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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 A P P L E Y I E L D R E L A T I O N S H I P S 1969  I N THE OKANAGAN BRITISH  IN  A R E A OF  COLUMBIA  by  EWON L E E  B.Sc,  Seoul National  University,  1964  A THESIS SUBMITTED I N P A R T I A L FULFILMENT THE R E Q U I R E M E N T S F O R THE DEGREE M A S T E R OF in  OF  OF  SCIENCE  t h e Department of  Agricultural  We  accept  required  this  thesis  Economics  as conforming  to the  standard  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 September,  1972  In presenting  this thesis  r e q u i r e m e n t s f o r an British freely that  in partial  advanced degree at the  Columbia, I agree t h a t available  permission  for reference for extensive  s c h o l a r l y p u r p o s e s may ment o r by  his  be  not  be  shall  and  study.  I further  g r a n t e d by  a l l o w e d w i t h o u t my  Date  September,  1972  the  Head o f my  agree for Depart-  that  for f i n a n c i a l gain  written  Columbia  of  make i t  this thesis  I t i s understood  Department of 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 V a n c o u v e r 8, Canada  University  Library  copying of  representatives.  the  the  copying or p u b l i c a t i o n of t h i s t h e s i s shall  f u l f i l m e n t of  permission.  ABSTRACT  The factors  purpose o f the study  i s t o determine  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  which  i n the  Okanagan a r e a d u r i n g t h e y e a r 1969. Regression quantify yield different it  analysis  i s u s e d i n an a t t e m p t t o  relationships.  tree-size  i s necessary  A comparison  c a t e g o r i e s i n order t o determine  to f i t separate  regression equations  o f u s i n g t h e d a t a f o r t h e t h r e e groups equation.  For this  p u r p o s e an E q u a l i t y  o f Slope  The outcome o f t h e t e s t  significant  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 fortree-size  t o c o m b i n e them i n t o  ratic  Test i s  categories.  coefficients i n  Hence i t i s f e a s i b l e  a r e employed:  two d i f f e r e n t  types  one i s a C o b b - D o u g l a s  i n t h e l o g a r i t h m s a n d t h e o t h e r i s a quad-  function. Both  f u n c t i o n s i n c l u d e a dependent  namely, y i e l d p e r acre  and s e v e n i n d e p e n d e n t  that i s , d e n s i t y , age, value o f f e r t i l i z e r of spray  applied, pruning  variable, variables;  a p p l i e d , value  and t h i n n i n g l a b o u r h o u r s ,  graphical  dummy, a n d  variables  a r e m e a s u r e d on a p e r - a c r e b a s i s e x c e p t  case  instead  one e q u a t i o n .  relationships  function linear  whether  shows t h a t t h e r e a r e no  For the r e g r e s s i o n a n a l y s i s , of y i e l d  among  i n a single regression  performed.  the e q u a t i o n s  i s made  tree-size  index.  These  o f a g e , g e o g r a p h i c a l dummy and t r e e - s i z e The  data, which  consists  geo-  independent i n the  index.  o f cross - section  informa-  ii  tion  f o r 1969 r e p r e s e n t s one h u n d r e d and n i n e t e e n s a m p l e  apple p l o t s . apple  I t was d e r i v e d f r o m p e r s o n a l i n t e r v i e w s w i t h  growers. The  from  q u a d r a t i c f u n c t i o n poses a problem  cross-terms  modify  i n the equation.  the f u n c t i o n  i n such  arising  I t was n e c e s s a r y t o  a manner t h a t t h e c r o s s - t e r m s  i n c l u d e d i n t h e 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 logical each and  or economic grounds.  type o f f u n c t i o n used  The r e g r e s s i o n r e s u l t s f o r  i n the analysis  estimates of coefficients  shown. into  on b i o -  are d i s c u s s e d  and r e l a t e d s t a n d a r d  errors  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 b r o k e n  apple v a r i e t y  groups because d i f f e r e n t  down  varieties of  a p p l e may w e l l h a v e 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 . trees  i n the s p e c i f i c plots  up o f a m i x t u r e  Apple  u n d e r s t u d y , h o w e v e r , a r e made  of varieties,  thus  i t i s extremely  difficult  to  draw a c l e a r map o f a c r e a g e s  o c c u p i e d by e a c h  In  attempting to obtain variety  data, notwithstanding the  mixture  of varieties  down u n d e r c e r t a i n yields  into  grade  i n stands, the o r i g i n a l  assumptions.  varietal,  c o n s t i t u e n t s s i m i l a r problems  yields  tree-size,  grades in  tests  apple  arise.  of differences  a r e made u n d e r s t a t e d c o n d i t i o n s f o r apple-grade,  These t e s t s differences  data i s broken  A l s o i n decomposing  Despite these d i f f i c u l t i e s , among a v e r a g e  variety.  i n average  and r e g i o n a l c a t e g o r i e s .  r e v e a l t h a t t h e r e a r e no apple y i e l d s  forvarieties,  and r e g i o n s . , b u t t h e r e a r e s i g n i f i c a n t  t h e case o f d i f f e r e n t  tree  sizes.  significant apple  differences  The r e s u l t s  o f these  iii  tests  are p r e s e n t e d i n Chapter V I . The  theoretical the  t o be  But, i n p r a c t i c e ,  empirical  arity The  framework,  form of f u n c t i o n  problem  Cobb-Douglas  problem.  the s e l e c t e d  function,  Apart from t h i s ,  a serious  tree-size conclusion  of view of s t a t i s t i c a l  h o w e v e r , does n o t c a u s e s u c h  the r e p r e s e n t a t i o n  the  Okanagan i n  1969.  a  i t s a p p l i c a t i o n brought i n  On  this evidence, a  basic  tentative  made i n f a v o u r o f t h e C o b b - D o u g l a s  for  inference.  except f o r the c o e f f i c i e n t of the  index v a r i a b l e . was  to  multicolline-  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 to the independent v a r i a b l e s  the  independent  i t does n o t c o n f o r m w e l l  s i t u a t i o n ; i t produces from the p o i n t  seems, w i t h i n  able to represent s a t i s f a c t o r i l y  apple y i e l d r e l a t i o n s h i p with  variables. the  quadratic  function  o f an a p p l e y i e l d r e l a t i o n s h i p i n  iv  TABLE OF  CONTENTS  CHAPTER I . II.  PAGE  INTRODUCTION  1  LITERATURE REVIEW OF APPLE BIOLOGY  6  Factors  S i z e o f Tree  7  Soil  8  Condition  Frequency  IV.  Condition  10  a t B l o s s o m Time  Pruning  12  Thinning  13  Spraying  13  Density  14  LITERATURE REVIEW OF S T A T I S T I C S  17  DATA  32  Conditions  V.  9  of Frost-injury  Unfavourable  III.  6  I n f l u e n c i n g Apple Production  o f Sampling  '. . .  Sampling Method  33  E M P I R I C A L RESULTS  45  Introduction  45  Results  from t h e Cobb-Douglas Model  54  Results  from t h e Q u a d r a t i c  55  Model  D i s c u s s i o n o f t h e R e s u l t s f r o m A p p l y i n g CobbD o u g l a s 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 ... VI.  32  TESTING FOR THE DIFFERENCE BETWEEN TWO  MEANS ..  63 65  Introduction  65  Outcome o f t - t e s t  68  Discussion  73  of t-test  V  CHAPTER VII.  PAGE  SUMMARY AND CONCLUSION  76  Summary .  76  Conclusion  78  BIBLIOGRAPHY APPENDIX  81 • •  85  vi  L I S T OF TABLES TABLE I. II. III. IV.  V.  PAGE Classification  of Rootstock  Vigour  86  V a r i a b l e s Used i n Models Estimated Simple  87  Linear Regression  .  88  Evaluation of Size-Controlling Effects of 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 T r e e S i z e i n Terms o f an I n d e x V a l u e  89  Tree-Size C l a s s i f i c a t i o n  of I n i t i a l  Equation  Total  Samp l e VI. VII. VIII.  IX. X.  38  Tree-Size C l a s s i f i c a t i o n  Input Data C o r r e l a t i o n M a t r i x f o r Simple L i n e a r Regress i o n w i t h One H u n d r e d and N i n e t e e n P a i r s o f Observations Covariance Table  Table  f o r Three T r e e - S i z e Groups  f o r S. i n Terms o f C! I  XI. XII.  o f Sample E n t e r p r i s e s 39  91 .  47 48  I  Results of Equality T r e e - S i z e Groups Significant  90  Slope  Coefficients  Test  f o r Three 92  a t .05 L e v e l f o r  Cobb-Douglas Model  93  XIII.  C o r r e l a t i o n M a t r i x f o r Cobb-Douglas Model  XIV.  S i g n i f i c a n t C o e f f i c i e n t s a t .05 L e v e l f o r Q u a d r a t i c Model 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 Involving only S i g n i f i c a n t Variables  96  O b s e r v e d and C a l c u l a t e d V a l u e s o f A p p l e Y i e l d s b a s e d on Q u a d r a t i c M o d e l I n v o l v i n g only S i g n i f i c a n t Variables  97  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 G r o u p s ....  69  XV. XVI.  XVII.  ...  94 95  vii  TABLE XVIII. XIX. XX.  PAGE R e s u l t s from t - t e s t f o r Average-Apple-Yield D i f f e r e n c e s Between Regions  70  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  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  viii  L I S T OF FIGURES FIGURE  PAGE  1.  Value  of Regression  i n Reducing V a r i a t i o n  2.  Polynomial Regression M u l t i p l e Regression  inY  27  as a S p e c i a l Case o f  53  A  3.  Range o f V a l u e s f o r P o s s i b l e fi>, A r o u n d O r i g i n When X i a n d % are Highly 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 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 (Across A l l T r e e - S i z e Groups)  6.  D i f f e r e n c e s i n A v e r a g e A p p l e 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)  2  7.  Y i e l d s Among  70 ....  D i f f e r e n c e s i n A v e r a g e A p p l e Y i e l d s Among Apple V a r i e t i e s (Across A l l T r e e - S i z e Groups)  .  71 72  ix  ACKNOWLEDGEMENTS  The generous  author wishes t o g r a t e f u l l y  assistance  supervisor, intellectual  acknowledge t h e  a n d g u i d a n c e p r o v i d e d by h i s t h e s i s  D r . M. J . D o r l i n g .  In addition,  the stimulating  e n v i r o n m e n t p r o v i d e d by t h e s t a f f  graduate students at the U n i v e r s i t y extremely motivating  o f B r i t i s h C o l u m b i a was  and r e w a r d i n g .  Whole-hearted A c t o n and s t a f f  and f e l l o w  gratitude  i n t h e Economics  i s e x t e n d e d t o Mr. K. B r a n c h , Canada  Department  o f A g r i c u l t u r e , V a n c o u v e r , w i t h o u t whose a s s i s t a n c e t h e s i s would n o t have been  possible.  this  CHAPTER I  INTRODUCTION  Today t h e t r e e f r u i t i n d u s t r y o f B r i t i s h is  c e n t r e d i n t h e Okanagan v a l l e y  mile s t r i p  from Vernon t o Osoyoos.  the B r i t i s h and  i n a narrow  Columbia apple  Similkame'en v a l l e y s .  crop  one-hundred-  Approximately  Approximately  4% o f t h e p r o v i n c i a l  i s produced i n the Creston  being  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 area.  ranging  2% o f t h e c r o p  area, with Creston  lower w i n t e r temperature  than  industry i n British  University producers  trees to withstand of fruit  the "backbone" o f the t r e e -  b y P r o f e s s o r M. J . D p r l i n g ,  Columbia, i n d i c a t e d t h a t 91% o f apple  i n t h e Okanagan d e r i v e d a l l t h e i r  from t r e e f r u i t .  1  trees,  Columbia.  conducted  of B r i t i s h  Lillooet,  areas.  other kinds  have l o n g been c o n s i d e r e d  A survey  During  British  Columbia apple  orchard  renovation:  trees,  The  f r o m V a n c o u v e r I s l a n d t o t h e Lower M a i n l a n d ,  Due t o t h e a b i l i t y o f a p p l e  fruit  valley  i s produced i n s c a t t e r e d pockets  K a m l o o p s , S a l m o n Arm, a n d G r a n d F o r k s  apples  94% o f  i s p r o d u c e d i n t h e Okanagan  total  remaining  Columbia  farming  t h e 1960's and e a r l y  growers c o n t i n u e d  o l d trees are being  revenue  1970's most  t o engage i n r e p l a c e d by y o u n g  a n d 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 O k a n a g a n A p p l e P r o d u c e r H i s Management A t t i t u d e a n d B e h a v i o u r , 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 , U.B.C., 1 9 6 8 .  2  are b e i n g accepted is  r e p l a c e d by more a c c e p t a b l e  ones.  The commonly  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 f o r m e r  g i v i n g way t o more d e n s e p l a n t i n g .  orchards  years  Newly p l a n t e d  with tree spacing of approximately  12 t o 16 f e e t  b e t w e e n rows a n d 5 t o 10 f e e t i n t h e row a r e b e c o m i n g commonplace. The r e p l a n t i n g p r o g r a m w h i c h B r i t i s h apple  growers have u n d e r t a k e n  apple  i n d u s t r y o f the Province  far  as t h e a b i l i t y  Columbia  should help t o place the i n a s t r o n g e r p o s i t i o n so  to produce c o m p e t i t i v e l y i s  concerned.  T h e r e has b e e n an i n c r e a s e d e m p h a s i s on costs  of production; e a r l i e r  thinning,  demands.  the  estimate y i e l d  sion  to changing  of this  study  matters  root-  l e d to  the purpose of which i s t o  relationships with special  reference to  a n d 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  of regres-  analysis. As  regression analyses  a p r e l i m i n a r y step, various simple  analyses  were a t t e m p t e d ;  linear  d i s c u s s i o n of these  c e n t r e on t h e a p p l e y i e l d p e r f o r m a n c e i n r e l a t i o n  to the b a s i c independent  factors  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 , t h e amount o f the  market  c o n t r i b u t o r s t o a c h i e v i n g some o f t h e s e  The n e e d f o r c o n s i d e r a t i o n o f t h e s e  initiation  density  i n response  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  seem l i k e l y  goals.  easier pruning,  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  to ensure c o m p e t i t i v e n e s s  stocks  fruiting;  lower  amount o f s p r a y , p r u n i n g  fertilizer,  and t h i n n i n g l a b o u r  hours.  