6 . 5 18 - 34 53 4 6 3 . 3 4 9 8 . 4 18 - 34 54 5 6 1 . 0 5 9 7 . 0 18 - 34 55 2 4 6 . 6 2 7 1 . 8 18 - 35 6 4 9 1 . 1 4 5 4 . 6 18 - 35 7 3 8 7 . 3 4 0 8 . 6 18 - 35 8 3 2 5 . 6 3 1 0 . 0 18 - 3 5 ' 11 3 5 8 . 1 3 8 7 . 6 18 - 35 12 ' 2 5 4 . 8 3 0 3 . 3 18 - 35 - 13 3 8 1 . 8 " 3 3 6 . 3 18 - 35 22 2 6 2 . 2 2 5 8 . 3 18 - 35 23 6 0 0 . 1 7 0 0 . 8 18 - 35 23 4 1 3 . 9 4 5 8 . 1 18 - 35 24 2 3 2 . 9 2 1 8 . 5 18 - 35 25 3 4 6 . 8 3 3 4 . 8 18 - 35 26 4 8 5 . 6 5 3 0 . 1 18 - 35 27 4 6 9 . 8 4 5 7 . 6 18 - 35 29 3 5 9 . 5 4 0 2 . 1 18 - 35 30 3 8 8 . 7 3 6 5 . 5 18 - 35 31 5 2 7 . 8 4 7 4 . 2 18 - 35 32 3 9 3 . 0 3 6 2 . 1 18 - 35 33 6 0 0 . 1 6 6 0 . 7 18 - 35 34 7 1 6 . 0 6 8 4 . 9 18 - 35 35 8 6 0 . 6 7 9 7 . 9 18 - 36 7 4 1 9 . 5 3 4 3 . 1 18 - 36 8 5 0 0 . 0 4 9 4 . 9 18 36 9 3 3 9 . 0 3 5 5 . 5 18 - 36 10 4 8 8 . 7 3 9 2 . 9 18 - 36 11 5 5 3 . 1 5 3 9 . 5 18 - 36 12 2 7 1 . 9 26-6.6 18 - 36 13 7 8 2 . 0 8 1 4 . 9 18 - 36 14 2 6 2 . 9 3 5 0 . 3 18 - 36 15 7 5 3 . 6 7 7 0 . 1 NOTE : THE VOLUMES ARE FOR L I V I N G TREES ONLY , DBH 1 7 . 5 CM+ I l l S A M P L E P R O B L E M D A T A P S U I D S S U I D W H O L E - S T E M V O L U M E , M 3 / H A X ( 1 9 6 4 ) Y ( 1 9 7 9 ) 18 - 36 16 4 5 1 . 0 4 7 5 . 7 18 - 36 17 3 5 9 . 3 3 3 3 . 8 18 - 36 18 1 7 3 . 8 2 0 2 . 7 18 - 36 . 20 6 5 7 . 2 7 7 3 . 1 18 - 37 17 2 5 . 9 6 1 . 7 18 - 37 19 6 9 5 . 9 7 8 5 . 5 18 - 37 20 5 6 4 . 8 6 0 1 . 2 18 - 37 23 7 2 2 . 4 6 6 5 . 2 18 - 37 27 4 7 9 . 9 4 4 6 . 9 18 - 37 29 3 1 3 . 1 2 6 0 . 5 18 - 37 30 3 6 5 . 6 3 3 5 . 4 18 - 37 32 8 1 6 . 9 8 7 2 . 0 18 - 37 33 8 0 . 1 1 1 6 . 3 18 - 37 34 6 9 6 . 7 6 4 4 . 2 18 - 37 35 5 1 . 8 3 9 . 5 18 - 37 36 8 9 6 . 1 8 6 4 . 5 18 - 37 37 7 9 4 . 2 8 2 0 . 1 18 - 37 38 3 4 6 . 2 2 8 8 . 4 18 - 37 39 4 0 6 . 5 3 3 4 . 1 18 - 37 40 7 7 . 4 1 2 4 . 8 18 - 37 41 1 6 5 . 6 8 9 . 3 18 - 37 42 5 9 9 . 5 6 3 6 . 7 18 - 37 43 3 2 7 . 6 2 8 4 . 2 19 - 8 '5 3 3 0 . 3 3 5 6 . 8 19 - 8 6 2 7 0 . 4 3 1 1 . 4 21 - 1 12 3 6 4 . 0 4 0 5 . 5 21 - 1 13 4 2 3 . 9 3 6 6 . 9 21 - 1 14 2 1 8 . 4 3 1 9 . 3 21 - 1 15 2 7 3 . 2 2 9 3 . 9 21 - 1 16 1 1 7 . 8 1 0 8 . 5 21 - 1 17 2 2 5 . 9 2 0 8 . 0 21 - 1 18 3 0 0 . 5 3 3 0 . 3 21 - 1 19 4 0 4 . 0 4 0 3 . 2 21 - 2 6 7 4 . 0 1 9 0 . 1 21 - 2 7 1 1 7 . 0 l 1 5 8 . 1 21 - 2 8 4 9 3 . 8 5 6 5 . 2 21 - 2 9 5 1 1 . 4 5 3 5 . 0 21 - 2 10 4 3 9 . 8 3 9 5 . 0 21 - 2 11 4 3 2 . 1 4 0 1 . 8 21 - 2 12 5 3 1 . 6 5 4 5 . 5 2.1 - 2 13 2 3 9 . 4 2 3 9 . 8 21 - 2 14 2 2 2 . 