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Study of apple yield relationships in 1969 in the Okanagan area of British Columbia Lee, Ewon 1972

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c A STUDY OF APPLE Y I E L D RELATIONSHIPS I N 1969 I N THE OKANAGAN AREA OF B R I T I S H COLUMBIA by EWON LEE B . S c , S e o u l N a t i o n a l U n i v e r s i t y , 1964 A THESIS SUBMITTED I N PAR T I A L FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e D e p a r t m e n t o f A g r i c u l t u r a l E c o n o m i c s We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA S e p t e m b e r , 1972 I n 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 r 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 purposes may be g r a n t e d by the Head o f my Depart-ment or 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 or 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 . Department o f A g r i c u l t u r a l Economics The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date September, 1972 ABSTRACT The purpose o f the s t u d y i s t o dete r m i n e w h i c h f a c t o r s c o n t r i b u t e d t o the p r o d u c t i o n o f apples i n the Okanagan a r e a d u r i n g the y e a r 1969. R e g r e s s i o n a n a l y s i s i s used i n an attempt t o q u a n t i f y y i e l d r e l a t i o n s h i p s . A comparison i s made among d i f f e r e n t t r e e - s i z e c a t e g o r i e s i n o r d e r t o determine whether i t i s n e c e s s a r y t o f i t s e p a r a t e r e g r e s s i o n e q u a t i o n s i n s t e a d o f u s i n g the d a t a f o r the t h r e e groups i n a s i n g l e r e g r e s s i o n e q u a t i o n . For t h i s purpose an E q u a l i t y o f Sl o p e T e s t i s per f o r m e d . The outcome o f the t e s t shows t h a t t h e r e are no s i g n i f i c a n t d i f f e r e n c e s among c o r r e s p o n d i n g c o e f f i c i e n t s i n the e q u a t i o n s f o r t r e e - s i z e c a t e g o r i e s . Hence i t i s f e a s i b l e t o combine them i n t o one e q u a t i o n . For the r e g r e s s i o n a n a l y s i s , two d i f f e r e n t types o f y i e l d r e l a t i o n s h i p s are employed: one i s a Cobb-Douglas f u n c t i o n l i n e a r i n the l o g a r i t h m s and the o t h e r i s a quad-r a t i c f u n c t i o n . B oth f u n c t i o n s i n c l u d e a dependent v a r i a b l e , namely, y i e l d p e r acre and seven independent v a r i a b l e s ; t h a t i s , d e n s i t y , age, v a l u e o f f e r t i l i z e r a p p l i e d , v a l u e o f s p r a y a p p l i e d , p r u n i n g and t h i n n i n g l a b o u r h o u r s , geo-g r a p h i c a l dummy, and t r e e - s i z e i n d e x . These independent v a r i a b l e s are measured on a p e r - a c r e b a s i s e x c e p t i n the case o f age, g e o g r a p h i c a l dummy and t r e e - s i z e i n d e x . The d a t a , w h i c h c o n s i s t s o f c r o s s - s e c t i o n i n f o r m a -i i t i o n f o r 1969 r e p r e s e n t s one hundred and n i n e t e e n sample apple p l o t s . I t was d e r i v e d from p e r s o n a l i n t e r v i e w s w i t h apple growers. The q u a d r a t i c f u n c t i o n poses a problem a r i s i n g from c r o s s - t e r m s i n the e q u a t i o n . I t was n e c e s s a r y t o modify the f u n c t i o n i n such a manner t h a t the c r o s s - t e r m s i n c l u d e d i n the r e g r e s s i o n e q u a t i o n were j u s t i f i e d on b i o -l o g i c a l or economic grounds. The r e g r e s s i o n r e s u l t s f o r each type o f f u n c t i o n used i n the a n a l y s i s are d i s c u s s e d and e s t i m a t e s o f c o e f f i c i e n t s and r e l a t e d s t a n d a r d e r r o r s shown. I t seems d e s i r a b l e t h a t d a t a s h o u l d be br o k e n down i n t o apple v a r i e t y groups because d i f f e r e n t v a r i e t i e s o f apple may w e l l have d i s t i n c t b e a r i n g c h a r a c t e r i s t i c s . A p p l e t r e e s i n the s p e c i f i c p l o t s under s t u d y , however, are made up o f a m i x t u r e o f v a r i e t i e s , thus i t i s e x t r e m e l y d i f f i c u l t t o draw a c l e a r map o f acreages o c c u p i e d by each v a r i e t y . In a t t e m p t i n g t o o b t a i n v a r i e t y d a t a , n o t w i t h s t a n d i n g the m i x t u r e o f v a r i e t i e s i n s t a n d s , the o r i g i n a l d a t a i s broken down under c e r t a i n a s s u m p t i o n s . A l s o i n decomposing apple y i e l d s i n t o grade c o n s t i t u e n t s s i m i l a r problems a r i s e . D e s p i t e t h e s e d i f f i c u l t i e s , t e s t s o f d i f f e r e n c e s among average y i e l d s are made under s t a t e d c o n d i t i o n s f o r v a r i e t a l , t r e e - s i z e , a p p l e - g r a d e , and r e g i o n a l c a t e g o r i e s . These t e s t s r e v e a l t h a t t h e r e are no s i g n i f i c a n t d i f f e r e n c e s i n average apple y i e l d s f o r v a r i e t i e s , apple grades and regions., but t h e r e are s i g n i f i c a n t d i f f e r e n c e s i n t h e case o f d i f f e r e n t t r e e s i z e s . The r e s u l t s o f t h e s e i i i t e s t s are p r e s e n t e d i n C h a p t e r V I . The q u a d r a t i c form o f f u n c t i o n seems, w i t h i n the t h e o r e t i c a l framework, to be a b l e t o r e p r e s e n t s a t i s f a c t o r i l y the a pple y i e l d r e l a t i o n s h i p w i t h the s e l e c t e d independent v a r i a b l e s . B u t , i n p r a c t i c e , i t does not conform w e l l t o the e m p i r i c a l s i t u a t i o n ; i t produces a s e r i o u s m u l t i c o l l i n e -a r i t y p r o b l e m from the p o i n t of view o f s t a t i s t i c a l i n f e r e n c e . The Cobb-Douglas f u n c t i o n , however, does not cause such a problem. A p a r t from t h i s , i t s a p p l i c a t i o n b r o u g h t i n almost a l l the c o e f f i c i e n t s c o r r e s p o n d i n g t o the b a s i c i ndependent v a r i a b l e s e x c e p t f o r the c o e f f i c i e n t o f the t r e e - s i z e i n d e x v a r i a b l e . On t h i s e v i d e n c e , a t e n t a t i v e c o n c l u s i o n was made i n f a v o u r o f the Cobb-Douglas f u n c t i o n f o r the r e p r e s e n t a t i o n o f an apple y i e l d r e l a t i o n s h i p i n the Okanagan i n 1969. i v TABLE OF CONTENTS CHAPTER PAGE I . INTRODUCTION 1 I I . LITERATURE REVIEW OF APPLE BIOLOGY 6 F a c t o r s I n f l u e n c i n g Apple P r o d u c t i o n 6 S i z e o f Tree 7 S o i l C o n d i t i o n 8 Frequency o f F r o s t - i n j u r y 9 U n f a v o u r a b l e C o n d i t i o n at Blossom Time 10 P r u n i n g 12 T h i n n i n g 13 S p r a y i n g 13 D e n s i t y 14 I I I . LITERATURE REVIEW OF STATISTICS 17 IV. DATA 32 C o n d i t i o n s o f Sam p l i n g '. . . 32 S a m p l i n g Method 33 V. EMPIRICAL RESULTS 4 5 I n t r o d u c t i o n 45 R e s u l t s from the Cobb-Douglas Model 54 R e s u l t s from t h e Q u a d r a t i c Model 55 D i s c u s s i o n o f the R e s u l t s from A p p l y i n g Cobb-Douglas and Q u a d r a t i c R e g r e s s i o n A n a l y s e s ... 63 V I . TESTING FOR THE DIFFERENCE BETWEEN TWO MEANS .. 65 I n t r o d u c t i o n 65 Outcome o f t - t e s t 68 D i s c u s s i o n o f t - t e s t 73 V CHAPTER PAGE VII. SUMMARY AND CONCLUSION 76 Summary . 76 Conclusion 78 BIBLIOGRAPHY 81 APPENDIX • • 8 5 v i LIST OF TABLES TABLE PAGE I . C l a s s i f i c a t i o n o f R o o t s t o c k V i g o u r 86 I I . V a r i a b l e s Used i n Models 87 I I I . E s t i m a t e d S i m p l e L i n e a r R e g r e s s i o n E q u a t i o n . 88 IV. E v a l u a t i o n o f S i z e - C o n t r o l l i n g E f f e c t s o f R o o t s t o c k , I n t e r m e d i a t e Framerwork S t o c k and S t r a i n o f S c i o n V a r i e t y on T o t a l Tree S i z e i n Terms o f an Index V a l u e 89 V. T r e e - S i z e C l a s s i f i c a t i o n o f I n i t i a l T o t a l Samp l e 38 V I . T r e e - S i z e C l a s s i f i c a t i o n o f Sample E n t e r p r i s e s 39 V I I . Input Data 90 V I I I . C o r r e l a t i o n M a t r i x f o r Si m p l e L i n e a r Regres-s i o n w i t h One Hundred and N i n e t e e n P a i r s o f O b s e r v a t i o n s 91 IX. C o v a r i a n c e T a b l e f o r Three T r e e - S i z e Groups . 47 X. T a b l e f o r S. i n Terms o f C! 48 I I X I . R e s u l t s o f E q u a l i t y S l o p e Test f o r Three T r e e - S i z e Groups 92 X I I . S i g n i f i c a n t C o e f f i c i e n t s at .05 L e v e l f o r Cobb-Douglas Model 93 X I I I . C o r r e l a t i o n M a t r i x f o r Cobb-Douglas Model ... 94 XIV. S i g n i f i c a n t C o e f f i c i e n t s at .05 L e v e l f o r Q u a d r a t i c Model 9 5 XV. C o r r e l a t i o n M a t r i x f o r Q u a d r a t i c Model I n v o l v i n g o n l y S i g n i f i c a n t V a r i a b l e s 96 XVI. Observed and C a l c u l a t e d V a l u e s o f Apple Y i e l d s based on Q u a d r a t i c Model I n v o l v i n g o n l y S i g n i f i c a n t V a r i a b l e s 97 X V I I . R e s u l t s from t - t e s t s f o r A v e r a g e - A p p l e - Y i e l d D i f f e r e n c e s R e l a t i n g t o T r e e - S i z e Groups .... 69 v i i TABLE PAGE X V I I I . R e s u l t s from t - t e s t f o r A v e r a g e - A p p l e - Y i e l d D i f f e r e n c e s Between Regions 70 XIX. R e s u l t s from t - t e s t s f o r A v e r a g e - A p p l e - Y i e l d D i f f e r e n c e s R e l a t e d to Grades 71 XX. R e s u l t s from t - t e s t f o r A v e r a g e - A p p l e - Y i e l d D i f f e r e n c e s R e l a t i n g to V a r i e t y 73 v i i i LIST OF FIGURES FIGURE PAGE 1. V a l u e o f R e g r e s s i o n i n Reducing V a r i a t i o n i n Y 27 2. P o l y n o m i a l R e g r e s s i o n as a S p e c i a l Case o f M u l t i p l e R e g r e s s i o n 53 A 3. Range o f Va l u e s f o r P o s s i b l e fi>, Around O r i g i n When X i and % 2 are H i g h l y C o l l i n e a r . 60 4. D i f f e r e n c e s i n Average Apple Y i e l d s Among T r e e - S i z e Groups 68 5. D i f f e r e n c e i n Average Apple Y i e l d s Between Regions ( A c r o s s A l l T r e e - S i z e Groups) 70 6. D i f f e r e n c e s i n Average Apple Y i e l d s Among Apple Grades ( A c r o s s A l l T r e e - S i z e Groups) .... 71 7. D i f f e r e n c e s i n Average A p p l e Y i e l d s Among App l e V a r i e t i e s ( A c r o s s A l l T r e e - S i z e Groups) . 72 i x ACKNOWLEDGEMENTS The a u t h o r wishes t o g r a t e f u l l y acknowledge the generous a s s i s t a n c e and guidance p r o v i d e d by h i s t h e s i s s u p e r v i s o r , Dr. M. J . D o r l i n g . I n a d d i t i o n , the s t i m u l a t i n g i n t e l l e c t u a l e nvironment p r o v i d e d by the s t a f f and f e l l o w g r aduate s t u d e n t s at the U n i v e r s i t y o f B r i t i s h C olumbia was e x t r e m e l y m o t i v a t i n g and r e w a r d i n g . W h o l e - h e a r t e d g r a t i t u d e i s extended t o Mr. K. A c t o n and s t a f f i n the Economics B r a n c h , Canada Department o f A g r i c u l t u r e , Vancouver, w i t h o u t whose a s s i s t a n c e t h i s t h e s i s would n o t have been p o s s i b l e . CHAPTER I INTRODUCTION Today the t r e e f r u i t i n d u s t r y of B r i t i s h Columbia i s c e n t r e d i n the Okanagan v a l l e y i n a narrow one-hundred-m i l e s t r i p from Vernon to Osoyoos. A p p r o x i m a t e l y 94% o f the B r i t i s h Columbia a p p l e crop i s produced i n the Okanagan and Similkame'en v a l l e y s . A p p r o x i m a t e l y 4% o f the p r o v i n c i a l t o t a l i s produced i n the C r e s t o n a r e a , w i t h C r e s t o n v a l l e y b e i n g the major o r c h a r d i n g d i s t r i c t o f t h a t a r e a . The r e m a i n i n g 2% o f the crop i s produce d i n s c a t t e r e d p o c k e t s r a n g i n g from Vancouver I s l a n d t o the Lower M a i n l a n d , L i l l o o e t , Kamloops, Salmon Arm, and Grand Forks a r e a s . Due t o the a b i l i t y o f ap p l e t r e e s t o w i t h s t a n d l o w e r w i n t e r t e m p e r a t u r e than o t h e r k i n d s o f f r u i t t r e e s , a p p l e s have l o n g been c o n s i d e r e d the "backbone" o f the t r e e -f r u i t i n d u s t r y i n B r i t i s h Columbia. A s u r v e y c o n d u c t e d by P r o f e s s o r M. J . D p r l i n g , 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 n d i c a t e d t h a t 91% o f apple p r o d u c e r s i n the Okanagan d e r i v e d a l l t h e i r f a r m i n g revenue from t r e e f r u i t . 1 D u r i n g the 1960's and e a r l y 1970's most B r i t i s h Columbia a p p l e growers c o n t i n u e d t o engage i n o r c h a r d r e n o v a t i o n : o l d t r e e s are b e i n g r e p l a c e d by young t r e e s , and i n many i n s t a n c e s , o b s o l e t e v a r i e t i e s and s t r a i n s *M. J . D o r l i n g , The Okanagan Apple P r o d u c e r H i s Manage-ment A t t i t u d e and B e h a v i o u r , Department o f A g r i c u l t u r a l E c onomics, U.B.C., 1968. 2 are b e i n g r e p l a c e d by more a c c e p t a b l e ones. The commonly a c c e p t e d t r e e s p a c i n g o f 30 f e e t x 30 f e e t o f former y e a r s i s g i v i n g way t o more dense p l a n t i n g . Newly p l a n t e d o r c h a r d s w i t h t r e e s p a c i n g o f a p p r o x i m a t e l y 12 t o 16 f e e t between rows and 5 t o 10 f e e t i n the row are becoming commonplace. The r e p l a n t i n g program which B r i t i s h Columbia ap p l e growers have u n d e r t a k e n s h o u l d h e l p t o p l a c e the ap p l e i n d u s t r y o f the P r o v i n c e i n a s t r o n g e r p o s i t i o n so f a r as t h e a b i l i t y t o produce c o m p e t i t i v e l y i s concerned. There has been an i n c r e a s e d emphasis on lower c o s t s o f p r o d u c t i o n ; e a r l i e r f r u i t i n g ; e a s i e r p r u n i n g , t h i n n i n g , and h a r v e s t i n g ; and e a s i e r s p r a y p e n e t r a t i o n , a l l t o ensure c o m p e t i t i v e n e s s i n response t o c h a n g i n g market demands. The s t a n d a r d i z e d M a i l i n g v i g o u r - c o n t r o l l i n g r o o t -s t o c k s seem l i k e l y c o n t r i b u t o r s t o a c h i e v i n g some of t h e s e g o a l s . The need f o r c o n s i d e r a t i o n o f t h e s e m a t t e r s l e d t o the i n i t i a t i o n o f t h i s s t u d y the purpose o f w h i c h i s t o e s t i m a t e y i e l d r e l a t i o n s h i p s w i t h s p e c i a l r e f e r e n c e t o d e n s i t y and o t h e r p r o d u c t i o n i n f l u e n c e s , by means o f r e g r e s -s i o n a n a l y s i s . As a p r e l i m i n a r y s t e p , v a r i o u s s i m p l e l i n e a r r e g r e s s i o n a n a l y s e s were a t t e m p t e d ; d i s c u s s i o n o f t h e s e a n a l y s e s c e n t r e on the a p p l e y i e l d p erformance i n r e l a t i o n t o the b a s i c independent f a c t o r s o f p r o d u c t i o n , namely, d e n s i t y p e r a c r e , age o f t r e e s , the amount of f e r t i l i z e r , the amount o f s p r a y , p r u n i n g and t h i n n i n g l a b o u r h o u r s . 3 The e m p i r i c a l r e s u l t s o f t h e s e a n a l y s e s are d i s c u s s e d i n C h a p t e r V and the r e s u l t s are shown i n T a b l e I I I i n the Appendix. Numerous e m p i r i c a l and l o g i c a l c r i t e r i a have been used i n the s e l e c t i o n o f a q u a d r a t i c f u n c t i o n i n the s t u d y : b e f o r e t h i s was a c c o m p l i s h e d , d i f f e r e n t a l g e b r a i c models were employed i n r e p r e s e n t i n g o b s e r v a t i o n a l d a t a . S e l e c t i o n o f a p a r t i c u l a r form o f f u n c t i o n was based m a i n l y on two c o n s i d e r a t i o n s : s i g n i f i c a n c e o f s t r u c t u r a l c o e f f i c i e n t s and b e s t f i t . -The b e s t f i t was i n d i c a t e d by the magnitude o f the c o e f f i c i e n t o f d e t e r m i n a t i o n , R 2, assuming t h a t the c o n d i t i o n o f n o r m a l l y and i n d e p e n d e n t l y d i s t r i b u t e d e r r o r s was not v i o l a t e d . The l o g i c a l reasons f o r s e l e c t i o n o f the q u a d r a t i c model are the f o l l o w i n g : (1) i t a l l o w s b o t h d e c l i n i n g and n e g a t i v e m a r g i n a l p r o d u c t i v i t y ( t h e s e c o n d i t i o n s are v e r y i m p o r t a n t from the a p p l e s t u d y ' s p o i n t of view because apple y i e l d i s assumed t o be s u b j e c t t o the law of d i m i n i s h i n g r e t u r n s ) ; (2) i t does not impose such s t r i c t r e s t r a i n t s on a y i e l d r e l a t i o n s h i p as the Cobb-Douglas and S p i l l m a n e q u a t i o n s ; (3) a maximum t o t a l y i e l d i s d e f i n e d . 2 In summary, d e c i d i n g b o t h the f u n c t i o n a l form, and w h i c h v a r i a b l e s t o omit and w hich t o r e t a i n was done on the b a s i s o f the l o g i c , i n c l u d i n g c o n s i d e r a t i o n o f p h y s i c a l and b i o l o g i c a l r e l a t i o n s and s t a t i s t i c a l p r o b a b i l i t y l e v e l s . I f the R 2 i s s a t i s f a c t o r y and the l o g i c o f the p r o d u c t i o n 2See E. 0. Heady and J . D i l l o n , A g r i c u l t u r a l P r o d u c t i o n  F u n c t i o n s , pp. 75-78, Ames, Iowa: Iowa S t a t e U n i v e r s i t y P r e s s , 1961. 4 s i t u a t i o n does n ot d i c t a t e t h a t the e x c l u d e d v a r i a b l e must be i n c l u d e d , the new r e g r e s s i o n e s t i m a t e s may be r e g a r d e d as s e r v i n g s a t i s f a c t o r i l y . The q u a d r a t i c f u n c t i o n was chosen by the above c r i t e r i a . More o f t e n t h a n n o t , s e l e c t i o n among a l g e b r a i c forms o f e q u a t i o n s i s no l e s s d i f f i c u l t t han d e c i s i o n w i t h r e s p e c t t o the s i g n i f i c a n c e l e v e l at w h i c h v a r i a b l e s w i l l be o m i t t e d from the q u a d r a t i c models b e i n g examined. T h i s s t u d y was d e s i g n e d t o e s t i m a t e y i e l d e q u a t i o n s and i d e n t i f y the most i m p o r t a n t c o n t r i b u t i n g f a c t o r s i n ap p l e p r o d u c t i o n i n the Okanagan a r e a o f B r i t i s h Columbia i n 1969. An a s s u m p t i o n was made t h a t a l l independent v a r i a b l e s were measured w i t h o u t e r r o r s and the dependent v a r i a b l e was a s t o c h a s t i c v a r i a b l e e x h i b i t i n g o b s e r v e d d i s -t u r b a n c e . S a m p l i n g methods and methods o f d e r i v a t i o n o f d a t a are o u t l i n e d i n Chapter IV. Chapter V i s devo t e d t o a p r e l i m i n a r y r e v i e w o f two d i f f e r e n t forms o f f u n c t i o n i n terms o f t h e i r r e l e v a n c y t o the s t u d y . The E q u a l i t y o f S l o p e T e s t i s a l s o r e v i e w e d t o ensure t h a t t h e r e can be j u s t i f i c a -t i o n f o r co m b i n i n g the t h r e e d i f f e r e n t t r e e - s i z e e q u a t i o n s i n t o a s i n g l e e q u a t i o n . The purpose o f t h e E q u a l i t y o f Sl o p e T e s t i s i n r e l a t i o n t o K l i n e a r r e g r e s s i o n e q u a t i o n s i n m independent v a r i a b l e s : Y^=b0+b^"Xj + + t> m x m> i = 1 , 2 , , k. 3 I t t e s t s the h y p o t h e s i s HQ : M = b? = 3As i s e x p l a i n e d i n Ch a p t e r I I I r e g r e s s i o n e q u a t i o n s are e x p r e s s e d i n the e a r l y t h e s i s c h a p t e r s as h a v i n g s m a l l 'x' ( d e v i a t i o n ) v a r i a b l e s . These forms are c o n v e n i e n t con-c e p t u a l i z a t i o n s . 5 = b.., j = 1, 2, , m. A r e g r e s s i o n e q u a t i o n Y = b n + b, x, + + b x i s found f o r each o f k groups O i l m m b f o f sample u n i t s and an F - t e s t i s c a r r i e d out t o de t e r m i n e whether d i f f e r e n c e s i n the e s t i m a t e d c o e f f i c i e n t s among groups i s due t o s a m p l i n g e r r o r s or r e a l d i f f e r e n c e s . The r e s u l t s o f the a n a l y s i s i n v o l v i n g a s e l e c t e d r e g r e s s i o n e q u a t i o n and r e l a t e d d i s c u s s i o n are p r e s e n t e d at the end o f Chapter V. In Chapter VI, numerous t e s t s are made c o n c e r n i n g d i f f e r e n c e s among average apple y i e l d s w i t h r e g a r d t o d i f f e r e n t t r e e s i z e , a p p l e v a r i e t i e s , a p p l e g r a d e s , and r e g i o n s as i m p o r t a n t s o u r c e s o f i n f l u e n c e . F i n a l l y , C h a p t e r V I I p r e s e n t s a summary o f the main c o n c l u s i o n s and i m p l i c a t i o n s o f the s t u d y . CHAPTER I I LITERATURE REVIEW OF APPLE BIOLOGY F a c t o r s I n f l u e n c i n g Apple P r o d u c t i o n There tends i n p r a c t i c e t o be two main a s p e c t o f an apple e n t e r p r i s e w h i c h focus i n t e r e s t : one i s b i o l o g i c a l , and the o t h e r i s economic. The p r i m a r y purpose o f t h i s s t u d y was t o o u t l i n e some o f the most i m p o r t a n t f a c t o r s a s s o c i a t e d w i t h the y i e l d performance o f s p e c i f i c apple e n t e r p r i s e p l o t s . There are many f a c t o r s i n f l u e n c i n g apple p r o d u c t i o n , perhaps too many t o p i n p o i n t them a l l . A h i g h l e v e l o f management i n o p e r a t i n g a p p l e o r c h a r d s may w e l l be c o n d u c i v e t o i n -c r e a s i n g the l e v e l o f p r o d u c t i o n . The same can be s a i d o f s i z e o f o p e r a t i o n , type o f machinery a v a i l a b l e , and so f o r t h . But t h e s e f a c t o r s are d i f f i c u l t t o q u a n t i f y and t h i s makes f i t t i n g a r e g r e s s i o n e q u a t i o n i n whi c h they are r e p r e s e n t e d i n f e a s i b l e . On account o f t h i s d i f f i c u l t y , a t t e n t i o n w i l l be c o n f i n e d o n l y t o the q u a n t i f i a b l e f a c t o r s o f p r o d u c t i o n . I t goes w i t h o u t s a y i n g t h a t t h i s p r o c e d u r e cannot be immune from a danger o f o v e r s i m p l i f i c a t i o n . In any s t u d y o f o r c h a r d p r o d u c t i o n , the f o l l o w i n g v a r i a b l e s , among o t h e r s , are i m p o r t a n t : 1 t r e e s i z e , s o i l 'See J . C. F o l g e r and S. M. Thomson, The Commercial A p p l e  I n d u s t r y o f No r t h A m e r i c a , ed. L. H. B a i l y , pp. 539-547. New Yo r k : M a c m i l l a n Co., 1921. See R. Bush, Tree F r u i t Growing, r e v i s e d by E. G. G i l b e r t . P r e p a r e d i n c o n j u n c t i o n and c o l l a b o r a t i o n w i t h the R o y a l H o r t i c u l t u r a l S o c i e t y . Penguin Books, 1962. See D. W. Ware, E. D. Woodward and H. W. T r e v o r , A Study o f  Apple P r o d u c t i o n i n the Okanagan V a l l e y o f B r i t i s h C o l u m b i a , Canada Department of A g r i c u l t u r e , M a r k e t i n g S e r v i c e : Economic D i v i s i o n Ottawa, January 1952. 