3  The  empirical  Chapter V  results  and  Appendix.  the  of  these  results  are  selection  b e f o r e t h i s was  of  are  discussed  shown i n T a b l e I I I i n  Numerous e m p i r i c a l  u s e d i n the  analyses  and  logical criteria  a quadratic  function  accomplished, d i f f e r e n t  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 of  a particular  form of  considerations: and  best  function  significance  of  best  coefficient  of  of normally  and  violated. model a r e  The the  f i t was  important  determination, R , independently  following:  from the  (2)  In  (1)  as  the  the  basis  and  biological relations 2  of  the  two  the  to  to  errors  the  of law  defined.  and  was  not quadratic  very apple  diminishing restraints  Spillman  functional  on  a  equations;  form,  r e t a i n was  consideration  statistical the  are  and  2  which to  logic, including  the  condition  of the  of  C o b b - D o u g l a s and  and  the  view because  impose s u c h s t r i c t  o m i t and  is satisfactory  magnitude of  (these conditions  summary, d e c i d i n g b o t h t h e  which variables  2  on  i t allows both d e c l i n i n g  subject  and  R  Selection  coefficients  for selection  apple study's point  i t does n o t  relationship  by  distributed  l o g i c a l reasons  a maximum t o t a l y i e l d i s  I f the  models  based mainly  assuming that  2  i s assumed t o be  returns);  (3)  study:  data.  structural  indicated  negative marginal productivity  yield  have been  fit.The  yield  was  the  i n the  algebraic  in  of  probability  l o g i c of  the  done  on  physical levels.  production  S e e 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. 7 5 - 7 8 , Ames, Iowa: Iowa S t a t e U n i v e r s i t y Press, 1961.  4  situation be  included,  serving by  does n o t d i c t a t e t h a t t h e new r e g r e s s i o n  satisfactorily.  t h e above  the excluded v a r i a b l e  e s t i m a t e s may be r e g a r d e d  The q u a d r a t i c  forms o f e q u a t i o n s  f u n c t i o n was c h o s e n  t h a n n o t , s e l e c t i o n among  i s no l e s s  difficult  respect to the s i g n i f i c a n c e l e v e l o m i t t e d from This and  the quadratic  than  at which  models b e i n g  s t u d y was d e s i g n e d  i d e n t i f y t h e most i m p o r t a n t  An  assumption  v a r i a b l e was a s t o c h a s t i c turbance.  Sampling  data are o u t l i n e d preliminary  i s also  tion  f o r combining  into  a single  Slope  Test  factors  i n apple  C o l u m b i a i n 1969.  dis-  IV. C h a p t e r  V i s devoted  to a  o f two d i f f e r e n t f o r m s o f f u n c t i o n i n  t o ensure  that  The E q u a l i t y  t h e r e c a n be  the three d i f f e r e n t t r e e - s i z e  equation.  o f Slope  justificaequations  The p u r p o s e o f t h e E q u a l i t y o f  i s i n r e l a t i o n to K l i n e a r regression variables:  1,2,  I t tests the hypothesis  3  equations  e x h i b i t i n g observed  i n m independent  3  examined.  methods a n d methods o f d e r i v a t i o n o f  reviewed  , k.  will  and t h e d e p e n d e n t  terms o f t h e i r r e l e v a n c y t o t h e s t u d y . Test  with  a l l independent  errors  variable  i n Chapter  review  decision  variables  contributing  was made t h a t  were m e a s u r e d w i t h o u t  algebraic  to estimate y i e l d  p r o d u c t i o n i n t h e Okanagan a r e a o f B r i t i s h  variables  as  criteria.  More o f t e n  be  must  Y^=b +b^"Xj 0  +  equations  + t> H : Q  M  x m  m  > i =  = b? =  As i s e x p l a i n e d i n Chapter I I I r e g r e s s i o n e q u a t i o n s a r e e x p r e s s e d i n t h e 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' (deviation) variables. These forms a r e c o n v e n i e n t conceptualizations .  5  = b.., j = 1, 2 , b  n  O  + b, x, i  +  , m.  + b x  l  m  of sample u n i t s  i s found  regression  the end o f Chapter made c o n c e r n i n g  Y =  groups f  determine  i n t h e e s t i m a t e d c o e f f i c i e n t s among errors  or r e a l  differences.  involving a  and r e l a t e d d i s c u s s i o n V.  selected  are presented at  I n C h a p t e r V I , numerous t e s t s a r e  differences  among a v e r a g e a p p l e y i e l d s  regard to d i f f e r e n t tree  size,  and  sources  r e g i o n s as i m p o r t a n t  ChapterVIIpresents  k  i s c a r r i e d out t o  r e s u l t s of the a n a l y s i s  equation  implications  f o r each o f  m  g r o u p s i s due t o s a m p l i n g The  equation b  and an F - t e s t  whether d i f f e r e n c e s  A regression  apple  v a r i e t i e s , apple  of influence.  with grades,  Finally,  a summary o f t h e main c o n c l u s i o n s a n d  of the study.  CHAPTER I I LITERATURE REVIEW OF APPLE BIOLOGY Factors  I n f l u e n c i n g Apple There tends  i n practice  an a p p l e e n t e r p r i s e w h i c h and  the o t h e r i s economic.  was t o o u t l i n e with  Production  focus i n t e r e s t :  the y i e l d performance  one i s b i o l o g i c a l ,  The p r i m a r y p u r p o s e  some o f t h e most i m p o r t a n t  T h e r e a r e many f a c t o r s  of specific  influencing  t o o many t o p i n p o i n t them a l l . in  t o be two m a i n a s p e c t o f  of this  factors associated  apple  enterprise  apple p r o d u c t i o n ,  A high level  size But  of production.  fitting  factors  are d i f f i c u l t  On a c c o u n t  of this  to i n -  a n d so f o r t h .  t o q u a n t i f y and t h i s they  c o n f i n e d only t o the q u a n t i f i a b l e  It  goes w i t h o u t s a y i n g t h a t t h i s  attention  factors  procedure  makes  are represented  difficulty,  be  from  available,  a r e g r e s s i o n equation i n which  infeasible.  perhaps  The same c a n be s a i d o f  o f o p e r a t i o n , type o f machinery these  plots.  o f management  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  creasing the l e v e l  study  will  of production.  cannot  be immune  a danger o f o v e r s i m p l i f i c a t i o n . In  variables,  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  among o t h e r s , a r e i m p o r t a n t :  1  tree s i z e ,  soil  'See J . C. F o l g e r and S. M. Thomson, The C o m m e r c i a l 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 , e d . L. H. B a i l y , pp. 539-547. New Y o r k : M a c m i l l a n Co., 1921. See R. B u s h , T r e e F r u i t G r o w i n g , r e v i s e d b y 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 t h e 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 . P e n g u i n B o o k s , 1962. See D. W. Ware, E. D. Woodward and H. W. T r e v o r , A S t u d y o f A p p l e P r o d u c t i o n i n t h e 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 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 O t t a w a , J a n u a r y 1952. :  7  c o n d i t i o n s , the frequency of f r o s t - i n j u r y , unfavorable ditions  at blossom time,  density. Size  pruning,  There a r e d i s c u s s e d  thinning, spraying  conand  i n turn.  o f t h e Tree A p p l e v a r i e t i e s a r e p r o p a g a t e d by means o f g r a f t -  ing or budding.  Therefore  i t i s n e c e s s a r y t o have  on w h i c h t o g r a f t o r b u d s c i o n - w o o d o f s e l e c t e d Most a p p l e t r e e s rootstock with use  varieties.  c o n s i s t o f two d i s t i n c t  and s c i o n v a r i e t y .  I n some i n s t a n c e s ,  t h e more t e n d e r a p p l e v a r i e t i e s , trees with  rootstocks  a winter-hardy  trunk  parts,  particularly  i t may be d e s i r a b l e t o  and/or framework.  Wood  o f 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 p u r p o s e and i f it  differs  from the r o o t s t o c k ,  mediate stock. stock  contain  iate stock  Consequently, trees with three  distinct  sections:  and s c i o n v a r i e t y .  consideration,  since  an  intermediate  rootstock,  Rootstocks  ultimate  i n t r o d u c t i o n of dwarfing  provide  tree  t o a d o p t new o r c h a r d  of rootstock  vigour  rootstocks  p l a n t i n g systems.  yield.  t h e most  and s p u r t h e oppor-  Classification  i s presented i n Table I i n the Appendix.  S e m i - s t a n d a r d , s e m i - d w a r f , and d w a r f t r e e s to combine h i g h  first  size.  type v a r i e t i e s showing compacter growth p r o v i d e d tunity  intermed-  are given  s i z e c o n t r o l l i n g roots  p r a c t i c a l means o f d e t e r m i n i n g The  i t i s r e f e r r e d t o as an i n t e r -  tree population  Tukey, E x t e n s i o n  p e r acre  Horticulturist  U n i v e r s i t y , h a s shown t h a t  the smaller  make i t p o s s i b l e  with  e a r l y and h i g h  at Washington tree  allowing  State larger  8  numbers o f t r e e s p e r early  yield.  acre  has  a greater  to s i m i l a r f a c t s i n h i s  "Many o l d , a i l i n g , o u t - o f - d a t e In r e p l a n t i n g t h e s e b l o c k s , g r o w e r has  smaller and  tree than large  Way  renovation.  in bringing  at  early fruit  have f o u n d t h a t  standard  l a r g e r or  produced.  Soil  also  required  i n new  e a s i e r t o h a n d l e t r e e s , more a t t r a c t i v e t o  because of reduced b e a r i n g  area, w i l l  t r e e s , but  l e a s t as  need  apple  3  trees,  produce l e s s f r u i t  as more t r e e s  l a r g e y i e l d s per  for  labour,  returns."  small  land,  can  per  be  acre  will  4  Condition The  p r e r e q u i s i t e o f an  well-drained. conditions  Soils  are  the  orchard  products  soil  of the  under w h i c h they have d e v e l o p e d .  involve mineral and  and  statement:  become i n c r e a s i n g l y c o n s c i o u s o f t h e  B r a s e and  be  orchards  capable of producing high  planted,  high  2  Fisher alludes  the  potential for  biological  materials phenomena.  as w e l l as  i s that  environmental These  topographic,  Well-drained  soils  i t be  conditions climatic  which  reflect  2  R. B. T u k e y , " I m p l i c a t i o n s o f E c o n o m i c s on O r c h a r d Managem e n t " , 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 o f t h e F i r s t B r i t i s h Columbia F r u i t Growers' Association-sponsored H o r t i c u l t u r a l Conference. pp. 59-60 (November 1 9 6 9 ) .  3  D. V. F i s h e r , H i g h - D e n s i t y O r c h a r d s f o r B r i t i s h C o l u m b i a 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 R e s e a r c h B r a n c h , Canada D e p a r t m e n t o f A g r i c u l t u r e , M a r c h 1966.  ''K. D. B r a s e and R. D. Way, R o o t s t o c k s and M e t h o d s u s e d f o r D w a r f i n g F r u i t T r e e s , New Y o r k S t a t e A g r i c u l t u r a l E x p e r i m e n t S t a t i o n , p. 783, 1959.  9  the  forces  of s o i l  classified believed  as z o n a l s o i l s .  and v e g e t a t i o n , a r e  The z o n a l d i s t i n c t i o n i s  t o be due t o a v a r i a b l e  relationship of  genesis, climate  m o i s t u r e and t e m p e r a t u r e  c h a r a c t e r i z i n g mountainous  country.  The  t h e l o w e r - p a r t s l o p e i n t h e Okanagan b e l o n g s t o t h e  Glenmore c l a y - l o a m f o r m a t i o n ; the s o i l b e l o n g s t o t h e Oyama l o a m y - s a n d sified  as d a r k b r o w n s o i l s The a p p l e t r e e  quires  a number  o f the upper  formation.  by K e l l y  and  part  Both are c l a s -  Spilsbury.  5  i n commercial p r o d u c t i o n also re-  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 , and of  zinc.  These elements  fertilizer  The F r e q u e n c y In mention in  soil  and s p r a y  are f r e q u e n t l y  applied  copper,  i n the form  compounds.  of Frost-injury considering  g e o g r a p h i c and c l i m a t i c  factors,  s h o u l d be made o f t h e i m p o r t a n c e o f f r o s t - i n j u r y  orchards i n c e r t a i n  areas r e c e i v e  locations.  Orchards  o c c a s i o n a l damage f r o m f r o s t .  a r e a s a r e more s u s c e p t i b l e  i n most Okanagan H o w e v e r , some  t h a n o t h e r s , and f o r t h e 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 quite v a r i a b l e .  established  a table  indicating  different  c o r r e s p o n d i n g t o a r e a s i n t h e O k a n a g a n as  Ware  frost-free periods follows:  6  5  C . C. K e l l y and R. H. S p i l s b u r y , " S o i l S u r v e y o f t h e O k a n a g a n and S i m i l k a m e e n V a l l e y o f B.C.", R e p o r t No. 3 o f B.C. S u r v e y . The B r i t i s h C o l u m b i a D e p a r t m e n t 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 , D o m i n i o n D e p a r t m e n t o f A g r i culture, pp. 2 0 - 7 1 , 1949.  6  D . W. Ware, O r g a n i z a t i o n and R e t u r n s o f S t o n e F r u i t and P e a r E n t e r p r i s e s i n t h e O k a n a g a n V a l l e y , B.C. 1 9 4 9 - 1 9 5 0 , D e p a r t ment o f A g r i c u l t u r e E c o n o m i c 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  Area  The  Frost-free Period  Kelowna  150  Summerland  176  Penticton  152  Oliver  162  Keremeos  188  frost-free period  as  defined  number o f days b e t w e e n t h e the  t e m p e r a t u r e o f 32°  c o n d i t i o n i n the are  averages  fall  obtained  F. was of the  for  above d a t a  date i n the  recorded  and  same y e a r .  f o r a ten-year  Unfavourable Conditions It  last  f o r the  i n Days  is  s p r i n g on the  The  first  a t B l o s s o m Time  in injury  W i n t e r t e m p e r a t u r e may  h a n d , some c o l d w i n t e r  spring.'  factor.  Temperature i n the  T e m p e r a t u r e s o f 26°  h o u r o r so  as  an  is  a most i m p o r t a n t  explicitly  enters  be  temperature  so  can  or  27°  F.  is  l e a f out  s p r i n g may  the  low  equation  On  normally  for periods  f o r an  to  required  a as  Thus,  v a r i a b l e which e i t h e r i m p l i c i t l y supply  as  tree.  a l s o be  c a u s e damage t o f l o w e r s .  