8 1 3 2 . 9 21 - 2 15 * 9 2 . 4 9 0 . 0 21 - 2 16 6 3 7 . 3 6 8 0 . 1 21 - 2 17 6 4 7 . 8 6 3 2 . 3 21 - 2 18 5 0 4 . 7 4 4 3 . 1 21 - 2 19 4 4 4 . 7 4 0 1 . 9 21 - 2 20 5 6 9 . 0 6 3 8 . 4 N O T E : T H E V O L U M E S A R E F O R L I V I N G T R E E S O N L Y , D B H 1 7 . 5 CM+ 112 SAMPLE PROBLEM DATA PSU ID SSU ID WHOLE-STEM VOLUME, M3/HA X ( 1 9 6 4 ) Y ( 1 9 7 9 ) 21 - 2 21 1 9 9 . 3 2 1 3 . 4 21 - 2 22 2 4 0 . 6 2 8 2 . 8 21 - 2 23 4 1 6 . 8 3 7 6 . 2 21 - 2 24 6 7 7 . 1 6 6 1 . 1 21 - 2 25 4 9 0 . 5 4 9 8 . 1 21 - 2 26 403 . 0 4 0 8 . 6 21 - 2 27 4 6 9 . 3 5 0 8 . 9 21 - 2 28 6 0 3 . 2 5 7 9 . 0 21 - 2 29 1 5 1 . 3 1 1 1 . 3 21 - 2 30 3 7 1 . 2 3 5 7 . 0 18 - 18 8 -- 1 3 8 . 4 18 - 18 9 -- • 1 1 0 . 3 18 - 19 6 — 7 7 . 3 18 - 19 7 -- 1 1 9 . 3 18 - 19 8 -- 6 6 . 4 18 - 19 9 -- 1 1 2 . 4 18 - 19 10 — 1 6 9 . 1 18 - 19 11 — 1 2 7 . 6 18 - 28 • 1 -- 2 . 5 18 - 28 2 — 7 8 . 4 18 - 28 3 — 10.3.7 18 - 28 4 — 6 . 4 18 - 28 5 — 6 0 . 2 18 - 28 6 4 7 . 9 18 - 30 1 — 9 6 . 4 18 - 30 2 — 0 . 0 18 - 30 3 — 9 1 . 6 18 - 30 5 — 1 3 4 . 9 18 - 30 6 1 0 2 . 3 18 - 30 7 — 6 1 . 4 18 - 32 39 — 0 . 0 18 - 32 40 — 4 . 1 18 - 38 17 — 3 8 1 . 9 18 - 39 13 — 1 2 8 . 3 18 - 39 14 — 7 7 . 0 18 - 39 15 — 8 5 . 7 18 - 40 10 -- 3 4 1 . 9 18 - 40 11 — 2 6 2 . 8 18 - 40 12 5 9 7 . 3 19 - 5 17 -- 7 0 . 5 19 - 5 18 — 1 5 2 . 9 19 - 5 19 — 1 3 2 . 5 19 - 5 20 — 1 6 8 . 9 19 - 5 21 — 1 3 3 . 4 19 - 5 22 — 2 0 0 . 1 19 - 5 23 — 1 4 6 . 5 19 - 8 7 — 5 .6 19 - 8 8 — 4 . 3 NOTE: THE VOLUMES ARE FOR LIVING TREES ONLY, DBH 17.5 CM+ 113 SAMPLE PROBLEM DATA PSU ID SSU ID WHOLE-STEM VOLUME, M3/HA X (1964) Y (1979) 19 - 8 9 -- 65.3 19 - 8 10 -- 101.1 19 -• 8 11 — 67.1 19 - 10 5 -- 45.2 19 - 10 6 — 11.9 19 - 12 29 — 506.7 19 - 12 30 — 429.5 19 - 15 17 0.0 19 - 15 18 -- 277.5 19 - 15 19 — 15.6 19 - 15 20 -- 499.2 19 - 15 21 -- 92.8 19 - 15 22 • — 88.4 19 - 17 1 461.3 19 - 17 2 — 3.2 19 - 17 3 — 20.7 19 - 17 4 -- 349.8 19 - 17 5 -- 386.4 21 - 3 3 -- 71.2 21 - 3 4 -- 26.8 21 - 3 5 -- 41.7 21 - 3 6 -- 68.1 x 21 - 3 7 — 4.6 21 - 3 8 -- 28.4 21 - 3 9 54.5 21 - 3 10 — 20.9 21 - 6 1 -- 87.2 21 - 6 2 -- 75.2 21 - 6 3 -- 53.6 21 - 6 5 25.3 21 - 6 6 — 8.5 21 - 6 7 — 90.1 21 - 6 8 191.4 NOTE: THE VOLUMES ARE FOR LIVING TREES ONLY, DBH 17.5 CM+ APPENDIX II SIMULATION OF REMEASUREMENT DATA 114 Remeasurement data were simulated using the Volume-Age curves (VACs) fitte d and currently used by the BCFS. The curves, Chapman-Richards generalization