7 c o n d i t i o n s , the f r e q u e n c y o f f r o s t - i n j u r y , u n f a v o r a b l e con-d i t i o n s a t b l o s s o m t i m e , p r u n i n g , t h i n n i n g , s p r a y i n g and d e n s i t y . There are d i s c u s s e d i n t u r n . S i z e o f the Tree Appl e v a r i e t i e s are p r o p a g a t e d by means o f g r a f t -i n g o r b u d d i n g . T h e r e f o r e i t i s n e c e s s a r y to have r o o t s t o c k s on w h i c h t o g r a f t or bud scion-wood o f s e l e c t e d v a r i e t i e s . Most a p p l e t r e e s c o n s i s t o f two d i s t i n c t p a r t s , r o o t s t o c k and s c i o n v a r i e t y . In some i n s t a n c e s , p a r t i c u l a r l y w i t h the more t e n d e r a p p l e v a r i e t i e s , i t may be d e s i r a b l e t o use t r e e s w i t h a w i n t e r - h a r d y t r u n k and/or framework. Wood of a w i n t e r - h a r d y v a r i e t y i s used f o r t h a t purpose and i f i t d i f f e r s from the r o o t s t o c k , i t i s r e f e r r e d t o as an i n t e r -mediate s t o c k . C o n s e q u e n t l y , t r e e s w i t h an i n t e r m e d i a t e s t o c k c o n t a i n t h r e e d i s t i n c t s e c t i o n s : r o o t s t o c k , i n t e r m e d -i a t e s t o c k and s c i o n v a r i e t y . R o o t s t o c k s are g i v e n f i r s t c o n s i d e r a t i o n , s i n c e s i z e c o n t r o l l i n g r o o t s p r o v i d e the most p r a c t i c a l means o f d e t e r m i n i n g u l t i m a t e t r e e s i z e . The i n t r o d u c t i o n o f d w a r f i n g r o o t s t o c k s and spur-type v a r i e t i e s showing compacter growth p r o v i d e d the oppor-t u n i t y t o adopt new o r c h a r d p l a n t i n g systems. C l a s s i f i c a t i o n o f r o o t s t o c k v i g o u r i s p r e s e n t e d i n T a b l e I i n the Appendix. S e m i - s t a n d a r d , semi-dwarf, and dwarf t r e e s make i t p o s s i b l e to combine h i g h t r e e p o p u l a t i o n p e r acre w i t h e a r l y and h i g h y i e l d . Tukey, E x t e n s i o n H o r t i c u l t u r i s t at Washington S t a t e U n i v e r s i t y , has shown t h a t the s m a l l e r t r e e a l l o w i n g l a r g e r 8 numbers o f t r e e s p e r a c r e has a g r e a t e r p o t e n t i a l f o r h i g h e a r l y y i e l d . 2 F i s h e r a l l u d e s t o s i m i l a r f a c t s i n h i s s t a t e m e n t : "Many o l d , a i l i n g , o u t - o f - d a t e o r c h a r d s r e q u i r e d r e n o v a t i o n . In r e p l a n t i n g t h e s e b l o c k s , and a l s o i n b r i n g i n g i n new l a n d , the grower has become i n c r e a s i n g l y c o n s c i o u s of the need f o r s m a l l e r e a s i e r t o h a n d le t r e e s , more a t t r a c t i v e t o l a b o u r , and c a p a b l e o f p r o d u c i n g h i g h e a r l y f r u i t r e t u r n s . " 3 Brase and Way have found t h a t s m a l l a p p l e t r e e s , because o f r e d u c e d b e a r i n g a r e a , w i l l produce l e s s f r u i t p e r t r e e than l a r g e s t a n d a r d t r e e s , but as more t r e e s can be p l a n t e d , l a r g e r or at l e a s t as l a r g e y i e l d s per a c r e w i l l be p r o d u c e d . 4 S o i l C o n d i t i o n The p r e r e q u i s i t e o f an o r c h a r d s o i l i s t h a t i t be w e l l - d r a i n e d . S o i l s are the p r o d u c t s o f the e n v i r o n m e n t a l c o n d i t i o n s under w h i c h they have d e v e l o p e d . These c o n d i t i o n s i n v o l v e m i n e r a l m a t e r i a l s as w e l l as t o p o g r a p h i c , c l i m a t i c and b i o l o g i c a l phenomena. W e l l - d r a i n e d s o i l s w h i c h r e f l e c t 2R. B. Tukey, " I m p l i c a t i o n s o f Economics on O r c h a r d Manage-ment", The 19 69 A p p l e Forum, P u b l i s h e d P r o c e e d i n g s of the F i r s t B r i t i s h Columbia F r u i t Growers' A s s o c i a t i o n - s p o n s o r e d H o r t i c u l t u r a l C o n f e r e n c e . pp. 59-60 (November 1969). 3D. V. F i s h e r , H i g h - D e n s i t y Orchards f o r B r i t i s h Columbia C o n d i t i o n s , Research S t a t i o n Summerland, B r i t i s h Columbia Research B r a n c h , Canada Department o f A g r i c u l t u r e , March 1966. ''K. D. B r a se and R. D. Way, R o o t s t o c k s and Methods used f o r D w a r f i n g F r u i t T r e e s , New York S t a t e A g r i c u l t u r a l Experiment S t a t i o n , p. 783, 1959. 9 the f o r c e s o f s o i l g e n e s i s , c l i m a t e and v e g e t a t i o n , are c l a s s i f i e d as z o n a l s o i l s . The z o n a l d i s t i n c t i o n i s b e l i e v e d t o be due t o a v a r i a b l e m o i s t u r e and tem p e r a t u r e r e l a t i o n s h i p c h a r a c t e r i z i n g mountainous c o u n t r y . The s o i l o f the l o w e r - p a r t s l o p e i n the Okanagan b e l o n g s t o the Glenmore c l a y - l o a m f o r m a t i o n ; the s o i l o f the upper p a r t b e l o n g s t o the Oyama loamy-sand f o r m a t i o n . Both are c l a s -s i f i e d as dark brown s o i l s by K e l l y and S p i l s b u r y . 5 The apple t r e e i n commercial p r o d u c t i o n a l s o r e -q u i r e s a number o f m i n e r a l e l e m e n t s , e.g., magnesium, p o t a s s i u m , manganese, c l a c i u m , s u l p h u r , i r o n , b o r o n , c o p p e r , and z i n c . These elements are f r e q u e n t l y a p p l i e d i n the form of f e r t i l i z e r and s p r a y compounds. The Frequency o f F r o s t - i n j u r y In c o n s i d e r i n g g e o g r a p h i c and c l i m a t i c f a c t o r s , mention s h o u l d be made o f the importance o f f r o s t - i n j u r y i n o r c h a r d s i n c e r t a i n l o c a t i o n s . Orchards i n most Okanagan areas r e c e i v e o c c a s i o n a l damage from f r o s t . However, some areas are more s u s c e p t i b l e than o t h e r s , and f o r the r e g i o n as a w h o l e , the m i c r o - c l i m a t e i s q u i t e v a r i a b l e . Ware e s t a b l i s h e d a t a b l e i n d i c a t i n g d i f f e r e n t f r o s t - f r e e p e r i o d s c o r r e s p o n d i n g t o areas i n the Okanagan as f o l l o w s : 6 5C. C. K e l l y and R. H. S p i l s b u r y , " S o i l Survey o f the Okanagan and Similkameen V a l l e y o f B.C.", Report No. 3 o f B.C. Survey. The B r i t i s h Columbia Department o f A g r i c u l t u r e i n c o o p e r a t i o n w i t h E x p e r i m e n t a l Farm S e r v i c e , Dominion Department o f A g r i -c u l t u r e , pp. 20-71, 1949. 6D. W. Ware, O r g a n i z a t i o n and Returns o f Stone F r u i t and Pear E n t e r p r i s e s i n the Okanagan V a l l e y , B.C. 1949-1950, Depart-ment o f A g r i c u l t u r e Economic D i v i s i o n , M a r k e t i n g S e r v i c e , Ottawa. 1952. pp. 5-6. 10 A r e a F r o s t - f r e e P e r i o d i n Days Kelowna 150 Summerland 176 P e n t i c t o n 152 O l i v e r 162 Keremeos 188 The f r o s t - f r e e p e r i o d as d e f i n e d f o r the above d a t a i s the number o f days between the l a s t date i n the s p r i n g on w h i c h the t e m p e r a t u r e o f 32° F. was r e c o r d e d and the f i r s t s i m i l a r c o n d i t i o n i n the f a l l o f the same y e a r . The f i g u r e s g i v e n are averages o b t a i n e d f o r a t e n - y e a r p e r i o d . U n f a v o u r a b l e C o n d i t i o n s a t Blossom Time I t i s w e l l known t h a t v a r i a t i o n s i n c l i m a t e account f o r c o n s i d e r a b l e v a r i a t i o n s i n crop y i e l d s . Temperature i s o f extreme importance at a l l seasons o f t h e y e a r i n the growing o f a p p l e s . W i n t e r t e m p e r a t u r e may be so low as t o r e s u l t i n i n j u r y t o the buds o r t h e wood o f the t r e e . On the o t h e r hand, some c o l d w i n t e r t e m p e r a t u r e i s r e q u i r e d t o ensure v e r n a l i z a t i o n so t h a t t r e e s l e a f out n o r m a l l y i n the s p r i n g . ' Temperature i n the s p r i n g may a l s o be a c r i t i c a l f a c t o r . Temperatures o f 26° o r 27° F. f o r p e r i o d s as s h o r t as an hour o r so can cause damage t o f l o w e r s . Thus, c l i m a t e i s a most i m p o r t a n t v a r i a b l e w h i c h e i t h e r i m p l i c i t l y o r e x p l i c i t l y e n t e r s any s u p p l y e q u a t i o n f o r an a g r i c u l t u r a l 11 c r o p . 7 I t i s d i f f i c u l t t o f i n d an a p p r o p r i a t e i n d e x f o r measuring the i n f l u e n c e o f c l i m a t e on a p p l e p r o d u c t i o n . The most c o n v e n i e n t measure might w e l l be t e m p e r a t u r e i f t h a t were summarized i n a c o n v e n i e n t form. For t h i s s t u d y a g e o g r a p h i c a l dummy v a r i a b l e was i n t r o d u c e d i n t o the a n a l y s i s w h i c h was i n t e n d e d t o r e p r e s e n t the i n f l u e n c e o f weather. The c o r r e l a t i o n e x i s t i n g between r a i n f a l l and s i z e o f a p p l e crop i n Nova S c o t i a has been found t o be n e g a t i v e by L o n g l e y l i m i t s o f the p o p u l a t i o n c o r r e l a t i o n c o e f f i c i e n t f o r the s e v e n - y e a r p e r i o d 1913-1929 i n c l u s i v e , i n v o l v i n g the May t o O c t o b e r p e r i o d , were e s t i m a t e d as -0 . 572 ± 0.110. 8 Hence, d e c r e a s i n g r a i n f a l l i n the summer months t e n d e d t o be a s s o c i a t e d w i t h an i n c r e a s i n g c r o p . I t i s o f i n t e r e s t to note t h a t d u r i n g the months i n w hich s p r a y i n g and d u s t i n g o p e r a t i o n s are done, the hours o f s u n s h i n e are a c r i t i c a l f a c t o r i n the p r o d u c t i o n of a p p l e s . A c o m b i n a t i o n o f more hours o f s u n s h i n e and l e s s r a i n f a l l d u r i n g the months o f May, June and J u l y r e s u l t s i n a more e f f e c t i v e p r o d u c t i o n o f t r e e f o l i a g e and b e t t e r c o n t r o l o f i n s e c t s and d i s e a s e s . 7 J . P. D o l l , "An A n a l y t i c a l Technique f o r E s t i m a t i n g a Weather Index from M e t e o r o l o g i c a l Measurements", J o u r n a l  o f Farm Economics, V o l . 49, No. 1, F e b r u a r y 1967. H. S. Lawrence, "The E f f e c t o f Weather on A g r i c u l t u r a l Out-p u t " : A Look at Methodology, J o u r n a l o f Farm Economics, V o l . 46, No. 1, F e b r u a r y 1964. A. K o u t s o g i a n n e - K o k k o v a , An E c o n o m e t r i c Study o f the L e a f  Tobacco Market o f Greece, pp. 164-166, A t h e n s , 1962. 8W. V. L o n g l e y , Some Economic A s p e c t s of the Apple I n d u s t r y  i n Nova S c o t i a . A T h e s i s f o r the Degree o f D o c t o r of P h i l -osophy, pp. 2 2 - 2 3 , Nova S c o t i a Department o f A g r i c u l t u r e B u l l e t i n No. 113, 1932. 12 P r u n i n g A good t r e e framework o f d e s i r e d s i z e , form and s t r e n g t h i s n e c e s s a r y t o o b t a i n the maximum number o f w e l l s p a c e d branches and spurs i n as s m a l l an a r e a as p o s s i b l e and s t i l l a l l o w the f r u i t p l e n t y o f space to grow. P r u n i n g admits l i g h t and a i r , a l l o w s e a s i e r s p r a y i n g and p i c k i n g , and thus improves f r u i t buds. P r u n i n g i s e s s e n t i a l t o a v o i d a bare-wood c o n d i t i o n and t o i n d u c e f r u i t i n g n e a r the t r u n k or main b r a n c h e s . C o r r e c t p r u n i n g h e l p s t o make p o s s i b l e the r i g i d c ordon shape o r the p e r manently dwarfed p y r a m i d t r e e , but i t must be employed i n the r i g h t way on the r i g h t v a r i e t y o f t r e e , p l a n t e d i n s u i t a b l e s o i l , i f the b e s t r e s u l t s are t o be e x p e c t e d . I t i s no use e x p e c t i n g a l l v a r i e t i e s o f t r e e t o conform t o the same s t a n d a r d ; one must adapt one's p r u n i n g t o t a k e advantage o f the n a t u r a l h a b i t o f the p a r t i c u l a r v a r i e t y . 9 Apple t r e e s may be s a i d t o pass through t h r e e d i s t i n c t p e r i o d s : (1) f o r m a t i v e p e r i o d , (2) t r a n s i t i v e p e r i o d , and (3) f r u i t i n g p e r i o d . 1 0 A p p r o p r i a t e p r u n i n g t r e a t m e n t changes m a t e r i a l l y w i t h each of t h e s e p e r i o d s . I t i s d u r i n g t h e f o r m a t i v e p e r i o d t h a t the t r e e devotes i t s e n e r g i e s t o wood growth. The p r o p e r s e l e c t i o n , d i s t r i b u t i o n and t r a i n i n g o f branches d u r i n g t h i s time d e t e r m i n e s the a b i l i t y o f the t r e e t o b e a r heavy crops o f f r u i t i n l a t e r y e a r s . A l l p r u n i n g d u r i n g t h e t r a n s i t i o n a l p e r i o d i s t o 9 B u s h , op. c i t . , pp. 115-116. 1 0 F o l g e r , et a l . , op. c i t . , p. 283. 13 d e v e l o p and m a i n t a i n a l i b e r a l s u p p l y o f f r u i t i n g wood, w e l l d i s t r i b u t e d t h r o u g h o u t t h e e n t i r e t r e e . T h i n n i n g The t h i n n i n g o f a p p l e s i s no more th a n a form of p r u n i n g . I f a l l the f r u i t on an a p p l e t r e e showing a heavy f r u i t s e t were a l l o w e d t o mature, s m a l l misshapen f r u i t s and l i m b breakage would r e s u l t . T h i s i s because f r u i t bud f o r m a t i o n t a k e s p l a c e e a r l y i n the season d u r i n g a p e r i o d o f extreme c o m p e t i t i o n between f r u i t buds and young f r u i t f o r a v a i l a b l e f o o d s u p p l i e s . E a r l y removal o f s u r p l u s f r u i t s removes much o f t h i s c o m p e t i t i o n . 1 1 S p r a y i n g Orchards must be s p r a y e d r e g u l a r l y and t h o r o u g h l y i n o r d e r t o p r o t e c t the f r u i t from s e r i o u s i n s e c t or d i s e a s e damage. Depending on the n a t u r e and e x t e n t o f the i n f e s t a -t i o n s , a p p l e s r e q u i r e from f o u r t o seven s p r a y s a y e a r . Among e n t o m o l o g i s t s , however, t h e r e are two s c h o o l s o f t h o u g h t . The one b e l i e v e s t h a t w h o l e s a l e l i q u i d a -t i o n by p o i s o n o u s s p r a y s i s d e s i r a b l e . In c o n t r a s t , the o t h e r s c h o o l r e g r e t s the massacre o f many b e n e f i c i a l i n s e c t s and hopes t h a t b i o l o g i c a l c o n t r o l w i l l prove s u p e r i o r . There no doubt e x i s t s a danger o f i n d u c i n g immune ra c e s of i n s e c t s ; t o d a y , t h e r e are s e v e r a l p e s t s w h i c h , h a v i n g i n the p a s t been exposed t o c e r t a i n s p r a y s , have now 1 1 I b i d . , op. c i t . , p. 283. 14 d e v e l o p e d a degree o f immunity. However, apple growers cannot e x p e c t t h e i r p a r t i c u l a r o r c h a r d s t o be f r e e from a t t a c k s o f i n s e c t s and d i s e a s e s w h i c h o c c u r e l s e w h e r e , and any who omit s p r a y i n g are u n l i k e l y t o produce m a r k e t a b l e f r u i t . Dens i t y Van Roechoudt has w r i t t e n : "At the end o f the s i x t h growing s e a s o n , on an a c r e b a s i s , t h e r e was a wide v a r i a t i o n i n y i e l d s from each o f the d i f f e r e n t p l a n t i n g con-c e p t s . The t r e e s p l a n t e d as hedgerows on M. V I I r o o t s t o c k at t h e d e n s i t y used had p r o d u c e d 25.9 times more f r u i t . The y i e l d was r e l a t e d t o the p l a n t i n g c o n c e p t , the number o f t r e e s p e r a c r e , the system o f p r u n i n g and t r a i n i n g f o l l o w e d and the type o f r o o t s t o c k u s e d . " 1 2 H a r r i s and Woods have r e p o r t e d from t h e i r i n v e s -t i g a t i o n s a t the Canada Department o f A g r i c u l t u r e E x p e r i m e n t a l Farm, S a a n i c h t o n , B.C., t h a t a p p l e t r e e s at h i g h e r d e n s i t y on M. IX r o o t s t o c k grow w e l l , produce h e a v i l y w i t h h i g h q u a n t i t y f r u i t at an age when s t a n d a r d t r e e s are f a r from b e i n g i n a s t a t e o f commercial p r o d u c t i o n . 1 3 I n t e n s i v e p l a n t i n g o f a p p l e t r e e s i m p l y i n g h i g h d e n s i t y p e r a c r e w i l l i n v o l v e a h i g h i n v e s t m e n t c o s t . Of p r i m a r y c o n s i d e r a t i o n , however, i s the a b i l i t y o f t h e crop 1 2 L . L. Van Roechoudt, Some F a c t o r s Which I n f l u e n c e the Use  of Dwarf and Semi-Dwarf Apple Trees f o r Commercial Orchards  i n the Okanagan V a l l e y o f B.C. U n p u b l i s h e d M a s t e r ' s T h e s i s , The 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 , 196 2. 1 3 J . H. H a r r i s and J . J. Woods, Dwarf Apple Trees on Vancouver  I s 1 a n d , E x p e r i m e n t a l Farm Research B r a n c h , S a a n i c h t o n , B.C., 1958. 15 to r e t u r n a p r o f i t on the investment. Smaller trees inher-e n t l y produce f r u i t at an e a r l i e r age; the l a r g e r number of trees per acre can r e s u l t i n a s i g n i f i c a n t l y higher y i e l d per acre. Tukey s t a t e s : "One of the most p o s i t i v e methods of i n c r e a s i n g y i e l d i n the e a r l y years of an orchard i s to increase the number of trees per acre, and i n c r e a s i n g the tree p o p u l a t i o n may be one of the most e f f e c t i v e means of cou n t e r a c t i n g the problem of obsolescence and r e p l a n t i n g o l d orchard s i t e s . " 1 1 * Tukey, op. c i t . , p. 58. CHAPTER I I I LITERATURE REVIEW OF STATISTICS . Modern s t a t i s t i c s are based upon p r o b a b i l i t y . There are a number o f c o n f l i c t i n g i d e a s about t h i s c o n c e p t , w h i c h i s fundamental f o r s c i e n t i f i c methodology. Some a u t h o r s h o l d t h a t p r o b a b i l i t y s t a t e m e n t s r e f e r t o a p r o p o s i t i o n and are hence l o g i c a l and n o t empir-i c a l . T h i s concept r e f e r s t o our r a t i o n a l degree o f b e l i e f i n a t h e o r y o r h y p o t h e s i s on the b a s i s o f e m p i r i c a l e v i d e n c e . Keynes, f o r example, expounds i n h i s t r e a t i s e on p r o b a b i l i t y as f o l l o w s : "What we know and what p r o b a b i l i t y we can a t t r i b u t e t o our r a t i o n a l b e l i e f s i s , t h e r e f o r e , s u b j e c t i v e i n the sense o f b e i n g r e l a t i v e t o the i n d i v i d u a l . But g i v e n the body o f p r e m i s e w h i c h our s u b j e c t i v e powers and c i r c u m s t a n c e s s u p p l y t o u s , and g i v e n the k i n d s o f l o g i c a l r e l a t i o n s upon w h i c h arguments can be based and w h i c h we have the c a p a c i t y to p e r c e i v e , the c o n c l u s i o n , w h i c h i t i s r a t i o n a l f o r us t o draw, s t a n d s t o t h e s e p r e m i s e s i n an o b j e c t i v e and w h o l l y l o g i c a l r e l a t i o n . Our l o g i c i s c o n c e r n e d w i t h d rawing c o n c l u s i o n s by a s e r i e s o f s t e p s o f c e r t a i n s p e c i f i e d k i n d s from a l i m i t e d body o f p r e m i s e s . " 1 A n o t h e r and e n t i r e l y d i f f e r e n t p r o b a b i l i t y con-c e p t r e f e r s to the r e l a t i v e f r e q u e n c y o f an e v e n t , as the XJ. M. Keynes: A T r e a t i s e on P r o b a b i l i t y , p. 18. 17 number of t r i a l s increases i n d e f i n i t e l y . The econometrician may, f o r i n s t a n c e , consider the r e l a t i v e frequency of business f a i l u r e s that i s , the percentage of businesses which f a i l each year. He may t a l k about the p r o b a b i l i t y of a business f a i l u r e as the l i m i t of the r e l a t i v e frequency of f a i l u r e s as the sample becomes l a r g e r and l a r g e r . Since the f i r s t p r o b a b i l i t y concept i s not yet u s e f u l f o r any but the s i m p l e s t problems of s t a t i s t i c a l i n f e r e n c e , only the second concept i s r e l e v a n t i n the case of s t a t i s t i c a l t e s t s i n the study. The fundamental purpose of r e g r e s s i o n a n a l y s i s i s to estimate the r e l a t i o n s h i p between the dependent and independent v a r i a b l e s . Once the r e l a t i o n s h i p between these v a r i a b l e s has been q u a n t i t a t i v e l y estimated, we may wish to know the goodness of f i t of the r e l a t i o n s h i p . I t i s impossible to estimate the r e l a t i o n s h i p between the v a r i a b l e s without f i r s t making some assumptions or deductions about the form of the r e l a t i o n s h i p . To i l l u s -t r a t e , consider a simple l i n e a r r e g r e s s i o n equation, = a + bx^ , where i = l , 2 , n and where x^=(X-X). 2 One advan-tage of measuring X^ as d e v i a t i o n s from t h e i r mean i s that the mathematics w i l l be s i m p l i f i e d because the sum of the new x values equals zero — that i s £x^=0 . This w i l l become convenient l a t e r on i n the proof of E(b) = 6 , var(b) 2 2 = a^/Ix^ . A l s o , i n the process of i n v e r t i n g , where X i s a matrix c o n s i s t i n g of a l l x observation and X' i s a 2 See R. J . Wonnacott and T. M. Wonnacott, Econometrics, pp. 245-246. John Wiley and Sons, Inc., 1970. 18 m a t r i x c o n s i s t i n g o f a l l x o b s e r v a t i o n s and %' the t r a n s p o s e o f % , the measurement o f x v a l u e s i n d e v i a t i o n form i s shown t o be v e r y c o n v e n i e n t . Suppose t h a t an ex p e r i m e n t c o u l d be r e p e a t e d many times a t a f i x e d v a l u e o f x. Then t h e r e would be o b s e r v e d some s t a t i s t i c a l f l u c t u a t i o n o f the Y v a l u e s c l u s t e r e d about a c e n t r a l v a l u e f o r m i n g a su b - p o p u l a -t i o n . The p r o b a b i l i t y f u n c t i o n o f Y f o r a g i v e n x, we s h a l l c a l l P ( Y / x ) . There w i l l be a s i m i l a r p r o b a b i l i t y f u n c t i o n f o r Y at any o t h e r e x p e r i m e n t a l l e v e l of x. C o n s e q u e n t l y , p r o b a b i l i t y f u n c t i o n s f o r Y^ at the v a r i o u s l e v e l s o f x^ w i l l be P ( Y i / x i ) . To keep the problem manageable, l e t t h e r e be a r e a s o n a b l e s e t o f assumptions about the r e g u l a r i t y o f t h e s e s u b - p o p u l a t i o n s . These assumptions may be w r i t t e n c o n c i s e l y as f o l l o w s : the random v a r i a b l e s Y^ are s t a t i s t i c a l l y i n d e p e n d e n t , w i t h mean a+3x^ and v a r i a n c e a*. On o c c a s i o n , i t i s u s e f u l t o d e s c r i b e the d e v i a t i o n o f Y^ from i t s ex-p e c t e d v a l u e as the e r r o r o r d i s t u r b a n c e term IL , where the IL are independent random v a r i a b l e s , w i t h mean 0 and v a r i a n c e a*r. No assu m p t i o n i s y e t made about the shape o f the d i s t r i b u t i o n o f IL p r o v i d e d i t has a f i n i t e v a r i a n c e . The e r r o r term may be r e g a r d e d as the sum o f two components: 1. Measurement E r r o r . In m e a suring crop y i e l d , t h e r e may be an e r r o r r e s u l t i n g from c a r e l e s s h a r v e s t i n g or i n a c c u r a t e w e i g h i n g . 2 . S t o c h a s t i c E r r o r . D i s r e g a r d i n g measurement e r r o r , t h e r e would s t i l l 19 be some u n p r e d i c t a b l e d i f f e r e n c e s i n y i e l d s , f o r example, i n an e x p e r i m e n t u s i n g the same r a t e o f f e r t i l i z e r a p p l i c a -t i o n . Assume t h a t the s i t u a t i o n i s such t h a t t h e r e are no l a r g e measurement e r r o r s i n the v a r i a b l e s . However, t h e r e are c e r t a i n v a r i a b l e s w h i c h ought t o appear i n the e q u a t i o n but have been l e f t o ut. O m i s s i o n o f the l a t t e r r e s u l t s i n r a t h e r l a r g e e r r o r s i n the e q u a t i o n s . 3 I f the e n t i r e p o p u l a t i o n s o f v a l u e s (x^,Y^) are known, i t i s p o s s i b l e t o compute the e x a c t v a l u e s o f the r e g r e s s i o n parameters a, 6 and o^. D e t e r m i n a t i o n o f l e a s t s q u ares i s the most a c c e p t a b l e method f o r f i t t i n g a s t r a i g h t l i n e . The method o f l e a s t squares r e q u i r e s t h a t the e s t i -mators (a,b) be s e l e c t e d i n such a way t h a t the sum o f the s q u a r e d d e v i a t i o n s o f Y^=a+bX^ from the f i t t e d r e g r e s s i o n l i n e be a minimum t h a t i s m i n i m i z e e ? = ( Y . - a - b x . ) 2 , i ^ i iJ ' where e i s the e r r o r term. For t e s t i n g h y p o t h e s e s , i t w i l l be n e c e s s a r y t o know how the e s t i m a t o r s a and b are d i s t r i -b u t e d around t h e i r p a r a m e t e r s , a and g. The l e a s t squares e s t i m a t o r s a and b are t h e n the b e s t l i n e a r u n b i a s e d e s t i -mators o f a and 8. That i s , t o sum up: E(a) = a Var (a) = c^/n E(b) = 6 Var (b) = tfy/£*-_ 3T. Haavelmo, "The P r o b a b i l i t y Approach i n E c o n o m e t r i c s " , E c o n o m e t r i c a , V o l . 