any  account  Temperature i s  t o t h e buds o r t h e wood o f t h e  t o e n s u r e v e r n a l i z a t i o n so t h a t t r e e s the  given  period.  v a r i a t i o n s i n crop y i e l d s .  growing of apples.  other  similar  figures  of extreme importance at a l l seasons of the y e a r i n  the  which  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  considerable  result  the  in  critical short climate or  agricultural  11  crop.  7  It measuring The  is difficult  the i n f l u e n c e  to find  of climate  on a p p l e p r o d u c t i o n .  i n a c o n v e n i e n t form.  a g e o g r a p h i c a l dummy v a r i a b l e was w h i c h was  size  correlation  coefficient  into  the  analysis  of weather.  e x i s t i n g between r a i n f a l l  ± 0.110.  8  and  of the p o p u l a t i o n  It  Hence, d e c r e a s i n g r a i n f a l l associated with  i s of i n t e r e s t  spraying  i n t h e summer crop.  d u r i n g t h e months  and d u s t i n g o p e r a t i o n s a r e d o n e , t h e f a c t o r i n the p r o d u c t i o n  A c o m b i n a t i o n o f more h o u r s o f s u n s h i n e  rainfall  results  inclusive,  an i n c r e a s i n g  to note that  hours o f sunshine are a c r i t i c a l apples.  1913-1929  correlation  t o O c t o b e r p e r i o d , w e r e e s t i m a t e d as  months t e n d e d t o be  i n which  limits  f o r the seven-year p e r i o d  i n v o l v i n g t h e May  less  introduced  study  o f a p p l e c r o p i n N o v a S c o t i a h a s b e e n f o u n d t o be  n e g a t i v e by L o n g l e y  -0 . 572  For t h i s  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 The  d u r i n g t h e months o f May,  June  and  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  better control of insects 7  index f o r  most c o n v e n i e n t m e a s u r e m i g h t w e l l be t e m p e r a t u r e i f  t h a t were summarized  of  an a p p r o p r i a t e  and  and  July  foliage  and  diseases.  J . P . D o l l , "An A n a l y t i c a l T e c h n i q u e 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 E c o n o m i c s , V o l . 4 9 , No. 1, F e b r u a r y 1967. H. S. L a w r e n c e , "The E f f e c t o f W e a t h e r on A g r i c u l t u r a l Output": A Look a t M e t h o d o l o g y , J o u r n a l o f Farm E c o n o m i c s , V o l . 4 6 , 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 S t u d y o f t h e L e a f T o b a c c o M a r k e t o f G r e e c e , pp. 1 6 4 - 1 6 6 , A t h e n s , 1962.  8  W. V. L o n g l e y , Some E c o n o m i c A s p e c t s o f t h e A p p l e 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 o f P h i l o s o p h y , pp. 2 2 - 2 3 , Nova S c o t i a D e p a r t m e n t o f A g r i c u l t u r e B u l l e t i n No. 1 1 3 , 1932.  12  Pruning A good t r e e f r a m e w o r k o f d e s i r e d s i z e , strength  i s n e c e s s a r y t o o b t a i n the  spaced branches  and  spurs  i n as  and  the  fruit  plenty  still  admits and  light  thus  avoid the  allow and  a i r , allows  improves  possible  rigid  pyramid t r e e , but  all  Correct  r e s u l t s are varieties  t o be  habit  A p p l e t r e e s may distinct periods: period,  and  (3)  (1)  be  i s during  energies and  years.  1 0  formative  t o wood g r o w t h .  of the  period,  Folger,  (2)  c i t . , pp.  e t a l . , op.  expecting  The  natural  three  each of t h e s e  pruning periods.  t r e e devotes i t s  proper s e l e c t i o n , d i s t r i b u t i o n  the  this  time determines fruit  transitional  115-116. c i t . , p.  one  transitive  Appropriate  1 0  p e r i o d t h a t the  during  use  i f the  same s t a n d a r d ;  t r e e t o b e a r heavy crops o f  A l l pruning  B u s h , op.  in suitable s o i l ,  on  9  formative  t r a i n i n g of branches during  ability  9  the  r i g h t way  advantage o f the  t r e a t m e n t changes m a t e r i a l l y w i t h It  dwarfed  s a i d to pass through  fruiting period.  near  permanently  of t r e e to conform to the  of the p a r t i c u l a r v a r i e t y .  picking,  t o make  I t i s no  to take  and  Pruning  helps  employed i n the  expected.  must a d a p t one's p r u n i n g  possible  fruiting  pruning  c o r d o n shape o r the i t must be  as  i s e s s e n t i a l to  to induce  r i g h t v a r i e t y of t r e e , p l a n t e d  best  area  o f s p a c e t o grow.  Pruning  or main b r a n c h e s . the  an  easier spraying  buds.  and  maximum number o f w e l l  small  a b a r e - w o o d c o n d i t i o n and  trunk  the  fruit  form  283.  the  in  later  period  i s to  13  develop well  and  maintain  a liberal  d i s t r i b u t e d throughout  supply  of f r u i t i n g  the e n t i r e  wood,  tree.  Thinning The pruning. fruit and  t h i n n i n g of apples  I f a l l the  fruit  on  i s no  an a p p l e  a form  limb breakage would r e s u l t . takes place e a r l y  This  fruits  fruits  i s because f r u i t  bud  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 available  of  t r e e showing a heavy  s e t were a l l o w e d t o mature, s m a l l misshapen  formation  for  more t h a n  food s u p p l i e s .  removes much o f t h i s  buds and y o u n g  fruit  E a r l y removal of s u r p l u s competition.  1 1  Spraying O r c h a r d s must be in  order  damage. tions,  t o p r o t e c t the D e p e n d i n g on  apples  sprayed  fruit  regularly  and  thoroughly  from s e r i o u s i n s e c t  the n a t u r e  and  extent  o f the  r e q u i r e from four to seven sprays  of thought.  t i o n by p o i s o n o u s  The  sprays  one  infesta-  two  believes that wholesale  is desirable.  disease  a year.  Among e n t o m o l o g i s t s , h o w e v e r , t h e r e a r e schools  or  liquida-  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 t h e m a s s a c r e 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 T h e r e no  races having 1 1  of i n s e c t s ;  control w i l l  doubt e x i s t s today,  there  prove s u p e r i o r .  a d a n g e r o f i n d u c i n g immune are s e v e r a l p e s t s  which,  i n t h e p a s t b e e n e x p o s e d t o c e r t a i n s p r a y s , have  Ibid.,  op.  c i t . , p.  283.  now  14  d e v e l o p e d a degree cannot  o f immunity.  expect t h e i r p a r t i c u l a r  However, a p p l e o r c h a r d s t o be  attacks  of insects  and d i s e a s e s w h i c h  any who  omit s p r a y i n g are u n l i k e l y  growers  free  from  occur elsewhere,  to produce  and  marketable  fruit. Dens i t y Van sixth  cepts.  of  in yields  The  "At t h e end o f t h e  y i e l d was  t h e r e was  as h e d g e r o w s on M.  used had produced  25.9  r e l a t e d to the p l a n t i n g  VII  t i m e s more  and t h e t y p e o f r o o t s t o c k Harris  tigations  F a r m , S a a n i c h t o n , B.C., on M.  IX r o o t s t o c k  quantity being  fruit  rootstock fruit.  training  1 2  from t h e i r  inves-  of A g r i c u l t u r e Experimental  that apple trees  at h i g h e r d e n s i t y  grow w e l l , p r o d u c e h e a v i l y w i t h h i g h  a t an age when s t a n d a r d t r e e s  in a state  of commercial  Intensive planting density per acre w i l l primary  used."  and Woods h a v e r e p o r t e d  a t the Canada Department  con-  c o n c e p t , t h e number  t r e e s p e r a c r e , t h e s y s t e m o f p r u n i n g and  followed  a wide  from each o f the d i f f e r e n t p l a n t i n g  trees planted  the density  The  has w r i t t e n :  g r o w i n g s e a s o n , on an a c r e b a s i s ,  variation  at  Roechoudt  involve  production.  of apple trees  are f a r from  1 3  implying high  a high investment cost.  c o n s i d e r a t i o n , however, i s the a b i l i t y  Of  of the crop  1 2  L . L. Van R o e c h o u d t , Some F a c t o r s W h i c h I n f l u e n c e t h e Use o f D w a r f and S e m i - D w a r f A p p l e T r e e s f o r C o m m e r c i a l O r c h a r d s i n t h e Okanagan V a l l e y o f B.C. Unpublished Master's Thesis, 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 , 196 2.  1 3  J . H. H a r r i s and J . J. Woods, D w a r f A p p l e T r e e s on V a n c o u v e r I s 1 a n d , E x p e r i m e n t a l Farm R e s e a r c h B r a n c h , S a a n i c h t o n , B.C., 1958.  15  t o r e t u r n a p r o f i t on t h e i n v e s t m e n t .  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 o f t r e e s p e r acre can r e s u l t i n a s i g n i f i c a n t l y h i g h e r per  acre.  Tukey s t a t e s :  yield  "One o f the most p o s i t i v e methods  o f i n c r e a s i n g y i e l d i n t h e e a r l y y e a r s o f an o r c h a r d i s t o increase  t h e number o f t r e e s p e r a c r e , and i n c r e a s i n g t h e  tree population counteracting  may be one o f the most e f f e c t i v e means o f  the problem o f o b s o l e s c e n c e and r e p l a n t i n g  old orchard s i t e s . " * 1 1  Tukey, op. c i t . , p. 58.  CHAPTER I I I  LITERATURE REVIEW OF  Modern s t a t i s t i c s  STATISTICS .  a r e b a s e d upon  probability.  There  a r e a number o f c o n f l i c t i n g i d e a s a b o u t  which  i s fundamental  for scientific  Some a u t h o r s h o l d refer  to a proposition  ical. in  we  methodology.  that p r o b a b i l i t y  t o our r a t i o n a l  a t h e o r y o r h y p o t h e s i s on t h e b a s i s  probability  can a t t r i b u t e  subjective  f o r example,  as f o l l o w s :  concept,  statements  and a r e h e n c e l o g i c a l and n o t  This concept r e f e r s  Keynes,  this  degree  empir-  of b e l i e f  of empirical evidence.  expounds i n h i s t r e a t i s e  "What we  know and w h a t  to our r a t i o n a l b e l i e f s  on  probability  i s , therefore,  i n the sense of being r e l a t i v e to the  individual.  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 t h e k i n d s o f logical w h i c h we it  relations  have the c a p a c i t y  is rational  an o b j e c t i v e  J.  M.  Our  logic is of steps of  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 . " and e n t i r e l y  to the r e l a t i v e  Keynes:  and  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  d r a w i n g c o n c l u s i o n s by a s e r i e s  Another  X  c a n be b a s e d  f o r us t o d r a w , s t a n d s t o t h e s e p r e m i s e s i n  specified  cept refers  arguments  and w h o l l y l o g i c a l r e l a t i o n .  concerned with certain  upon w h i c h  different  probability  f r e q u e n c y o f an e v e n t , as  A T r e a t i s e on P r o b a b i l i t y ,  p.  18.  conthe  1  17  number o f t r i a l s i n c r e a s e s i n d e f i n i t e l y . may,  The  econometrician  f o r i n s t a n c e , c o n s i d e r the r e l a t i v e f r e q u e n c y  business f a i l u r e s which  fail  t h a t i s , the percentage  each y e a r .  He may  of  of businesses  t a l k about the p r o b a b i l i t y  a b u s i n e s s f a i l u r e as the l i m i t o f 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 . the f i r s t p r o b a b i l i t y concept  i s not y e t u s e f u l  the s i m p l e s t problems o f s t a t i s t i c a l second  concept  of  Since  f o r any  but  i n f e r e n c e , o n l y the  i s r e l e v a n t i n the case o f s t a t i s t i c a l  tests  i n the s t u d y . The  fundamental  purpose o f r e g r e s s i o n a n a l y s i s  t o e s t i m a t e the r e l a t i o n s h i p between the dependent independent variables  variables.  know the goodness of f i t o f the  e s t i m a t e d , we may  wish to  relationship.  I t i s i m p o s s i b l e t o e s t i m a t e the  or d e d u c t i o n s  and  Once the r e l a t i o n s h i p between these  has been q u a n t i t a t i v e l y  between the v a r i a b l e s  relationship  w i t h o u t f i r s t making some  assumptions  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, + bx^  , where i = l , 2 ,  tage o f measuring  n and where x^=(X-X).  2  One  = a advan-  X^ as d e v i a t i o n s from t h e i r mean i s t h a t  the mathematics w i l l be s i m p l i f i e d because the sum new  is  x v a l u e s equals zero —  o f the  t h a t i s £x^=0 . T h i s w i l l  become c o n v e n i e n t l a t e r on i n the p r o o f o f E(b) = 6 , v a r ( b ) 2  2  = a^/Ix^  .  A l s o , i n the p r o c e s s of i n v e r t i n g  is a matrix consisting 2  o f a l l x o b s e r v a t i o n and X'  , where X is a  See R. J . Wonnacott and T. M. Wonnacott, E c o n o m e t r i c s , 245-246. John W i l e y and Sons, I n c . , 1970.  pp.  18  matrix consisting of a l l x observations o f % , t h e measurement o f x v a l u e s shown t o be v e r y c o u l d be r e p e a t e d  convenient. many t i m e s  t h e r e w o u l d be o b s e r v e d Y values tion. call for  P(Y/x).  some s t a t i s t i c a l  o f x. Then  f l u c t u a t i o n of the forming  a  sub-popula-  f u n c t i o n o f Y f o r a g i v e n x , we  level  o f x.  f u n c t i o n s f o r Y^ a t t h e v a r i o u s  shall  function  Consequently, levels  o f x^  be P ( Y / x ) . i  To reasonable  i  k e e p t h e p r o b l e m m a n a g e a b l e , l e t t h e r e be a  s e t o f assumptions  sub-populations. follows:  about t h e r e g u l a r i t y  o f these  T h e s e a s s u m p t i o n s may be w r i t t e n c o n c i s e l y  t h e random v a r i a b l e s Y^ a r e s t a t i s t i c a l l y  independent, it  at a fixed value  experiment  T h e r e w i l l be a s i m i l a r p r o b a b i l i t y  probability  as  S u p p o s e t h a t an  Y a t any o t h e r e x p e r i m e n t a l  will  i n d e v i a t i o n form i s  c l u s t e r e d about a c e n t r a l v a l u e  The p r o b a b i l i t y  a n d %' t h e t r a n s p o s e  w i t h mean a+3x^ a n d v a r i a n c e a * .  