of the Von Bertalanffy's growth function, take the form -b.CA-b.) b_ V = b ^ l - e 2 4 ] 3 where 3 V = stand volume in m A = stand age in 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 in British 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 = V +. (V - V ) + £ r n p c where V = volume per ha at- the plot in 1979 as estimated using the appropriate n VAC (with A = A^ '= (stand age at the plot in 1964) + 15) V = volume per ha at the plot as actually measured in 1964 P ' V = volume per ha at the plot in 1964 as estimated using the appropriate c VAC ( w i t h 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 and variance.1927•21 (= estimated variance between P plot volumes). 115 The calculation is shown graphically as below Volume/ha (m3) Age (years) The plot volume was allowed to grow at the rate dictated by the VAC, and the random element £ accounted for natural disasters, such as windfall, etc. For example, plot number 18-015-31 in good site in growth type I in forest inventory zone F: V = 650.6 m3/ha V P = 479.7227 [ 1 - 2 . 7 1 8 2 8 " 0 ' 0 3 5 7 ( 1 7 ° - 0 ) ] 8 ' 6 5 8 3 c = 470.2 m3/ha V = 479.7227 [1-2.71828"0'0357(185-0) 8.6583 n = ^7^.1 m3/ha and V r = 474.1 + (650.6 - 470.2) + 7.7 662.3 m /ha The simulated results were in agreement with the results obtained from a remeasurement pilot study in 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 plots, and actually remeasured (according to the 1964 standards) during the summer of 1980. The correlation between the 1980 measurements and the 1964 measurements and that between the 1979 simulated measurements and the 1964 measurements were not significantly different from each other. The results of the pilot study and the derived st a t i s t i c s are summarized below: A. Data Plot ID 18-34-35 18-35-34 18-35-35 18-36-10 18-36-11 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 18-31-31 ' 314.6 307.6 18-33-24 390.7 315.2 18-34-28 476.3 510.0 315.5 383.1 684.2 716.0 877.4 860.6 562.6 488.7 596.2 553.1 117 Plot ID Whole stem Volume (living trees only, dbh 17.5cm+) 1980 1964 18-37-34 21-01-14 21-01-15 21-01-18 21-02-06 21-02-13 21-07-74 692.0 249.3 241.1 307.3 106.0 362.4 646.1 696.7 218.4 273.2 300.5 74.0 239.4 625.8 B. Statistics Volume in 1980 Volume in 1964 mean 399.90 375.23 standard deviation 201.98 209.08 Correlation between 1980 and 1964 measurements, £ = 0.97. 

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