12, 1944, Supplement. H. B. Mann and A. Wald, "On the S t a t i s t i c a l Treatment o f L i n e a r S t o c h a s t i c D i f f e r e n c e E q u a t i o n s " , E c o n o m e t r i c a , V o l 11, p. 173, 1943. 20 where E and Var s t a n d f o r e x p e c t e d v a l u e and v a r i a n c e r e -s p e c t i v e l y . These p r o p e r t i e s have been p r o v e d w i t h t h e use o f Gauss-Markov Theorem w i t h o u t making any assumption about the shape o f the d i s t r i b u t i o n o f the e r r o r term. 1* S i n c e the s l o p e c o e f f i c i e n t b i s u s u a l l y o f more i n t e r e s t t o us than the i n t e r c e p t c o e f f i c i e n t a, we s h a l l c o n c e n t r a t e on the s l o p e . P r o o f o f E(b) = 3 and Var (b) = c^/Ex? a l o n e i s as f o l l o w s . The f o r m u l a f o r b may be r e w r i t t e n as: where Thus , where b = Z ( x i / K ) Y i (3-1) K = Ex? . (3-2) b = Ew.Y. = WiYi + w 2Y 2+ w Y • (3-3) i i n n w i = x ^ K (3-4) From the t h e o r y o f l i n e a r t r a n s f o r m a t i o n s , i t f o l l o w s t h a t : E(b) = WjECYx) + w 2 E ( Y 2 ) + w n E ( Y n ) = ^ E ^ ) (3-5) N o t i n g t h a t the v a r i a b l e s Y^ are assumed i n d e p e n d e n t , i t f o l l o w s t h a t V a r ( b ) = w?Var Y j + w|Var Y 2 + w* (3-6) Var Y =Ew?Var Y. n I I U s i n g the mean from (3-5) and E(Y^)=a+8x^ as assumed p r e -v i o u s l y , t h e n E(b) = Zw i(a+6x i) = aZvr ^  + fcEw^ and n o t i n g e q u a t i o n (3-4) , th e n MVonnacott and Wonnacott, op. c i t . , pp. 48-51 21 E(b) = (a/k)Zx i + ( 3 / k ) Z ( X i ) x , But, s i n c e Zx^ i s zero, then 2 . E(b) = 0 + (B/k)Ex i . Furthermore, from equation (3-2) E(b) = 3 From equation (3-6) and from Var(Y^)=a^ as assumed p r e v i o u s l y , Var(b) = Zw 2a 2 = £(x 2/k 2)a 2 = ( a 2 / k 2 ) Z x 2 Again, n o t i n g equation (3-2), Var(b) = a 2/Ex? R e c a l l i n g the assumption that values are s t a t i s t i c a l l y independent and also that b i s a l i n e a r combination of a l l Y. (that i s b = Ex-Y./Ex 2), i t fo l l o w s I ^ l I iJ ' that the shape of the b d i s t r i b u t i o n w i l l a l so be normal. The n o r m a l i t y assumption of the e r r o r term i s r e q u i r e d only f o r s m a ll sample e s t i m a t i o n s . Without assuming that the Y^ are normally d i s t r i b u t e d , as sample s i z e i n c r e a s e s , the d i s t r i b u t i o n of b w i l l u s u a l l y approach n o r m a l i t y , t h i s can be j u s t i f i e d by a g e n e r a l i z e d form of the C e n t r a l L i m i t Theorem. I f we have s p e c i f i e d the form of the d i s t r i b u t i o n of the e r r o r terms i n our r e g r e s s i o n model, then the method of l e a s t squares i s j u s t i f i e d by the method of maximum l i k e l i h o o d (which could also have been used to ob t a i n e s t i -mators a and 8). For g e n e r a l i t y , suppose that, we have a sample of s i z e n. We wish to know: P(Y 1,Y 2----Y n) (3-7) 22 That i s , we wish to know the l i k e l i h o o d or p r o b a b i l i t y dens-i t y of the sample we observed, expressed as a f u n c t i o n of the p o s s i b l e p o p u l a t i o n values o f a, 3 and a* . T h e r e f o r e , f i r s t c o n s i d e r the p r o b a b i l i t y d e n s i t y of the f i r s t value o f Y. which i s 1 P ( Y l ) = y 2 ^ e ( S ) ( Y l " ( a + 3 X l ) ) ' ( 3 - 8 ) where e = 2 . 7 1 8 2 8 T h i s i s simply the normal d i s t r i b u t i o n o f Ylt with i t s mean ( a + 3 X j ) and v a r i a n c e (a^) s u b s t i t u t e d i n t o the a p p r o p r i a t e p o s i t i o n s . The independence of the Y^ values j u s t i f i e s m u l t i p l y i n g a l l these p r o b a b i l i t y d e n s i t i e s t o g e t h e r to f i n d the j o i n t p r o b a b i l i t y d e n s i t y : P ( Y i ,Y2 , ----Y n) = ( 3 - 9 ) ( ,=L=- e-Osa*) ( Y j - C a + S x J ) 2 ) / —L=r - (%a 2) (Y 2 - (a+3x 2)) 2) 1 V2TI02 6 7 Tf O 6 7 v y v y = J T / _ L _ - (iga 2) (Y.-(a+3x.)) 2 ) \/2TT0y where T F r e p r e s e n t s the product o f n f a c t o r s . Using the f a m i l i a r r u l e f o r e x p o n e n t i a l s , the product of equation ( 3 - 9 ) can be expressed as f o l l o w s : P(Y ,Y , Y J = (-1— W 2 - ( % a 2 ) ( Y - ( a + 3 x . ) ) 2 ( 3 - 1 0 ) y . R e c a l l i n g t h a t with the observed Y^ s p e c u l a t i o n i s made concerning the values o f a, 3 and o 2 t then, to emphasize t h i s , the equation ( 3 - 1 0 ) i s renamed the l i k e l i h o o d f u n c t i o n : L ( a , B l 0 p . ( ^ r « e - ^ ' v - ^ ) ! t s . n ) y 2 3 T h e r e f o r e , the q u e s t i o n i s : Which v a l u e s o f a and 3 make L l a r g e s t ? The o n l y p l a c e t h a t a and 3 appear i s i n the exponent. Moreover, m a x i m i z i n g a f u n c t i o n w i t h a n e g a t i v e exponent i n v o l v e s m i n i m i z i n g the a l g e b r a i c magnitude o f the exponent. D e s i g n a t i n g our e s t i m a t o r s as a and b, the pro b l e m i s t o s e l e c t v a l u e s f o r thes e t h a t m i n i m i z e ( Y i - a - b x . ) 2 . (3-12) The c o n c l u s i o n t h a t f o l l o w s i s t h a t maximum l i k e l i h o o d e s t i m a t e s are i d e n t i c a l t o l e a s t squares e s t i m a t e s when the r e g r e s s i o n model has a n o r m a l l y d i s t r i b u t e d e r r o r . So f a r the independent v a r i a b l e x has assumed a g i v e n s e t o f f i x e d v a l u e s . However, i n many c a s e s , x cannot be c o n t r o l l e d i n t h i s manner. Thus i f we are exa m i n i n g the e f f e c t o f r a i n f a l l on y i e l d , i t must be r e c o g n i z e d t h a t x ( i . e . , r a i n f a l l ) i s a random v a r i a b l e , c o m p l e t e l y o u t s i d e our c o n t r o l . The method of l e a s t squares i s s t i l l v a l i d whether x i s a f i x e d o r a random v a r i a b l e , p r o v i d e d t h a t we assume t h a t the d i s t r i b u t i o n o f x does not depend on a, 3 , and a 2 , and t h a t the e r r o r terms are n o r m a l l y d i s t r i b u t e d and independent o f the x's (3-13) Of t h e s e a s s u m p t i o n s , we must emphasize the independence o f x and U. I t can be shown t h a t the maximum l i k e l i h o o d and l e a s t squares e s t i m a t e s c o i n c i d e and may be a p p l i e d r e g a r d -l e s s o f whether the independent v a r i a b l e x i s f i x e d o r 24 random, p r o v i d e d x i s independent o f the e r r o r and parameters i n the e q u a t i o n b e i n g e s t i m a t e d . The l i k e l i h o o d o f our sample now i n v o l v e s the p r o b a b i l i t y o f o b s e r v i n g b o t h x and Y. T h e r e f o r e , i f t h e x^ are in d e p e n d e n t , the l i k e l i h o o d f u n c t i o n i s L = P ( x 1 } P ( Y 1 / x 1 ) P ( x 2 ) P ( Y 2 / x 2 ) (3-14) S i n c e the e r r o r terms are c o n s i d e r e d n o r m a l , L = P ( X i ) ^ e - d a p ( Y l - a - 3 x l ) 2 P ( x 2 ) _ l _ v y i^^y e ' t e a p ( Y 2 - a - B x 2 )2 (3-15) C o l l e c t i n g t he e x p o n e n t s , e ^ _ V ~ I ~ L = P ( x , ) - - - - / 1 xn/2 Q - ( ^ a2) Z ( Y , - a - 3 x . )2 (3-16) \2TJO2- ) y S i n c e a c c o r d i n g t o e q u a t i o n (3-13), P(x) does n ot depend on the parameters a, 3, and , the problem o f m a x i m i z i n g t h i s l i k e l i h o o d f u n c t i o n reduces t o the m i n i m i z a t i o n o f the exponent i n e q u a t i o n (3-11). I t i s o f i n t e r e s t t o note what would happen i f the independent v a r i a b l e x i s c o r r e l a t e d w i t h the e r r o r terms. R e c o n s i d e r the model, Y = a + 3 x + U (3-17) By t a k i n g the c o v a r i a n c e s o f x w i t h each o f the v a r i a b l e s i n the e q u a t i o n , the f o l l o w i n g r e s u l t s , 5 S = S + S (3-18) xy xx xu v J In o r d e r t o e s t i m a t e 3, S i s d i v i d e d by S ( v a r i a n c e o f ' xy 1 xx v 5 W o n n a c o t t and Wonnacott, op. c i t . , pp. 149-157 25 x) such t h a t S /S = b + S /S (3-19) xy xx xu xx v From the o b s e r v a t i o n o f x and Y, S and S are e a s i l y ' xy xx 1 c a l c u l a t e d . However, U i s u n o b s e r v a b l e , so t h a t S cannot be e v a l u a t e d . T h e r e f o r e , i f we can assume t h a t S i s s m a l l enough t o n e g l e c t , we w i l l o b t a i n the e s t i m a t o r S /S = b (3-20) xy xx v J We r e c o g n i z e t h i s as the l e a s t squares e s t i m a t o r . 6 That i s , from e q u a t i o n ( 3 - 2 0 ) , the l e a s t squares e s t i m a t o r i s j u s t i -P P f i e d under c o n d i t i o n s t h a t S 0 w h i l e S ^ nonzero xu + xx •* p (where ^ i s d e f i n e d as approaches i n p r o b a b i l i t y as n + °°) . We have so f a r d e a l t w i t h r e g r e s s i o n a n a l y s i s r e l e v a n t t o t h i s s t u d y . B u t , i n t e r e s t may a l s o f o c u s on c o r r e l a t i o n a n a l y s i s t h a t i s the degree to wh i c h v a r i a b l e s are r e l a t e d o r a s s o c i a t e d . S i mple c o r r e l a t i o n a n a l y s i s y i e l d s o n l y one c o e f f i c i e n t and i n d e x number d e s i g n e d t o g i v e an immediate p i c t u r e o f how c l o s e l y two v a r i a b l e s move t o g e t h e r . In c o r r e l a t i o n a n a l y s i s , cause and e f f e c t r e l a t i o n s are u n i m p o r t a n t . A d i s t i n c t i o n between r e g r e s s i o n a n a l y s i s and c o r r e l a t i o n a n a l y s i s must be made to a v o i d c o n f u s i o n w h i c h may a r i s e from the s u b t l e t y o f the p r o p o s i t i o n s i n v o l v e d i n b o t h a n a l y s e s . In r e g r e s s i o n a n a l y s i s , a l l the independent v a r i a b l e s are assumed f i x e d . They do not o c c u r i n a proba-b i l i s t i c way. On the o t h e r hand, c o r r e l a t i o n a n a l y s i s i s 6 As n o t e d b e f o r e b = ZYx/Ex 2 = Zyx/Exx = \J Eyx/n-1 / ^ X T A T T = s y x / s x x . 26 c o n c erned m a i n l y w i t h random v a r i a b l e s . Independent v a r i -a b l e s must have a r e s p e c t i v e p r o b a b i l i t y d i s t r i b u t i o n . In view o f t h e s e d i f f e r e n c e s , r 2 v a l u e s can be a d e q u a t e l y i n t e r p r e t e d o n l y i n a c o r r e l a t i o n a n a l y s i s . Y e t , s i n c e c o r r e l a t i o n and r e g r e s s i o n are so c l o s e l y r e l a t e d mathemat-i c a l l y , c o r r e l a t i o n o f t e n becomes a u s e f u l a i d i n r e g r e s s i o n a n a l y s i s . S p e c i f i c a l l y , c o n s i d e r the r e l a t i o n between the e s t i m a t e d c o r r e l a t i o n c o e f f i c i e n t r , and the e s t i m a t e d r e g r e s s i o n s l o p e c o e f f i c i e n t b. I t was shown t h a t b = E x y / E x 2 (3-21) N o t i n g t h a t b o t h x and y are d e f i n e d as d e v i a t i o n s , then r = Exy/ E x 2 E y 2 (3-22) Then b / r =\fT^4~2/^2 = V Ey 2 / E x 2 (3-23) I f we now d i v i d e b o t h the numerator and denominator i n s i d e the square r o o t s i g n by ( n - 1 ) , b / r = x / ( E y 2 / n - l ) / E x 2 / n - l ) = S y / S x (3-24) o r b = r ( S /S x) (3-25) T h i s c l o s e c o r r e s p o n d e n c e between b and r w i l l be o f utmost i m p o r t a n c e i n the subsequent argument as t o w h i c h t o o l i s the more p o w e r f u l — r e g r e s s i o n or c o r r e l a t i o n a n a l y s i s . C o n s i d e r f i t t i n g a r e g r e s s i o n l i n e t o the s c a t t e r o f o b s e r v a t i o n s ( x ^ , Y ^ ) . T h i s i s r e p r e s e n t e d i n F i g u r e 1, where Y^ = the r e g r e s s i o n e s t i m a t e o f Y^ . 27 Y 0 X i x Figure 1. The value of r e g r e s s i o n i n reducing v a r i a t i o n i n Y . Now, the best p r e d i c t i o n of a Y without knowing x would be the average observed value (Y) . At x^, i t i s c l e a r from t h i s diagram that we would make a very large e r r o r namely (Y^ - Y) the d e v i a t i o n of Y^ from i t s mean. However, once the r e g r e s s i o n equation has been c a l -c u l a t e d , we p r e d i c t Y to be Y^ and t h i s reduces the e r r o r , s i n z e (Y^ - Y) which i s a l a r g e part of the d e v i a t i o n has now been "explained". Therefore, t h i s leaves only a r e l a -t i v e l y s m a l l "unexplained" d e v i a t i o n (Y^ - Y ^ ). T o t a l d e v i a t i o n of Y i s the sum: ( Y T - Y) = (Y\ - Y) + (Y. - Y\ ) , f o r any i ( 3-26 ) I t f o l l o w s that E ( Y I - Y ) 2 = E ( Y I - Y) 2 + E ( Y I - Y\) 2 ( 3-27 ) where v a r i a t i o n i s defined as the sum of the squared devia-t i o n s . Since (Y. - Y) = y. = bx., i t i s convenient to , v l J ' I l ' r e w r i t e equation ( 3-27 ) as 28 £ (Y^ - Y ) 2 = b 2 E x ? + Z ( Y i - Y \ ) 2 (3-28) The f a c t t h a t e x p l a i n e d v a r i a t i o n i s the v a r i a t i o n a c c o u n t e d f o r by the e s t i m a t e d r e g r e s s i o n c o e f f i c i e n t b i s now c l a r i -f i e d , by the above e q u a t i o n . The p r o c e d u r e o f decomposing t o t a l v a r i a t i o n and the a n a l y s i s o f i t s components i s c a l l e d " a n a l y s i s o f v a r i a n c e a p p l i e d t o r e g r e s s i o n " . From the f o r e g o i n g , a n u l l h y p o t h e s i s t e s t on B may be c o n s t r u c t e d . The q u e s t i o n i s t h e n , whether the r a t i o o f the e x p l a i n e d v a r i a n c e t o u n e x p l a i n e d v a r i a n c e i s s u f f i c i e n t l y l a r g e t o r e j e c t the h y p o t h e s i s t h a t Y i s u n r e l a t e d t o x. S p e c i f i c a l l y , a t e s t o f the h y p o t h e s i s HQ."3= 0 i n v o l v e s f o r m i n g the r a t i o "F" e q u a l to v a r i a n c e e x p l a i n e d by r e g r e s s i o n d i v i d e d by the u n e x p l a i n e d v a r i a n c e e q u a l t o : b 2 ( I x ? / S 2 ) (3-29) where S 2 i s the sample v a r i a n c e o f Y. I t must be emphasized t h a t t h i s i s j u s t an a l t e r n a t i v e way o f t e s t i n g the n u l l h y p o t h e s i s w rith the use o f the " t - d i s t r i b u t i o n " : c a l c u l a t e d " t " = b/v/ S z/Zx? (3-30) For the " t - d i s t r i b u t i o n " t o be s t r i c t l y v a l i d , the s t r o n g a ssumption i s made t h a t the d i s t r i b u t i o n o f Y^ i s n o r m a l . Note t h a t the "F" and " t " d i s t r i b u t i o n s are r e l a t e d , g e n e r a l l y , as f o l l o w s : F = t 2 , where t h e r e i s one degree o f freedom i n the numerator o f F. The v a r i a t i o n i n Y w i l l now be r e l a t e d t o r . I t f o l l o w s from e q u a t i o n (3-25) t h a t b = r y ' E Y j / E x 2 . Then, s u b s t i t u t i n g t h i s v a l u e f o r b i n e q u a t i o n (3-28) 29 Z(Y i - Y ) 2 = r 2 E y ? + E(Y i - Y ^ 2 (3-31) N o t i n g t h a t y? i s by d e f i n i t i o n (Y^ - 7 ) 2 , t he s o l u t i o n f o r 2 r i s r 2 = [ E ( Y i - Y ) 2 - E (Y^ - Y.^]/Z(Y. - Y ) 2 (3-32) F i n a l l y the numerator can be r e - e x p r e s s e d by n o t i n g e q u a t i o n ( 3 - 2 7 ) . Thus r 2 = E(Y\ - Y ) 2 / E ( Y i - Y) 2 (3-33) w h i c h i s the e x p l a i n e d v a r i a t i o n o f Y d i v i d e d by the t o t a l v a r i a t i o n o f Y . C o m p l i c a t i o n s a r i s e as soon as more than two v a r i a b l e s a re i n t r o d u c e d i n t o the e q u a t i o n . To i l l u s t r a t e , c o n s i d e r a s i m p l e t h r e e v a r i a b l e example. Thus, o f our e s t i m a t e d r e g r e s s i o n e q u a t i o n i s Y = a + bx + c z , t h e n R 2 = E ( Y i - Y ) 2 / E ( Y i - Y ) 2 (3-34) w h i c h i s the e x p l a i n e d v a r i a t i o n o f Y d i v i d e d by the t o t a l v a r i a t i o n o f Y . Note t h a t t h i s c a l c u l a t i o n i s i d e n t i c a l t o r 2 i f t h e r e i s o n l y one independent v a r i a b l e . I f t h e r e i s more than one independent v a r i a b l e , t h e n the numerator r e p r e s e n t s the v a r i a t i o n o f Y e x p l a i n e d by a l l independent v a r i a b l e s . Thus, as a d d i t i o n a l e x p l a n a t o r y v a r i a b l e s are added t o t h e model, we can i m m e d i a t e l y see how h e l p f u l t h e s e v a r i a b l e s are i n i m p r o v i n g our e x p l a n a t i o n o f Y by w a t c h i n g how f a s t R 2 i n c r e a s e s i n e q u a t i o n ( 3 - 3 4 ) . F i n a l l y , i t has been p r o v e d t h a t e q u a t i o n (3-28) can be g e n e r a l i z e d i n the m u l t i p l e r e g r e s s i o n case t o : 30 t o t a l v a r i a t i o n = v a r i a t i o n explained by ( X j ,x2 + additional v a r i a t i o n explained by x n + unexplained v a r i a t i o n . (3-35) This statement can be used to construct the r a t i o "F" = additional variance explained by x n divided by unexplained variance. (3-36) It is now appropriate to summarize the differences between the regression and c o r r e l a t i o n models. The two models d i f f e r in the assumptions made about the independent variables. The regression model makes few assumptions about the independent variables, but the more r e s t r i c t i v e c o r r e l a t i o n model requires that the independent variables be random variables, forming with Y a multivariate normal d i s t r i b u t i o n . The regression model may be used to describe the f e r t i l i z e r - y i e l d problemwhere f e r t i l i z e r a pplication is assumed fix e d on the one hand, or gives r i s e to a b i v a r i a t e normal population of f e r t i l i z e r and y i e l d on the other. However, the c o r r e l a t i o n model describes only the l a t t e r . It i s true that r 2 can be calculated even when f e r t i l i z e r is f i x e d , as an ind i c a t i o n of how e f f e c t i v e l y regression reduces v a r i a t i o n ; but r cannot he used for inferences about the population parameter, p. In addition, regression answers more i n t e r e s t i n g questions. Like c o r r e l a t i o n , i t not only indicates i f two variables move together; but also estimates how. Moreover, i t can be shown that a key issue i n c o r r e l a t i o n analysis the test of the n u l l 31 h y p o t h e s i s HQ:p = 0 can be answered d i r e c t l y from r e g r e s s i o n a n a l y s i s by t e s t i n g the e q u i v a l e n t n u l l h y p o t h e s i s H q : 3 = 0. Thus, r e j e c t i o n o f 6 = 0 i m p l i e s r e j e c t i o n o f p = 0, and the c o n c l u s i o n must be t h a t c o r r e l a t i o n does e x i s t between f e r t i l i z e r and y i e l d . S i n c e r e g r e s s i o n answers a b r o a d e r and more i n t e r e s t i n g s e t o f q u e s t i o n s , as w e l l as some c o r r e l a t i o n q u e s t i o n s , i t becomes the more comprehensive t e c h n i q u e . To ~sum up, w h i l e s i m p l e c o r r e l a t i o n a n a l y s i s c o r r e s p o n d s t o s i m p l e r e g r e s s i o n a n a l y s i s , the p a r t i a l c o r r e l a t i o n a n a l y s i s c o r r e s p o n d s t o m u l t i p l e r e g r e s s i o n a n a l y s i s . R e c a l l i n g how the m u l t i p l e r e g r e s s i o n c o e f f i c i e n t b e s t i m a t e s how Y i s r e l a t e d t o x i f z were c o n s t a n t , the p a r t i a l c o r r e l a t i o n c o e f f i c i e n t r i s a s i m i l a r c o n c e p t . xy, z I t e s t i m a t e s the degree t o w h i c h x and Y move t o g e t h e r i f z were h e l d c o n s t a n t . R e j e c t i o n of the h y p o t h e s i s t h a t 3 = 0 i s e q u i v a l e n t t o r e j e c t i n g the n u l l h y p o t h e s i s t h a t p = 0 . Hence, m u l t i p l e r e g r e s s i o n w i l l n ot o n l y answer xy , z i t s own s e t o f q u e s t i o n s , but a l s o p a r t i a l c o r r e l a t i o n q u e s t i o n s as w e l l . CHAPTER IV DATA C o n d i t i o n s o f Sampling I n A p r i l 1969, B.C. Tree F r u i t s L t d . s u p p l i e d the Economics B r a n c h , C.D.A., Vancouver and the Department o f A g r i c u l t u r a l Economics, U.B.C. w i t h c u r r e n t s u r v e y d a t a , l i s t i n g a p p l e t r e e numbers a c c o r d i n g t o y e a r o f p l a n t i n g , r o o t s t o c k c a t e g o r y and v a r i e t y f o r i n d i v i d u a l growers i n the Okanagan and C r e s t o n A reas o f B r i t i s h Columbia. Three major d i f f i c u l t i e s i n u s i n g the s u r v e y d a t a f o r s a m p l i n g purposes can be c i t e d : 1) R o o t s t o c k c a t e g o r i e s w h i l e g e n e r a l l y i n d i c a t i n g t r e e s i z e would no doubt f a i l t o do so i n the case o f i n t e r m e d i a t e s t o c k s and spur s t r a i n s ( u n l e s s growers t h e m s e l v e s c o r r e c t e d f o r t h i s f a c t o r ) . 2) No d a t a were shown f o r a s t a n d a r d r o o t s t o c k c a t e g o r y . S e m i - s t a n d a r d , semi-dwarf and dwarf r o o t s t o c k c a t e g o r i e s were i n c l u d e d . 3) In showing d a t a f o r an i n d i v i d u a l grower no d i s t i n c t i o n was made between t r e e s i n an homo-geneous o r c h a r d a r e a and t r e e s i n an i n t e r p l a n t e d o r c h a r d a r e a . In o r d e r t o meet the o b j e c t i v e s o f t h i s s t u d y , i t was n e c e s s a r y t o s e l e c t a r e p r e s e n t a t i v e sub-sample o f ap p l e 33 e n t e r p r i s e s f o r each o f the t r e e - s i z e c a t e g o r i e s : s t a n d a r d , s e m i - s t a n d a r d , semi-dwarf, d w a r f . 1 Moreover, the t e c h n i c a l -i t i e s o f c o s t i n g e n t e r p r i s e s made i t e s s e n t i a l t h a t homogen-eous e n t e r p r i s e p l o t s s h o u l d be s e l e c t e d and c o s t e d a p a r t from the r e s t o f o r c h a r d f r u i t on c o o p e r a t i n g farms. Thus, from the f o r e g o i n g e x p l a n a t i o n , i t i s obvious t h a t the s u r v e y d a t a p r e c l u d e d the i d e a l p o p u l a t i o n enumeration o f growers and e n t e r p r i s e s , t h e r e b y , c o n s i d e r a b l y r e s t r i c t i n g s a m p l i n g s o p h i s t i c a t i o n . N e v e r t h e l e s s , i t was a c c e p t e d t h a t i n view o f t h e r e b e i n g no a l t e r n a t i v e d a t a s o u r c e , the e x i s t i n g s u r v e y d a t a c o u l d p r o v i d e an enumeration w h i c h a l t h o u g h hot i d e a l , would a t l e a s t l e a d t o a b e t t e r sample ( w i t h time and s t a f f a v a i l a b l e ) than any a l t e r n a t i v e p r o c e d u r e which d i s p e n s e d w i t h p o p u l a t i o n d a t a and attem p t e d random s e l e c -t i o n . T h i s f a c t w i l l be more a p p r e c i a t e d when i t i s r e c a l l e d t h a t i n 1966, 4,271 census farms were r e c o r d e d i n the Okanagan census d i v i s i o n . Most o f the s e were p r o d u c i n g a p p l e s but o n l y a s m a l l p r o p o r t i o n were i n a p o s i t i o n t o h e l p w i t h the st u d y . S a m p l i n g Method The s u r v e y d a t a f o r i n d i v i d u a l a p p l e growers per-m i t t e d the f o l l o w i n g s a m p l i n g method when b r o k e n down by x I t s h o u l d be made c l e a r t h a t t r e e - s i z e c a t e g o r i e s r e f l e c t the e f f e c t s o f i n t e r m e d i a t e s t o c k s and spur s t r a i n s of S c i o n v a r i e t i e s where the s e are p r e s e n t . In the common case o f j u s t r o o t s t o c k and s c i o n o c c u r r i n g , t r e e - s i z e c a t e g o r y becomes synonymous w i t h r o o t s t o c k c a t e g o r y . 