On o c c a s i o n ,  i s u s e f u l t o d e s c r i b e t h e d e v i a t i o n o f Y^ f r o m i t s ex-  pected  value  as t h e e r r o r o r d i s t u r b a n c e t e r m  IL , w h e r e  t h e IL a r e i n d e p e n d e n t random v a r i a b l e s , w i t h mean 0 a n d v a r i a n c e a* . r  No a s s u m p t i o n i s y e t made a b o u t t h e s h a p e o f  the  distribution  The  e r r o r t e r m may be r e g a r d e d 1.  i t has a f i n i t e  variance.  as t h e sum o f two c o m p o n e n t s :  Measurement E r r o r . In  resulting 2.  o f IL p r o v i d e d  measuring crop y i e l d ,  t h e r e may be an e r r o r  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  weighing.  Stochastic Error. D i s r e g a r d i n g measurement e r r o r , t h e r e w o u l d  still  19  be some u n p r e d i c t a b l e in  differences  in yields,  for  example,  an e x p e r i m e n t u s i n g t h e same r a t e o f f e r t i l i z e r  tion.  Assume t h a t t h e s i t u a t i o n  l a r g e measurement e r r o r s are  i s such t h a t t h e r e are  i n the v a r i a b l e s .  c e r t a i n v a r i a b l e s w h i c h ought  but have been l e f t rather  out.  large errors If  applica-  However, t h e r e  t o appear  i n the e q u a t i o n  Omission o f the l a t t e r  i n the e q u a t i o n s .  results  are  known, i t i s p o s s i b l e  t o compute t h e e x a c t v a l u e s o f t h e  r e g r e s s i o n parameters  a , 6 and o^.  Determination of  s q u a r e s i s t h e most a c c e p t a b l e method f o r f i t t i n g The  method o f l e a s t  mators  ( a , b ) be  selected  squared deviations line  be  squares  requires  i n s u c h a way  that i s minimize  where e i s the e r r o r term. be n e c e s s a r y t o know how  E(a) Var  That  =  a  (a)  =  E(b) = V a r (b) 3  straight  the  estiof the  regression  e?=(Y.-a-bx.) , 2  ^  i  J  i  the e s t i m a t o r s  '  g.  a and b a r e d i s t r i The  least  squares  a and b a r e t h e n t h e b e s t l i n e a r u n b i a s e d  o f a and 8.  least  For t e s t i n g hypotheses, i t w i l l  b u t e d a r o u n d t h e i r p a r a m e t e r s , a and  mators  a  t h a t t h e sum  i  estimators  that  o f Y^=a+bX^ f r o m t h e f i t t e d  a minimum  in  3  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^)  line.  no  i s , t o sum  esti-  up:  c^/n  6 = tfy/£*-_  T . H a a v e l m o , "The P r o b a b i l i t y A p p r o a c h 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 . 1 2 , 1944, Supplement. H. B. Mann and A. W a l d , "On t h e S t a t i s t i c a l T r e a t m e n t o f Linear Stochastic Difference Equations", Econometrica, Vol 1 1 , p. 1 7 3 , 1943.  20  w h e r e E and V a r s t a n d f o r e x p e c t e d spectively.  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 G a u s s - M a r k o v Theorem w i t h o u t  m a k i n g any a s s u m p t i o n a b o u t  the  shape o f t h e d i s t r i b u t i o n  the  slope c o e f f i c i e n t b i s usually  than  the i n t e r c e p t  o f t h e e r r o r term. *  Since  1  coefficient  o f more i n t e r e s t  a , we  shall  t o us  concentrate  on  P r o o f o f E ( b ) = 3 and V a r (b) = c ^ / E x ? a l o n e i s  the s l o p e . as  v a l u e and v a r i a n c e r e -  follows.  The f o r m u l a  b = Z(x /K)Y i  f o r b may be r e w r i t t e n a s : (3-1)  i  where  K = Ex?  .  (3-2)  Thus , b = Ew.Y.  = WiYi  + w Y + 2  w Y  2  i i  •  (3-3)  n n  where w  i  = x^K  (3-4)  From t h e t h e o r y o f l i n e a r E(b)  transformations, i t follows that:  = WjECYx) + w E ( Y ) + 2  w  2  n  N o t i n g t h a t t h e v a r i a b l e s Y^ follows  (3-5)  a r e assumed i n d e p e n d e n t , i t  = w ? V a r Y j + w|Var Y Var Y  t h e mean f r o m  viously,  n  that Var(b)  Using  E(Y ) = ^ E ^ )  n  +  2  w*  =Ew?Var Y. I  I  ( 3 - 5 ) and E ( Y ^ ) = a + 8 x ^ as assumed p r e -  then E(b)  = Z w ( a + 6 x ) = aZvr ^ +  and n o t i n g e q u a t i o n  (3-6)  i  i  fcEw^  (3-4) , t h e n  M V o n n a c o t t and W o n n a c o t t , op. c i t . ,  pp.  48-51  21  E(b) = ( a / k ) Z x  i  + (3/k)Z( )x, X i  B u t , s i n c e Zx^ i s z e r o , then E(b) = 0 + ( B / k ) E x2  .  .  i  F u r t h e r m o r e , from e q u a t i o n (3-2) E(b) = 3 From e q u a t i o n (3-6) and from V a r ( Y ^ ) = a ^ as assumed p r e v i o u s l y , Var(b) = Zw a 2  2  = £(x /k )a 2  2  = (a /k )Zx  2  2  2  2  Again, noting equation (3-2), Var(b) = a /Ex? 2  R e c a l l i n g the assumption t h a t statistically  v a l u e s are  independent and a l s o t h a t b i s a l i n e a r  c o m b i n a t i o n o f a l l Y. ( t h a t i s b = E x - Y . / E x ) , i t f o l l o w s 2  I  ^  l  I  i ' J  t h a t t h e shape o f t h e b d i s t r i b u t i o n w i l l a l s o be n o r m a l . The n o r m a l i t y for  assumption o f t h e e r r o r term i s r e q u i r e d  s m a l l sample e s t i m a t i o n s .  only  Without assuming t h a t t h e Y^  are n o r m a l l y 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 , t h e d i s t r i b u t i o n o f 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 Theorem.  by a g e n e r a l i z e d form o f the C e n t r a l L i m i t  I f we have s p e c i f i e d t h e form o f t h e 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 likelihood  by t h e method o f maximum  (which c o u l d a l s o have been used t o o b t a i n  esti-  mators a and 8 ) . For g e n e r a l i t y , suppose that, we have a sample o f s i z e n.  We w i s h t o know: P(Y ,Y ----Y ) 1  2  n  (3-7)  22  That  i s , we w i s h  ity  t o know t h e l i k e l i h o o d  o f t h e sample we o b s e r v e d ,  the p o s s i b l e first  or p r o b a b i l i t y  expressed  population values  as a f u n c t i o n o f  o f a , 3 and a* .  c o n s i d e r the p r o b a b i l i t y  dens-  Therefore,  d e n s i t y o f the f i r s t  value o f  Y. w h i c h i s 1  P(Yl) = y 2 ^ e S " (  i s simply  (a+3Xj)  the normal  and v a r i a n c e  distribution  (a^) s u b s t i t u t e d  positions.  The i n d e p e n d e n c e  multiplying  a l l these  find  ' (3-8)  (a+3Xl))  e = 2.71828  where This  )(Yl  the j o i n t  with  lt  into  densities  justifies  together to  density:  P ( Y i ,Y , ----Y ) = 2  (3-9)  n  ( ,=L=- - O s a * ) ( Y j - C a + S x J ) ) / 2  e  1  V2TI02  6  —L=r Tf O  7  y  v  i t s mean  the appropriate  o f t h e Y^ v a l u e s  probability  probability  of Y  - (%a ) ( Y - ( a + 3 x ) ) ) 2  2  2  2  6  7  y - (iga ) ( Y . - ( a + 3 x . ) ) ) v  =  J T / _ L _  2  2  \/2TT0y  where T F r e p r e s e n t s t h e p r o d u c t familiar  rule  f o r e x p o n e n t i a l s , the product  c a n be e x p r e s s e d P(Y  ,Y ,  Y J  that  concerning this,  L ( a  = (-1—  with  W2  B l 0 p  .  (  -(%a )(Y - ( 2  of equation ( 3 - 9 )  a +  (3-10)  2  Y^ s p e c u l a t i o n i s made  o f a, 3 and o  (3-10)  2 t  then,  t o emphasize  i s renamed t h e l i k e l i h o o d  ^ r « e - ^ ' v - ^ ) y  3x.))  .  the observed  the values  the equation  ,  Using the  as f o l l o w s :  y Recalling  of n factors.  !  t s  function:  .  n )  23  Therefore,  the question  is:  o f a and 3 make  Which v a l u e s  The o n l y p l a c e t h a t a a n d 3 a p p e a r i s i n t h e  L largest? exponent.  Moreover, maximizing  a function with  a negative  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 t h e exponent.  D e s i g n a t i n g o u r e s t i m a t o r s as a and b , t h e  problem i s t o s e l e c t values  (Y  The  i  f o r these  - a - bx.)  are i d e n t i c a l  to least  r e g r e s s i o n model has a n o r m a l l y So  .  controlled  i n this  effect  of r a i n f a l l  (i.e.,  rainfall)  our  control.  (3-12)  error.  x has a s s u m e d a  i t must be r e c o g n i z e d t h a t x completely  squares  isstill  and  i n d e p e n d e n t o f t h e x's  x a n d U.  valid  o f x does n o t d e p e n d on a , 3 ,  , and t h a t t h e e r r o r terms a r e n o r m a l l y  assumptions,  outside  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  a  these  when t h e  Thus i f we a r e e x a m i n i n g t h e  The m e t h o d o f l e a s t  and  less  estimates  distributed  i s a random v a r i a b l e ,  assume t h a t t h e d i s t r i b u t i o n  least  likelihood  H o w e v e r , i n many c a s e s , x c a n n o t  manner.  on y i e l d ,  whether x i s a f i x e d  2  squares  f a r the independent v a r i a b l e  given s e t of fixed values.  Of  minimize  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  estimates  be  2  that  distributed (3-13)  we must e m p h a s i z e t h e i n d e p e n d e n c e o f  I t c a n b e shown t h a t t h e maximum l i k e l i h o o d and  squares  estimates  c o i n c i d e a n d may be a p p l i e d r e g a r d -  o f whether the independent v a r i a b l e  x i s fixed or  24  x i s independent o f the e r r o r  random, p r o v i d e d in  the equation being  s a m p l e now i n v o l v e s Y.  estimated.  The l i k e l i h o o d o f o u r  the p r o b a b i l i t y of observing both  i f t h e x^ a r e i n d e p e n d e n t ,  Therefore,  and p a r a m e t e r s  x and  the l i k e l i h o o d  function i s L = P(x }P(Y1/x )P(x2)P(Y2/x2) 1  Since L  =  P  the error  (  X  i  ^  )  terms a r e c o n s i d e r e d  - d a p  e  normal,  (Yl-a-3x ) P(x2)  _ l _  2  l  y  v  (3-14)  1  i^^y e'teap (Y2-a-Bx2)2  Collecting  the exponents, 1  L = P(x,)----/  \2TJO2-  y  Since  according  xn/2 Q - ( ^ a 2 ) Z ( Y , - a - 3 x . ) 2 e ^ ~ I ~ ) _  t o equation  the p a r a m e t e r s a , 3, and this  l i k e l i h o o d function  exponent i n e q u a t i o n It the  (3-16)  V  ( 3 - 1 3 ) , P ( x ) does n o t d e p e n d on , the problem of maximizing  reduces t o t h e m i n i m i z a t i o n o f the  (3-11).  i s o f i n t e r e s t t o n o t e what w o u l d happen i f  independent v a r i a b l e  terms.  (3-15)  Reconsider  x i s correlated with the error  t h e model,  Y = a + 3 x + U By  taking  in  the equation,  In o r d e r  the covariances  o f x w i t h each o f t h e v a r i a b l e s  the following r e s u l t s , S = S + S xy xx xu  t o estimate  (3-17)  5  v  J  3, S i s d i v i d e d by S (variance o f ' xy xx 1  5  (3-18)  v  W o n n a c o t t a n d W o n n a c o t t , op. c i t . , p p . 149-157  25  x)  such  that S  xy  /S  = b + S /S xu xx  xx  From t h e o b s e r v a t i o n  (3-19)  v  o f x a n d Y, S and S are e a s i l y ' xy xx 1  calculated.  H o w e v e r , U i s u n o b s e r v a b l e , so t h a t S  c a n n o t be e v a l u a t e d . is  Therefore,  i f we c a n assume t h a t S  s m a l l enough t o n e g l e c t , we w i l l S /S = b xy xx  We r e c o g n i z e  this  from e q u a t i o n  (3-20),  (where ^ i s d e f i n e d  that S  P 0 xu +  variables  study.  analysis  analysis y i e l d s only  that  P ^ nonzero x x •*  regression  analysis  Simple c o r r e l a t i o n and i n d e x  number  an i m m e d i a t e p i c t u r e o f how c l o s e l y two I n c o r r e l a t i o n a n a l y s i s , cause and  r e l a t i o n s are unimportant. A d i s t i n c t i o n between r e g r e s s i o n  correlation  a n a l y s i s must be made t o a v o i d  a n a l y s i s and confusion  may a r i s e f r o m t h e s u b t l e t y o f t h e p r o p o s i t i o n s both analyses. variables  In r e g r e s s i o n  On t h e o t h e r  As n o t e d b e f o r e  which  involved i n  a n a l y s i s , a l l the independent  a r e assumed f i x e d .  b i l i s t i c way. 6  on  i s t h e degree t o which  one c o e f f i c i e n t  v a r i a b l e s move t o g e t h e r . effect  is justi-  B u t , i n t e r e s t may a l s o f o c u s  are r e l a t e d o r associated.  designed t o give  S  That i s ,  6  estimator  while  J  as a p p r o a c h e s i n p r o b a b i l i t y as n + °°) .  We h a v e s o f a r d e a l t w i t h  correlation  v  the l e a s t squares  p  to this  the estimator (3-20)  as t h e l e a s t s q u a r e s e s t i m a t o r .  f i e d under c o n d i t i o n s  relevant  obtain  They do n o t o c c u r i n a p r o b a hand, c o r r e l a t i o n a n a l y s i s i s  b = ZYx/Ex  2  = Z y x / E x x = \J E y x / n - 1 /  ^XTATT  = s  y x  /s  x x  .  26  concerned mainly ables  with  random v a r i a b l e s .  Independent  vari-  must h a v e 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 .  view of these d i f f e r e n c e s , r interpreted  only  correlation  and  ically,  can  be  adequately  in a correlation analysis. regression  are  so  Yet,  Specifically,  estimated  consider  the  correlation coefficient  regression  slope  c o e f f i c i e n t b. b = Exy/Ex  t h a t b o t h x and  regression  r e l a t i o n between  r , and I t was  the  the  estimated  shown  that (3-21)  2  y are  r = Exy/  since  c l o s e l y r e l a t e d mathemat-  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  analysis.  Noting  values  2  In  defined  Ex Ey 2  as  deviations,  then (3-22)  2  Then  b/r I f we the  now  =\fT^4~ /^ 2  = V Ey  2  2  / Ex  d i v i d e b o t h t h e n u m e r a t o r and  s q u a r e r o o t s i g n by b/r  (3-23)  2  denominator i n s i d e  (n-1),  = /(Ey /n-l)/Ex /n-l) 2  2  x  = S /S y  (3-24)  x  or b = r ( S /S )  (3-25)  x  This  close  c o r r e s p o n d e n c e b e t w e e n b and  i m p o r t a n c e i n the  s u b s e q u e n t a r g u m e n t as  the  —  more p o w e r f u l Consider  of observations w h e r e Y^  = the  regression fitting  (x^,Y^). regression  be  of  utmost  to which t o o l  is  or c o r r e l a t i o n a n a l y s i s .  a regression This  r will  line  i s represented  estimate  o f Y^  .  t o the  scatter  i n Figure  1,  27  Y  0  x  X i  F i g u r e 1. value of r e g r e s s i o n i n reducing  The  the b e s t p r e d i c t i o n of a Y w i t h o u t knowing  Now, x would be  the average o b s e r v e d v a l u e  c l e a r from t h i s diagram t h a t we namely (Y^  error mean.  (Y^  sinze now  - Y)  However, once the  - Y)  regression  e q u a t i o n has  - Y)  = (Y\  large  been c a l -  (Y^  error,  deviation  T h e r e f o r e , t h i s leaves  d e v i a t i o n of Y i s the  i t is  t h i s reduces the  t i v e l y small "unexplained" deviation  T  At x^,  d e v i a t i o n of Y^ from i t s  the  which i s a l a r g e p a r t of the  been " e x p l a i n e d " .  (Y  (Y) .  would make a v e r y  p r e d i c t Y t o be Y^ and  c u l a t e d , we  v a r i a t i o n i n Y.  only  - Y^).  has  a rela-  Total  sum: - Y)  + (Y.  - Y\)  , f o r any  i (3-26)  It follows E(Y  I  that - Y)  = E(Y  2  I  - Y)  where v a r i a t i o n i s d e f i n e d tions.  S i n c e, (Y.l - Y)J v  rewrite equation (3-27)  2  + E(Y  I  as the sum  = ' y. = bx., I l ' as  - Y\)  2  (3-27)  o f the s q u a r e d  devia-  i t i s c o n v e n i e n t to  28  £ (Y^ - Y ) The  2  = b Ex? 2  + Z(Y  fact that explained variation  f o r by t h e e s t i m a t e d  total  variation  and t h e a n a l y s i s o f i t s components  is  test  on B may  be  to unexplained  variance  the hypothesis  a test  of the hypothesis  "F"  equal  the  unexplained  constructed.  to variance  i s s u f f i c i e n t l y large to  t h a t Y i s u n r e l a t e d t o x.  Specifically,  HQ."3= 0 i n v o l v e s f o r m i n g  the r a t i o  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  variance  equal to:  b (Ix?/S ) 2  S  2  (3-29)  2  i s t h e s a m p l e v a r i a n c e o f Y.  that this hypothesis  i s just  an a l t e r n a t i v e  way  I t must be of t e s t i n g  emphasized  the n u l l  w ith t h e use o f the " t - d i s t r i b u t i o n " : r  c a l c u l a t e d " t " = b/v/ S / Z x ?  (3-30)  z  For  t h e " 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 s s u m p t i o n i s made t h a t t h e d i s t r i b u t i o n o f Y^ N o t e t h a t t h e " F " and " t " d i s t r i b u t i o n s g e n e r a l l y , as f o l l o w s :  F = t  2  , where  o f f r e e d o m i n t h e n u m e r a t o r o f F. now  From  i s then, whether the r a t i o of the e x p l a i n e d  reject  where  clari-  applied to regression".  foregoing, a n u l l hypothesis  variance  i s the v a r i a t i o n accounted  The p r o c e d u r e o f d e c o m p o s i n g  c a l l e d "analysis of variance  The q u e s t i o n  (3-28)  2  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  f i e d , by t h e above e q u a t i o n .  the  - Y\)  i  be r e l a t e d  to r.  are  related,  there  i s one  variation  I t f o l l o w s from equation  b = ry'EYj/Ex Then, s u b s t i t u t i n g  The  2  t h i s value  i s normal.  degree  in Y  (3-25) t h a t  . f o r b i n equation  will  (3-28)  29  Z(Y Noting r  2  - Y)  i  = r E y ? + E(Y  2  - Y^  2  t h a t y ? i s by d e f i n i t i o n  i  (3-31)  2  (Y^ - 7 ) ,  the s o l u t i o n f o r  2  is r  Finally  =[E(Y  2  i  - Y)  2  - E (Y^  - Y.^]/Z(Y.  t h e n u m e r a t o r c a n be r e - e x p r e s s e d  (3-27).  - Y)  (3-32)  2  by n o t i n g  equation  Thus r  2  = E(Y\ - Y ) / E ( Y  which i s the explained v a r i a t i o n variation  - Y)  2  (3-33)  2  i  o f Y d i v i d e d by t h e t o t a l  of Y . Complications  variables  arise  are introduced  consider a simple estimated  as s o o n as more t h a n  into the equation.  t h r e e v a r i a b l e example.  regression equation R  2  = E(Y  - Y) /E(Y  illustrate,  Thus, o f our  i s Y = a + bx + c z , then  2  i  To  two  i  - Y)  (3-34)  2  which i s the explained v a r i a t i o n  o f Y d i v i d e d by t h e t o t a l  variation  calculation  r  2  of Y .  i fthere  more t h a n  Note t h a t t h i s  i s o n l y one i n d e p e n d e n t v a r i a b l e .  one i n d e p e n d e n t v a r i a b l e , t h e n  represents  the v a r i a t i o n  variables.  it  how  the numerator  T h u s , 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 i n improving  watching  fast R  2  I f there i s  o f Y e x p l a i n e d by a l l i n d e p e n d e n t  a d d e d t o t h e m o d e l , we c a n i m m e d i a t e l y these  i s identical to  increases  s e e how  helpful  o u r e x p l a n a t i o n o f Y by  i n equation  has b e e n p r o v e d t h a t e q u a t i o n  v a r i a b l e s are  (3-34).  Finally,  ( 3 - 2 8 ) c a n 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 e x p l a i n e d by ( X j ,x + a d d i t i o n a l v a r i a t i o n explained  by x  n  +  2  unexplained  variation.  (3-35)  This statement can be used to c o n s t r u c t a d d i t i o n a l variance  e x p l a i n e d by x  n  the r a t i o "F" =  d i v i d e d by  unexplained  variance.  (3-36) It  i s now a p p r o p r i a t e  to summarize the d i f f e r e n c e s  between the r e g r e s s i o n and c o r r e l a t i o n models. models d i f f e r variables. the  i n the assumptions made about the independent  The r e g r e s s i o n model makes few assumptions about  independent v a r i a b l e s , but the more  c o r r e l a t i o n model r e q u i r e s be  restrictive  that the independent v a r i a b l e s  random v a r i a b l e s , forming with Y a m u l t i v a r i a t e normal  distribution. the  The two  The r e g r e s s i o n model may be used t o d e s c r i b e  fertilizer-yield  problemwhere f e r t i l i z e r  assumed f i x e d on the one hand, or gives normal p o p u l a t i o n  of f e r t i l i z e r  r i s e to a b i v a r i a t e  and y i e l d on the o t h e r .  However, the c o r r e l a t i o n model d e s c r i b e s  only  It  i s t r u e that r  is  f i x e d , as an i n d i c a t i o n o f how e f f e c t i v e l y  2  application is  can be c a l c u l a t e d even when  the l a t t e r . fertilizer regression  reduces v a r i a t i o n ; but r cannot he used f o r i n f e r e n c e s about the p o p u l a t i o n  parameter, p.  answers more i n t e r e s t i n g q u e s t i o n s . not  only  In a d d i t i o n , r e g r e s s i o n Like c o r r e l a t i o n , i t  i n d i c a t e s i f two v a r i a b l e s move t o g e t h e r ; but  a l s o 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 t e s t o f the n u l l  31  h y p o t h e s i s H Q :p  =  0  can  regression  analysis  Hq:3  =  Thus, r e j e c t i o n  =  and  0,  0.  the  answered d i r e c t l y  testing of  the 6 =  c o n c l u s i o n must be  between f e r t i l i z e r b r o a d e r and  by  be  and  yield.  more i n t e r e s t i n g  some c o r r e l a t i o n  equivalent n u l l 0 implies  that  hypothesis  rejection  correlation  Since regression set  from  does  exist  answers  a  o f q u e s t i o n s , as w e l l  q u e s t i o n s , i t becomes t h e  more  p  of  as  comprehensive  technique. To ~sum  up,  while simple c o r r e l a t i o n  corresponds to simple regression  analysis,  correlation  to m u l t i p l e  analysis. b  analysis Recalling  e s t i m a t e s how  partial  corresponds how  the  Y is related  correlation  multiple to  analysis  the  partial regression  regression  coefficient  x i f z were c o n s t a n t ,  coefficient r  is a similar  the  concept.  xy, z It  estimates  the  if  z were h e l d  d e g r e e t o w h i c h x and  constant.  Rejection  of  Y move t o g e t h e r the  hypothesis  3 = 0 i s e q u i v a l e n t to r e j e c t i n g the n u l l h y p o t h e s i s 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 not o n l y xy, z its  own  set  questions  as  of q u e s t i o n s , but well.  also  partial  that that answer  correlation  CHAPTER I V  DATA Conditions  of Sampling In A p r i l  1 9 6 9 , B.C. T r e e F r u i t s  Ltd. supplied the  E c o n o m i c s B r a n c h , C.D.A., V a n c o u v e r a n d t h e D e p a r t m e n t o f Agricultural listing  E c o n o m i c s , U.B.C. w i t h  a p p l e t r e e numbers a c c o r d i n g  rootstock  category  Areas o f B r i t i s h  Three major d i f f i c u l t i e s f o r sampling purposes  to year of p l a n t i n g , growers i n the  Columbia.  i n using  the survey  data  generally indicating  s i z e w o u l d no d o u b t f a i l  case o f i n t e r m e d i a t e (unless  data,  c a n be c i t e d :  Rootstock categories while tree  survey  and v a r i e t y f o r i n d i v i d u a l  Okanagan and C r e s t o n  1)  current  stocks  t o do s o i n t h e  and s p u r s t r a i n s  growers themselves c o r r e c t e d  for this  factor). 2)  No d a t a w e r e shown f o r a s t a n d a r d category. rootstock  3)  S e m i - s t a n d a r d , s e m i - d w a r f and d w a r f categories  In showing data  were  included.  f o r an i n d i v i d u a l  d i s t i n c t i o n was made b e t w e e n t r e e s geneous o r c h a r d orchard  necessary  area  and t r e e s  g r o w e r no i n an homo-  i n an i n t e r p l a n t e d  area.  In order was  rootstock  t o meet t h e o b j e c t i v e s o f t h i s  to select a representative  study, i t  sub-sample o f apple  33  enterprises  f o r each o f t h e t r e e - s i z e  semi-standard, ities  semi-dwarf, dwarf.  1  categories: standard,  Moreover, the t e c h n i c a l -  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  eous e n t e r p r i s e p l o t s  t h a t homogen-  s h o u l d be s e l e c t e d and c o s t e d  from  the r e s t of orchard  fruit  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  on c o o p e r a t i n g f a r m s .  enterprises,  sophistication. of  survey  N e v e r t h e l e s s , i t was a c c e p t e d  staff  available)  which  that  This  fact w i l l  random  time which selec-  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  i n 1 9 6 6 , 4,271 c e n s u s f a r m s w e r e r e c o r d e d  census d i v i s i o n .  although hot  any a l t e r n a t i v e p r o c e d u r e  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 a t t e m p t e d tion.  t h a t i n view  l e a d t o a b e t t e r sample ( w i t h than  sampling  data source, the e x i s t i n g  d a t a c o u l d p r o v i d e an e n u m e r a t i o n  i d e a l , would a t l e a s t and  o f growers  thereby, considerably r e s t r i c t i n g  t h e r e b e i n g no a l t e r n a t i v e  Thus,  that the survey  data precluded the i d e a l p o p u l a t i o n enumeration and  apart  Most o f t h e s e were p r o d u c i n g  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  i n t h e Okanagan apples but  to help with the  study. Sampling  Method The s u r v e y  data f o r i n d i v i d u a l  m i t t e d the f o l l o w i n g sampling x  apple  growers  method when b r o k e n  per-  down b y  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 t h e 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 s e a r e p r e s e n t . I n t h e common c a s e 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  rootstock categories: 1)  Growers were l i s t e d a) T h e i r h a v i n g apple  according t o :  a minimum number (66)  trees i n the semi-dwarf  b) T h e i r h a v i n g apple  o r more  category.  a minimum number (100) o r more  t r e e s i n t h e dwarf c a t e g o r y , where  t h e y h a d n o t p r e v i o u s l y q u a l i f i e d u n d e r a) above . c) T h e i r h a v i n g apple  a minimum number (33) o r more  trees i n the semi-standard  category,  where t h e y had n o t p r e v i o u s l y q u a l i f i e d u n d e r a) o r b) a b o v e . G r o w e r s who e n t e r e d  these  lists  w e r e known t o be i n p o s s e s -  s i o n o f a minimum number o f a p p l e (corresponding  reasonably  trees of d i s t i n c t  well with tree size).  mark them as t h a t much more l i k e l y selection, bearing ing  i n mind t h e h i g h  and t h e n e e d t o c o s t i n d i v i d u a l  to qualify frequency  type  This  would  f o r sample of interplant-  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,  density, variety  Also  and g r o w i n g  t h a t growers l i s t e d  practice.  i t was assumed  i n t h e manner a l r e a d y e x p l a i n e d w o u l d  make i t p o s s i b l e f o r a s u b - s a m p l e o f s t a n d a r d enterprises 2)  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  W i t h i n each o f t h e t h r e e group l i s t s above, g e o g r a p h i c a l two  levels.  Firstly  sub-groupings  tree-size  sub-samples. outlined  i n 1)  w e r e made a t  a c c o r d i n g t o N. O k a n a g a n  ( W e s t b a n k and n o r t h w a r d ) ,  S. O k a n a g a n  (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 with 3)  regard to constituent d i s t r i c t s .  District horticulturalists sure  that l i s t s  o f growers r e f e r r e d  entities  ( i . e . , no d o u b l e  business  s t r u c t u r e was p e r m i t t e d ) .  they h e l p e d up-date l i s t s that had 4)  w e r e c o n s u l t e d t o make  a very  to managerial  counting of a single Furthermore,  w h e n e v e r i t was known  r e c e n t change i n o w n e r s h i p  o r tenancy  occurred.  On t h e 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 t h e inevitable decided ing  drop-out  rate f o r cooperators ,  t o o b t a i n 140 a p p l e  i t was  enterprises f o r cost-  i n 1 9 6 9 , e a c h one c o n f o r m i n g  t o homogeneity  conditions. K n o w l e d g e o f a p p l e p r o d u c t i o n i n t h e O k a n a g a n and Creston  areas  l e d t o t h e 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 the s m a l l t o t a l 130  c a t e g o r y w o u l d be a d e q u a t e t o r e p r e s e n t  number o f s u c h  enterprises.  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  mately  The  allocated  remaining  i n approxi-  e q u a l numbers t o s t a n d a r d , s e m i - s t a n d a r d , a n d s e m i -  dwarf c a t e g o r i e s . Since the study costings which  made i t n e c e s s a r y  d a t a t o be c o l l e c t e d , accounts  required detailed enterprises f o rassociated total  i t was d e c i d e d t o l i m i t  t o a r o u n d 100 i n o r d e r t o e n s u r e  worker time.  the t o t a l  adequate  farm farm  field-  Up t o two e n t e r p r i s e s w e r e 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  d e d u c e d t h a t many  36  cooperators would s e t t l e 5)  I t i s now  relevant  stratification and  district  above. along  The the  Rootstock Category  as  to discuss the  basis  the  already  described  Area  N.  