34 r o o t s t o c k c a t e g o r i e s : 1) Growers were l i s t e d a c c o r d i n g t o : a) T h e i r h a v i n g a minimum number (66) or more app l e t r e e s i n the semi-dwarf c a t e g o r y . b) T h e i r h a v i n g a minimum number (100) or more app l e t r e e s i n the dwarf c a t e g o r y , where they had not p r e v i o u s l y q u a l i f i e d under a) above . c) T h e i r h a v i n g a minimum number (33) or more apple t r e e s i n the s e m i - s t a n d a r d c a t e g o r y , where they had not p r e v i o u s l y q u a l i f i e d under a) or b) above. Growers who e n t e r e d t h e s e l i s t s were known t o be i n posses-s i o n o f a minimum number o f ap p l e t r e e s o f d i s t i n c t type ( c o r r e s p o n d i n g r e a s o n a b l y w e l l w i t h t r e e s i z e ) . T h i s would mark them as t h a t much more l i k e l y t o q u a l i f y f o r sample s e l e c t i o n , b e a r i n g i n mind the h i g h f r e q u e n c y o f i n t e r p l a n t -i n g and the need t o c o s t i n d i v i d u a l e n t e r p r i s e s o f a homo-geneous n a t u r e w i t h r e g a r d t o t r e e - s i z e c a t e g o r y , age, d e n s i t y , v a r i e t y and growing p r a c t i c e . A l s o i t was assumed t h a t growers l i s t e d i n the manner a l r e a d y e x p l a i n e d would make i t p o s s i b l e f o r a sub-sample o f s t a n d a r d t r e e - s i z e e n t e r p r i s e s t o be s e l e c t e d a l o n g w i t h o t h e r sub-samples. 2) W i t h i n each o f the t h r e e group l i s t s o u t l i n e d i n 1) above, g e o g r a p h i c a l s u b - g r o u p i n g s were made at two l e v e l s . F i r s t l y a c c o r d i n g t o N. Okanagan (Westbank and n o r t h w a r d ) , S. Okanagan (southward 35 from Westbank) and C r e s t o n a r e a s , and s e c o n d l y w i t h r e g a r d t o c o n s t i t u e n t d i s t r i c t s . 3) D i s t r i c t h o r t i c u l t u r a l i s t s were c o n s u l t e d t o make su r e t h a t l i s t s o f growers r e f e r r e d t o m a n a g e r i a l e n t i t i e s ( i . e . , no double c o u n t i n g o f a s i n g l e b u s i n e s s s t r u c t u r e was p e r m i t t e d ) . F u r t h e r m o r e , they h e l p e d up-date l i s t s whenever i t was known t h a t a v e r y r e c e n t change i n ownership o r tenancy had o c c u r r e d . 4) On the b a s i s o f f i e l d - w o r k e r a v a i l a b i l i t y and the i n e v i t a b l e drop-out r a t e f o r c o o p e r a t o r s , i t was d e c i d e d to o b t a i n 140 apple e n t e r p r i s e s f o r c o s t -i n g i n 1969, each one c o n f o r m i n g t o homogeneity c o n d i t i o n s . Knowledge o f app l e p r o d u c t i o n i n the Okanagan and C r e s t o n areas l e d t o the c o n c l u s i o n t h a t 10 e n t e r p r i s e s i n the dwarf t r e e - s i z e c a t e g o r y would be adequate t o r e p r e s e n t the s m a l l t o t a l number o f such e n t e r p r i s e s . The r e m a i n i n g 130 e n t e r p r i s e s were c o n s i d e r e d b e s t a l l o c a t e d i n a p p r o x i -m a t e l y e q u a l numbers t o s t a n d a r d , semi - s t a n d a r d , and semi-dwarf c a t e g o r i e s . S i n c e the study r e q u i r e d d e t a i l e d e n t e r p r i s e s c o s t i n g s w h i c h made i t n e c e s s a r y f o r a s s o c i a t e d t o t a l farm d a t a t o be c o l l e c t e d , i t was d e c i d e d t o l i m i t the t o t a l farm a c c o u n t s t o around 100 i n o r d e r to ensure adequate f i e l d -w o r k e r t i m e . Up t o two e n t e r p r i s e s were p e r m i t t e d f o r each c o o p e r a t o r , a l t h o u g h i t was c o r r e c t l y deduced t h a t many 36 c o o p e r a t o r s would s e t t l e f o r one e n t e r p r i s e . 2 5) I t i s now r e l e v a n t t o d i s c u s s the purpose o f s t r a t i f i c a t i o n on the b a s i s o f e n t e r p r i s e , a r e a and d i s t r i c t as r e f e r r e d t o i n S e c t i o n 1 - 3 above. The breakdown o f a p p l e grower numbers a l o n g the l i n e s a l r e a d y d e s c r i b e d i s g i v e n below. R o o t s t o c k C a t e g o r y Grower P o p u l a t i o n by A r e a (a) N. Okanagan S. Okanagan C r e s t o n (b) Semi-dwarf 109 181 Dwarf 11 + 9 + S e m i - s t a n d a r d 139 + 81 + (a) F o r Dwarf and S e m i - s t a n d a r d c a t e g o r i e s , the numbers o f growers were i n excess o f f i g u r e s shown and t h i s i s i n d i c a t e d by p l u s s i g n s . See S e c t i o n 1. above. (b) No breakdown f o r the C r e s t o n a r e a i s g i v e n , s i n c e a s m a l l number o f e n t e r p r i s e s , i n v o l v e 4 g r o w e r s , was s e l e c t e d by judgment. I t seemed r e a s o n a b l e t o e x p e c t t h a t s a m p l i n g w i t h i n t h e semi-dwarf and dwarf c a t e g o r i e s by means of random l i s t i n g o f growers ( i n v o l v i n g s u b s t i t u t i o n p r o c e d u r e and, i f neces-s a r y , e x h a u s t i o n o f l i s t s ) would a c h i e v e random s e l e c t i o n o f sub-samples a c r o s s areas f o r the f o u r c a t e g o r i e s o f apple 2The d i s c u s s i o n concerns i n i t i a l s e l e c t i o n o f e n t e r p r i s e s . L a t e r i n the s t u d y , a few i n i t i a l s i n g l e e n t e r p r i s e s underwent p a r t i t i o n i n g t o s a f e g u a r d homogeneity c o n d i t i o n s and f a c i l i t a t e a n a l y s i s . M o d i f i c a t i o n o f t h i s t y pe c o u l d l e a d t o a grower e v e n t u a l l y c o n t r i b u t i n g more than two e n t e r p r i s e s . 37 e n t e r p r i s e s . 3 O b t a i n i n g more than one e n t e r p r i s e from a grower w o u l d not a f f e c t randomness p r o v i d i n g a l l growers c o n t a c t e d were g i v e n an e q u a l chance o f c o o p e r a t i n g . Ad-m i t t e d l y , the p a r e n t p o p u l a t i o n o f growers was somewhat reduced i n t h i s case and i t might be argued more comprehen-s i v e l y i n terms o f the e n t e r p r i s e c o n s t i t u e n c y shown by i n d i v i d u a l growers. However, i t was s t i l l t hought s a t i s -f a c t o r y from the s t a n d p o i n t o f u s e f u l , s t a t i s t i c a l i n f e r e n c e and i t h e l d hopes o f b e i n g h i g h l y e f f i c i e n t i n terms o f f i e l d - w o r k . U n f o r t u n a t e l y , i t soon became c l e a r t h a t growers c o n t a c t w i t h i n the semi-dwarf and dwarf c a t e g o r i e s showed a h i g h i n c i d e n c e o f e i t h e r a) i n a b i l i t y t o help,- or b) u n w i l l -i n g n e s s t o h e l p , even when p o s s i b l e . In f a c t , t h e former was the more i m p o r t a n t owing t o the l a c k o f homogeneous e n t e r p r i s e u n i t s . W ith t h i s e x p e r i e n c e i n mind, the d e c i s i o n was made t o e x t e n d c o n t a c t s t o the s e m i - s t a n d a r d c a t e g o r y l i s t i n g . I n f a c t , the v e r y h i g h r a t e o f s u b s t i t u t i o n on randomized l i s t i n g s f o r the Okanagan areas meant t h a t p r a c -t i c a l l y a l l growers l i s t e d were c o n t a c t e d t o a c h i e v e s u f f i -c i e n t numbers o f c o o p e r a t o r s f o r each o f the sub-samples. Because o f the r e l a t i v e l y s m a l l acreage o f a p p l e s i n the C r e s t o n a r e a , a judgment sample o f e n t e r p r i s e s i n v o l v i n g f o u r growers was o b t a i n e d t h e r e w i t h the h e l p o f the 3 I f e x p e r i e n c e showed t h a t e x h a u s t i o n o f l i s t s would be u n n e c e s s a r y , a p r o c e d u r e was d e v i s e d f o r m a i n t a i n i n g a r e a r e p r e s e n t a t i o n i n sub-samples. 38 D i s t r i c t H o r t i c u l t u r i s t . The f i n a l breakdown o f ap p l e e n t e r p r i s e s composing the i n i t i a l t o t a l sample drawn i n the s p r i n g and summer o f 1969 was as f o l l o w s : TABLE V T r e e - S i z e C l a s s i f i c a t i o n o f I n i t i a l T o t a l Sample No. of E n t e r p r i s e s f o r E n t e r p r i s e C a t e g o r y Okanagan and C r e s t o n Areas S t a n d a r d 37 S e m i - s t a n d a r d 60 Semi-dwarf 38 Dwarf 7 TOTAL 14 2 However, i t s h o u l d be made c l e a r t h a t the f i n a l sample o f apple p l o t s used i n the s t u d y (n=119) n e c e s s i t a t e d d e l e t i o n from the above l i s t where d a t a p r o v e d u n s a t i s f a c t o r y as w e l l as some p a r t i t i o n i n g of e n t e r p r i s e d a t a t o ensure t h a t homogeneity c o n d i t i o n s were met. An attempt was made t o c a t e g o r i z e t r e e s i n t o f o u r t r e e - s i z e groups i n accordance w i t h a method s u g g e s t e d by Dr. D. F i s h e r , Summerland Research S t a t i o n . 4 T h i s c l a s s i f i -*It was s u g g e s t e d t h a t f o u r i n f l u e n c i n g f a c t o r s , e.g., r o o t -s t o c k , i n t e r m e d i a t e s t o c k , s c i o n v a r i e t y and s o i l type be c o n s i d e r e d i n o r d e r t h a t t r e e s i z e c o u l d be r e p r e s e n t e d by an i n d e x v a l u e . For example, g o l d e n d e l i c i o u s on s t a n d a r d i n t e r m e d i a t e s t o c k on s e e d l i n g r o o t s t o c k on poor s o i l -1.0x1.0x1.0x0.60 = 0.60. T h i s i n d e x v a l u e would c a t e g o r i z e the above example as semi-dwarf i n t r e e s i z e . S i n c e no a c c u r a t e i n f o r m a t i o n on s o i l types was a v a i l a b l e f o r the s t u d y , o n l y the f i r s t t h r e e f a c t o r s mentioned above have been t a k e n i n t o a c c o u n t . F u r t h e r d e t a i l s o f d e r i v i n g the i n d e x are shown i n T a b l e IV i n the Appendix. 39 c a t i o n o f a p p l e t r e e was s u c c e s s f u l l y c a r r i e d out. However, v a r i e t y c l a s s i f i c a t i o n was not s a t i s f a c t o r i l y a c h i e v e d because a r i g o r o u s attempt t o group t r e e s i n t o a p p r o p r i a t e v a r i e t y l e d t o a r b i t r a r y c l a s s i f i c a t i o n . T h i s f r u s t r a t i n g e x p e r i e n c e stems l a r g e l y from the f a c t t h a t a s i n g l e p l o t , f o r example, based on t r e e - s i z e c l a s s i f i c a t i o n underwent a f u r t h e r p a r t i t i o n i n g i n o r d e r t o ensure a r i g o r o u s v a r i e t y c a t e g o r i z a t i o n . C o n s e q u e n t l y , c l a s s i f i c a t i o n a c c o r d i n g t o v a r i e t y r e s u l t e d i n the sample s i z e b e i n g expanded more r a p i d l y than when c l a s s i f i c a t i o n o f t r e e s i z e was done. The f i n a l sample breakdown o f apple e n t e r p r i s e s , based on t r e e - s i z e i s shown below: TABLE VI T r e e - S i z e C l a s s i f i c a t i o n o f Sample E n t e r p r i s e s No. o f E n t e r p r i s e s f o r T r e e - S i z e C a t e g o r y Okanagan and C r e s t o n Areas S t a n d a r d 23 S e m i - s t a n d a r d 62 Semi-dwarf 2 8 Dwarf 6 TOTAL 119 I t s h o u l d be n o t e d t h a t t r e e - s i z e c a t e g o r i e s r e f l e c t the e f f e c t s o f i n t e r m e d i a t e s t o c k s and s p u r s t r a i n s o f s c i o n v a r i e t i e s where t h e r e are p r e s e n t . In the common case o f j u s t r o o t s t o c k and s c i o n o c c u r r i n g , t r e e - s i z e c a t e g o r y becomes synonymous w i t h r o o t s t o c k c a t e g o r y as s t a t e d p r e v i o u s l y . 40 For each e n t e r p r i s e s e l e c t e d i n the s t u d y , the f o l l o w i n g i n f o r m a t i o n was o b t a i n e d : 5 1. Weight o f app l e y i e l d 2. D e n s i t y o f apple t r e e s 3. Age o f t r e e s 4. Cost o f s p r a y a p p l i e d 5. Cost o f f e r t i l i z e r a p p l i e d 6. Labour hours spent on p r u n i n g and t h i n n i n g 7. T r e e - s i z e i n d e x S i n c e b o t h h i r e d and f a m i l y l a b o u r were employed i n p r u n i n g and t h i n n i n g o p e r a t i o n s , and not a l l apple p r o d u c e r s i n the st u d y managed t o keep an u p - t o - d a t e r e c o r d o f l a b o u r h o u r s , t h e r e i s l i k e l y t o have been some memory b i a s i n r e c o r d i n g p r u n i n g and t h i n n i n g h o u r s . I n o r d e r t o c a l c u l a t e d a t a on a p e r acre b a s i s r e l e v a n t t o t a l e n t e r p r i s e d a t a were d i v i d e d by c o r r e s p o n d i n g t o t a l a c r e a g e s . Tree age and v a l u e s o f dummy v a r i a b l e s r e p r e s e n t i n g area d i f f e r e n c e s r e q u i r e d no such m o d i f i c a t i o n . I m p l i c i t i n t h i s p r o c e d u r e i s an ass u m p t i o n t h a t a l l independent v a r i a b l e s had n o t h i n g t o do w i t h v a r i a t i o n i n ac r e a g e . The ou t p u t v a r i a b l e and n o n - l a n d i n p u t v a r i a b l e s e n t e r i n t o the a n a l y s i s as co-e f f i c i e n t s o r q u a n t i t i e s p e r a c r e . Independent v a r i a b l e s used i n the r e g r e s s i o n a n a l y s i s are as f o l l o w s : 6 1. A p p l e y i e l d (Y) p e r acre measured i n pounds. 2. D e n s i t y (D) measured i n terms o f number o f t r e e s per acre (range i n s t u d y 48 - 605). 5A copy o f the i n f o r m a t i o n s h e e t i s p r e s e n t e d i n the Appendix. 6 A f u l l l i s t o f independent v a r i a b l e s ( e x c e p t the dummy v a r i a b l e ) i s g i v e n i n T a b l e V I I i n the Appendix. 41 Age (A) o f t r e e s measured i n y e a r s (range i n s t u d y 4 - 55). I t i s o f i n t e r e s t t o note t h a t apple t r e e s of one to t h r e e y e a r s o f age, which were i n c l u d e d i n the i n i t i a l s a m p l i n g , d i d not be a r any r e c o g n i z a b l e amount o f f r u i t f o r t h e y e a r i n which the s t u d y was cond u c t e d . F e r t i l i z e r (F) measured i n $ c o s t p e r a c r e . Data r e g a r d i n g the amount o f f e r t i l i z e r used was thought t o be l e s s r e l i a b l e than the c o s t e s t i m a t e s o b t a i n e d from growers. Spray (S) measured i n $ c o s t p e r acre f o r the same reasons as above. In f a c t , amounts o f s p r a y a c t u a l l y r e p o r t e d were so heterogeneous t h a t i t was v i r t u a l l y i m p o s s i b l e t o d e r i v e a m e a n i n g f u l i n t e r p r e t a t i o n . P r u n i n g and t h i n n i n g hours (P) measured i n t o t a l hours p e r a c r e spent on the s e p r a c t i c e s and i n c l u d e h i r e d and o t h e r f a m i l y l a b o u r h o u r s . T r e e - s i z e i n d e x (T) c a l c u l a t e d a c c o r d i n g t o a method s u g g e s t e d by F i s h e r , as e x p l a i n e d on Page 38. Dummy v a r i a b l e (G) used f o r s e v e r a l p u r p o s e s . The Okanagan a r e a was d i v i d e d i n t o N o r t h and South r e g i o n s j u s t n o r t h o f Summerland. T h i s d i v i s i o n was made because o f e n v i r o n m e n t a l d i f f e r e n c e s i n the two r e g i o n s , which were assumed t o account f o r some v a r i -a t i o n i n ap p l e y i e l d s . D i f f e r e n c e s o b s e r v e d between the N o r t h and South Okanagan r e g i o n s i n c l u d e v a r i a t i o n s i n s o i l type and weather o b s e r v a t i o n s f o r the y e a r 42 under s t u d y . A c c o r d i n g to the " C l i m a t e o f B r i t i s h C o lumbia R e p o r t " f o r 1968 - 1969, s l i g h t l y d i f f e r e n t mean t e m p e r a t u r e s f o r the two r e g i o n s were r e g i s t e r e d d u r i n g the p e r i o d May 1968 t o May 1969. The average t e m p e r a t u r e s were 44°F. and 48°F. i n the N o r t h and South r e g i o n s r e s p e c t i v e l y . These t e m p e r a t u r e s were r e c o r d e d i n the growing p e r i o d , w h i c h i s d e f i n e d as the number o f days w i t h an average d a i l y t e m p e r a t u r e above 4 3 & F . 7 The Report a l s o showed t h a t i n 1969 t h e r e were s l i g h t d i f f e r e n c e s between the two r e g i o n s i n p r e c i p i t a t i o n f o r the months May t o O c t o b e r i n c l u s i v e . L o n g l e y used the May t o October p e r i o d and found t h e r e e x i s t e d a n e g a t i v e r e l a t i o n s h i p between r a i n f a l l and a p p l e p r o d u c t i o n f o r the A n n a p o l i s V a l l e y i n Nova S c o t i a . 8 The average p r e c i p i t a t i o n i n the n o r t h e r n r e g i o n o f the Okanagan ran from 1.09 t o 1.23 i n c h e s f o r the i n d i c a t e d p e r i o d , whereas i n the s o u t h e r n r e g i o n a r e l a t i v e l y low average r a i n f a l l o f 0.10 t o 0.92 i n c h e s was r e p o r t e d f o r the same p e r i o d . V a r i a t i o n s i n y i e l d due to weather f a c t o r s may be f u r t h e r c l a s s i f i e d a c c o r d i n g t o d i r e c t o r i n d i r e c t a c t i o n o f the c a u s a l agent. I t i s v e r y l i k e l y t h a t such weather components as h u m i d i t y , l i g h t and a i r movements d i r e c t l y 7The C l i m a t e of B r i t i s h Columbia - T a b l e s o f Temperature, P r e c i p i t a t i o n s , and Sunshine Report f o r 1969 - 1970, P r o v i n c e o f B r i t i s h Columbia Department o f A g r i c u l t u r e , pp. 9-10. e L o n g l e y . , op. c i t . , pp. 22-23. 43 i n f l u e n c e y i e l d s . M oreover, owing t o the r e l a t i o n s h i p s and i n t e r r e l a t i o n s h i p s p r e s e n t i n weather components, y i e l d s w i l l be i n d i r e c t l y a f f e c t e d . The i n t e n s i t y o f c e r t a i n i n s e c t i n f e s t a t i o n s and p l a n t d i s e a s e s , f o r i n s t a n c e , i s a f f e c t e d by weather. The e f f e c t o f weather on app l e y i e l d can a l s o v a r y w i t h the l e v e l o f f e r t i l i z e r , s o i l t y p e , c u l t u r a l p r a c t i c e s , and many o t h e r f a c t o r s . Because o f t h i s c o m p l e x i t y , the f o l l o w i n g assumptions were made: (1) non-weather i n f l u e n c e i s u n c o r r e l a t e d w i t h weather i n f l u -ence; (2) a l l v a r i a t i o n s i n y i e l d due t o non-weather i n f l u -ences are n o r m a l l y d i s t r i b u t e d w i t h an e x p e c t e d v a l u e o f ze r o and a f i n i t e v a r i a n c e . Data were s p l i t i n t o two p a r t s o f a p p r o x i m a t e l y e q u a l s i z e . F i f t y - f o u r e n t e r p r i s e p l o t s out o f a t o t a l o f one hundred and n i n e t e e n were a s s i g n e d t o the n o r t h Okanagan r e g i o n and the r e m a i n i n g s i x t y - f i v e e n t e r p r i s e p l o t s t o the s o u t h Okanagan r e g i o n . R e g i o n a l d i f f e r e n c e s i n apple y i e l d s , as e x p l a i n e d e a r l i e r , can t h e o r e t i c a l l y be p a r t l y e x p l a i n e d by dummy v a r i a b l e s . A dummy v a r i a b l e i s o n l y an i n d i c a t o r v a r i a b l e . I t has o n l y two n u m e r i c a l v a l u e s . I n the case o f the Okanagan r e g i o n s '1' was a s s i g n e d t o any e n t e r p r i s e p l o t i n the s o u t h and '0' was a s s i g n e d t o any e n t e r p r i s e p l o t i n the n o r t h . M o d i f i c a t i o n was n e c e s s a r y i n the use o f '0' and '1' when the Cobb-Douglas f u n c t i o n was used t o e s t i m a t e the y i e l d r e l a t i o n s h i p . The v a l u e z e r o becomes a problem i n the p r o c e s s o f l o g a r i t h m i c t r a n s f o r m a t i o n s , because Ln 0 approaches -°° . The a l t e r n a t i v e p a i r o f v a l u e s , 0.1 and 10, 44 were t h e r e f o r e s u b s t i t u t e d f o r '0' and '1', r e s p e c t i v e l y . These two i n d i x e s were employed to represent a y i e l d v a r i a -t i o n , i f any, which may be due mainly to d i f f e r e n c e s i n l o c a t i o n s . Of course, any p a i r of numbers would serve the purpose e q u a l l y as w e l l as 0 and 1. But the magnitude of c o e f f i c i e n t s would vary depending on the values taken by the dummy v a r i a b l e s . Hence, i n t e r p r e t a t i o n of c o e f f i c i e n t s d e r i v e d from a c e r t a i n p a i r of numbers i s bound to d i f f e r from some other p a i r of numbers. CHAPTER V EMPIRICAL RESULTS I n t r o d u c t i o n B e f o r e a t t e m p t i n g the m u l t i p l e r e g r e s s i o n a n a l y s i s , a s i m p l e r e g r e s s i o n a n a l y s i s was p e r f o r m e d o f ap p l e y i e l d on each independent v a r i a b l e , namely, d e n s i t y p e r a c r e , age o f t r e e s , t h e c o s t o f f e r t i l i z e r used per a c r e , the c o s t o f s p r a y used p e r a c r e , and p r u n i n g and t h i n n i n g l a b o u r hours p e r a c r e . The t h i n g t o note i s t h a t i t seems c o n c e p t u a l l y v e r y l i k e l y t h a t d e n s i t y p e r acre may be h i g h l y c o r r e l a t e d w i t h t r e e - s i z e i n d e x . Whether t h e s e two independent v a r i -a b l e s are c o r r e l a t e d can r e a d i l y be checked by the i n s p e c t i o n o f the c o r r e l a t i o n m a t r i x . The c o r r e l a t i o n m a t r i x i s g i v e n i n T a b l e V I I I i n the Appendix. N o t w i t h s t a n d i n g t h e p r o b a b i l -i t y o f such a c o r r e l a t i o n o c c u r r i n g , the s i m p l e r e g r e s s i o n o f d e n s i t y p e r acr e on t r e e - s i z e i n d e x was t r i e d . P r i o r t o r u n n i n g s i m p l e l i n e a r r e g r e s s i o n a n a l y s e s , d a t a were grouped a c c o r d i n g t o t r e e - s i z e c l a s s i f i c a t i o n : t h a t i s , s t a n d a r d , s e m i - s t a n d a r d , and semi-dwarf. On t h e s e c l a s s i f i e d d a t a , c o r r e s p o n d i n g s i m p l e l i n e a r r e g r e s s i o n a n a l y s e s were p e r f o r m e d w i t h r e s p e c t to each i n d i v i d u a l independent v a r i a b l e . F i n a l l y , d a t a were lumped t o g e t h e r , which p e r m i t t e d s i m p l e l i n e a r r e g r e s s i o n a n a l y s e s t o be p e r f o r m e d on a l l o v e r a l l se t o f d a t a . I n the l i g h t o f s i g n i f i c a n t r e g r e s s i o n co-e f f i c i e n t s and the b e s t ' f i t ' c r i t e r i o n , the s i m p l e l i n e a r 46 r e g r e s s i o n model, r e g a r d l e s s o f whether d a t a were d i s a g g r e -g a t e d o r n o t , f a i l e d t o i n d i c a t e any s t r o n g a p p l e y i e l d r e l a t i o n s h i p s . The e m p i r i c a l r e s u l t s from the s i m p l e l i n e a r r e g r e s s i o n a n a l y s i s i s shown i n T a b l e I I I i n the Appendix. The i m p l i c a t i o n from t h e s e r e s u l t s may be t h a t a p p l e y i e l d r e l a t i o n s h i p s e x i s t w i t h s e v e r a l v a r i a b l e s c o n s i d e r e d s i m u l t a n e o u s l y , and t h e r e f o r e any apple y i e l d r e l a t i o n s h i p might w e l l take a c u r v e l i n e a r form r a t h e r t han a s t r a i g h t l i n e . T h i s r a t i o n a l e paved the way f o r m u l t i p l e r e g r e s s i o n a n a l y s i s w h i c h i s d i s c u s s e d l a t e r i n the c h a p t e r . The m u l t i p l e r e g r e s s i o n r o u t i n e o f the "UBC TRIP" computer program wTas used t o p r o v i d e l e a s t squares r e g r e s s i o n e s t i m a t e s . 1 A n o t h e r program was used f o r the E q u a l i t y o f S l o p e T e s t t o see whether the d i f f e r e n c e s i n r e g r e s s i o n c o e f f i c i e n t s among t r e e - s i z e groups c o u l d be a s c r i b e d t o s a m p l i n g e r r o r s or t o d i f f e r e n c e s among g r o u p s . 2 To i l l u s t r a t e , a s i n g l e v a r i a b l e - o f - c l a s s i f i c a t i o n i n the form ( X ^ i , Y ^ i ) , ( X ^ 2 Y ^ 2 ) , ( X ^ 3 , Y^ 3) i s p r e s e n t e d , where X and Y r e p r e s e n t f e r t i l i z e r a p p l i e d and app l e y i e l d , f o r i n s t a n c e . The f i r s t s u b s c r i p t i denotes the number o f o b s e r v a t i o n s i n each group and the * J . H. B j e r r i n g and P. Seagraves, UBC TRIP ( T r i a n g u l a r Regres-s i o n Package) Vancouver: U.B.C, Computing C e n t r e , Nov. 1970. B i l l Coshow, UBC BMDX 64: G e n e r a l L i n e a r H y p o t h e s i s , U.B.C., Computing C e n t r e , August 1971. 2 C h i n h L e - D i n h , UBC SLTEST: E q u a l i t y o f S l o p e T e s t , U.B.C., Computing C e n t r e , June 1971. 