S.  Okanagan  Okanagan 109  1-3  Creston  9+  139 +  81 +  Semi-standard categories,  the  i n d i c a t e d by  See  signs.  breakdown f o r the  Section  sary,  of sub-samples  this  of  is  above. since  a  4 growers,  was  to expect that  sampling w i t h i n  the  dwarf c a t e g o r i e s  by  means o f random  ( i n v o l v i n g s u b s t i t u t i o n procedure  exhaustion  numbers  judgment.  seemed r e a s o n a b l e  of growers  1.  Creston area i s given,  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  s e m i - d w a r f and  (b)  181  11 +  plus  below.  (a)  f i g u r e s shown and  s e l e c t e d by  area  i s given  by  D w a r f and  of  of e n t e r p r i s e ,  growers were i n e x c e s s o f  (b) No  2  purpose  Grower P o p u l a t i o n  Semi-standard  It  2  r e f e r r e d to i n Section  lines  Dwarf  For  enterprise.  b r e a k d o w n o f a p p l e g r o w e r numbers  Semi-dwarf  (a)  on  f o r one  and,  listing i f neces-  o f l i s t s ) w o u l d a c h i e v e random s e l e c t i o n across  areas  f o r the  four  categories  of  apple  The 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 of e n t e r p r i s e s . L a t e r i n t h e 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 to 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 p e c o u l d l e a d t o a g r o w e r e v e n t u a l l y c o n t r i b u t i n g more t h a n two enterprises.  37  enterprises.  3  O b t a i n i n g more t h a n  grower would not  affect  one  e n t e r p r i s e from  randomness p r o v i d i n g a l l growers  c o n t a c t e d w e r e 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 .  m i t t e d l y , the parent  p o p u l a t i o n o f g r o w e r s was  reduced  and  sively  in this  i n terms o f the  individual factory and  case  growers.  a  i t m i g h t be  Ad-  somewhat  a r g u e d more c o m p r e h e n -  e n t e r p r i s e c o n s t i t u e n c y shown  H o w e v e r , i t was  still  thought  satis-  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 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  by  inference  i n terms  of  field-work. U n f o r t u n a t e l y , i t s o o n became c l e a r c o n t a c t w i t h i n the high  incidence of e i t h e r  ingness was  s e m i - d w a r f and  a) i n a b i l i t y  t o help,- o r b) In f a c t ,  the  With  was  c o n t a c t s t o the s e m i - s t a n d a r d  listing.  In f a c t ,  randomized l i s t i n g s tically  this  the v e r y h i g h  relatively  category  rate of s u b s t i t u t i o n meant t h a t to achieve  f o r each o f the  on pracsuffi-  sub-samples.  s m a l l acreage of apples  i n the  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 g r o w e r s was  3  former  i n mind, the d e c i s i o n  f o r the Okanagan areas  c i e n t numbers o f c o o p e r a t o r s  Creston  experience  a l l growers l i s t e d were c o n t a c t e d  Because of the  unwill-  o w i n g t o t h e l a c k o f homogeneous  enterprise units. made t o e x t e n d  growers  d w a r f c a t e g o r i e s showed a  t o h e l p , e v e n when p o s s i b l e .  t h e more i m p o r t a n t  that  o b t a i n e d there w i t h the h e l p of  the  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 w o u l d 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  District  Horticulturist. The  the  initial  1969  f i n a l breakdown o f apple e n t e r p r i s e s  total  composing  s a m p l e drawn i n t h e s p r i n g and summer o f  was as f o l l o w s : TABLE V Tree-Size  Classification  Enterprise Standard  37  Semi-standard  60  Semi-dwarf  38  Dwarf  7  TOTAL  14 2 be made c l e a r t h a t  apple p l o t s used i n the study f r o m t h e above l i s t  the f i n a l  tree-size  sample o f  (n=119) n e c e s s i t a t e d  as some p a r t i t i o n i n g o f e n t e r p r i s e  An  Sample  deletion  w h e r e 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  homogeneity c o n d i t i o n s  Dr.  Total  No. o f E n t e r p r i s e s f o r Okanagan and C r e s t o n A r e a s  Category  However, i t s h o u l d  well  of I n i t i a l  data t o ensure  that  w e r e met.  a t t e m p t was made t o c a t e g o r i z e  groups i n accordance w i t h  trees  into  four  a method s u g g e s t e d b y  D. F i s h e r , S u m m e r l a n d R e s e a r c h S t a t i o n .  4  This  classifi-  *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 t y p e 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 . F o r e x a m p l e , 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 p o o r s o i l 1 . 0 x 1 . 0 x 1 . 0 x 0 . 6 0 = 0.60. T h i s i n d e x v a l u e w o u l d c a t e g o r i z e t h e above e x a m p l e as s e m i - d w a r f 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 t y p e s was a v a i l a b l e f o r t h e s t u d y , o n l y t h e f i r s t t h r e e f a c t o r s m e n t i o n e d above h a v e been t a k e n i n t o a c c o u n t . Further d e t a i l s o f d e r i v i n g the i n d e x a r e shown i n T a b l e I V i n t h e A p p e n d i x .  39  cation  of  variety  apple tree  was  successfully  c l a s s i f i c a t i o n was  because a r i g o r o u s variety  l e d to  experience  attempt to  arbitrary  stems l a r g e l y  f o r e x a m p l e , b a s e d on further  not  carried  out.  However,  s a t i s f a c t o r i l y achieved group t r e e s  into  appropriate  classification.  This  frustrating  from the  tree-size  fact  that  a single  plot,  c l a s s i f i c a t i o n underwent  p a r t i t i o n i n g i n order to ensure a rigorous  variety  categorization.  Consequently, c l a s s i f i c a t i o n according  variety  resulted  i n the  rapidly  t h a n when c l a s s i f i c a t i o n o f The  b a s e d on  final  tree-size  tree  sample breakdown of  s i z e was apple  of  No. o f E n t e r p r i s e s O k a n a g a n and C r e s t o n  Category  Semi-standard  62  Semi-dwarf  28  Dwarf  6  TOTAL  119  scion  case of  s h o u l d be effects  of  noted that  intermediate  v a r i e t i e s where t h e r e just  rootstock  tree-size  and  are  scion  stocks present.  occurring,  c a t e g o r y becomes synonymous w i t h r o o t s t o c k stated  Enterprises  23  the  previously.  enterprises,  VI  Standard  reflect  done.  i s shown b e l o w :  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  It  to  s a m p l e s i z e b e i n g e x p a n d e d more  TABLE  Tree-Size  a  for Areas  categories and In  spur the  strains common  tree-size category  as  40  For each e n t e r p r i s e s e l e c t e d i n t h e s t u d y , t h e following  i n f o r m a t i o n was o b t a i n e d :  1.  Weight o f apple  2.  Density o f apple  3.  Age o f t r e e s  4.  Cost  o f spray a p p l i e d  5.  Cost  of f e r t i l i z e r applied  6.  Labour hours  7.  Tree-size  yield trees  spent  on p r u n i n g  producers of  and f a m i l y l a b o u r were  i n t h e s t u d y managed t o k e e p an u p - t o - d a t e there  i slikely  i n recording pruning  calculate  and t h i n n i n g h o u r s .  is  an a s s u m p t i o n  to  do w i t h v a r i a t i o n  i n acreage.  or quantities per acre.  1.  Apple  2.  Density  6  yield  acre  acreages.  i n this  T r e e age  procedure  variables  The o u t p u t  enter into  used i n t h e r e g r e s s i o n a n a l y s i s  per  Implicit  t h a t a l l independent  input variables  efficients  total  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 s u c h m o d i f i c a t i o n .  5  In order to  d a t a on a p e r a c r e 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  v a l u e s o f dummy v a r i a b l e s  non-land  record  t o h a v e b e e n some memory  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 and  employed  and t h i n n i n g o p e r a t i o n s , and n o t a l l a p p l e  labour hours,  bias  and t h i n n i n g  index  Since both h i r e d in pruning  5  had n o t h i n g  v a r i a b l e and  the analysis  as c o -  Independent  variables  a r e as f o l l o w s :  6  (Y) p e r a c r e m e a s u r e d i n p o u n d s .  (D) m e a s u r e d i n t e r m s o f number o f t r e e s (range  i n study  48 - 6 0 5 ) .  A copy o f t h e 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 t h e Appendix. A f u l l l i s t o f i n d e p e n d e n t v a r i a b l e s ( e x c e p t t h e dummy v a r i a b l e ) i s given i n Table V I I i n the Appendix.  41  Age  (A) o f t r e e s m e a s u r e d i n y e a r s  4 - 55).  I t i s of interest  (range  i n study  t o note  t h a t apple t r e e s  o f one t o t h r e e y e a r s o f a g e , w h i c h  were i n c l u d e d i n  the i n i t i a l  sampling, d i d not bear  amount o f f r u i t  f o r the year  any r e c o g n i z a b l e  i n which  t h e s t u d y was  conducted. Fertilizer  (F) measured i n $ c o s t p e r a c r e .  r e g a r d i n g t h e amount o f f e r t i l i z e r t o be l e s s from Spray  reliable  Data  u s e d was  thought  than the cost estimates o b t a i n e d  growers. (S) m e a s u r e d i n $ c o s t p e r a c r e  reasons  as a b o v e .  In f a c t ,  amounts o f s p r a y  r e p o r t e d w e r e so h e t e r o g e n e o u s impossible  t h a t i t was  to derive a meaningful  P r u n i n g and t h i n n i n g h o u r s hours  p e r acre spent  hired  and o t h e r f a m i l y  f o r t h e same  virtually  interpretation.  (P) m e a s u r e d i n t o t a l  on t h e s e p r a c t i c e s labour  and i n c l u d e  hours.  Tree-size  index  suggested  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  (T) c a l c u l a t e d  (G) u s e d  j u s t n o r t h o f Summerland.  r e g i o n s , which  in soil  and S o u t h  type  i n t o North  The  and South  regions  T h i s d i v i s i o n was made differences  i n t h e two  w e r e assumed t o a c c o u n t  a t i o n i n apple y i e l d s . the North  a c c o r d i n g t o a method  f o r several purposes.  Okanagan a r e a was d i v i d e d  because o f environmental  actually  f o r some  D i f f e r e n c e s observed  Okanagan  vari-  between  regions include variations  and w e a t h e r o b s e r v a t i o n s f o r t h e y e a r  42  under  study.  A c c o r d i n g to the " C l i m a t e of  C o l u m b i a R e p o r t " f o r 1968  - 1969,  mean t e m p e r a t u r e s f o r t h e two d u r i n g t h e p e r i o d May  1968  British  slightly  different  r e g i o n s were  t o May  1969.  registered  The  average  t e m p e r a t u r e s w e r e 44°F. and 48°F. i n t h e N o r t h South  regions respectively.  recorded  These t e m p e r a t u r e s were  i n the growing p e r i o d , which  t h e number o f days w i t h above 4 3 F . &  The  7  t h e r e were s l i g h t in precipitation  an a v e r a g e  Report  and  i s defined  daily  as  temperature  a l s o showed t h a t  in  1969  d i f f e r e n c e s b e t w e e n t h e two r e g i o n s f o r t h e months May  to October  inclus ive. L o n g l e y u s e d t h e May there existed  to October p e r i o d  average p r e c i p i t a t i o n  O k a n a g a n r a n f r o m 1.09  i n Nova  rainfall  8  i n the n o r t h e r n r e g i o n of the  t o 1.23  o f 0.10  and  Scotia.  inches f o r the  indicated  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 average  found  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  apple 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 The  and  t o 0.92  i n c h e s was  low  reported for  t h e same p e r i o d . Variations further classified o f the c a u s a l  in yield  e  to weather  according to direct  agent.  I t i s very l i k e l y  c o m p o n e n t s as h u m i d i t y , l i g h t 7  due  f a c t o r s may  or i n d i r e c t t h a t such  and a i r movements  action weather  directly  The Climate of B r i t i s h Columbia - Tables of Temperature, P r e c i p i t a t i o n s , and S u n s h i n e R e p o r t f o r 1969 - 1 9 7 0 , P r o v i n c e o f B r i t i s h Columbia Department of A g r i c u l t u r e , pp. 9-10. L o n g l e y . , op.  cit.,  pp.  22-23.  be  43  influence yields.  M o r e o v e r , owing t o t h e r e l a t i o n s h i p s and  interrelationships  present  will  affected.  be i n d i r e c t l y  insect  infestations  The e f f e c t  also vary with the l e v e l  cultural this  practices,  The i n t e n s i t y  of certain  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 w e a t h e r . can  i n weather components, y i e l d s  o f w e a t h e r on a p p l e  of fertilizer,  a n d many o t h e r  complexity, the following  factors.  soil  yield  type,  Because o f  a s s u m p t i o n s w e r e made:  (1) n o n - w e a t h e r 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 w e a t h e r  influ-  ence;  influ-  (2) a l l v a r i a t i o n s  ences a r e n o r m a l l y z e r o and a f i n i t e of approximately out  of a total  i ny i e l d  due t o n o n - w e a t h e r  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 variance.  equal  size.  Data were s p l i t Fifty-four  o f one h u n d r e d a n d n i n e t e e n w e r e a s s i g n e d t o  enterprise plots  t o the south  Regional  d i f f e r e n c e s i n apple y i e l d s ,  A dummy 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 Okanagan r e g i o n s the south the n o r t h . and  as e x p l a i n e d  e x p l a i n e d by dummy  i s o n l y an i n d i c a t o r  values.  variable.  I n t h e case o f t h e  ' 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  a n d '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 M o d i f i c a t i o n was n e c e s s a r y  i n t h e u s e o f '0'  ' 1 ' when t h e C o b b - D o u g l a s f u n c t i o n was u s e d t o e s t i m a t e  the y i e l d in  sixty-five  Okanagan r e g i o n .  c a n t h e o r e t i c a l l y be p a r t l y  variables.  two p a r t s  enterprise plots  t h e n o r t h Okanagan r e g i o n and t h e r e m a i n i n g  earlier,  into  relationship.  