47 second s u b s c r i p t 1, 2, and 3 denote c o r r e s p o n d i n g groups: 1 r e p r e s e n t s a s t a n d a r d a p p l e group, e t c . N a t u r a l l y t h e s e p r o c e d u r e s e x t e n d t o more th a n a s i n g l e v a r i a b l e - o f - c l a s s i f i -c a t i o n . Suppose a q u e s t i o n a r i s e s as t o whether the r e g r e s -s i o n l i n e s c o r r e s p o n d i n g t o each group are t o be r e g a r d e d as the same. To answer t h e q u e s t i o n a d e q u a t e l y r e q u i r e s con-s t r u c t i o n o f the c o v a r i a n c e t a b l e , as shown below. I t w i l l be c o n v e n i e n t t o denote the q u a n t i t i e s i n T a b l e IX by i n d i -v i d u a l l e t t e r s . TABLE IX C o v a r i a n c e T a b l e f o r the Three T r e e - S i z e Groups W i t h i n each group E x z Exy E y z Ey 1 * C C C C' x x x x y j y y ! y y x C C C C' x x 2 x y 2 y y 2 y y 2 C C C C' xx 3 xy 3 yy 3 yy 3 1 2 3 Among means W i t h i n groups T o t a l C C C C' xxm xym yym yym C C C C' xxw xyw yyw yyw C . C + C «. C x x t x y t y y t y y t The d e f i n i t i o n s o f the q u a n t i t i e s t o be computed are as f o l l o w s : C v , C v , C v r e p r e s e n t the c o m p u t a t i o n EX 2 - ( E X ) 2 / n X X j X A 2 XX 3 f o r groups 1, 2, 3. *"xyi * ^ x y 2 ' ^ x y 3 r e P r e s e n t t n e c o m p u t a t i o n EXY - EXEY/n f o r groups 1, 2, 3. C y y i » C y y 2 , Cyy 3 r e p r e s e n t the c o m p u t a t i o n EY 2 - ( E Y ) 2 / n f o r groups 1, 2, 3. 48 The q u a n t i t i e s i n the column E y ' 2 are computed by the f o r m u l a EY 2 - ( E X Y ) 2 / E X 2 . The q u a n t i t i e s S i , S 2 , S 3 , S 4 are d e f i n e d i n terms o f C! i n T a b l e X. 1 51 = the sum o f squares o f Y v a l u e s from the r e g r e s s i o n l i n e i n each group, t o t a l l e d f o r a l l groups. 5 2 = the v a r i a t i o n among r e g r e s s i o n c o e f f i c i e n t s o f the d i f f e r e n t groups 5 3 = the sum o f squares o f d e v i a t i o n s o f the means from the r e g r e s s i o n l i n e o f the means w i t h r e g a r d t o Y v a l u e s . Sn = the square o f the d i f f e r e n c e between c o e f f i c i e n t s w i t h i n groups (bv,) and c o e f f i c i e n t s among means (b ) , and S t = S x + S 2 + S 3 + Sn (see Ta b l e XI i n A p p e n d i x ) . TABLE X Ta b l e f o r S i i n terms o f D e f i n i t i o n s o f S. 1 D.F Sj = c» . y y i k ( n - 2) S 2 = C* yyw S k - 1 S 3 = C 3 yym k - 2 S4 = c -y y t C ' - C' yyw yym 1 T o t a l S t = y y t kn - 2 R e f e r r i n g t o T a b l e X, n = the number o f o b s e r v a t i o n s and k = the number o f groups. T h e r e f o r e , a t e s t o f whether one r e -g r e s s i o n l i n e can be used f o r a l l o b s e r v a t i o n s can be f o r m u l a t e d as f o l l o w s : 49 S 2 + S 3 + S H F 2(k - 1) S i k ( n - 2) The E q u a l i t y o f S l o p e T e s t was used f o r t h r e e d i f f e r e n t t r e e - s i z e g r o u p s , each group c o n t a i n i n g twenty-seven independent v a r i a b l e s . I t was used o n l y f o r the q u a d r a t i c f u n c t i o n because o f the p r i o r i t y g i v e n t o t h a t f u n c t i o n as e x p l a i n e d i n Chapter I. The r e s u l t s s u p p o r t the h y p o t h e s i s t h a t t h e r e are no d i f f e r e n c e s i n c o r r e s p o n d i n g r e g r e s s i o n c o e f f i c i e n t s among the groups S t a n d a r d v e r s u s S e m i - s t a n d a r d , S t a n d a r d v e r s u s Semi-dwarf, and Semi - s t a n d a r d v e r s u s Semi-Dwarf. The c a l c u l a t e d F = 0.29 and t a b u l a t e d F Q 5 (D.F.: 30, 55) = 1.67. Approximate v a l u e s are t a k e n because the T a b l e f o r the F - t e s t i n Snedecor's S t a t i s t i c a l Method does not g i v e a v a l u e w i t h 35 and 54 degrees o f freedom, the ones r e l e v a n t t o the a n a l y s i s . The r e s u l t s from the F - t e s t are p r e s e n t e d i n T a b l e XI i n the Appendix. U s i n g t r e e - s i z e i n d e x , the f o l l o w i n g c a t e g o r i z a t i o n seemed t o be r e a s o n a b l e : S t a n d a r d t r e e f a l l i n g i n the range 0.97 - 1.00; S e m i - s t a n d a r d i n 0.61 - 0.88; Semi-dwarf t r e e i n 0.25 - 0.60; and Dwarf t r e e i n 0.20 - 0.21. An i m p o r t a n t t h i n g t o note i s t h a t no account was t a k e n s e p a r a t e l y o f the dwarf t r e e group i n the s t u d y . T h i s group c o n s i s t e d o f o n l y s i x e n t e r p r i s e s . As s u c h , i t seemed 50 p r a c t i c a l to merge the group i n with the semi-dwarf group. Further d e t a i l s regarding the Equality of Slope Test w i l l dealt with l a t e r . The results of the Equality of Slope tests for tree-size groups were obtained as follows: (1) regres-sion equation for each sample; (2) the twenty-seven common slope c o e f f i c i e n t s ; (3) F-ratio and i t s p r o b a b i l i t y . This information i s given in Table XI i n the Appendix. On the evidence of no difference among regression c o e f f i c i e n t s the three tree-size groups were combined so that a single regression equation might be f i t t e d . Thus, two basic re-gression models were applied to the o v e r a l l enterprise data. The two basic models used in the ensuing regression analysis are as follows: one is a Cobb-Dougla.s function l i n e a r i n logarithms; Ln Y = Ln a + B 2 Ln D + 6 3 Ln A + 8 4 Ln F + B i 5 Ln S + B6 Ln P + B? Ln G + B 8 Ln T + Ln V . A l l that i s necessary now i s a simple renaming of the terms i n t h i s equat ion: Y = Ln Y = Y i e l d Bi = Ln a X i = Ln D = Log dens i t y x 2 = Ln A = Log age x 3 = Ln F = Log cost of f e r t i l i z e r X„ = Ln S = Log cost of spray x 5 = Ln P = Log hours i n pruning and thinning x 6 = Ln G = Log geographical dummy x 7 = Ln T = Log tree-size index e = Ln V , where X i , X 2 , , X 7 represent independent v a r i 51 a b l e s used i n the s t u d y . Assume t h a t V i s d i s t r i b u t e d so t h a t Q = Ln V s a t i s f i e s the assumptions made e a r l i e r ; namely, Ln V % N ( 0 , a 2 ) . S u b s e q u e n t l y , the l o g a r i t h m i c e q u a t i o n appears as a f a m i l i a r l i n e a r model. In m a t r i x n o t a t i o n , V s v//5 + p 1 X l , 2 > > X i , 8 3 i e i Y 2 1 X 2 , 2 > j X 2 , 8 32 e 2 Y 3 1 ' X 3 ) 2 , , X 3 , 8 e 3 • = • • • + • 3 8 Y 1 1 9 1 X l l 9 , 2 , , X i l 9 , 8 e 1 1 9 Assume t h a t E (6) = O , Cov (Q ) = a 2S , i . e . The i m p o r t a n t r e q u i r e m e n t o f t h i s l o g a r i t h m t r a n s -f o r m a t i o n i s t h a t the e r r o r term i n the n a t u r a l form e q u a t i o n i s m u l t i p l i c a t i v e . I f t h i s assumption i s u n w a r r a n t e d , then 52 the model w i l l r e q u i r e s p e c i a l t r e a t m e n t which i s beyond the c o n c e r n o f t h i s s t u d y . The o t h e r model used i s a q u a d r a t i c f u n c t i o n , Y = S i + B 2 D + 8 3 A + 6n F + 6 5 S + Be P + B 7 G + B 8 T + B 9 D2 + B i o A 2 + B11 F 2 + B12 S 2 + B i s P 2 + B m DA + B i s DF + B i e DS + B1 7 DP + B1 s DT + B19 AF + B 2 0 AS + B 21 AP + B 2 2 AT + B23 FS + B 2 i FP + B 2 5 FT + 826 SP + B 2 7 ST + B 2e PT + e , (where D, A, F, S, P, G, and T r e f e r t o d e n s i t y , age, c o s t o f f e r t i l i z e r ' , c o s t o f s p r a y s , p r u n i n g and t h i n n i n g h o u r s , g e o g r a p h i c a l dummy and t r e e - s i z e i n d e x r e s p e c t i v e l y ) . A g a i n , a l l o b s e r v a t i o n s can be s t a c k e d i n t o a column v e c t o r as f o l l o w s : Y = X/3 + e , where E ce) = 0 , Cov ( 6 ) = a Yi 1 x i >2 , — " , X1 , 2 e Bi e i Y 2 1 X2 ,2 , - - -• _ , X 2 , 2 8 B2 e 2 = ; + • B 2 £ Y 1 19 1 X 11 9 , 2 , • " , X 11 9 , 2 8 e 1 19 When one v a r i a b l e i s used t o o b t a i n s e v e r a l r e -g r e s s o r s , as i n t h i s model, a q u e s t i o n may a r i s e as t o whether m u l t i c o l l i n e a r i t y becomes a problem. For example, D^ and D? are f u n c t i o n a l l y dependent ( i . e . , one i s the square o f the o t h e r ) ; t h e y are not l i n e a r l y dependent ( i . e . , one i s n o t , s a y , t w i c e the o t h e r ) . 53 G e o m e t r i c a l l y , the c o - o r d i n a t e p o i n t s (D, D 2) l i e on a c u r v e as shown i n F i g u r e 2 below; the i m p o r t a n t t h i n g however, i s t h a t they do not l i e on a s t r a i g h t l i n e . Thus, the p r o blem o f m u l t i c o l l i n e a r i t y may or may not be a v o i d e d a c c o r d i n g t o the degree o f c u r v a t u r e i n v o l v e d . F i g u r e 2. P o l y n o m i a l r e g r e s s i o n as a s p e c i a l case o f m u l t i p l e r e g r e s s i o n . The o u t p u t o f the T r i p program f o r b o t h r e g r e s s i o n models i n c l u d e d : (1) the e s t i m a t e d r e g r e s s i o n c o e f f i c i e n t s ; (2) the s t a n d a r d e r r o r o f each c o e f f i c i e n t ; (3) t h e F - r a t i o and a s s o c i a t e d p r o b a b i l i t y f o r each r e g r e s s i o n c o e f f i c i e n t ; (4) the s t a n d a r d e r r o r of the e s t i m a t e , Y; ( f ) the c o e f f i c i e n t o f m u l t i p l e d e t e r m i n a t i o n , R 2; and (6) the c o r r e l a t i o n m a t r i x . In showing data, s u b s e q u e n t l y s t a n d a r d e r r o r s o f the r e g r e s s i o n c o e f f i c i e n t s are shown i n p a r e n t h e s i s . The a s s o c i a t e d p r o b a b i l i t y o f the F - r a t i o f o r each c o e f f i c i e n t i s shown below each s t a n d a r d e r r o r . The r e s u l t s o f the 54 e s t i m a t e d s t e p w i s e Cobb-Douglas r e g r e s s i o n e q u a t i o n are g i v e n i n T a b l e X I I i n the Appendix and the c o r r e l a t i o n m a t r i x f o r the Cobb-Douglas f u n c t i o n i s shown i n T a b l e X I I I i n the Appendix. The r e s u l t s o f the e s t i m a t e d q u a d r a t i c model are p r e s e n t e d on page 57 and the s t e p - w i s e r e g r e s s i o n e q u a t i o n i s shown i n T a b l e XIV i n the Appendix. I t s c o r r e l a t i o n m a t r i x appears i n T a b l e XV i n the Appendix. R e s u l t s from the Cobb-Douglas Model The e s t i m a t e d o v e r a l l e n t e r p r i s e r e g r e s s i o n equa-t i o n i n l o g a r i t h m s i s : Y = 1.5514 + 0.5713 D + 1.2263 A + 0.22263 F + 0.2485 S + (1.4674) (0.2331) (0.2226) (0.0965) (0.1053) 0.1967 P + 0.1069 G + 0.1712 T . R 2 = 0.4147, where (0.0748) (0.0423) (0.2455) a l l v a r i a b l e s are e x p r e s s e d i n l o g a r i t h m i c form. The r e g r e s s i o n c o e f f i c i e n t s f o r a l l v a r i a b l e s e x c e p t t h a t f o r t r e e s i z e were found s i g n i f i c a n t l y d i f f e r e n t from z e r o at the 5% l e v e l o f p r o b a b i l i t y . A p p r o x i m a t e l y 401 o f t o t a l v a r i a t i o n i n crop y i e l d (Y) has been acc o u n t e d f o r by t h e independent v a r i a b l e s . The v a l u e o f R 2 i s not improved i n the s t e p w i s e r e g r e s s i o n e q u a t i o n when o n l y those independent v a r i a b l e s w h i c h make a s i g n i f i c a n t c o n t r i b u t i o n t o apple y i e l d are i n c l u d e d . The r e s u l t s o f the s t e p w i s e r e g r e s s i o n and c o r r e s p o n d i n g c o r r e l a t i o n m a t r i x are shown i n T a b l e XII and X I I I r e s p e c t i v e l y i n the Appendix. In view o f the f a c t t h a t p r i m a r y i n t e r e s t i n t h e 55 a p p l e s t u d y i s i n the r e g r e s s i o n model r a t h e r t h a n the c o r r e l a t i o n model, the m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t R cannot be c o n s i d e r e d s t r i c t l y as an e s t i m a t e o f the p o p u l a -t i o n c o r r e l a t i o n between the independent v a r i a b l e s and the dependent v a r i a b l e . T h i s i s because the independent v a r i a b l e s i n the r e g r e s s i o n model are o b s e r v e d i n terms o f g i v e n v a l u e s and n o t a m u l t i v a r i a t e normal d i s t r i b u t i o n . Even s o , R does p r o v i d e a summary s t a t i s t i c t o measure the good-ness o f f i t o f the o b s e r v e d p o i n t s t o the r e g r e s s i o n p l a n e . 3 R e s u l t s from t h e Q u a d r a t i c Model B e f o r e p r o c e e d i n g w i t h the combined d a t a , s e v e r a l p o i n t s s h o u l d be made i n o r d e r to c l a r i f y the u n d e r l y i n g con-c e p t s i n v o l v e d i n t h e employment o f E q u a l i t y o f S l o p e T e s t . F i r s t l y , the T e s t i s p a r t o f " A n a l y s i s o f C o v a r i -ance", the p r i m a r y c o n c e r n o f w h i c h i s to f i n d out whether a s i n g l e r e g r e s s i o n l i n e i s s t a t i s t i c a l l y v a l i d i n r e p r e s e n t -i n g a y i e l d r e l a t i o n s h i p . C o n s e q u e n t l y , the a n a l y s i s does not produce R 2 v a l u e s . S e c o n d l y , the a n a l y s i s i s i n c a p a b l e o f a u t o m a t i c a l l y e l i m i n a t i n g the i n s i g n i f i c a n t v a r i a b l e s and p e r f o r m i n g the E q u a l i t y o f S l o p e T e s t w i t h o n l y the r e m a i n i n g s i g n i f i c a n t v a r i a b l e s . T h i r d l y , t h e most d i f f i -c u l t p r o b lem i s i n d e c i d i n g w h i c h v a r i a b l e s are t o be r e t a i n e d , and w h i c h are t o be o m i t t e d from the model. G e n e r a t i o n o f i n n u m e r a b l e terms from v a r i a b l e s 3M. E z e k i e l and K. A. Fox, Methods o f C o r r e l a t i o n and Regres-s i o n A n a l y s i s , 3 r d e d i t i o n . pp. 2 70-281 , 1965. 56 s q u a r e d or c o m b i n a t i o n s o f the seven b a s i c independent v a r i -a b l e s i s p o s s i b l e : G 2, T 2, GT, e t c . But from the s u b j e c t i v e p o i n t o f v i e w , the v a r i a b l e s w h i c h have been e x c l u d e d would seem t o be l a c k i n g any l o g i c a l b a s i s , v a r i a b l e s i n c l u d e d are j u s t i f i e d i n t h a t they are c a p a b l e o f h e l p i n g t o r e p r e -s e n t a b i o l o g i c a l phenomenon, i . e . , the law o f d i m i n i s h i n g r e t u r n s i n the case o f t h e sq u a r e d terms. Even s o , sq u a r e d terms l i k e G 2 and T 2 can i n no way ap p e a l t o the senses by whi c h s u b j e c t i v e judgment i s made. For the same r e a s o n , some o f the c r o s s - t e r m s do not appear i n the model. The "UBC SLTEST" was used f o r the purpose o f E q u a l i t y o f Sl o p e T e s t . The r e s u l t o f the t e s t w i t h r e s p e c t t o each r e g r e s s i o n e q u a t i o n i s as f o l l o w s : 1. R e g r e s s i o n E q u a t i o n f o r S t a n d a r d T r e e - S i z e Group. Y = 989.300 + 1212 D + 15.060 A + 1457 F + 3374 S + 2742 P + 1466 G - 1043 T - 29.360 D 2 - 235.600 A 2 + 18.130 F 2 + 2.353 S 2 - 2.143 P 2 - 118.700 DA + 55.200 DF + 15.330 DS - 24.650 DP + 7484 DT + 95.960 AF + 62.690 AS - 30.760 AP - 12.980 AT - 7.246 FS - 2.846 FP - 8113 FT + 2.474 SP - 7018 ST + 11.130. 2. R e g r e s s i o n E q u a t i o n f o r Semi- S t a n d a r d T r e e - S i z e Group. Y = - 7.650 - 668.4000 D + 2.9720 A - 4246 F + 2872 S - 2515 P - 878.4000 G - 2.2360 T - 0.8332 D 2 - 265.7000 A + 15.6500 F 2 + 1.3470 S 2 + 2.5170 P 2 + 1.7850 DA - 9.5550 - 6487 DS + 9.4690 DP + 1263 DT + 24.4000 AF - 36.0300 AS + 17.3800 AP - 2.7030 AT + 30.4000 FS - 15.6700 FP 57 + 5115 FT - 9.0710 SP - 2186 ST + 1863 PT . 3. R e g r e s s i o n E q u a t i o n f o r Semi-Dwarf T r e e - S i z e Group. Y = - 12.5100 - 162 D + 2793 A - 1895 F = 781.200 S + 1057 P + 491.500 G + 50.2600 T + 0.6151 D 2 + 202.7000 A 2 - 3.264 F 2 + 3.6610 S 2 - 0.1218 P 2 + 11.7900 DA + 1.9570 DF - 1.1880 DS - 0.9559 DP - 418.6000 DT - 57.0900 AF + 38.4700 AS - 29.0100 AP - 2.445 AT - 8.5100 FS - 5.8330 FP + 8 2 9 8 FT + 1.6100 SP - 3448 ST - 1442 PT . An F - t e s t was p e r f o r m e d on the t w e n t y - s e v e n co-e f f i c i e n t s h e l d i n common by each r e g r e s s i o n e q u a t i o n . The r e s u l t i n d i c a t e s t h a t t h e r e are no s i g n i f i c a n t d i f f e r e n c e s i n comparable r e g r e s s i o n c o e f f i c i e n t s among the t h r e e t r e e - s i z e groups at the 5% l e v e l o f s i g n i f i c a n c e . Data from the t e s t are p r e s e n t e d i n T a b l e XI i n the Appendix. T h e r e f o r e on the b a s i s o f t h i s r e s u l t the t h r e e s e p a r a t e samples were combined i n t o one sample. A q u a d r a t i c r e g r e s s i o n e q u a t i o n was then e s t i m a t e d as f o l l o w s : Y = - 3.200 + 96.5140 D + 3527.7863 A - 306.9661 F + 365.2424 S (7.818) (210.5007) (4881.8270) (984.6400) (451.9521) - 698.2985 P - 231.1846 G + 3.7550 T - 0.004814 D 2 - 18.7206 A 2 (379.0393) (665.2497) (8.189) (0.2094) (32.5372) - 6.9504 F 2 - 1.5232 S 2 + 0.6457 P 2 + 6.5565 DA + 1.4649 DF (3.8344) (1.1533) (0.3758) (11.9919) (2.1995) - 1.3623 DS + 1.1930 DP - 218.4573 DT - 26.8412 AF - 3.0749 AS (1.1785) (0.8984) (166.1315) (54.1606) (0.9068) - 2.0330 DS + 2610.1141 AT + 6.1613 FS + 0.1162 FP + 1004.4861 I (10.5395) (4545.6588) (3.4245) (3.2792) (901.3124) + 0.5085 SP + 170.2734 ST + 511.4871 GT . R 2 = 0.7534 (0.8558) (414.9117) (344.7191) 58 The f a c t t h a t e l e v e n r e g r e s s o r c o e f f i c i e n t s i n the above e q u a t i o n were not s i g n i f i c a n t at the 5 p e r cent l e v e l can i m m e d i a t e l y be checked by o b s e r v i n g t h a t t h e s t a n d a r d e r r o r s i n p a r e n t h e s e s o f the c o e f f i c i e n t s exceeded v a l u e s o f the c o r r e s p o n d i n g c o e f f i c i e n t s . N o n - s i g n i f i c a n c e i s a l s o t r u e o f o t h e r c o e f f i c i e n t s i n the e q u a t i o n but s t e p w i s e r e g r e s s i o n a t a l a t e r s t a g e w i l l s e l e c t the s i g n i f i c a n t v a r i a b l e s f o r a f i n a l e q u a t i o n a n a l y s i s . A q u e s t i o n at t h i s s t a g e may a r i s e as t o why t h i s k i n d o f s i t u a t i o n has o c c u r r e d . The f i r s t n e c e s s a r y s t e p t o t a k e i s t o examine whether any o f the a p p r o p r i a t e as-sumptions made i n c o n n e c t i o n w i t h e s t i m a t i n g the q u a d r a t i c f u n c t i o n have been v i o l a t e d . T h e r e f o r e , i n i t i a l l y the c o r r e l a t i o n matrix 1* must be i n v e s t i g a t e d t o see i f m u l t i -c o l l i n e a r i t y might have caused problems. A c l o s e i n s p e c t i o n shows t h a t t h e r e are a number of n e a r - l i n e a r c o m b i n a t i o n s formed between independent v a r i a b l e s and r e g r e s s o r s (no l i n e a r dependence was shown between independent v a r i a b l e s ) most o f w h i c h have been g e n e r a t e d i n the p r o c e s s o f e i t h e r s q u a r i n g an independent v a r i a b l e or i n t e r a c t i n g one i n d e -pendent v a r i a b l e w i t h a n o t h e r . These o c c u r r e n c e s are a d i r e c t v i o l a t i o n o f the a s s u m p t i o n t h a t a r e g r e s s o r D 2 , f o r example, i s f u n c t i o n a l l y b u t not l i n e a r l y dependent on D. However, i f the curve segment on w h i c h the c o o r d i n a t e p o i n t and D? i n F i g u r e 2 l i e s i s c l o s e t o the shape o f a s t r a i g h t l i n e segment, t h e r e can be problems o f m u l t i -"*See T a b l e XV i n the Appendix. 59 c o l l i n e a r i t y . I f D. and D? form an almost n e a r - l i n e a r combina-1 1 t i o n , the v a r i a b l e x, 2 and x 1 ) 9 i n terms o f m a t r i x w i l l be almost l i n e a r l y dependent: 1 X l ,2 > X i , 9 , - - _ _ , X i , 2 8 1 X l 1 9, 2 > " > X i i 9, g , - " _ " , X i l 9 , 2 8 Such m u l t i c o l l i n e a r i t y r e s u l t s i n e x t r e m e l y l a r g e e n t r i e s i n the i n v e r s e m a t r i x (jot) ~ 1 • S i n c e a 2 (X'}< ) ~ 1 i s the c o v a r i a n c e m a t r i x f o r the , 5 we t h e r e f o r e o b t a i n v e r y l a r g e c o v a r i a n c e s , and hence b r o a d c o n f i d e n c e i n t e r v a l s . The m u l t i c o l l i n e a r i t y p r o b l e m may be c l e a r l y v i s u a l i z e d , g e o m e t r i c a l l y , i n F i g u r e 3. But t o keep the geometry manageable, an e l l i p s o i d t h a t d e l i m i t s most o f the £ i ' s , the s o - c a l l e d " e l l i p s o i d o f c o n c e n t r a t i o n " i s shown. For the independent e r r o r s assumed e a r l i e r , the e l l i p s o i d i s s i m p l y a s p h e r e . T h i s sphere o f Y o b s e r v a t i o n s i s c e n t e r e d at the mean E ( Y )> which i s i n the p l a n e g e n e r a t e d by K i and X 2 • F i g u r e 3 shows what happens when r e g r e s s o r s X \ and X 2 & r e n ° t o r t h o g o n a l m u t u a l l y ( p e r p e n d i c u l a r ) but c o l l i n e a r , the i n t e r v a l o f p i ' s i s d i s p e r s e d on b o t h s i d e s o f the o r i g i n . The p o i n t e s t i m a t e may be p o s i t i v e , but t h e r e i s a good chance i t may be n e g a t i v e . 5 J . J o h n s t o n , E c o n o m e t r i c Methods, New Y o r k : M c G r a w - H i l l , p. 110, 1960. 60 F i g u r e 3 Range o f v a l u e s f o r p o s s i b l e (Si's around o r i g i n when %i and X 2 are h i g h l y c o l l i n e a r . A l t h o u g h F i g u r e 3 shows t h a t the t r u e $1 i s n o t z e r o , t h i s i s v e r y d i f f i c u l t t o e s t a b l i s h s t a t i s t i c a l l y . U s u a l l y Ho(£i = 0) w i l l n o t be r e j e c t e d under c o n d i t i o n s where t h e r e i s a huge s t a n d a r d e r r o r o f 0 i . I f , t h e r e f o r e , any v a l u e s o f t h e s e r e g r e s s o r s i n the c o r r e l a t i o n m a t r i x are c l o s e t o , s a y , Jo. 8J , the r e g r e s s i o n a n a l y s i s s h o u l d be c a r r i e d out w i t h one o f t h e h i g h l y c o r r e l a t e d v a r i a b l e s o m i t t e d . I t i s , however, e x t r e m e l y d i f f i c u l t t o d e c i d e w h i c h r e g r e s s o r s t o omit and w h i c h t o r e t a i n because t h o s e r e g r e s s o r s i n c l u d e d i n t h e e q u a t i o n have been s e l e c t e d on the b a s i s o f l o g i c p h y s i c a l or b i o l o g i c a l r e l e v a n t t o the p r o d u c t i o n p r o c e s s b e i n g examined. Under th e s e c i r c u m s t a n c e s i t i s p o s s i b l e t o t e s t , by s t e p w i s e r e g r e s s i o n , whether or not each o f the r e g r e s s o r s (and f o r t h a t m a t t e r o t h e r independent v a r i a b l e s ) i s making a s i g n i f i c a n t c o n t r i b u t i o n t o e x p l a i n i n g v a r i a t i o n i n y i e l d . The f o r w a r d s t e p w i s e r e g r e s s i o n q u a d r a t i c e q u a t i o n a c t u a l l y 61 s e l e c t e d the f o l l o w i n g v a r i a b l e s at the 5 p e r ce n t l e v e l o f s i g n i f i c a n c e : Y = 5733.2578 + 2739.7249 A + 398.6005 S - 1096.3665 P - 8.6445 F 2 + 0.9406 P 2 + 3.5579 DF - 2.2731 DS + 1.9925 DP - 2671.2339 AT + 2.8387 FS + 866.9939 PT . R 2 = 0.7212 . Once m u l t i c o l l i n e a r i t y becomes a p r o b l e m , even s t e p w i s e r e g r e s s i o n would not h e l p r e s o l v e i t . S t e p w i s e r e g r e s s i o n i s d e s i g n e d t o s e l e c t independent v a r i a b l e s l e a s t l i n e a r l y combined i n the f i r s t p l a c e , and ne x t l e s s l i n e a r l y combined and so on i n the o r d e r o f independent v a r i a b l e s l a i d out i n r e g r e s s i o n e q u a t i o n . I t f o l l o w s t h a t f o r w a r d r e g r e s s i o n does not n e c e s s a r i l y c o i n c i d e w i t h backward r e g r e s s i o n . i f independent v a r i a b l e s are c o l l i n e a r . There-f o r e , s e l e c t e d independent v a r i a b l e s may d i f f e r a c c o r d i n g t o the r e g r e s s i o n r o u t i n e i n s t u r c t i o n , i . e . , f o r w a r d s or backwards. Moreover, i f some independent v a r i a b l e s are l i n e a r l y dependent on the o t h e r ones, the v a l u e o f R 2 be-comes d u b i o u s . C o o r d i n a t e p o i n t s o f l i n e a r l y - d e p e n d e n t i ndependent v a r i a b l e s are not s p r e a d out but c l u s t e r e d i n n e a r l y l i n e a r f a s h i o n i n d i m e n s i o n a l space i n v o l v e d , and thus the d e t e r m i n a t i o n o f a m e a n i n g f u l r e g r e s s i o n s u r f a c e by l e a s t squares method i s r e n d e r e d t h a t much more d i f f i c u l t . 62 There is a point that should be made about the d i s t r i b u t i o n of the apple-yield dependent variable with respect to fixed values of an independent variable i n the regression equation. From the fact that the error term e is assumed normally d i s t r i b u t e d with mean = 0 and variance = a 2 , i t follows that the random variable, apple y i e l d , for s p e c i f i e d values of the independent variables, is also assumed normally d i s t r i b u t e d with mean = 3 i + B2X.2 + B 3X 3 + 3 HX„ + 65X5- + 8 6X 6 + B 7X 7 and variance = a 2 , i f a regres-sion model in regard to apple production is constructed i n the following manner: Y = 3 2 + 3 2X 2 + 3 3X 3 + 3wX4 + 3sX 5  + 3eX 6 + 37X7 + e . One hundred and nineteen observations are large enough to validate this assumption. . I f , however, the assumption is not met, parametric methods which have been employed are no longer adequate. There are two alterna-tiv e ways to deal with this s i t u a t i o n . One is perhaps to transform the raw data using logarithmic or square-root transformations, etc. The other way to cope with this problem is to apply non-parametric s t a t i s t i c s in which case the technique i s e n t i r e l y beyond the scope of the study. The d e t a i l s of results from stepwise regression are tabulated and shown in Table XIV i n the Appendix. Table XVI in the Appendix shows both observed values and the corresponding values of apple-crop y i e l d on the basis of the above equation involving the selected independent variables with s i g n i f i c a n t regression c o e f f i c i e n t s . 63 D i s c u s s i o n o f the R e s u l t s from A p p l y i n g Cobb-Douglas and  Q u a d r a t i c R e g r e s s i o n A n a l y s e s On a p r i o r i grounds, t h e r e s u l t s o f the Cobb-Douglas f u n c t i o n show e x p e c t e d s i g n s f o r the r e g r e s s i o n co-e f f i c i e n t s o f the seven r e g r e s s i o n c o e f f i c i e n t s e s t i m a t e d . Only t h a t o f the t r e e - s i z e v a r i a b l e s was not found s i g n i f i -c a n t at the 5 p e r cen t l e v e l o f p r o b a b i l i t y . U s i n g the q u a d r a t i c f u n c t i o n , o n l y the t h r e e b a s i c independent v a r i a b l e s , age o f t r e e ; c o s t o f s p r a y ; and p r u n i n g and t h i n n i n g hours were s i g n i f i c a n t at the 5 p e r cen t l e v e l . W h i l e the Cobb-Douglas f u n c t i o n i n t h i s case does not produce a m u l t i c o l l i n e a r i t y p r o b l e m , the q u a d r a t i c f u n c t i o n has shown e v i d e n c e o f m u l t i c o l l i n e a r i t y as i n d i c a -t e d by i n s p e c t i o n o f the c o r r e l a t i o n m a t r i x . I f the m u l t i -c o l l i n e a r i t y c o n d i t i o n e x i s t s between two v a r i a b l e s , i t i s v e r y d i f f i c u l t t o e s t a b l i s h the l e v e l of s t a t i s t i c a l s i g -n i f i c a n c e o f c o e f f i c i e n t s . T h e r e f o r e the i n f l u e n c e on c r o p y i e l d o f one v a r i a b l e may be e r r o n e o u s l y a t t r i b u t e d t o the o t h e r . I t would seem r e a s o n a b l e t o say t h a t t h i s s t u d y p r o v i d e d i n s u f f i c i e n t e v i d e n c e f o r c h o o s i n g between the two models. P r e f e r e n c e f o r the q u a d r a t i c f u n c t i o n o v er the Cobb-Douglas may be s t a t e d on the p u r e l y d e d u c t i v e o r t h e o r e t i c a l grounds t h a t i n t e r a c t i o n o f f a c t o r s can be at work. However, t h i s k i n d o f s e l e c t i o n o f the q u a d r a t i c f u n c t i o n i s made on the same grounds as e x p l a i n e d by Hume's p h i l o s o p h i c a l i n s i g h t s : 64 "When we give the preference to one set of arguments above another, we do nothing but decide from our fee l i n g concerning the sup e r i o r i t y of t h e i r influence." CHAPTER VI TESTING FOR THE DIFFERENCE BETWEEN TWO MEANS I n t r o d u c t i o n In c o n d u c t i n g the t e s t s i t was n e c e s s a r y t o use o v e r a l l unweighted p e r - a c r e means f o r s p e c i f i e d g r o u p i n g s o f a p p l e p l o t s . For i n s t a n c e , s i n c e the s i z e s o f apple e n t e r p r i s e s were i n i t i a l l y d e t e r m i n e d more by c o n v e n i e n c e o f r e c o r d k e e p i n g than by the r e l a t i o n s h i p t hey have t o the t o t a l a c r e s o f e n t e r p r i s e s on the farms (not r e a d i l y d e f i n -a b l e ) , i t became p r a c t i c a b l e t o c a l c u l a t e a p e r - a c r e average f o r each v a r i a b l e on each p l o t (the unweighted mean). These c o u l d then a l l o w an o v e r a l l p e r - a c r e mean t o be c a l c u l a t e d f o r a p a r t i c u l a r v a r i a b l e a c r o s s any sample group. W h i l e t h e s e types o f average are p e r m i s s i b l e , t h e i r i n t e r p r e t a t i o n i s l i m i t e d s t r i c t l y t o the narrow c o n t e x t i n which they were d e r i v e d . Hence, the t e s t s may show d i f f e r e n c e s or no d i f f e r -ences among means, but i t must be remembered t h a t the means r e l a t e t o i n d i v i d u a l farmer performance i n apple y i e l d s where each farmer has a w e i g h t o f one. A l s o i t i s i m p o r t a n t t o r e a l i z e t h a t a d i f f e r e n c e i s w i t h r e g a r d t o the samples f o r 1969 and t h e s e samples may show q u i t e d i f f e r e n t d i s t r i -b u t i o n s w i t h r e g a r d t o age, d e n s i t y , c o s t o f f e r t i l i z e r , c o s t o f s p r a y , p r u n i n g and t h i n n i n g l a b o u r h o u r s , t r e e - s i z e i n d e x , and g e o g r a p h i c a l l o c a t i o n . T h e r e f o r e , the d i f f e r e n c e o r l a c k o f d i f f e r e n c e i n the type o f means used can e a s i l y 66 be seen as a r e s u l t o f i n f l u e n c e s o f the above v a r i a b l e s . In c o n c l u s i o n , i t can be s a i d t h a t the r e s u l t s b e a r v e r y c a r e f u l i n t e r p r e t a t i o n and are t o be seen o n l y as s l i g h t l y e x t e n d i n g our i n s i g h t s i n t o the y i e l d r e l a t i o n s h i p p i c t u r e a l r e a d y s t u d i e d i n the p r e v i o u s c h a p t e r s . A t - t e s t i s used t o t e s t the n u l l h y p o t h e s i s t h a t two samples come from p o p u l a t i o n s w i t h the same mean: c o n s e q u e n t l y , t h i s t e s t s whether two samples are " s i g n i f i -c a n t l y d i f f e r e n t " i n t h i s r e g a r d . The o r d i n a r y method o f making a t e s t o f s i g n i f i c a n c e f o r the d i f f e r e n c e between means o f two independent samples assumes t h a t the two popu-l a t i o n v a r i a n c e s are the same. 1 I t has been assumed about the ap p l e c r o p y i e l d Y, t h a t a sample mean Y 1 } i s n o r m a l l y d i s t r i b u t e d around the p o p u l a t i o n mean, u, as f o l l o w s : Y j ^ N ( u j , o 2 / n j ) , where a 2 r e p r e s e n t s the v a r i a n c e o f the p o p u l a t i o n , and n the s i z e o f the sample drawn. S i m i l a r l y , Y 2 ^ N ( u 2 ,o2/n2). Independence o f the two s a m p l i n g p r o c e d u r e s w i l l ensure t h a t the two random v a r i a b l e s Y1 and Y 2 are independent: ( Y ! - Y 2 ) ^ N ( u i -y 2 , a 2 / n 1 + a 2 / n 2 ) . When p o p u l a t i o n v a r i a n c e a 2 i s unknown, i t must be e s t i m a t e d : Sf, = ( ) ( x M - X O 2 + E1?2 (X 2 - X 2 ) 2 ) ] , P n i + n 2 - 2 1 - 1 1 1 - 1 1 where S2, = p o o l e d v a r i a n c e . The f o r m u l a f o r the t - t e s t i s : 1G. W. Snedecor and W. G. Cochran, S t a t i s t i c a l Method, 6th Ed., pp. 114-115, 1969. 67 c a l c u l a t e d t = Y i - Y 2 / \l S p ( l / n i + l / n 2 ) . There may e x i s t s i t u a t i o n s i n which the assumption t h a t o\ = o\ i s s u s p e c t . I f s o , the f o r m u l a f o r the v a r i -ance o f (Y"i - Y 2 ) i n independent samples s t i l l h o l d s , namely, ol^ = o\/ri\ + c 2 / n 2 . When a 2 i s unknown, the u n b i a s e d e s t i m a t o r S 2 i s s u b s t i t u t e d . The o r d i n a r y t v a l u e i s r e p l a c e d by the s t a t i s t i c : t ' = Y i - Y2y/\j S1 /n 1 + S 2 / n 2 . T h i s q u a n t i t y does not f o l l o w s t u d e n t ' s t - d i s t r i b u t i o n when n i = n 2 . 2 I f , however, sample s i z e o f each group i s e q u a l , t and t ' become i d e n t i c a l . On the o t h e r hand, i f the samples are not o f e q u a l s i z e , o n l y approximate degrees o f freedom w i l l be c a l c u l a t e d by the f o l l o w i n g f o r m u l a : ( S ! / n i + S l / n 2 ) 2 / [ ( S 1 / n 1 ) 2 / n 1 - l + ( S 2 /n 2) / n 2 - 2) ] . 3I t s h o u l d be n o t e d t h a t the y i e l d measurements are o b t a i n e d on a p e r - a c r e b a s i s , and t - t e s t s t h r o u g h o u t are p e r f o r m e d on t h i s b a s i s . As mentioned e a r l i e r i n the t h e s i s , s a m p l i n g was not c o n d u c t e d from the t h r e e s e p a r a t e t r e e - s i z e group popu-l a t i o n s . R a t h e r , sub-samples o f t r e e - s i z e and o t h e r group e n t e r p r i s e s were o b t a i n e d from the sample o f apple p r o d u c e r s drawn from a s i n g l e p o p u l a t i o n . In the a n a l y s e s which f o l -low, sample s i z e s o f groups are unequal but t h i s f e a t u r e i s p e r m i t t e d by the computer program. The t - t e s t r o u t i n e o f the TRIP program has t h r e e 2 I b i d . p. 115. 3R. E. W a l p o l e , I n t r o d u c t i o n to S t a t i s t i c s , The M a c M i l l a n Co., C o l l i e r - M a c M i l l a n L i m i t e d , London. pp. 230-231, 1968. 68 d i f f e r e n t f ormulae at i t s d i s p o s a l : Formula ( 1 ) : the o n l y a s s u m p t i o n made about the p a r e n t popu-l a t i o n s i n the d e r i v a t i o n o f t h i s f o r m u l a i s n o r m a l i t y . Formula ( 2 ) : t h i s i s a s p e c i a l f o r m u l a used when t h e r e are d i f f e r e n c e s i n the d a t a p a i r e d s c o r e s ( o f no conc e r n i n t h i s s t u d y ) . Formula ( 3 ) : t h i s i s a more s e n s i t i v e v e r s i o n o f Formula ( 1 ) . Formula (3) i s v a l i d o n l y when the p o p u l a t i o n v a r i a n c e s are e q u a l . In f a c t , u s e r s can r e q u e s t the t - t e s t t o use Formula (1) i f i t f i n d s the sample v a r i a n c e s s i g n i f i c a n t l y d i f f e r e n t , and t o use Formula (3) when t h a t i s not the ca s e . Outcome o f T - t e s t B e f o r e showing a t e s t f o r average y i e l d d i f f e r e n c e s , i t i s d e s i r a b l e t o have a p i c t u r e o f d i f f e r e n c e s among average y i e l d s f o r the s p e c i f i c c a t e g o r i e s s t u d i e d . The r e s p e c t i v e f i g u r e s below are p r e s e n t e d t o s e r v e t h i s purpose and f o l l o w -i n g each has a t a b l e showing d e t a i l s o f the r e l e v a n t t e s t . T r e e - S i z e Group S t a n d a r d Semi - S t a n d a r d Semi - Dwarf Average A p p l e Y i e l d by T r e e - S i z e Group |/S,362.«i Sample S i z e 23 62 34 10,000 20,000 30,000 40,000 Y i e l d (pounds) p e r acr e F i g u r e 4 D i f f e r e n c e s i n average a p p l e y i e l d s among t r e e - s i z e groups 69 The r e s u l t s from t - t e s t s c o n c e r n i n g F i g u r e 4 are shown i n Ta b l e X V I I . TABLE XVII R e s u l t s from t - t e s t s f o r a v e r a g e - a p p l e - y i e l d d i f f e r e n c e s r e l a t i n g t o t r e e - s i z e groups. T r e e - S i z e Group C a l c u l a t e d T-value D.F. T-Prob. F-Prob. Formula Used S t a n d a r d v s . Semi - S t a n d a r d 2.132 83 0 .034 0 .163 (3) S t a n d a r d v s . Semi - Dwarf 2.450 55 0 .016 0 . 730 (3) Semi - S t a n d a r d v s . Semi-Dwarf 0. 199 94 0 . 823 0 . 225 (3) I f the T - P r o b a b i l i t y i s l e s s t h a n 0.05, i t i s u s u a l l y c o n c l u d e d t h a t the sample means are s i g n i f i c a n t l y d i f f e r e n t . I f the F - P r o b a b i l i t y i s l e s s t h a n 0.05, i t i s u s u a l l y c o n c l u d e d t h a t sample v a r i a n c e s are s i g n i f i c a n t l y d i f f e r e n t and t h e r e f o r e f o r m u l a (3) i s i n a p p r o p r i a t e f o r c a l c u l a t i n g t . A c c o r d i n g t o t h e s e c r i t e r i a , the average a p p l e y i e l d d i f f e r e n c e between S t a n d a r d and Semi-Standard t r e e - s i z e groups i s s i g n i f i c a n t l y d i f f e r e n t . The same i s t r u e between S t a n d a r d and Semi-Dwarf t r e e - s i z e groups. But t h i s was not the case w i t h S e m i - S t a n d a r d and Semi-Dwarf t r e e - s i z e groups where the d i f f e r e n c e between means was found not t o be s i g n i f i c a n t . The t h r e e c o r r e s p o n d i n g p a i r s o f sample v a r i a n c e s were found not t o show s i g n i f i c a n t d i f f e r e n c e s and thus f o r m u l a (3) was used t h r o u g h o u t the t -70 t e s t Region Average Apple Y i e l d by Region Okanagan N o r t h Okanagan South 23. 68V. </ 20, Stf-o. o 2o,ooo Y i e l d (pounds) per acre Sample S i z e 54 65 40, 000 F i g u r e 5 D i f f e r e n c e i n average apple y i e l d s between r e g i o n s ( a c r o s s a l l t r e e - s i z e groups) The outcome o f the t - t e s t c o n c e r n i n g F i g u r e 5 are shown i n T a b l e X V I I I . TABLE X V I I I R e s u l t s from t - t e s t f o r a v e r a g e - a p p l e - y i e l d d i f f e r e n c e between r e g i o n s Regions C a l c u l a t e d T -value D.F. T-Prob. F-Prob. Formula Used Okanagan N o r t h v s . Okanagan South 0 . 581 73 0 .570 0.0 CD There i s no s i g n i f i c a n t d i f f e r e n c e i n average y i e l d between the Okanagan N o r t h and Okanagan South r e g i o n s . But t h e i r sample v a r i a n c e s are s i g n i f i c a n t l y d i f f e r e n t . 71 Grade Average A p p l e Y i e l d by Grade S amp1e S i z e E x t r a Fancy Fancy Cee C u l l 10, 000.0 2, iio.il 2, 87/.OS 5,006 10,000 iS.OQO Y i e l d (pounds) p e r acre F i g u r e 6 D i f f e r e n c e s i n average apple y i e l d s among app l e grades ( a c r o s s a l l t r e e - s i z e groups) The outcome o f t h e t - t e s t c o n c e r n i n g F i g u r e 6 are shown i n Tabl e XIX. TABLE XIX R e s u l t s from t - t e s t s f o r a v e r a g e - a p p l e - y i e l d d i f f e r e n c e s r e l a t e d t o grades Grade E x t r a Fancy v s . Fancy E x t r a Fancy v s . Cee E x t r a Fancy v s . C u l l Fancy v s . Cee Fancy v s . C u l l Cee v s . C u l l C a l c u l a t e d T-value 3. 353 6 .472 5. 367 5. 448 3. 354 •0.777 D.F, 159 134 184 191 210 170 T-Prob 0. 001 0.0 0 . 0 0.0 0 . 001 0. 444 F-Prob 0 . 0 0.0 0.0 0.1 0 .006 0.0 Formula Used CD CD ( i ) ( i ) CD ( i ) The d i f f e r e n c e s between average y i e l d s f o r p a i r s o f grades are s i g n i f i c a n t w i t h the e x c e p t i o n o f Cee v e r s u s C u l l . The sample v a r i a n c e s are a l s o s i g n i f i c a n t l y d i f f e r e n t f o r a l l 72 p a i r s o f grade c o u p l i n g s . V a r i e t y Average Appl e Y i e l d by V a r i e t y Sample S i z e Golden D e l i c i o u s Red D e l i c i o u s S p a r t a n M c i n t o s h F i g u r e 7 D i f f e r e n c e s i n average a p p l e y i e l d s among apple v a r i e t i e s ( a c r o s s a l l t r e e - s i z e groups) As has been mentioned i n the i n t r o d u c t o r y c h a p t e r , the breakdown o f d a t a i n t o v a r i e t i e s has r e s u l t e d i n an a r b i t r a r y m a n i p u l a t i o n o f d a t a . When sub-sampling'was c a r r i e d o u t , i t was done i n accordance w i t h the t r e e - s i z e g r oups, and hence a s i n g l e e n t e r p r i s e c o u l d sometimes i n -v o l v e d i f f e r e n t k i n d s o f apple v a r i e t i e s . The r i g o r o u s attempt t o group d a t a by v a r i e t y has c o n t r i b u t e d t o e n l a r g e d sample s i z e i n some i n s t a n c e s i m p l y because i t i n v o l v e d p a r t i t i o n i n g o r i g i n a l e n t e r p r i s e s . I n t h i s p r o c e d u r e the u n d e r l y i n g assumptions o f e n s u r i n g a n a l -y s i s were not thought t o be i n f r i n g e d s e r i o u s l y , a l t h o u g h some c a u t i o n i n a c c e p t a n c e o f the r e s u l t s i s thought neces-s a r y . The r e s u l t s o f the t - t e s t s on v a r i e t y y i e l d d a t a are g i v e n i n Table XX. 2l, 377.7 21, 73 B. 0 26,?2*. 7 /£>, 6/0.3 36 42 40 31 10,000 20,000 30,000 40,000 Y i e l d (pounds) p e r acre 73 TABLE XX R e s u l t s from t - t e s t f o r average - a p p l e - y i e l d d i f f e r e n c e s r e l a t i n g t o v a r i e t y V a r i e t y C a l c u l a t e d T -value D.F. T-Prob. F-Prob. Formula Used Golden D e l i c i o u s v s . Red D e l i c i o u s -0.084 76 0 . 893 0 .552 (3) Golden D e l i c i o u s v s . S p a r t a n -0.747 56 0 . 465 0 . 0 (1) Golden D e l i c i o u s v s . M c i n t o s h 1.166 62 0 . 247 0.045 (1) Red D e l i c i o u s v s . S p a r t a n -0.719 52 0 .482 0 . 0 (1) Red D e l i c i o u s v s . M c i n t o s h 1. 336 71 0 .182 0 .129 (3) S p a r t a n v s . M c i n t o s h 1.451 49 0.149 0.0 (1) V a r i e t a l d i f f e r e n c e s i n mean v a l u e s o f crop y i e l d are not s i g n i f i c a n t . F u r t h e r m o r e , the sample v a r i a n c e s f o r Golden D e l i c i o u s and Red D e l i c i o u s are not s i g n i f i c a n t l y d i f f e r e n t , as was the case a l s o f o r Red D e l i c i o u s and M c i n t o s h . For a l l o t h e r p a i r s o f v a r i e t i e s , the v a r i a n c e s were s i g n i f i c a n t l y d i f f e r e n t . D i s c u s s i o n o f the t - t e s t The s i g n i f i c a n t d i f f e r e n c e s i n average y i e l d s between the S t a n d a r d and Semi-Standard groups and between the S t a n d a r d and Semi-Dwarf groups was s u r p r i s i n g because the t r e e - s i z e i n d e x v a r i a b l e had been found not s t a t i s t i c a l l y 74 s i g n i f i c a n t as was dropped from the r e g r e s s i o n e q u a t i o n s . The r e a s o n f o r t h i s may be found i n t h e i n t e r p r e -t a t i o n o f the c h a r a c t e r i s t i c s of the t - t e s t i n r e l a t i o n t o r e g r e s s i o n a n a l y s i s . The t - t e s t may be seen as a s i m p l e r e g r e s s i o n on dummy v a r i a b l e s r e p r e s e n t i n g the t h r e e d i f f e r e n t t r e e - s i z e groups. The f o l l o w i n g s i t u a t i o n can be d e p i c t e d : Dummy v a r i a b l e s S t a n d a r d group 0 0 0 Semi - S t a n d a r d group 1 0 0 Semi-Dwarf group 0 1 0 A dummy v a r i a b l e f o r the S t a n d a r d group i s not needed because S e m i - S t a n d a r d and Semi-Dwarf groups r e f l e c t d i f f e r e n t i a l s measured from the S t a n d a r d group base. T h i s s i t u a t i o n can e a s i l y be v i s u a l i z e d i n m a t r i x n o t a t i o n : Y i 0 1 0 Y 2 0 0 1 0 1 0 Y 1 1 9 0 0 1. 3i B 3 Dummy v a r i a b l e s are not the o n l y means o f a d j u s t -i n g d a t a . A n o t h e r method i s t o d e v i s e a s c a l e f o r t r e e - s i z e as has been done e a r l i e r i n t h i s s t u d y . The d e s i r e d r e l a t i o n o f crop y i e l d t o the t r e e - s i z e v a r i a b l e can now be e s t i m a t e d 75 by a s i m p l e r e g r e s s i o n o f apple y i e l d on t r e e - s i z e i n d e x . T h i s l a t t e r method i s advantageous because i t i s not neces-s a r y to assume d i s c r e t e s h i f t s . Thus adjustment f o r any o b s e r v a t i o n v a r i e s from one o b s e r v a t i o n t o a n o t h e r . T h i s same i d e a was u t i l i z e d when h y p o t h e s i z i n g the E q u a l i t y o f S l o p e s among the t r e e - s i z e groups. I t i s c r u c i a l t o remember t h a t the p r o d u c t i o n r e l a t i o n s h i p s under e x a m i n a t i o n by the t - t e s t are t h e o r e t i c a l l y i m p l i e d w i t h i n the framework o f the m u l t i p l e r e g r e s s i o n model u s i n g seven independent v a r i -a b l e s and twenty r e g r e s s o r s . The t - t e s t o f the type used has l i m i t e d a p p l i c a -t i o n i n the d a t a c o n t e x t o f t h e s t u d y . I t i s a m e a n i n g f u l method d e r i v i n g i n f o r m a t i o n under the c o n d i t i o n o f s i n g l e v a r i a b l e - c l a s s i f i c a t i o n where o t h e r i n f l u e n c e s are h e l d c o n s t a n t . I t i s p r e c i s e l y our i n a b i l i t y t o h o l d o t h e r i n f l u -ences c o n s t a n t when t e s t i n g the y i e l d d i f f e r e n c e s f o r p a i r s o f c a t e g o r i e s , w h i c h r e n d e r s t h e r e s u l t s from the t - t e s t l e s s p o w e r f u l than t h o s e o b t a i n e d from m u l t i p l e r e g r e s s i o n . N e v e r t h e l e s s the t e s t s are o f i n t e r e s t because they do p r o -v i d e f u r t h e r i n s i g h t s by way o f e i t h e r a f f o r d i n g a s l i g h t l y d i f f e r e n t a t t a c k o r even a d d i n g t o the o v e r a l l a n a l y s i s , e.g., d i f f e r e n c e s among grades. CHAPTER V I I SUMMARY AND CONCLUSION The highest achievement would be to grasp that whatever we call a "fact" is already theory. Goet^e Summary The o b j e c t i v e o f t h i s s t u d y was t o e s t i m a t e r e g r e s -s i o n r e l a t i o n s h i p s between ap p l e y i e l d s and c e r t a i n i n f l u -e n c i n g f a c t o r s f o r the Okanagan a r e a o f B r i t i s h Columbia i n 1969 . Two types o f e q u a t i o n were employed t o r e p r e s e n t y i e l d r e l a t i o n s h i p s . These were the Cobb-Douglas and Q u a d r a t i c forms. The e x p l a n a t o r y v a r i a b l e s used i n r e g r e s s i o n were as f o l l o w s : (1) d e n s i t y p e r a c r e , (2) age o f t r e e , (3) the c o s t o f f e r t i l i z e r a p p l i e d p e r a c r e , (4) the c o s t o f sp r a y a p p l i e d p e r a c r e , (5) p r u n i n g and t h i n n i n g l a b o u r hours p e r a c r e , (6) g e o g r a p h i c a l dummy v a r i a b l e , and (7) t r e e - s i z e i n d e x . When a Cobb-Douglas f u n c t i o n was f i t t e d t o a l l sample o b s e r v a t i o n s ( a c r o s s t r e e - s i z e g r o u p s ) , the independent v a r i a b l e s were found s i g n i f i c a n t at 5 per cent l e v e l o f p r o b a b i l i t y w i t h the e x c e p t i o n o f t r e e - s i z e i n d e x . On the o t h e r hand, a Q u a d r a t i c f u n c t i o n i n v o l v i n g t w e n t y - e i g h t 77 independent terms, i n c l u d e d o n l y e l e v e n terms as b e i n g s i g -n i f i c a n t at the 5 per c e n t l e v e l o f p r o b a b i l i t y . Here the s e l e c t e d v a r i a b l e s were as f o l l o w s : (1) age o f t r e e , (2) c o s t o f s p r a y per a c r e , (3) p r u n i n g and t h i n n i n g l a b o u r hours p e r a c r e , (4) s q u a r e d f e r t i l i z e r c o s t per a c r e term, (5) s q u a r e d p r u n i n g and t h i n n i n g hours per a c r e term, (6) c r o s s - t e r m between d e n s i t y and f e r t i l i z e r c o s t per a c r e , (7) c r o s s - t e r m between d e n s i t y and s p r a y c o s t p e r a c r e , (8) c r o s s - t e r m between d e n s i t y and p r u n i n g and t h i n n i n g l a b o u r hours per a c r e , (9) c r o s s - t e r m between age o f t r e e and t r e e - s i z e i n d e x , (10) c r o s s - t e r m between f e r t i l i z e r and s p r a y c o s t s p e r a c r e , and (11) c r o s s - t e r m between p r u n i n g and t h i n n i n g l a b o u r hours per a c r e and t r e e - s i z e i n d e x . In view o f the p r o p e r t i e s o f a Q u a d r a t i c f u n c t i o n and the economic t h e o r y o f p r o d u c t i o n , a c h o i c e was made i n f a v o u r o f i t r e l a t i v e to the Cobb-Douglas f u n c t i o n t o r e p r e s e n t the a p p l e - y i e l d r e l a t i o n s h i p . But the Q u a d r a t i c f u n c t r e s u l t e d i n a c o m p l i c a t e d problem a r i s i n g from s q u a r i n g independent v a r i a b l e s to be used as r e g r e s s o r s . The use o f an independent v a r i a b l e a l o n g w i t h i t s s q u a r e d term may have g e n e r a t e d some c o l l i n e a r i t y . I f two independent v a r i -a b l e s cause a m u l t i c o l l i n e a r i t y p r o b lem i t i s e x t r e m e l y d i f f i c u l t t o deduce the i n f l u e n c e o f one o f the v a r i a b l e s on the dependent v a r i a b l e because i t might w e l l be t h a t the o t h e r v a r i a b l e has e q u a l i n f l u e n c e . 