the process  approaches  The v a l u e  z e r o becomes a p r o b l e m  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  -°° .  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 1 0 ,  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 r e p r e s e n t a y i e l d v a r i a t i o n , i f any, w h i c h may be due m a i n l y t o d i f f e r e n c e s i n locations.  Of c o u r s e , any p a i r o f numbers w o u l d s e r v e the  purpose e q u a l l y  as w e l l as 0 and 1.  But the magnitude o f  c o e f f i c i e n t s would v a r y depending on the v a l u e s 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 o f 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 o f numbers i s bound t o d i f f e r from some o t h e r p a i r o f numbers.  CHAPTER V  EMPIRICAL  RESULTS  Introduction Before a simple  attempting  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 ,  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 a p p l e  yield  on  e a c h i n d e p e n d e n t v a r i a b l e , n a m e l y , d e n s i t y p e r a c r e , age o f t r e e s , the cost o f f e r t i l i z e r used per acre, the cost of spray per  u s e d p e r a c r e , and p r u n i n g  acre.  very  The t h i n g t o n o t e i s t h a t i t seems  likely  t h a t d e n s i t y per acre  with tree-size ables  index.  are c o r r e l a t e d can r e a d i l y  Table  ity  o f such a c o r r e l a t i o n  running  simple  according  corresponding respect  Finally, linear set  linear  Notwithstanding  regression  o f data.  efficients  In the l i g h t  and t h e b e s t  Prior to  d a t a were  grouped  that i s , standard,  On t h e s e  classified  independent  data,  were  performed  variable.  d a t a were lumped t o g e t h e r , w h i c h p e r m i t t e d  regression analyses  i s given  the p r o b a b i l -  i n d e x was t r i e d .  regression analyses  t o each i n d i v i d u a l  vari-  be c h e c k e d by t h e i n s p e c t i o n  classification:  and s e m i - d w a r f . simple  two i n d e p e n d e n t  regression analyses,  to tree-size  semi-standard,  with  linear  be h i g h l y c o r r e l a t e d  o c c u r r i n g , the simple  on t r e e - s i z e  hours  conceptually  The c o r r e l a t i o n m a t r i x  V I I I i n the Appendix.  of d e n s i t y per acre  may  Whether these  of the c o r r e l a t i o n matrix. in  and t h i n n i n g l a b o u r  simple  t o be p e r f o r m e d on a l l o v e r a l l of s i g n i f i c a n t  ' f i t ' criterion,  r e g r e s s i o n cothe simple  linear  46  regression  model, r e g a r d l e s s  gated or not, relationships. linear  failed  t o i n d i c a t e any s t r o n g  The e m p i r i c a l  regression  Appendix.  analysis  considered  apple  yield  r e s u l t s from t h e simple  i s shown i n T a b l e  relationships exist with simultaneously,  r e l a t i o n s h i p might w e l l than a s t r a i g h t l i n e .  the  disaggre-  I I I i n the  The i m p l i c a t i o n f r o m t h e s e r e s u l t s may be t h a t  apple y i e l d  multiple  o f whether d a t a were  regression  several  and t h e r e f o r e  any a p p l e  take a c u r v e l i n e a r This  variables  form  yield  rather  r a t i o n a l e p a v e d t h e way f o r  a n a l y s i s which  i s discussed  later i n  chapter. The  multiple  regression  routine  c o m p u t e r p r o g r a m w as u s e d t o p r o v i d e  l e a s t squares  T  estimates.  o f t h e "UBC T R I P " regression  1  A n o t h e r p r o g r a m was u s e d f o r t h e E q u a l i t y T e s t t o see whether t h e d i f f e r e n c e s among t r e e - s i z e g r o u p s c o u l d or  to differences  (X^ , 3  be a s c r i b e d  among g r o u p s .  variable-of-class ification  i n regression  2  3  and a p p l e y i e l d ,  f o rinstance.  i d e n o t e s t h e number o f o b s e r v a t i o n s  errors  a single  ( X ^ i , Y ^ i ) , (X^  Y ^ ) i s p r e s e n t e d , where X and Y r e p r e s e n t  applied  coefficients  to sampling  To i l l u s t r a t e ,  i n t h e form  o f Slope  The f i r s t  2  Y^ ), 2  fertilizer subscript  i n e a c h g r o u p and t h e  * J . H. B j e r r i n g and P. S e a g r a v e s , UBC T R I P ( T r i a n g u l a r R e g r e s s i o n P a c k a g e ) V a n c o u v e r : U . B . C , C o m p u t i n g C e n t r e , Nov. 1 9 7 0 . B i l l Coshow, UBC BMDX 64: G e n e r a l L i n e a r 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 Computing C e n t r e , June 1971.  H y p o t h e s i s , U.B.C.,  o f S l o p e T e s t , U.B.C.,  47  1,  second s u b s c r i p t 1 represents  2,  and 3 denote c o r r e s p o n d i n g g r o u p s :  a standard apple group, e t c .  p r o c e d u r e s e x t e n d t o more t h a n a s i n g l e cation. sion the  Suppose a q u e s t i o n  lines  arises  these  variable-of-classifi-  as t o w h e t h e r  the regres-  c o r r e s p o n d i n g t o e a c h g r o u p a r e t o be r e g a r d e d as  same.  To a n s w e r  struction  the question  adequately requires  o f t h e c o v a r i a n c e t a b l e , as shown b e l o w .  be c o n v e n i e n t t o d e n o t e t h e q u a n t i t i e s vidual  Naturally  con-  It will  i n T a b l e IX by  indi-  letters. TABLE I X C o v a r i a n c e Table f o r t h e Three T r e e - S i z e Ex  W i t h i n each  C C C  Among means  C  Within  C  groups  Total  C  as f o l l o w s :  C  , C  XXj  v  , C  C  C yy! C yy C yy 3 C yym C yyw C «. yyt  1  xx  x  xx  2  xyj  C xy C xy 3 C xym C xyw C xyt 2  xx 3 xxm xxw . xxt  +  C' yy C' yy C' yy 3 C' yym C' yyw C yyt x  2  2  t o be c o m p u t e d  2  - (EX) /n 2  XX3  XA2  *"xyi * ^ x y ' ^ x y 3 2  g r o u p s 1,  r  e  P  r  e  s  e  groups y y i  Ey *  z  r e p r e s e n t the computation EX  v  for  C  Ey  d e f i n i t i o n s of the q u a n t i t i e s  are v  Exy  group  1 2 3  The  z  Groups  » Cyy , 2  Cyy  3  n  t  t  1,  n  2,  3.  c o m p u t a t i o n EXY - EXEY/n f o r  e  2,  3.  represent the computation EY for  groups  1,  2,  3.  2  - (EY) /n 2  48  The  quantities  EY  - (EXY) /EX  2  i n t h e column  2  The  Ey'  2  a r e c o m p u t e d by t h e f o r m u l a  .  2  quantities  Si, S , S , S 2  3  are defined  4  i n terms  o f C! i n T a b l e X. 1  51  = t h e sum o f s q u a r e s o f Y v a l u e s f r o m t h e r e g r e s s i o n i n each  52  group, t o t a l l e d  = the v a r i a t i o n different  53  f o r a l l groups.  among r e g r e s s i o n c o e f f i c i e n t s  = t h e sum o f s q u a r e s o f d e v i a t i o n s line  o f t h e means w i t h  o f t h e means f r o m t h e regard to Y values.  = the square o f the d i f f e r e n c e between within  groups  and S  = S  t  o f the  groups  regression Sn  line  x  (b ,) and c o e f f i c i e n t s v  + S  + S  2  coefficients among means (b ) ,  + Sn ( s e e T a b l e XI i n A p p e n d i x ) .  3  TABLE X Table Definitions S j = c» . yyi S = C* yyw S = Cyym  forS  i  i n terms o f D.F  o f S. 1  k ( n - 2) k - 1  S  2  k - 2  3 3  S4  = cy y t - C '  yyw  Total  S  t  =  - C' yym  1  kn - 2  yyt  R e f e r r i n g t o T a b l e X, n = t h e number o f o b s e r v a t i o n s and k = t h e number o f g r o u p s . gression  line  f o r m u l a t e d as  Therefore, a test  c a n be u s e d follows:  of whether  one r e -  f o r a l l o b s e r v a t i o n s c a n be  49  S  + S  2  F  2(k  +  3  -  S  H  1)  Si k(n The different  Equality  tree-size  seven independent quadratic  - 2)  o f S l o p e T e s t was  groups,  each group c o n t a i n i n g  variables.  I t was  The  results  support  among t h e versus  groups  Standard  S e m i - d w a r f , and  calculated  F = 0.29  Approximate values  and  and  XI  The  Semi-standard,  Q 5  i n the  55)  0.97  Semi-standard  0.25  - 0.60; An  =  f o r the  give a value  The 1.67. F-test with  the  F - t e s t are p r e s e n t e d  in  Appendix.  reasonable:  - 1.00;  the  Standard  Semi-Dwarf.  (D.F.: 30,  M e t h o d does n o t  from  seemed t o be  taken  versus  are taken because the Table  results  are  regression coefficients  tabulated F  Using t r e e - s i z e  in  that there  54 d e g r e e s o f f r e e d o m , t h e ones r e l e v a n t t o  analysis. Table  I.  Semi - s t a n d a r d v e r s u s  i n Snedecor's S t a t i s t i c a l 35  given to that  the h y p o t h e s i s  differences i n corresponding  twenty-  used only f o r the  f u n c t i o n because o f the p r i o r i t y  f u n c t i o n as e x p l a i n e d i n C h a p t e r  no  used f o r three  and  i n d e x , the Standard i n 0.61  tree f a l l i n g - 0.88;  D w a r f t r e e i n 0.20  important  following  t h i n g to note  -  categorization i n the  Semi-dwarf t r e e 0.21.  i s t h a t no  account  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 .  group c o n s i s t e d of o n l y s i x e n t e r p r i s e s .  range  As  such,  was This  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 E q u a l i t y of Slope Test w i l l dealt with l a t e r .  The r e s u l t s of the E q u a l i t y of Slope t e s t s  for t r e e - s i z e groups were obtained as f o l l o w s :  (1) regres-  s i o n equation f o r each sample; (2) the twenty-seven common slope c o e f f i c i e n t s ; (3) F - r a t i o and i t s p r o b a b i l i t y . information i s given i n Table XI i n the Appendix.  This  On the  evidence of no d i f f e r e n c e among r e g r e s s i o n c o e f f i c i e n t s the three t r e e - s i z e groups were combined so that a s i n g l e regression equation might be f i t t e d .  Thus, two b a s i c re-  gression models were a p p l i e d to the o v e r a l l e n t e r p r i s e data. The two b a s i c models used i n the ensuing 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 :  Ln Y = Ln a + B  2  Ln D + 6 3 Ln A + 8 4  Ln P + B? Ln G + B  8  Ln T + Ln V .  l i n e a r i n logarithms; Ln F + Bi  5  Ln S + B  6  one i s a Cobb-Dougla.s f u n c t i o n  All  that i s necessary now i s a simple renaming of the terms i n t h i s equat i o n : Y  = Ln Y = Y i e l d  Bi  = Ln a  Xi  = Ln D = Log dens i t y  x2  = Ln A = Log age  x3  = Ln F = Log cost of f e r t i l i z e r  X„  = Ln S = Log cost of spray  x5  = Ln P = Log hours i n pruning and t h i n n i n g  x6  = Ln G = Log geographical dummy  x7  = Ln T = Log t r e e - s i z e  e = Ln V , where X i , X  2  index ,  , X represent independent v a r i 7  51  ables  used i n the study.  that  Q  Assume t h a t V i s d i s t r i b u t e d  = Ln V s a t i s f i e s t h e a s s u m p t i o n s made e a r l i e r ;  n a m e l y , Ln V % N ( 0 , a ) .  Subsequently, the logarithmic  2  e q u a t i o n appears V  notation,  v//5  s  X  1  Y  3  1  •  '  ,2  2  X  3 ) 2  l i n e a r model.  > Xi,8  >  X  j  ,  ,  2  ,8  X 3 ,  3i  ei  32  e  2  e  3  8  1 1 9  1  X l l 9 ,  Assume t h a t E (6)  +  •  •  3  Y  is  i s that  multiplicative.  matrix  2  ,  = O  ,  •  8  e  X i l 9 , 8  , Cov (Q ) = a S 2  ,  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 formation  In  p  +  Xl,2>  2  =  as a f a m i l i a r  1 Y  •  so  119  i.e.  logarithm  the e r r o r term i n the n a t u r a l I f this  assumption  trans-  form e q u a t i o n  i s unwarranted,  then  52  t h e model w i l l the concern  of this  + B  +  A  Bio  D + 83  2  DS  + B1  B23  FS  + B11  2  7  DP  A + F  A,  geographical  F +  6n  + B12  2  S  +  S + Be  65  + Bis  2  AF  B19  FT  + 826  F, S,  G,  and  P,  P  P  +  SP  +  observations  ST  + Bis AP  DF  stacked  index into  D  +  PT  +  thinning  2  Bie  + B 2 2 AT  + B2e  and  =  T + B9  + ,  e  T r e f e r t o d e n s i t y , age,  tree-size  c a n be  DA  + B 21  B27  cost of sprays, pruning  dummy and  G + B8  B7  + Bm  2  + B 2 0 AS  FP + B 2 5  of f e r t i l i z e r ' ,  all  which i s beyond  study.  + B1 s DT  + B2i  (where D,  treatment  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  The Si  require special  cost hours,  respectively).  a column v e c t o r  Again, as  follows:  Y=  X/3  +  e  , where E  Yi  1  x  Y2  1  X  i >2 , 2  —  ,2 , - - -•  ce) =  0  ", _  ,  ,  Cov  X1  , 2  e  Bi  ei  X2  ,2  8  B2  e  ;  =  = a  (6)  2  •  +  B2£ Y 1  1  19  X 11  When one g r e s s o r s , as  9  ,  2 , •"  ,X  11 9 ,  e  2 8  in this  m o d e l , a q u e s t i o n may  are  functionally  dependent  square of the  o t h e r ) ; they  are not  one  and  D?  i s not, say,  19  v a r i a b l e i s used to o b t a i n s e v e r a l rearise  w h e t h e r 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 . D^  1  twice  the  other).  (i.e.,  linearly  as  For one  to  example,  i s the  dependent  (i.e.,  53  Geometrically, on  the co-ordinate  points  (D, D ) l i e 2  2 below; the important  a c u r v e as shown i n F i g u r e  thing  h o w e v e r , i s t h a t t h e y do n o t l i e on a s t r a i g h t l i n e . the  problem o f m u l t i c o l l i n e a r i t y  according  t o the degree o f c u r v a t u r e  Figure Polynomial  may  regression  o r may n o t be  (2) t h e s t a n d a r d and  2.  as a s p e c i a l c a s e o f m u l t i p l e  (1) t h e e s t i m a t e d  regression  e r r o r o f each c o e f f i c i e n t ;  e r r o r of the estimate,  of m u l t i p l e determination,  R ; 2  Y;  regression  coefficients  shown b e l o w e a c h s t a n d a r d  coefficients;  coefficient;  and (6) t h e c o r r e l a t i o n m a t r i x . errors of  a r e shown i n p a r e n t h e s i s .  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 is  regression  ( f ) the c o e f f i c i e n t  In showing data, s u b s e q u e n t l y s t a n d a r d the  regression.  (3) t h e F - r a t i o  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  (4) t h e s t a n d a r d  avoided  involved.  The o u t p u t o f t h e T r i p p r o g r a m f o r b o t h models i n c l u d e d :  Thus,  error.  