78 Conclus i o n L o o k i n g back t o the r e g r e s s i o n model, s e v e r a l assumptions were made so t h a t i n f e r e n c e s from e s t i m a t e d r e g r e s s i o n e q u a t i o n s c o u l d be made. Suppose t h a t the f o l l o w i n g f u n c t i o n a l r e l a t i o n s h i p e x i s t s i n r e g a r d t o apple y i e l d Y = B i + B 2 X 2 + B 3 X 3 + BnX H + B 5 X 5 + B&XG + B 7 X 7 + e . A s t r o n g a ssumption must be made about e: namely, e ^ N ( 0 , a 2 ) . In t h i s model i t i s i m p l i e d t h a t a l l independent v a r i a b l e s are t r e a t e d as f i x e d . The o n l y random v a r i a b l e i n the model i s Y whi c h i s deduced from the f a c t t h a t e i s a random v a r i a b l e . A n o t h e r i m p o r t a n t assumption made i n t h e model i s t h a t a l l independent v a r i a b l e s are indepe n d e n t o f one a n o t h e r , and o f t h e e r r o r term e. A l t h o u g h the terms " r a n -dom v a r i a b l e " and "independence" t e s t our powers o f compre-h e n s i o n , t h e y do i n f a c t c o r r e s p o n d t o e m p i r i c a l l y d e t e r m i n a b l e f e a t u r e s i n c e r t a i n a c t u a l p r o c e s s e s as a consequence o f v a r i o u s r u l e s employed by s t a t i s t i c i a n s . Treatment o f a l l independent v a r i a b l e s as t a k i n g f i x e d v a l u e s i m p l i e s t h a t the o n l y e r r o r a l l o w e d was an e r r o r i n the e q u a t i o n due t o the o m i s s i o n o f some i n p u t f a c t o r s . In f a c t , e r r o r i n the measurement o f the i n c l u d e d i n p u t v a r i a b l e s i s e x t r e m e l y l i k e l y . The e r r o r may be due t o "human element" i n v o l v e d f o r i n s t a n c e , m i s t a k e s may o c c u r i n the c o l l e c t i o n and r e c o r d i n g o f d a t a . The o b s e r v e d v a l u e s o f the v a r i a b l e s are n o t 79 s t r i c t l y comparable because o f l a c k o f homogeneity w h i c h i s the case w i t h f e r t i l i z e r c o s t , s p r a y c o s t , and p r u n i n g and t h i n n i n g l a b o u r h o u r s . T h e r e f o r e o b s e r v a t i o n e r r o r s are n e c e s s a r i l y p r e s e n t i n the d a t a . I t i s p o s s i b l e t h a t some method o f a d j u s t i n g the d a t a t o t a k e account of h e t e r o g e n e i t y might be u s e d , but even so i t i s d i f f i c u l t t o c o n t r i v e and most c e r t a i n l y would l e a v e something t o be d e s i r e d . In the case o f the s t u d y i t i s assumed t h a t measurement e r r o r i s not o f s e r i o u s p r o p o r t i o n . As l o n g as the independent v a r i a b l e s are not o r t h o g o n a l , X i • & 2 = O f o r example, s t e p w i s e r e g r e s s i o n cannot y i e l d a s a t i s f a c t o r y r e s u l t . 1 In view o f the m e t h o d o l o g i c a l issues' i n the f o r e -g o i n g s t a t i s t i c a l a n a l y s i s , i t i s a p p r o p r i a t e t o say t h a t the e m p i r i c a l r e s u l t s o b t a i n e d are i n c o n c l u s i v e . In t h i s c o n n e c t i o n i t i s t o be s u g g e s t e d t h a t to meet a l l the as-sumptions r e q u i r e d f o r d e r i v i n g an apple y i e l d e q u a t i o n , a c o n t r o l l e d e x p e r i m e n t would be n e c e s s a r y . W i t h i n the theo-r e t i c a l framework, p r e f e r e n c e may be g i v e n t o the Q u a d r a t i c form o f y i e l d r e l a t i o n s h i p o v e r t h a t summarized by a Cobb-Douglas f u n c t i o n . However, the former f u n c t i o n has posed a s e r i o u s s t a t i s t i c a l p r o b l e m ( m u l t i c o l l i n e a r i t y ) , w hich has a l r e a d y been d i s c u s s e d at some l e n g t h . N e e d l e s s t o s a y , i t i s hoped t h a t any f u t u r e s t u d y under s i m i l a r c i r c u m s t a n c e s to the one w h i c h has been con-Mv'onnacott, et a l . , E c o n o m e t r i c s , pp. 309- 312. 80 d u c t e d w i l l be i n a b e t t e r p o s i t i o n t o use the Cobb-Douglas f u n c t i o n o ver s e l e c t e d r e l e v a n t ranges o f d a t a and, t h e r e b y , a v o i d the m u l t i c o l l i n e a r i t y p r o b l e m as met i n the Q u a d r a t i c a n a l y s i s . S i n c e the r e g r e s s i o n t h e o r y g i v e s r i s e t o the most d i f f i c u l t c o n c e p t u a l p a r t o f the t h e s i s , t h i s b r i e f summary and c o n c l u s i o n s has h e l p e d t o remind the r e a d e r o f the v e r y r e a l o b s t a c l e s to the type o f a n a l y s i s u n d e r t a k e n . In the case o f the t e s t s o f s i g n i f i c a n c e f o r d i f f e r e n c e s among means, the c o n c e p t u a l framework i s somewhat e a s i e r t o u n d e r s t a n d , a l t h o u g h the a n a l y s i s r e s t s on d e f i n i t e assump-t i o n s . These as w e l l as the r e s u l t s o f t h a t a n a l y s i s o c c u r i n a p r e c e d i n g c h a p t e r , and t h e r e f o r e no attempt i s made t o r e p e a t t h e summary a l r e a d y g i v e n . 81 BIBLIOGRAPHY 82 BIBLIOGRAPHY B j e r r i n g , J . H. and S e a g r a v e s , P. UBC TRIP ( T r i a n g u l a r R e g r e s s i o n Package) Vancouver: U.B.C., Computing C e n t r e , November 1970. B r a s e , K. D. and Way, R. D. " R o o t s t o c k s and Methods used f o r D w a r f i n g F r u i t T r e e s , " New York S t a t e A g r i c u l t u r a l E x periment S t a t i o n , 783, 1959. Bush, R. Tree F r u i t Growing, r e v i s e d by E. G. G i l b e r t p r e p a r e d i n c o n j u n c t i o n and c o l l a b o r a t i o n w i t h the Ro y a l H o r t i c u l t u r a l S o c i e t y , Penguin Books, 1962. C h i n h L e - D i n h . UBC SLTESTS: E q u a l i t y o f S l o p e T e s t , U.B.C, Computing C e n t r e , June 1971. Coshow, W". UBC BMDX64: G e n e r a l L i n e a r H y p o t h e s i s , U.B.C, Computing C e n t r e , August 1971. D o l l , J . P. "An A n a l y t i c a l Technique f o r E s t i m a t i n g a Weather Index from M e t e o r o l o g i c a l Measurements," J o u r n a l o f Farm Economics, V o l . 49, No. 1, F e b r u a r y 1967. D o r l i n g , M. J . The Okanagan Apple P r o d u c e r H i s Management A t t i t u d e and B e h a v i o u r , Department o f A g r i c u l t u r a l Economics, U.B.C, 1968. E z e k i e l , M. and Fox, K. A. Methods o f C o r r e l a t i o n and  R e g r e s s i o n A n a l y s i s , 3 r d Ed., 1963. F i s h e r , D. V. H i g h - D e n s i t y Orchards f o r B r i t i s h Columbia  C o n d i t i o n s " Research S t a t i o n Summerland, B r i t i s h C olumbia R e s e a r c h B r a n c h , Canada Department o f A g r i c u l t u r e , March 1966. F o l g e r , J . C. and Thomson, S. M. The Commercial A p p l e  I n d u s t r y o f N o r t h A m e r i c a , Ed" L~. FT B a i l y . New Yor k : M a c M i l l a n Co. , 1921. Haavelmo, T. "The P r o b a b i l i t y Approach i n E c o n o m e t r i c s , " E c o n o m e t r i c a , V o l . 12, 1944. Supplement. H a r r i s , J . H. and Woods, J . J . D w a r f i n g Apple Trees on Vancouver I s l a n d , E x p e r i m e n t a l Farm Research B r a n c h , S a a n i c h t o n , W. C , 1958. Heady, E. 0. and D i l l o n , J . A g r i c u l t u r a l P r o d u c t i o n F u n c t i o n s , Ames, Iowa: Iowa S t a t e U n i v e r s i t y P r e s s , 1961. J o h n s t o n , J . E c o n o m e t r i c Methods, New Y o r k : M c G r a w - H i l l , 1960. 83 K e l l y , C. C. and S p i l s b u r y , R. H. S o i l Survey o f the Okanagan and Similkameen V a l l e y o f B. C , Report No. 3 o f B. C. Survey. The B r i t i s h Columbia Department o f A g r i c u l t u r e i n C o o p e r a t i o n w i t h E x p e r i m e n t a l Farm S e r v i c e , Dominion Department o f A g r i c u l t u r e , 1949. Keynes, J . M. A T r e a t i s e on P r o b a b i l i t y . K o u t s o g i a n n e - K o k k o v a , A. An E c o n o m e t r i c Study o f the L e a f Tobacco Market of Greece, A t h e n s , 19 62. Lawrence, H. S. "The E f f e c t o f Weather on A g r i c u l t u r a l Output": A Look at Methodology, J o u r n a l o f Farm. Economics, V o l . 46, No. 1, Fe b r u a r y 1964. L o n g l e y , W. V. Some Economic A s p e c t s o f the Apple I n d u s t r y i n Nova S c o t i a , A T h e s i s f o r the Degree o f Doc t o r o f P h i l o s o p h y , Nova S c o t i a Department o f A g r i c u l t u r e B u l l e t i n No. 113, 1932. Mann, H. B. and Wald, A. "On the S t a t i s t i c a l Treatment o f L i n e a r S t o c h a s t i c D i f f e r e n c e E q u a t i o n s , " E c o n o m e t r i c a , Vo1. 11, 19 4 3. Snedecor, G. W. and Cochran, W. G. S t a t i s t i c a l Method, 6th Ed., 1969. Tukey, R. B. " I m p l i c a t i o n s o f Economics on O r c h a r d Management," The 1969 Apple Forum, P u b l i s h e d P r o c e e d i n g s of the F i r s t B r i t i s h Columbia F r u i t Growers' A s s o c i a t i o n s p o n s o r e d H o r t i c u l t u r a l C o n f e r e n c e . November 1969. Van Roechoudt, L. L. Some F a c t o r s which I n f l u e n c e the Use o f Dwarf and Semi-dwarf Apple Trees f o r Commer-c i a l Orchards i n the Okanagan V a l l e y o f B. C , U n p u b l i s h e d M a s t e r ' s T h e s i s , The U n i v e r s i t y o f B. C. , 1962 . W a l p o l e , R. E. I n t r o d u c t i o n t o S t a t i s t i c s , The M a c M i l l a n Co., C o l l i e r - M a c M i l l a n L i m i t e d , London, 1968. Ware, D. W., Woodward, E. D. and T r e v o r , H. W. A Study  of Apple P r o d u c t i o n i n the Okanagan V a l l e y o t  B r i t i s h C o l u m b i a , Canada Department o f A g r i c u l t u r e , M a r k e t i n g S e r v i c e - Economic D i v i s i o n Ottawa, J a n u a r y 1952. Ware, D. W. O r g a n i z a t i o n and Returns o f Stone F r u i t and Pear E n t e r p r i s e s i n the Okanagan V a l l e y , B. C., 1949 - 1950, Department o f A g r i c u l t u r e Economic D i v i s i o n , M a r k e t i n g S e r v i c e , Ottawa, 1952. 84 Wonnacott, R. J . and Wonnacott, T. M. E c o n o m e t r i c s , John W i l e y and Sons, I n c . , 1970. APPENDIX TABLE I CLASSIFICATION OF ROOTSTOCK VIGOUR S i z e Group Dwarf Semi-dwarf Semi - s t a n d a r d S t a n d a r d R o o t s t o c k M. IX M.26, M.M.106, M.VII, M.IV M.I I , M . M . I l l , M.M.104 S e e d l i n g , M.XVI, M.XXV, M.M.109 U l t i m a t e Tree S i z e i n R e l a t i o n t o Tree S i z e on S e e d l i n g Roots 1/5 to 1/4 1/3 2/3 t o 3/4 F u l l s i z e Tree Anchorage Poor Poor t o F a i r F a i r t o Good Good Sou r c e : H i g h - d e n s i t y o r c h a r d s f o r B.C. c o n d i t i o n s , R esearch S t a t i o n , Summerland, B.C., Research B r a n c h , Canada Department of A g r i c u l t u r e . March, 1966. 87 TABLE I I VARIABLES USED IN MODELS V a r i a b l e Meaning 1 APPLE YIELD PER ACRE 2 DENSITY PER ACRE 3 AGE OF TREE 4 THE AMOUNT OF FERTILIZER APPLIED PER ACRE 5 THE AMOUNT OF SPRAY APPLIED PER ACRE 6 PRUNING AND THINNING LABOUR HOURS SPENT PER ACRE 7 GEOGRAPHICAL DUMMY 8 TREE SIZE INDEX 9 SQUARE OF VARIABLE 2 10 SQUARE OF VARIABLE 3 11 SQUARE OF VARIABLE 4 12 SQUARE OF VARIABLE 5 13 SQUARE OF VARIABLE 6 14 CROSS TERM BETWEEN VARIABLE 2 AND VARIABLE 3 15 CROSS TERM BETWEEN VARIABLE 2 AND VARIABLE 4 16 CROSS TERM BETWEEN VARIABLE 2 AND VARIABLE 5 17 CROSS TERM BETWEEN VARIABLE 2 AND VARIABLE 6 18 CROSS TERM BETWEEN VARIABLE 2 AND VARIABLE 8 19 CROSS TERM BETWEEN VARIABLE 3 AND VARIABLE 4 20 , CROSS TERM BETWEEN VARIABLE 3 AND VARIABLE 5 21 CROSS TERM BETWEEN VARIABLE 3 AND VARIABLE 6 2 2 CROSS TERM BETWEEN VARIABLE 3 AND VARIABLE 8 2 3 CROSS TERM BETWEEN VARIABLE 4 AND VARIABLE 5 24 CROSS TERM BETWEEN VARIABLE 4 AND VARIABLE 6 2 5 CROSS TERM BETWEEN VARIABLE 4 AND VARIABLE 8 26 CROSS TERM BETWEEN VARIABLE 5 AND VARIABLE 6 27 CROSS TERM BETWEEN VARIABLE 5 AND VARIABLE 8 2 8 CROSS TERM BETWEEN VARIABLE 6 AND VARIABLE 8 88 TABLE ESTIMATED SIMPLE LINEAR I I I REGRESSION EQUATION1 Equation f o r the Standard-Tree Group 1. VAR 2. VAR 3. VAR 4. VAR 5 VAR : 1 = 1 = 1 = 1 = 1 = 27750 (6726) 33950 (7118) 2770 (5360) 30400 (S603) 24090 i (4496) • 14.15 VAR (75.43) • 192 VAR (240) • 12 6.6 VAR (310.5) • 22.96 VAR (68.12) 60.50 VAR (41.62) S i g n i f i c a n c e of b Estimate at .05 Level NON. SIG. NON. SIG. NON. SIG. NON. SIG. NON. SIG. Equation f o r the Semi-Standard-Tree Group 0.0017 0.0296 0.0078 0.0054 0.0914 R2 1. VAR 1 = 15240 (3957) - 15. (21. 21 VAR 2 3) 2 NON. SIG. 0. .0085 2 . VAR 1 = - 3239 (4434) + 2114 VAR (413.5) .3 SIG. 0. ,3034 3. VAR 1 = 14810 (2289) + 149 (61 . 1 VAR .55) 4 SIG. 0. .0891 4 . VAR 1 = 14010 (3651) + 67. (51. 60 VAR 18) 5 SIG. 0. 0236 5 . VAR 1 = 106.1 (168.9) + 69. (219. 40 VAR 4) 8 NON. SIG. 0. 0017 Equation f o r the Semi -Dwarf-Tree Group r2 1. VAR 1 = 7688 (5664) + 41. (19. 7 3 VAR 6S) 2 SIG. 0. 1235 2 . VAR 1 = 12520 (7226) + 616. (702 . 1 VAR 2) 3 NON. SIG. 0. .0235 3. VAR 1 = 13110 (2709) + 171. (46. 1 VAR 07) 4 SIG. 0. .3012 4. VAR 1 = 12330 (2993) + ' 60. (18. 21 VAR 07) 5 SIG. 0. .2575 5 . VAR 1 = 13620 (3056) + 59. (22. 62 VAR 11) 6 SIG. 0. .1851 6. VAR 1 = 15260 (9345) + 9189 VAR (26200) 8 NON. SIG. 0. .0038 Equation T o t a l Data R2 1. VAR 1 = 18800 (2639) + 7.220 VAR (12.67) 2 NON. SIG. 0, .0028 2 . VAR 1 = 13800 (2431) + 479. (150 . 2 VAR 3 SIG. 0. .0795 3. VAR 1 = 16920 (1650) + 145. (39. 2 VAR 26) 4 SIG. 0, .1047 4 . VAR 1 = 16410 (1921) + 51. (18. 79 VAR 03) 5 SIG. 0. .0659 5. VAR 1 = 15530 (1852) + 68. (18. 50 VAR 13) 6 SIG. 0, .1087 6. VAR1 = 341.0 (27.74) - 245. (37. 5 VAR 81) 8 SIG. 0. .2649 'Data i n brackets r e f e r to regression c o e f f i c i e n t standard e r r o r s . 89 TABLE IV EVALUATION OF SIZE-CONTROLLING EFFECTS OF ROOTSTOCK, INTERMEDIATE FRAMEWORK STOCK AND STRAIN OF SCION VARIETY ON TOTAL TREE SIZE IN TERMS OF AN INDEX VALUE Index V a l u e A. S t a n d a r d v i g o u r c l o n a l and s e e d l i n g r o o t s 1 . 0 B. S t a n d a r d v i g o u r framework v a r i e t y 1 .0 C. S t a n d a r d v i g o u r s c i o n v a r i e t y 1 .0 R e d u c t i o n i n t r e e s i z e by r o o t s t o c k i n r e l a t i o n t o A We would r a t e s e m i - s t a n d a r d s t o c k s such as M.11 M . M . I l l A. 2 at 0 . 75 M.M.104 at 0 . 85 We would r a t e semi-dwarf s t o c k s such as M.M.106 at 0 . 50 M. IV at 0 .40 M.VII at 0 . 33 M. 26 at 0 .25 We would r a t e dwarf r o o t s t o c k s such as M. IX at — M.VIII at 0 . 20 R e d u c t i o n i n t r e e s i z e by framework s t o c k i n r e l a t i o n t o B We would r a t e s i z e c o n t r o l l i n g e f f e c t o f an i n t e r m e d i a t e s t o c k such as H a r a l s o n ( o n l y one on our s t u d y ) a t 0.75 R e d u c t i o n i n t r e e s i z e as a r e s u l t of the use o f a sp u r s t r a i n o f the s c i o n v a r i e t y i n r e l a t i o n t o C  We would r a t e r e d u c t i o n o f t r e e s i z e by use o f sp u r t y p e s t r a i n s at 0.75 A p p l i c a t i o n By c o m b i n i n g d i f f e r e n t f a c t o r s under A, B, and C, a t r e e - s i z e i n d e x v a l u e can be e s t a b l i s h e d . For example - Spur D e l i c i o u s on s t a n d a r d v i g o u r i n t e r m e d i a t e framework s t o c k on M.IV = 0.74 x 1.0 x 0.40 = 0.30 M c i n t o s h s t a n d a r d v i g o u r i n t e r m e d i a t e framework s t o c k on M.VIII = 1.0 x 1.0 x 0.33 = 0. 33 TAR1.: VI I INPUT RATA (,ri:B Al'Kl- CY APPl.r l!\ TI'UI'1'1 SI; PLOTS)1 n • 1  '_> , TrocSie ("roup: O.:::JI: OS 48 .00 41.00 7 .060 0.2S091: OS 48 .JO 5 5.00 6 . ::o 4951 . 46 .Hi :r.mi 2li .51 0.1-351: ns 4 3 . .Ml . i to. . " 5 0. >>r't.| OS 4S .00 4 .s.00 14 . *>:  o.:o?t>i:. u5 4H .JO 5*1.00 ti .500 0.^5831: OS li)4 3.0(1 8 .900 0.1481M- OS 3 .00 8. POO 925 1 . 5 3 . :.o .13.00 5 . *I0 0 . .'tf 1 01: KTJ 54 :: .oo ;i. . 0" 0. : tf 141- 0 i 7 3 . :o :4 .oo :o. . S.i o.6:::i-. os 70 ..'0 ::. oo 7 . .400 o.2:ssn os 58 .00 27.00 22 .62 0.3S91E OS 108 .0 IS.00 38 .56 0.2S3SL OS 108 .0 12.00 38. 56 0.1712H OS 108 .0 15.00 6. 00 o.SJj;i: os 108 19.00 23. . 34 0.40131: OS 108 .0 17.00 2. 540 Q.4o02i; OS 70 .00 19.00 17. ,89 0.4948E 05 48 .CO :: .oo 10 .10 o.4ci':i": os 73 .CO :s.oo 3. 690 0. 1S74E OS 36 . i"0 39.00 5. 910 0.1137t 05 48.00 45.00 1. ,80 Tree-Si:e Croup: Semi-Standard 4444. 118 .0 6.00 ft .630 393.0 118 .0 5.00 11. 80 292. 605 .0 9.00 1, .380 1723. 290 .0 4.00 7. 170 0.1214E 05 218 .0 8.00 18. 67 0.1S67F. OS 108. .0 12.00 IS. 00 0.2643E OS 108. 0 11.00 17, 24 0.3101E OS 194 .0 13.00 45. 39 279.0 194 .0 4.00 18. 16 SSOO. 108 .0 6.00 4. 730 6.34S6E 05 108 .0 11.00 77. 22 0.2621E 05 161 .0 14.00 18. 83 0.31SSE 05 134 .0 14.00 15. 57 O.1210E OS 86 .00 11.00 - 3, .720 8666. 194 .0 6.00 1. ,750 132.0 132 .0 15.00 17, 45 0.2777E OS 132 .0 15.00 19. 40 4OS0. 108 .0 6.00 9 .570 0.1S61E 05 108. 0 7.00 8, 640 4202. 97.00 10.00 16. 46 1071. 218 .0 5.00 14, 30 1285. 218 .0 5.00 14, 47 0.4S97E 05 90 .00 22.00 13, 8S Q.1224E OS 203 .0 13.00 9. ,560 0.5S3OE OS 218 .0 14.00 19. ,46 0.2BS8E 05 218 .0 11.00 33. ,64 0.16131 OS 108 .0 12.00 5. ,780 0.2S65K 0 5 108. 0 13.00 9. 720 0.2370E OS 108. .0 19.00 13. 12 •0.2032E OS 108 .0 13.00 13. 12 0.2044C OS 95 .00 22.00 . 7. 620 3468:- 272 .0 5. DO 5 . 780 9346. 97 .00 12.00 6 .310 0.1I33E 05 129 .0 10.00 14 .07 0.7674E OS 216 .0 13.00 4 .540 0.S228E 05 60S .0 7.00 . 227 .00.15S2E OS 104 .0 8.00 2 .60 0.1S30E 05 108 .0 9.00 18 .14 0.4671E OS 108 .0 IS.00 9. 150 0.3763E 05 186 .0 10.00 25 .09 6S43. 108 .0 5.00 7. 330 4232. 108 .0 8.00 6 .170 0.4479E 05 135 .0 12.00 12 .50 0.41S5E 05 150 .0 12.00 13 .90 0.279SE 05 107 .0 14.00 3 .320 0.1013E 0 5 279 .0 14.00 103 .3 0.1160E OS 194 .0 14.00 71 .86 0.12B0E 05 108 .0 6.00 17 .18 0.2354E 05 108 .0 6.00 17, .17 0.27101 05 97 .00 11.00 S. 110 7642. 218.0 7.00 0 .6800 9559. 218 .0 6.00 0. 6800 0.2139E OS 108 .0 12.00 • 20 .25 0.1192E 05 142 .0 S.00 30. 18 0.I316E 05 10 8 .0 8.00 9.680 0.1176E 05 108 .0 8.00 9. 680 998.0 108 .0 6.00 0 .6000U-7347. 1S1 .0 6.00 10, .56 5894. 1S1 .0 6.00 10. 56 2425. 70 .00 7.00 5, 450 S95S. 134 .0 5.00 9, .440 2198. 134. 0 5.00 9. 440 Tree-Sic Group: Semi-Dwarf 0.1630E 05 . 151, 0 10.00 18, 43 0.14861: OS 418. 0 5.00 20, 06 0.4172E OS 161 , ,0 12.00 13. 90 749S. 218. 0 8.00 28, 45 5883. 218. .0 8.00 • 30. ,044983. 218. .0 7.00 18, .68 O. 509OIi OS 34S. 0 9.00 271. 0 0.4346F. OS 388. 0 9 .00 50. 39 0.47021: OS 171. 0 22.00 16. 61 0.2710P. 05 218. 0 14.00 95, .09 0.1077» 05 339. 0 8.00 57. 7S 1995. 97. 00 13.00 7. 160 0.1376F. 0 5 4S3. ,0 S.00 7. ,030 0.26S9E 05 453. ,0 8.00 6. 820 BOOS. 201. 0 10.00 0. 1000C-0.1293E OS 97. 00 13.00 14, 43 O.6330E 05 605. 0 7.00 134. 7 928S. 108.0 13.00 23. 25 777.0 108. 0 13.00 23. 26 9934. 108. 0 11.00 54. 40 0.2384E 05 108. 0 8.00 22. 00 14 36. 290. 0 18.00 2. 870 ]417. 388. 0 4 .00 3. 00 3182. 142. 0 7.00 7. 260 8386. 272. (J S.00 27. 20 0.21611; 05 l'JI. 0 7.00 4 . 430 0,11211! 05 108.0 9.00 12. 7 3 G, 289011 05 145. 0 9.00 12. 61 Tree - SI in Hr onj>: Mwnr1 0.1077J! 05 3 39. 0 8.00 S7. 75 0.2227): 05 134 . 0 16.00 7. 320 976.0 3d 3. 0 5.00 0. lOOfl!-O.70h«!: 05 33 8. 0 9.00 P.  Mil 0.23'J'.I. Oi 3K7. 0 7.00 !!>fl 0.2'J'GI. 0.', 3«a. 0 7.00 'JO 42.38 112.9 64 . 7J OS,4 I 100.0 58.28 75.20 75.20 19.65 46.44 114.0 55.71 65.06 24.68 24.68 240.0 74. 15 57.55 25.59 29.24 104.6 70.03 1.80 17,84 18.19 34.86 13.66 34.22 60.00 100. 7 45.71 5.390 35.50 28.70 172.5 46.77 64.36 122.2 40.02 153.2 25.00 14.81 71.88 40.79 41.25 66.46 44.95 33.12 116.4 40.57 54.00 58.50 58.SO 78.48 77.14 49.59 47.63 10.00 145.0 99.38 22.46 58.58 31 .93 12.10 51.43 23.98 26.63 30.10 107.9 75.OS 105.2 105.2 43.47 33.07 33.0 7 30.66 49.91 46.60 46.SO 40.00 73.98 73.98 16.36 19.81 19.81 99.90 03.35 78.66 62.60 66.08 79.41 780.0 257. 0 70.35 60.21 182.0 21.66 75.75 76.17 82.51 27.36 217.0 39 .4 2 39.54 85.71 127.0 20 .0? 13.32 104 .9 43.52 123.7 78.02 97 . 75* 182.0 4 1.00 54. 7^  39. 14 1 .00 t.00 8.00 24 .00 108.0 5 8.00 4 3.00 io. on 21,00 21.00 17.00 108.0 106.0 116.0 3S.00 166.0 166.0 100.0 56.00 105.0 4.00 34S.0 34.00 57.00 96.00 Z3.00 18.00 52.00 '46.00 13.00 63.00 13.00 53.00 17.00 17.00 107.0 ' 52.00 43.00 240.0 67.00 19.00 107.0 34.00 86.00 17.00 5.00 6.00 72.00 31.00 1.00 2.00 54.00 92.00 160.0 87.00 25.00 53.00 23.00 28.00 50.00 57.00 1.00 19.00 176.0 84.00 32.00 30.00 78.00 87.00 63.00 19S.0 136.0 44.00 44.00 41.00 19.00 19.00 81.00 30.00 96.00 96.00 8.00 12.00 12.00 S4.00 4.00 4.00 56.00 61.00 509.0 364 .0 62.00 96.00 305.0 254 .0 65.00 1.00 1 .00 15.00 19.00 32.00 9.00 24.00 38.00 37.00 37.00 37.00 89.00 34.00 3.00 22.00 14.00 12.00 30.00 81.00 1.00 30.00 32.00 3 3.00 1 T.I). 0 1 50.0 1.000 1.000 1.000 1 .OHO 1.000 1 .000 1.000 1.000 1.000 1.000 1.000 1.000 1 .000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.8500 0.8500 0.7500 0.7S00 0.8500 0.7500 0.7500 0.7500 0.8500 0.7500 0.8500 0.7S00 0.7500 0.750O 0.8S00 0.7400 0.7000 0.7500 0.7500 0.7500 0.8500 0.8500 0.7500 0.7S00 0.7S00 0.7500 0.7SO0 0.8500 0.8800 0.7500 0.7500 0.7500 O.75O0 0.8300 0.7500 0.8500 0.7500 0.7500 0.7500 0.7500 0.8100 0.7500 0.6700 0.6100 0.7800 0.7S0O 0.7500 0.8400 0.8400 0.7S00 0.6SO0 0.6700 0.7500 0.7500 0.7SO0 0.7SOO 0.7500 0.8500 0.6400 .0.8400 0.7500 . 0.7500 0.4000 0.2500 0.3300 0.3300 0. 3300 0.5000 0. 3100 0. 4000 0.3300 0.4000 0.3300 0.2700 0.3300 0.3800 0.3600 0.3300 0.5000 0.3300 0.3300 0.3300 0 . 5900 0. SCJOO 0.2500 0. 5900 0.3300 0.3300 0.3300 0.3300 0.2000 0.2000 0 .2000 0.2100 0. 200(1 0.2000 119 <il,.-«-rtf»i i,.ni I«t;l] JI1 1J< !'.(«.<• - of I rri'Ji'iti TABLE V I I I CORRELATION MATRIX FOR SIMPLE LINEAR REGRESSION WITH ONE HUNDRED AND NINETEEN PAIRS OF OBSERVATIONS 1 V a r i a b l e VAR1 VAR 2 VAR 5 VAR4 VAR 5 VAR6 VAR 8 VAR1 1.0000 VAR2 0.0526 1.0000 VAR3 0.2820 -0.4374 1.0000 VAR4 0.3235 0.4166 -0.0920 1.0000 VAR5 0.2566 0.2495 -0.0094 0.6882 1.0000 VAR6 0.3298 0.0072 0.0481 0.2444 0.2745 1.0000 VAR8 0.1370 -0.5147 0.4314 -0.1310 -0.1726 -0.0669 1.0000 'See T able I I f o r d e f i n i t i o n o f v a r i a b l e s . 92 TABLE XI RESULTS OF EQUALITY OF SLOPE TEST FOR THREE TREE-SIZE GROUPS Dependent V a r i a b l e (VAR1) i s Apple Y i e l d Per A c r e ( Q u a d r a t i c E q u a t i o n ) 1 S l o p e C o e f f i c i e n t s f o r P o o l e d Data 2 S C P ( 1 ) 90 . 190 SCP(2) = 3430. 676 SCP(3) = -107. 634 SCP(4) 408. 662 SCP(5) = -666. 409 SCP(6) 88. 904 SCP(7) = -5393. 094 SCP(8) 0. 021 SCP(9) -19. 444 SCP(10) = -7. 001 S C P ( l l ) -1. 511 SCP(12) = 0 . 633 SCP(13) 7. 174 SCP(14) = . 1. 463 SCP(15) -1. 393 SCP(16) •- 1. 050 SCP(17) = -225. 679 SCP(18) = -33. 096 SCP(19) -4. 074 SCP(20) = -1. 760 SCP(21) = -2255. 386 SCP(22) - 5. 768 SCP(23) 0 . 214 SCP (24) = 905 . 375 SCP(25) 0. 681 SCP(26) = 145. 471 SCP(27) 478. 763 T e s t f o r H y p o t h e s i s o f Slo p e C o e f f i c i e n t s F = 0.29 D.F. N 2 = 5 4 D.F. N i = 35 PROB. = 1.00 ( T h e r e f o r e H 0 i s accepted) JSee T a b l e I I f o r d e f i n i t i o n o f v a r i a b l e s . 2SCP s t a n d s f o r Slo p e C o e f f i c i e n t s f o r p o o l e d d a t a ( a c r o s s t r e e - s i z e g r o u p s ) . TABLE X I I SIGNIFICANT COEFFICIENTS AT .05 LEVEL FOR COBB-DOUGLAS MODEL1 Dependent V a r i a b l e (VAR1) i s Apple Y i e l d P e r A c r e Constant VAR2 VAR3 VAR4 VAR5 VAR6 VAR 7 1.8530 0.4932 1.2182 0.2240 0.2533 0.1978 0.1062 ST. E r r o r (1.3990) (0.2040) (0.2218) (0.0962) (0.1049) (0.0746) (0.0422) 'F' V a l u e 0.0165 0.0000 0.0206 0.0166 0.0089 0.0128 'See T a b l e I I f o r d e f i n i t i o n o f v a r i a b l e s TABLE X I I I CORRELATION MATRIX FOR COBB-DOUGLAS MODEL1 VAR1 VAR2 VAR3 VAR4 VAR5 VAR6 VAR7 VAR8 VAR1 1.0000 VAR2 -0.0999 1.0000 VAR3 0.4406 -0.6386 1.0000 VAR4 0.3182 0.1268 0.0467 1.0000 VAR5 0.3140 0.0382 0.1118 0.3783 1.0000 VAR6 0.3294 -0.1278 0.2321 0.1351 0.0496 1.0000 VAR7 0.0588 0.0429 -0.1590 -0.1216 -0.1468 -0.0054 1.0000 VAR 8 0.1043 -0.5622 0.3399 -0.0355 0-.0390 0.0863 -0.0471 1. 0000 :See T a b l e I I f o r d e f i n i t i o n o f v a r i a b l e s . 95 TABLE XIV SIGNIFICANT COEFFICIENTS AT .05 LEVEL FOR QUADRATIC MODEL1 Dependent V a r i a b l e (VAR1) i s Apple Y i e l d P e r Acre C o n s t a n t VAR 3 VAR 5 VAR 6 VAR11 VAR13 VAR 15 VAR16 VAR17 VAR2 2 VAR2 3 VAR2 8 5733.2578 2739.7249 398.6005 1096.3665 -8 . 6445 0.9406 3. 5579 -2.2731 1.9925 2671.2339 2.8387 866.9939 ST. E r r o r (7318.3136) (1162.8074) (85.1116) (192.8729) (2.8635) (0.2208 (1.0176) (0.4338) (0.4324) (1094.0824) (0.9108) (143.6294) F' V a l u e 0.0193 0 .0000 0.0000 0.0033 0.0001 0. 0008 0.0000 0.0000 0 .0156 0.0025 0.0000 'See T a b l e I I f o r d e f i n i t i o n o f v a r i a b l e s . TABLE XV CORRELATION MATRIX FOR QUADRATIC MODEL INVOLVING ONLY SIGNIFICANT VARIABLES1 Variable VAR1 VAR2 VAR 3 VAR4 VAR 5 VAR6 VAR7 VAR8 VAR9 VAR10 VAR11 VAR12 VAR13 VAR14 VAR1 1, .0000 VAR2 0. .0526 1. .0000 VAR 5 0. .2820 -0. ,4374 1, .0000 VAR4 0. . 3235 0. .4166 -0 .0920 1. .0000 VAR5 0, .2566 0, .2495 -0 .0094 0, .6882 1, .0000 VAR 6 0. , 3298 0. .0072 0, .0481 0, .2444 0 , .2745 1, . 0000 VAR 7 -0. .0255 0. .0627 -0, . 2136 -0, .0539 -0, .1389 -0, .1480 1, .0000 VAR8 0. .1370 -0, .5147 0 .4314 -0, .1310 -0. .1726 -0. .0669 -0, .0594 1 .0000 VAR 9 0, .1169 0. .9559 -0. .2964 ' 0, .4453 0. ,2347 0. ,0036 0, .0687 -0. .3863 1. .0000 VAR 10 0 . 1631 -0 , .3535 0. .9545 -0, .1004 -0. ,0072 -0. .0242 -0, .1942 0. , 3867 -0. ,2238 1. ,0000 VAR 11 0 , .2945 0. . 3853 -0. ,0787 0. ,9321 0. , 7473 0. .2362 -0, .0275 -0. .1100 0. 4218 -0. ,0690 1, .0000 VAR12 0. . 2098 0. , 1882 -0. .0409 0. ,6780 0. .8997 0. , 2959 -0, .1120 -0. .1594 0. ,1595 -0, ,0355 0. , 8080 1. ,0000 VAR13 0. .2410 0. .0242 0 .0026 0. . 1890 0 . 2385 0, .9122 -0 .1793 -0 .1409 0. .0042 -0, ,0369 0 .1951 0 .2579 1. .0000 VAR14 0, .3697 0. .6630 0, . 1975 0, .4039 0, .2733 0, .1194 -0, . 1019 -0, .2557 0. .6756 0, .1542 0 .3413 0. .1875 0. .0782 1, .0000 VAR15 0. . 3277• 0. . 5443 -0, .1236 0. ,9258 0, .5881 0, .1536 0, .0111 -0. .1300 0, .6131 . -0, .1037 0, .9012 0. . 5355 0, .1239 0, .4471 VAR 16 0 , .2754 0, .5247 -0. .1565 0. .8092 0. .9104 . 0, .2475 -0 .0896 -0, .2855 0. .5203. -0, .1294 0, . 8445 0. . 8621 0. ,2241 0, .4188 VAR 17 0, .2729 0, .4303 -0, .1411 0. .4837 0. ,4974 0, . 7927 -0, .0982 -0, .3255 0. , 3872 -0, .1456 0 .4895 0. .5151 0, . 7150 0 , .4018 VAR18 0. .0952 0. . 7061 -0, . 3124 0. ,4063 0, .1047 -0. .0807 0, .0303 0, .1226 0. .7370 -0, .2613 0, . 3815 0. .0425 -0 , .0887 0, . 5165 VAR 19 0. .3658 0, .2117 0, .1840 0, ,9153 0. ,6274 0, .2789 -0, .1048 0. ,0015 0. ,2514 0, . 1321 0, .8078 0. .6170 0. ,1948 0, .4274 VAR 20 0. . 3264 -0. ,0980 0, .5966 0. , 3901 0. ,7189 0. .1916 -0, .2376 0. , 1537 -0. ,0325 0. , 59 77 0, .4393 0. .5903 • 0, .1455 0. ,2876 VAR 21 0. .3801 -0. ,2155 0, .4123 0. .1095 0. .1407 0. , 8489 -0, .1912 . 0 , .1959 -0, ,1514 0. .3058 0, .1019 0. . 1506 0. ,7488 0. , 1068 VAR 2 2 .0. .2501 -0. .4666 0. ,9654 -0. 1158 -0. ,0416 0. ,0178 -0. , 1814 0, ,6151 -0. .3114 0, .9399 -0. ,0930 -0. .0679 -0 . ,0375 0. ,0854 VAR2 3 0. .2368 0. .2507 -0. .0579 0. , 775 7 0. . 8677 0. ,2872 -0, .0909 -0, .1612 0. ,2386 -0. ,0512 0, .8894 0. .9827 0. ,2504 0. .2342 VAR 2 4 0, .2446 0. , 2141 -0, .0468 0. , 7816 0. 8169 0, .4706 -0, .0862 . -0, .1343 0. .1882 -0, .0581 0, .8562 0. .9342 0, .3980 0, . 2441 VAR2 5 0 , .3274 0. .3410 -0. .0313 0, ,8813 0. .3845 0, .1681 0. .0220 0 .1148 0. ,4166 -0. .0522 0 , .7753 0, .3319 0, .0884 0, . 3548 VAR2 6 0. . 2490 0. .1630 -0. .0306 0, ,6509 0. ,8737 0, .5046 -0, ,1433 -0. ,1589 0. ,1297 -0, .0428 0 .7648 0, .9606 • 0. , 4554 0, .1952 VAR2 7 0. .2884 0, .0245 0. .2160 0, ,4741 0. . 7931 0. .1704 -0, .0825 0, ,2728 0. .0786 0, .1949 0 , .5103 0. ,5747 0. .0955 0, .1S65 VAR2 8 0. .3500 -0. ,2026 0. , 2055 0. 1317 0. ,0954 0. , 8251 -0 , .0897 0. , 3093 -0, ,1439 0, .0923 0 , .0972 0. .1003 0. ,6411 0. .0043 Variable VAR15 VAR16 VAR17 VAR18 VAR19 VAR 20 VAR21 VAR22 VAR2 3 VAR24 VAR 2 5 VAR 2 6 VAR 2 7 VAR2 8 VAR15 1. , 0000 VAR 16 0, .7750 1. .0000 VAR17 0. .4254 0. , 5727 1, .0000 VAR18 0, .5597 0. . 3268 0, .2004 1, .0000 VAR 19 0. . 7556 0, ,6666 0, .4265 0, , 2320 1. .0000 VAR20 0. . 2985 0. , 5130 0. .2488 -0 , .1139 0. .5352 1. ,0000 VAR21 0, .0311 0. .0673 0. .4787 -0. .1902 0. .2520 0. 2899 1, ,0000 VAR2 2 -0, . 1333 -0. ,1888 -0, .1927 -0, ,2343 0. . 1451 0. , 5609 0, , 3943 1. ,0000 VAR 2 3 0 .6532 0. . 8843 0, . 5218 0 .1296 0. .6970 0, .5412 0 .1404 -0, .0844 1 .0000 VAR24 0. .6159 0, .8238 0, .6662 0, .1196 0, . 7480 0, .5149 0 .2934 -0. .0728 0, .9527 1, ,0000 VAR2 5 0, .8788 0. .5187 0, .3192 0. .5422 0, . 8180 0. .2269 0 .1085 -0, .0063 0. .4600 0. ,5144 1, .0000 VAR 2 6 0. .4986 0, . 8290 0 .6793 0, .0236 0, .6115 0. .5791 0 .3215 -0. ,0641 0 .9371 0. ,9527 0. .3272 VAR2 7 0. ,4401 0. , 5922 0, .2543 0, .1833 0, .4918 0, .7S76 0 .1756 0. .2594 0, . 5280 0. .4980 0, .3914 VAR28 0. .0490 0. ,0376 0, .4281 -0. .0596 0. .2338 0. , 1508 0, .9011 0. .2432 0, .0965 0, .2710 0. .1943 0000 5610 2645 1.0000 0.1892 'See Table II for d e f i n i t i o n of v a r i a b l e s . TABLE XV 1.0000 CORRELATION MATRIX FOR QUADRATIC MODEL INVOLVING ONLY SIGNIFICANT VARIABLES 97 TABLE XVI OBSERVED AND CALCULATED VALUES OF APPLE YIELDS BASED ON QUADRATIC MODEL INVOLVING ONLY SIGNIFICANT VARIABLES No. O b s e r v e d C a l c u l a t e d R e s i d u a l No. O b s e r v e d C a l c u l a t e d Res i d u a l 1. 22220 . 25870. - -3650.0 61. 15304 . 12966. 2338. 4 2. 25090. 14013. . 11077. 62. 46710. 31555. 15155. 3. 4951.0 34297. -29346. 63. 37630. 19907. 17723. 4. 17350. 31585. -14235. 64. 6543.0 5165.8 1377. 2 5. 36760. 25195. 11565. 65. . 4232.0 10588. -6356. ,2 6. 20760. 19789. 971.32 66. 44790. 21210. 23580. 7. 23830. 13026. 10804. 67. 41550. 22056. 19494. 8. 14800. 13026. 1774.4 68. 27950. 22411. 5538. 9 9. 9251. 0 21225. -11974 . 69. 10130. 18988 . -8858. ,4 10. 28100. 31564. -3464.2 70. 11600. 25000 . -13400. 11. 28140. 35365. -7225.2 71. 12800 . 10261. 2539. 1 12. 62220 . 33262. 28958. 72. 23540. 10259 . 13281. 13. 22550 . 32531. -9980.8 73. 27100. 15886. 11214. 14. 35910. 26331. 9578.7 74. 7642.0 9317.1 . -1675. 1 15. 28350. 22101. 6249.2 75. 9559.0 7146.7 2412. 3 16. 17120. 27863. -10743. 76. 21390 . 20684. 705. ,68 17. 54470. 31001. 23469 . 77. 11920 . 14588. -2667. 8 18. 40130. 29709. 10421. 78. 13160. 15346. -2185. 6 19. 46020. 24938. 21082. 79. 11760. 15346. -3585. 6 20. 49480. 46341. 3139.2 80. 998.0 4169.1 -3171. ,1 21. 46020. 30649 . 15371. 81. 7347.0 7592 . 3 -245. .27 22. 18740 . 28916. -10176. 82. 5894.0 7115.3 -1221. ,3 23. 11370. 29489. -18119. 83. 2425.0 9676.1 -7251. ,1 24. 44 44 .-0 690 3. 5 -2459 .5 -84. 595 5.0 3866.3 2088. ,7 25. 393.0 5115.1 -4722 .1 85. 2198.0 3866.3 -1668. ,3 26. 2992.0 26570. -23578. 86. 16000. 17459. -1458. ,8 27. 1723.0 6211.3 -4488.3 87. 14880. 11282. 3597. ,9 28. 12140. 14213. -2072.7 88. 41720. 45607. -3887. .1 29. 15670. 20162. -4492.3 89. 7495.0 20675. -13180. 30. 26430. 17032. 9398.0 90. 5883.0 14922. -9039. ,S 31. 31010 . 26450. 4560.5 91. 4983.0 15094 . -10111. 32. 279.0 " 4730.9 -4451.9 92. 50900. 50890. 9. ! 5859 33. 5500.0 5677.7 -177.65 93. 43460. 23871. 19589. 34. 34560. 17805. 16755. 94. 47020. 34947. 12073. 35. 26210. 26874. -664.44 95. 27100. 34165. -7065. , 5 36. 31550. 22932. 8617 . 7 96. . 10770. 25954. -1S184, 37. 12100 . 31349. -19249. 97. 1995.0 16300. -14305. 38. 8666.0 15275. -6608.6 98. 13760. 8904.6 4855, .4 39. 132.0 23189. -23057. 99. 26590. 18478. 8112. ,4 40. 27770. 28988. -1217.9 100 . 8005.0 13991. -5986, .5 41. 4050.0 7442.4 -3392.4 101. 12930. 17348. -4417. ,9 42. 15610. 12657. 2952.9 102. 63300. 47204. 16096, 43. 4202.0 14711. -10509. 103. 9285.0 18777. -9491, !s 44. 1071.0 6545.6 - 54 74 .6 104 . 777.0 18779. -18002, 45. 1285.0 6640 . 8 -5355 . 8 105. 9934 .0 17388 -7453, . 7 46. 45970. 31403. 14567. 106. 23840. 15579. 8261 .2 47. 12240. 22657. -10417. 107. 1436.0 34242. -32806, 48. 55300. 24749 . 30551. 108. 1417.0 3261 .1 -1844, . 1 49. 28580 . 24561. 4018.6 109. 3182.0 9171.4 -5989, .4 50. 16190 . 18802. -2611 . 8 110. 8386.0 7242.5 1143 . 5 51. 25650. 23354. 2296 . S 111. 21610 . 8799.6 12810, 52. 23700. 34929. -11229 . 112. 11280. 12355. -1075 . 3 53. 20320. 22872. -2551.9 113. 28900. 16522. 12378 54. 20442. 28198. -7755.9 114 . 10770. 24339. -13569 SS. 3468. 9224.4 -5756.4 115. 22270. 23062. -792, .12 56. 9346 .0 16222 . -6876.1 116. 976.0 8091.6 -7115 .6 57. 11330. 15743. -4412 .6 117. 20880. 19102. 1778, .1 58. 76740. 23925. 52815. 118. 23930 . 22415. 1515 .1 59 . 52280 . 60396 . -8115.6 119. 29660. 22435. 7224 .7 60. 15520. 7730.5 7789.5 A u t o c o r r e l a t i o n C o e f f i c i e n t 0.024 Durbin-Watson D S t a t i s t i c 1.948 INFORMATION SHEETS (COMPLETE SET OF APPLE - ENTERPRISE - DATA SHEETS USED IN PART FOR THESIS STUDY 1) d e s i g n e d by the Economics Branch - Vancouver, Canada Department o f A g r i c u l t u r e , B.C. ENTERPRISE TREE FRUIT RECORD Economics Branch, Canada Department of Agriculture 6660 N. W. Marine Drive, Vancouver 8, B. C„ Study Tear Name: , Record No. P.O. Address: . Date Taken: District: Taken by: Marketing Point: Check by; j Orel-1 Acres Land E lard Value/Acre escripti Other Acres on and Value Improved Value/Acre Unimproved Acres :Value/Acre Waste Acres Total Acres Owned Rented Additional Acres Owned Land Suitable for Orchard LAND IMPROVEMENTS LAND PURCHASES & SALES Page1- 2 Clearing Breaking Disc and Harrow Picking Roots Stones Acres Cover description Hours - Farm tractor - Unpaid Labor ' Cost - Hired tractor - Hired labor Other costs Total Costs Purchases Sales Price Paid Received Owing Yes ; No Purchased Land Cropped j i Sold Land Cropped ! ' Date of Purchase or Sale i Orchard Other im-i Un-proved .! improved Acres Purchased Acres Sold Expenditures on New Construction, Improvements and Repairs Associated with Land C A S H C O S T S Value of Farm Contributions To t a l Cach Cost Hired Mach. Mater'1 Cost,? Hired Labor j T o t a l Farm Tractor I'taber1!! Unpaid Labor Value Repairs Hrs, Costs Urr, Cost s Ce.sh Hrs., Cost Value Hr:;... \ Value rrrg„ Ditches ! Drainage Farm Reads i Fences Other j i ! : 1 . i 1 1 i i 1 Inventory of Buildings Page - 3 Year B u i l t • Replacement Cost Beg. of Year Typ- of Cone T t Cash New Bldg Equip & I'-Iaterial Ccct , Imp,. Hir?d Labor Value of 37 Fcrm Tractor arm Cont I-iatr'l ributions Unpaid Labor Total Costr Re-pairs Value | Value •Hrs.!Value i Machine Shed _ • _ _ . t i Tool Shed i •1 :i • i i Garage i! ;j ! i Pump House 1! ! i J i Pickers Cabins | i i " "f 1 i i 1 i ! i 1 j i : ! i 1 1 i ! ! . I 1 = i - l } j j 1 i ! ; * Road Side Stand \ t 1 i j I i 1 I 1 ! ! ! STANDARD APPLES TREE FRUIT INVENTORY Page - k V a r i e t y Spacing Trees/ Acre Planted 1969 1 to $ Yrs. 6 to 10 Yrs. 11 to 20 Yrs, Over 20 Yrs, T o t a l Trees Est. } I n t e r - j Winter 69 Acres j Plants Killed 1 Dam. Red D e l i c i o u s Golden D e l i c i o u s Mcintosh Winesap Newton Rome Beauty S p a r t x i TREE FRUIT INVENTORY Page - $ V a r i e t y Spacing Trees/ Acre Planted 1969 Planted! Planted 1968 | 1967 Planted 1966 Planted 1965-61 Over j T o t a l |Est. ! Winter I969 ! 5 Yrs. ? Trees jAcres 1 ! ' K i l l e d ] Dam,, Serai-Standard Apples 1 1 i : i ! i i i • 1 ! ' t 1 ! i 1 ! ; 1 i J \ ! ! i i | 1: 1 1 , j ! ! i ! j 1 ! i i -i 1 ;i Semi-Dwarf Apples 1 i i i 1 i \ 1 ! ! : i ! ' i ! 1 i ! i • : ! • ! 1 i j ! 1 • i ! ! \ ii i i 1 . \ \ 1 t . -\ I j '• \ i i : ; i i ] f \ t r * ! ! Dwarf Apples j ;. ! ! | 1 i i 1 1 ; ! ! . ! I i ! . i 1 i • * • i l l ! ! 1 1 : < - f I i 1 1 ! •. i ! : i i } i ! • • i ! 1 • 1 i 1 ! i t ' ' j [-! i 1 1 • : 1 ! * : 1 I i i i 1 TREE FRUIT INVENTORY Page - 6 Variety Spaaing 1 Trees/ Planted I969 1 to j 6 to £ Yrs.i 10 Yrs. 11 to 20 Yrs. Over 20 Yrs. Total Trees Est. Winter I969 Acre Acres IKil led . Damaged „ Fsars i .... i . 1- -~™ Peaches 1 1 i i 1 * 1 j . . . . j 1 ; i ! i ! ! ] . 1 " 1 • 1 1 1 i 1 ' 1 .. ... p. — _ ! i ! 1 TREE FRUIT INVENTORY Page - 7 Variety- Spacing Trees/j Planted Acre .; I969 1 to $ Yrs. 6 to 10 Yrs. 11 to 20 Yrs. Over 20 Yrs Total , Trees Est. Acres Winter I969 I Killed i Damaeed i Apricots \ ! | i 1 j i j 1 i 1 1 i ' ; 1 1 : ! 1 i i 1 1 1 5 i I Cherries j ! ! > i . i 1 j ! | j • j j I 1 i • Plums & Prunes J ! i i ! ! 1 ! 1 t i 1 } 1 1 1 *l 1 j i t _________ i 1 1 ! 1 1 PHYSICAL-DATA - RE SPECIFIC APPLE PLOTS - - j! PRODUCTION AND RECEIPTS FROM SPECIFIC APPLE PLOTS Page - 8~ Plot of Inten- Plot of Standard! Intensive Standard sive planting Planting H V a r i e t y r Variety- U Type of Pack ; Main Root Stock \:- Extra Fancy: Large f I Size of Block Ac. Medium ! 1 1 i Spacing j Small ! • Total No. of Trees | j. Fancy: Large ! j Trees per A Cre : Medium ; i Year Planted Small Winter Damage 69 ! ; 1 | j; Cee \ i Trees K i l l e d No. i ; ! !: Cuiis ! ; . Trees Damaged No. • . A \ \ r Receipts . i Ai r Drainage Good 1 I • Extra Fancy i i F a i r Fancy j t Poor Cee i 1 1 D i r e c t i o n of Slope 1 Rebates j Degree of Slope Level C u l l Returns I 1 Slig h t }l Total Receipts Moderate ji i !!, Farm Sales: Lbs. I Steep : 1 u 1 Receipts ! S o i l Tvoe 1 Home Use: Lbs. j Cover Crop j i Yes Valu° No GRADE AND SIZE OF FRUIT DELIVERED TO THE PACKim HOUSE Page - 9 Name of Packing House -A P P L E S Variety Type of Pack Extra Fancy Fancy Cees j Culls ! Large Medium Small Large Medium Small •-1 - - t- . j I \ • -j !  I i i Apple Receipts Variety 1 T y p e ° f variety j p a c k Extra Fancy Fancy Cee Rebates Cull Return Total [ Cull Receipts ' Charge j I i f ; ; 1 j I 1 i i i i l 1 ; 1 Variety- Type of r a c k GRADE AND SIZE OF FRUIT DELIVERED TO THE PACICING HOUSE P E A R S Extra Fancy I Fancy Mediiim Small Lar^e Small C G G S Culls Pear Receipts Variety t T-Type of Pack Extra Fancy Fancy Cee Rebates Total Receipts Cull Charge -( H I f GRADE AND SIZE RECEIPTS Page - 11 Plums and Prunes Variety- ( Select 1 1 i No. 1 No. 2 C u l l s Select No. 1; No. 2 Rebates T o t a l Receipts C u l l Charge i ; i Peaches V a r i e t y Domestic Grade Domestic Grade T o t a l Receipts C u l l Charge No. 1 No.. 2 uUXIS i i No. 1 No. 2 Kebaties I i j I 1 l i i • Apricots V a r i e t y Domestic Grade Domestic Grade Tot a l Receipts C u l l Charge No. 1 No. 2 No. 3 ouxis r No. 1 No. 2 No. 3 1U_JU ct Uco i 1 i I 1 i i i i I 1 t i C h e r r i e s V a r i e t y No. 1 Orchard Run Cu l l s i No. 1 Orchard Run debates T o t a l Receipts C u l l Charge i j 1 ! ! • • r i : 1 1 1 i FRUIT SOLD AND USED ON FARM Farm Sales Used on Farm |. k'axm Sales Used on l''ar,n Pounds Receipts Pounds Value Pounds Receipts Pounds Value Apples ; Peaches !; Apricots Pears )'• Cherries Plums and Prunes • f: Totals XXX XXX FERTILIZER USED Kind of Total Orchard \ Intensive Apple Plot c, , , . , ; Other Spe c i f i e d Standard Apples F r u i t F e r t i l i z e r ; Quantity Cost i-ILbs/Tree Quantity Costs jibs/Tree !; | Quantity Costs : Quan. Costs !| 1! || ii j; i; j; ii j j! ii | ii I; t' i • |i \ : N ! V: j • 1: • \ ll SPRAY MATERIAL Page - 13 Kind of Sprav Total Orchard ! Intensive Apple Plot Standard Apple Plot Other Sp e c i f i c Fruit ' Quantity Costs Quantity Costs Quantity Costs Quantity Costs Boron \ i Zinc 1 t i i Magnesium i t I t i 1 ! Manganese i I Iron ! Urea D i n i t r o c r e s o l I 1 Naphthalene acetmide i i ! (Amid Thin) ! 1 I ! Sevin 1 1 1 Triethanolamine Salt 1 of 2 , U, 5 - T.P. i •i Naphthalene Acetic J ;l Acid (N.A.A.) Alar Dormant Spray-Moras tan Keithane Tedion i 1 i Ethion l i II jj Kava thane i i | Movocide i i i D Lmethoate i i 1 I i (Cygon, Roger) 11 i i Total. ; X X X X X X 1 , • ! X X X i Kind of Spray T o t a l Orchard I Intensive Apple P l o t Standard Apple Plot Other S p e c i f i c F r u i t Quantity Costs Quantity Costs Quantity Costs Quantity Costs Guthion Perth ane Supreme and Superior | Type O i l s i I 1 D.D.T. | Parathion j D i a z i n i o n l Ii fl Thiodan i Para dichlorobenzene ! 1 Lime Sulphur ! Glyodin-Dodine \ ! J (Glyodax) ! i ! Dodine (Cyprex) Dichlone (Phygon) ! ] Ferbam i & Maneb < i | ! 1 Z i r i m ! n i i 1 Captan 1 « Bordeaux i i i ! I Bot ran i 1 1 r | i 1 1 i Malathion I i ! ! 1 1 i Wettable Sulphur i l 1 I 1 i Paste Sulphur 1 I F i x e d Copper i •  — i i i 1 1 j 1 ! i 1 j i ' i i i ! ii ! i I! ! | i 1 1 i I > COST OF SEEDS AND PLANTS Page - Ih Total Orchard Intensive Apple Plot j Quantity- J - -Cost i Quantity Cost H ' Quantity Cost j Quantity Cost F r u i t Trees j ] ! ! i Grass and Plants To t a l j Standard Apple Plot jOther S p e c i f i c F r u i t CUSTOM WORK Rate Total Oi Received -chard Paid Paid Intensive {Paid Standard Apple Plot Apple Plot Paid Other S p e c i f i c F r u i t Plowing D i s c i n g Mowing i i \ Raking \ j D i t c h i n g \ i : Spraying j Hauling | i !• 1 i !; ! Other trucking i | i T o t a l XXX j 1 I-i 1 i GENERAL MACHINERY AND EQUIPMENT - *S Total Orcha?-! / Share to j Intensive j Apple Plot • i cnare to i ftlic.re to ' No. Begin of Yr. Value Purchases Sales Cost of Repairs Star/., arc' Apple Plot 'Other Spe-c i f i c F r u i t I r r i g a t i o n - ! | I t ! I i I Orchard i 1 ! 1 i • & Mask Spray Costumes i i i ! j Pruning Equipment 1 j 1 1 Props j i Ladders Picking Bags I Orchard Boxes j i 1 1 i Equipment - i Pickers Cabin ' -1 1 I i Ditcher j i j i i j i 1 ! | t | j i ; 1 i i i : 1 1 1 i Sub Total XXX j j 1 ! • i 1 I GENERAL MACHINERY AND EQUIPMENT - Continued Ea.g«.- 16-T o t a l Orchard ( Share to j Intensive j Apple P l o t i Share to Share to Standard '. Other Spe-Apple Plot i c i f i c F r u i t 'Beg. of Year N o- jValue Purchases Sales ' l C 0 S t ° f ! i a i e s 'Repairs i Plow : i ! ! i i i ii !| D i s k i i IJ i! II Row Crop or F i e l d C u l t i v a t o r t i ii Harrows i * • it ,1 i • Mower 1 i i i! Rake i : i ! i j ! j I Hand Sprayer ! ! r T r a i l e r j '< i i! i ii ;i I: ; 1 , I i Wagon ! | 1 t ! | ii ; '! 1 j i Chain Saw j i I I I \ '. t ! E l e c t r i c Motors ! ! I 1 1 i Small Tools and i ! Garden Tools j j i I i i 1 ! • : l '• i < ! ; ; •> i ! i i •i ' i ' i ! . ; T o t a l ', 1 1 i . !-i i ! ;ge -Car •' Garden Truck i Tractor •Tractor | • Sprayer G i r a f f e ( Roto j S q u i r r e l j Mower f etc. i I Year i ! : : i Make j , i : 1 j Size i i i ! l i l i e s f o r Year i 1 1 • Miles to Farm i i Hours Used ! ! 1 i Value Beg. o f Yr. ! '• : . i i Purchase Price i 1 ; : ! ! Sales Price 1 ! ; ; ' I : Operating Costs i Cost of Fuel i i i i O i l and Grease " ; ; • ! • ! f ; Repairs j : ! i T i r e s 1 j : Licence • : ' t ' •' 1 *: 1 Insurance | | i ! • 1 i 1 ! 1 j ! ! •: ! 1 i T o t a l Operating \ Costs t I ! i ' < '. i ' , Proportion of Totals ! ' ! \ Intensive Apple P l o t j j \ Standard Apple Plot f ! • i Other S p e c i f i c F r u i t j i i i N. B. I f gross f i g u r e s only a v a i l a b l e on costs estimate cost of car operation or t r u c k i f used instead of car. LABOR RECORD Page - 18 Rate of Wages Total Orchard jj Intensive Apple Plot j Standard Apple Plot j Other Specific Fruit Total Hours Total Wages Board ! Total l Hours Total Board Total Hon rs Tot"! Board 1 Total Hours Total; Board Wao-e Hired: Month | i i I i I 1 i i 1 Day 1 i 1 1 i i | I 1 Piece Work i ! | ; i Family - Daughl :er i ! • j Son (Age ) 1 .Son (Age ) i Wife I t I i i t I i i Operator i 1 i i 1 i 1 i i 1 LABOR INPUTS RE PLOTS Intensive Apple Plot ! Standard Apple Plot Hired i Family 1 Operator j i i i Hired Family j Operator : 1 ! ! Pruning, Grafting, Repairing and Removing trees 1 ! 1 1 1 i Cultivating ! i i Mowing 1 1 I; i i 3 \ F e r t i l i z a t i o n 1 i i Spraying i i Thinning, propping j i , i Irrigating | [ | i _ _ ! Picking j | i ;| i i — — — — — — — — — — — — — — — — »—— 1 Distributing & hauling boxes to packing house j \ !; 1 ! Collecting and storing boxes ! J' <• i .! 1 i ! i j FARM LIABILITIES • .; Page - 19 , Borrowed From Purpose Amt. Owing Deg. of Yr. Borrowed During the Year Paid During Year Owing End of Year Amt. Term Rate f Princ. I n t . P r o v i n c i a l Gov't | Farm Credit i Farm Improvement i 1 V.L.A. Bank ! 1 i i C r edit Union Mortgage Co. 1 t ! Finance Co. ! i Machinery Co. 1 i Other | t > i j 1 Current Borrowing i t i 1 ! . 1 Bank j Credit Union I Other t 1 1 ! 1 ! I i I 1 i t 1 I 1 1 ..... 1 " RECEIPTS' Current Receipts Total Orchard Intensive Apple Plob oir . i ' . ' l i i rc l Apple Plot Oth Soocifio F r u i t • F r u i t : Apples Pears Plums and Prunes 1 i l Peaches i i 1 Apricots • ! Cherries Farm Sales Custom Work Non-Farm Earnings i Other i i Total i I 1 , C a p i t a l Receipts — 1 1 Real Estate Sales j 1 1 i Power Equipment Sale? 1 | General Equipment S S I P -i 1 Gthcr 1 I 1 i Total - i i I OF FAMILY " " v Pago' - 20 Sox | j Months-A-c j ct i i Operator | i Wife | xxx i Children: 1 j i i ! 1 i 2 ! t 1 1 1 3 1 u ! 1 i $ i 1 i I I i 6 ! ! i 1 i 1 7 ; 8 1 i I ! ? ! i i Others j I I i i i Year Operator Started on this Farm Acres Orchard Acres Improved Acres Unimproved I I EXPENSES Page - 2 1 Total jj Intensive Orchard i; Apple Plot Spaniard Anple F nt Other F r u i t Total Orchard Intensive Apple Plot Standard Apple Plot Other Spe c i f i c Fruit Current Expenses | Current Expenses ,, I Cash Rent E l e c t r i c ( T J to farm) 1 Land Taxes. P h o n e t o farm) 1 I r r i g a t i o n - x | £ e r Freight & Express Water T o l l Accounting E l e c t r i c i t y Interest(current) Gas and O i l Membership Fees F i r e Insurance Orchard Box Rental H a i l Insurance i Other Repairs - Land j Buildings j C u l l Charges i Plants &T. Seeds , ! Purchased \ F e r t i l i z e r } Spray Material j j Total Current Operating Costs of j a l l Equipment j C a p i t a l Expenses ! Labor: Mages i Un« Ir.v.n New Cons't Lr.-d C.P.P. 1 B u i l d i n g Liab,. In3, Land Ir^prcveir. ?rvb r, j ___________ p , • Custom Wcrk j ! Power Equip, Fur,. Meed Spr-.vs 1 Geu-val Equip, Pr^. Small Haivlware ! Ofcbr.r Mien. O i l & Greaco| I Total C a p i t a l j 1 . 1 1 LABOR TIME SHEET Date Description of Work Intensive Apple Plot Semi-Standard or Standard Apple Plot Full-time Employee and Operator Family Labor Full-time Employee and Operator Family Labor ' Casual Hired Adult j Under j 15 Yrs. Hired Adult I Under ;15 Yrs Hours Hours I ! I 1 1 ' ! ! i r 1 j L. . 1 ; , i i < ..... i 1 i i " " " " j 1 1 1 i I ! 5 i i i Ii i i I  . f i 1 i 1 1 CASfl EXPENSES Date Description (including custom work) Intensive Apple Plot Standard o? Semi-Standard Apple Plot Quantity or Hrs Cost Quantity or Hrs". Cost • * Y i CASH RECEIPTS Dace Kind of Apple Intensive Apple Plot Semi-Standard or Standard Apple Plot Pounds Grade " C u T l s Pounds Total Receipts C u l l Charges i Pounds Grade C u l l s Pounds T o t a l Receipts C u l l Charges U i NET ESTABLISHMENT COST IN I969 OF AN INTENSIVE APPLE PLOT PLANTED IN 19 Kind of root stock Land value per acre No. of trees i n plot Value of irrigation system Spacing Cost of trees Area planted - acres Est, value of equipment used Receipts from crop sales Est, value of equipment chargeable to plot Est, operating cost of equipment to plot Total Orchard Size - Acres Est. of taxes Date Quantify Hours 1/ Description of Item Custom Work Fer-t i l i z e r Irriga-tion Prun-Thitn-ning Spray-ing Tree Replace-ment Weed Con-t r o l Mow-ing Pick-ing & Hauling Sundry JJoliars .. _ 1 1 1/ I f operator or family labor - Do not put in value but indicate item applicable with "U-". NET ESTABLISHMENT COST IN I969 OF AN INTENSI VE APPLE PLOT PLANTED IN Cont'd Date Quantity Hours 1/ Description of Item Custom Work Fer-t i l i z e r Irriga-tion Pruning Thin-ing Spray-ing Tree Replace-ment Weed Con-t r o l Mow-ing Pick-ing & Hauline Sundry i Dollars 1/ If operator or family labor - Do not put in value but indicate item applicable with "K", 

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