The  coefficient  The r e s u l t s o f t h e  54  estimated stepwise given i n Table  Cobb-Douglas r e g r e s s i o n e q u a t i o n are  X I I i n t h e A p p e n d i x and t h e c o r r e l a t i o n  f o r t h e C o b b - D o u g l a s f u n c t i o n i s shown i n T a b l e Appendix. presented is  The r e s u l t s  m a t r i x appears Results  from  regression equation  XIV i n t h e Appendix.  i n Table  Its correlation  XV i n t h e A p p e n d i x .  t h e Cobb-Douglas Model  The e s t i m a t e d o v e r a l l tion  i n the  o f t h e e s t i m a t e d q u a d r a t i c model a r e  on page 57 and t h e s t e p - w i s e  shown i n T a b l e  XIII  matrix  e n t e r p r i s e r e g r e s s i o n equa-  i n logarithmsi 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 . (0.0748) (0.0423) (0.2455) all  variables  are expressed  R  from 401  t h a t f o r t r e e s i z e were found  z e r o a t t h e 5% l e v e l of t o t a l  variation  f o r by t h e i n d e p e n d e n t  which  of probability.  2  Approximately  (Y) h a s b e e n  i s not improved  e q u a t i o n when o n l y t h o s e  The r e s u l t s  respectively In view  accounted  i n the stepwise  independent  contribution  of the stepwise  variables  t o apple y i e l d are 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 a r e shown i n T a b l e XIII  different  variables.  make a s i g n i f i c a n t  included.  for a l l variables significantly  i n crop y i e l d  The v a l u e o f R regression  = 0.4147, where  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 except  2  X I I and  i n the Appendix. o f the f a c t that primary  interest  i n the  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 correlation c a n n o t be tion  the m u l t i p l e  considered s t r i c t l y  correlation  coefficient  variable.  This i s because  so,  and n o t a m u l t i v a r i a t e n o r m a l  o f f i t o f the o b s e r v e d p o i n t s  Results  cepts  s h o u l d be made i n o r d e r t o c l a r i f y involved  a single  3  a yield  line  2  values.  automatically  valid  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  remaining s i g n i f i c a n t  of  out  Covariwhether  i n represent-  C o n s e q u e n t l y , the a n a l y s i s  Secondly, the a n a l y s i s  and p e r f o r m i n g t h e E q u a l i t y  c u l t problem  i s to f i n d  is statistically  i s i n d e c i d i n g which  variables  T h i r d l y , t h e most variables  the diffi-  a r e t o be  r e t a i n e d , and w h i c h a r e t o be o m i t t e d f r o m t h e G e n e r a t i o n o f i n n u m e r a b l e terms  does  i s incapable  of Slope Test w i t h only  variables.  con-  of Slope Test.  the Test i s p a r t o f " A n a l y s i s  relationship. R  data, several  the u n d e r l y i n g  i n t h e employment o f E q u a l i t y  regression  not produce  3  good-  t o the r e g r e s s i o n p l a n e .  ance", the p r i m a r y concern of which  of  Even  from t h e Q u a d r a t i c Model  Firstly,  ing  variables  t o measure the  B e f o r e p r o c e e d i n g w i t h the combined points  the  of given  distribution.  R does p r o v i d e a summary s t a t i s t i c  ness  and  the independent  the r e g r e s s i o n model are o b s e r v e d i n terms  values  R  as an e s t i m a t e o f t h e p o p u l a -  c o r r e l a t i o n between the independent v a r i a b l e s  dependent in  model,  from  model. variables  M. E z e k i e l and K. A. F o x , M e t h o d s o f C o r r e l a t i o n sion Analys i s , 3rd e d i t i o n . pp. 2 70-281 , 1965.  and  Regres-  56  squared ables  or combinations  i spossible:  p o i n t o f view,  2  which  G  i n that they are capable  and T  2  The  t o each  of helping to repre-  phenomenon, i . e . , t h e l a w o f d i m i n i s h i n g o f t h e squared 2  terms.  Even s o , squared  c a n i n no way a p p e a l t o t h e s e n s e s b y  some o f t h e c r o s s - t e r m s  1.  basis, variables included  s u b j e c t i v e j u d g m e n t i s made.  Equality  But from t h e s u b j e c t i v e  2  any l o g i c a l  i n t h e case  terms l i k e  vari-  t h e v a r i a b l e s w h i c h have been e x c l u d e d w o u l d  sent a b i o l o g i c a l returns  b a s i c independent  G , T , GT, e t c .  seem t o be l a c k i n g are j u s t i f i e d  o f t h e seven  F o r t h e same  reason,  do n o t a p p e a r i n t h e m o d e l .  "UBC SLTEST" was u s e d  o f Slope Test.  The r e s u l t  f o r the purpose o f o f the test with respect  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 :  Regression Equation  f o r Standard  Tree-Size  Group.  Y = 989.300 + 1212 D + 15.060 A + 1457 F + 3374 S + 2742 P + 1466 G - 1043 T - 29.360 D + 2.353 S  2  - 2.143 P  2  2  - 235.600 A  2  + 18.130 F  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 + 1 1 . 1 3 0 . 2.  Regression Equation  Y = - 7.650  f o r Semi-Standard  Tree-Size  Group.  - 668.4000 D + 2.9720 A - 4246 F + 2872 S  - 2515 P - 878.4000 + 15.6500 F  2  G - 2.2360 T - 0.8332 D  + 1.3470 S  2  + 2.5170 P  2  2  - 265.7000 A  + 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 3.  FT - 9.0710 SP - 2186 ST + 1863 PT .  Regression Equation  Y = - 12.5100  Group.  - 162 D + 2793 A - 1895 F = 781.200  + 491.500 - 3.264 F  f o r Semi-Dwarf T r e e - S i z e  G + 50.2600 + 3.6610 S  2  - 1.1880 DS + 38.4700  T + 0.6151 D - 0.1218 P  2  - 0.9559 DP  AS  - 29.0100  + 11.7900  - 418.6000 AP  + 202.7000  2  2  S + 1057 P  DT  - 2.445 AT  A  2  DA + 1.9570  - 57.0900  DF  AF  - 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 efficients result  p e r f o r m e d on t h e t w e n t y - s e v e n co-  h e l d i n common by e a c h r e g r e s s i o n e q u a t i o n .  indicates  t h a t t h e r e a r e 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 tree-size  The  g r o u p s a t t h e 5% l e v e l Data from the t e s t  the Appendix.  Therefore  three separate  samples were  among t h e t h r e e  of significance.  are presented  i n T a b l e XI i n  on t h e b a s i s o f t h i s  r e s u l t the  c o m b i n e d i n t o one s a m p l e .  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  t h e n e s t i m a t e d as  A  follows:  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 (379.0393) (665.2497) (8.189) (0.2094) - 6.9504 F (3.8344)  2  - 1.5232 S (1.1533)  2  - 1.3623 DS + 1.1930 DP (1.1785) (0.8984)  + 0.6457 P (0.3758)  2  - 18.7206 A (32.5372)  2  + 6.5565 DA + 1.4649 DF (11.9919) (2.1995)  - 218.4573 DT (166.1315)  - 2.0330 DS + 2610.1141 AT (10.5395) (4545.6588)  2  - 26.8412 AF - 3.0749 AS (54.1606) (0.9068)  + 6.1613 FS + 0.1162 FP + 1004.4861 I (3.4245) (3.2792) (901.3124)  + 0.5085 SP + 170.2734 ST + 511.4871 GT (0.8558) (414.9117) (344.7191)  .  R  2  = 0.7534  58  The  fact  that eleven  above e q u a t i o n w e r e n o t can  immediately  errors the  be  corresponding  significant  c h e c k e d by  i n parentheses  regressor coefficients  o f the  observing  for a final  i n the  equation  A q u e s t i o n at t h i s k i n d o f s i t u a t i o n has to  cent  standard  exceeded values  e q u a t i o n but select  occurred.  may  The  t a k e i s t o e x a m i n e w h e t h e r any  stepwise  the  significant  arise  as t o why  f i r s t necessary  of the  1  collinearity  Therefore,  shows t h a t t h e r e  are  linear  d e p e n d e n c e was  independent v a r i a b l e  pendent v a r i a b l e w i t h another. direct  violation  o f the  D.  However, i f the D?  a straight  line  "*See T a b l e  XV  curve  i n the process or i n t e r a c t i n g  of one  These o c c u r r e n c e s  not  linearly  segment on w h i c h t h e  i n Figure 2 l i e s  (no  either inde-  are  a  Appendix.  dependent  on  coordinate  i s c l o s e t o the shape  s e g m e n t , t h e r e c a n be  i n the  regressors  2  example, i s f u n c t i o n a l l y but  and  combinations  assumption that a regressor D ,  for  point  i f multi-  shown b e t w e e n i n d e p e n d e n t v a r i a b l e s )  most o f w h i c h h a v e b e e n g e n e r a t e d s q u a r i n g an  quadratic  A close inspection  a number o f n e a r - l i n e a r  f o r m e d b e t w e e n i n d e p e n d e n t v a r i a b l e s and  as-  i n i t i a l l y the  i n v e s t i g a t e d t o see  might have c a u s e d p r o b l e m s .  this step  appropriate  s u m p t i o n s 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 t h e  c o r r e l a t i o n m a t r i x * must be  of  analysis.  stage  f u n c t i o n have been v i o l a t e d .  level  Non-significance i s also  r e g r e s s i o n at a l a t e r stage w i l l variables  5 per  that the  coefficients  coefficients.  true of other c o e f f i c i e n t s  at the  i n the  problems of  of  multi-  59  collinearity. I f D.  and  1  t i o n , the  variable  almost l i n e a r l y  D?  f o r m an a l m o s t n e a r - l i n e a r c o m b i n a -  1  x,  and  2  x  i n terms o f m a t r i x w i l l  1 ) 9  dependent:  1  >  ,2  Xl  1  X l 1 9, 2 > "  Xi  , 9 , - -  _  _  > X i i 9, g , - "  _  "  Such m u l t i c o l l i n e a r i t y entries the  i n the  covariance  large  matrix  visualized,  f o r the  and  For is  the  X\  multicollinearity  there 5  J. p.  ) ~  1  3.  But  is very  intervals.  be  clearly  t o keep  e l l i p s o i d that delimits  This  a t t h e mean E X 2 •  X2 &  collinear, o f the  (X'}<  therefore obtain  p r o b l e m may  geometrically, i n Figure  a sphere.  and  and  2  large  the  most o f  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.  simply  K i  we  5  i n extremely  Since a  1  ,  , Xil9,28  i n d e p e n d e n t e r r o r s assumed e a r l i e r , t h e  centered by  ~ •  X i , 2 8  ,  hence b r o a d c o n f i d e n c e  g e o m e t r y m a n a g e a b l e , an £ i ' s , the  results  (jot)  inverse matrix  covariances, The  be  re  the  origin.  n  ( Y  Figure °t  The  sphere of Y o b s e r v a t i o n s )>  is  which i s i n the plane  3 shows what h a p p e n s when  orthogonal  interval  ellipsoid  mutually  generated regressors  (perpendicular)  o f p i ' s i s d i s p e r s e d on b o t h  point estimate  i s a good c h a n c e i t may  be  may  be  positive,  but  sides but  negative.  J o h n s t o n , E c o n o m e t r i c M e t h o d s , New 110, 1960.  York:  McGraw-Hill,  60  Figure  3  Range o f v a l u e s f o r p o s s i b l e (Si's a r o u n d o r i g i n when %i a n d X 2 a r e h i g h l y c o l l i n e a r . Although is H  very  o(£i  $1 i s n o t z e r o ,  to establish s t a t i s t i c a l l y .  i s a huge s t a n d a r d therefore,  c o r r e l a t i o n matrix  regression highly  the true  where  error of 0i . any v a l u e s  are close  analysis should  o f these regressors i n  t o , s a y , Jo. 8J , t h e  be c a r r i e d o u t w i t h  c o r r e l a t e d v a r i a b l e s omitted.  extremely d i f f i c u l t  one o f t h e  I t i s , however,  to decide which regressors  t o omit and  which t o r e t a i n because those regressors  included  equation  of logic  h a v e b e e n s e l e c t e d on t h e b a s i s  or b i o l o g i c a l examined. by  stepwise  (and  to the production  i nthe physical  process  i t i s possible  r e g r e s s i o n , whether o r n o t each o f the  a significant The  relevant  Under t h e s e c i r c u m s t a n c e s  f o r that  this  Usually  0) w i l l n o t be r e j e c t e d u n d e r c o n d i t i o n s  If, the  3 shows t h a t  difficult  =  there  Figure  matter other  independent v a r i a b l e s )  being to test, regressors  i s making  contribution to explaining variation i n yield.  forward stepwise  regression  quadratic  equation  actually  61  selected  the f o l l o w i n g v a r i a b l e s  at the 5 per cent l e v e l o f  significance: Y = 5733.2578 + 2739.7249 A + 398.6005 S - 1 0 9 6 . 3 6 6 5 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 , e v e n  stepwise regression regression linearly  would not help  resolve  i t .  i s designed t o s e l e c t independent v a r i a b l e s  combined i n the f i r s t  place,  and n e x t l e s s  c o m b i n e d a n d so on i n t h e o r d e r o f i n d e p e n d e n t laid  out i n r e g r e s s i o n  regression  Stepwise  equation.  does n o t n e c e s s a r i l y  I t follows coincide  r e g r e s s i o n . i f independent v a r i a b l e s fore, selected  to the r e g r e s s i o n  routine  forward  backward  are c o l l i n e a r .  independent v a r i a b l e s  may  linearly  variables  that  with  least  differ  There-  according  i n s t u r c t i o n , i . e . , forwards or  backwards. M o r e o v e r , i f some i n d e p e n d e n t v a r i a b l e s a r e linearly  d e p e n d e n t on t h e o t h e r o n e s , t h e v a l u e o f R  comes d u b i o u s .  Coordinate points  independent v a r i a b l e s in nearly and  linearly-dependent  l i n e a r fashion  i n d i m e n s i o n a l space of a meaningful  involved,  regression  by l e a s t s q u a r e s method i s r e n d e r e d t h a t  difficult.  be-  are not spread out but c l u s t e r e d  thus the determination  surface  of  2  much more  62  There i s a p o i n t that s h o u l d be made about the d i s t r i b u t i o n of the a p p l e - y i e l d dependent v a r i a b l e w i t h r e s p e c t to f i x e d values of an independent r e g r e s s i o n equation. is  assumed normally  = a  2  variable  From the f a c t t h a t the e r r o r term e d i s t r i b u t e d w i t h mean = 0 and v a r i a n c e  , i t f o l l o w s t h a t the random v a r i a b l e , apple  f o r s p e c i f i e d values of the independent assumed normally  yield,  v a r i a b l e s , i s also  d i s t r i b u t e d w i t h mean = 3 i + B2X.2 + B X 3  + 3 X„ + 65X5- + 8 X H  i n the  6  6  + B X 7  7  and v a r i a n c e = a  3  , i f a regres-  2  s i o n model i n regard to apple p r o d u c t i o n i s c o n s t r u c t e d i n Y = 32  the f o l l o w i n g manner: +  3eX6  +  37X7 + e .  One  + 3 X 2  i s not met,  been employed are no  + 3 X 3  longer adequate.  The  other way  observations  There are two One  alterna-  i s perhaps to  to cope with  statistics  d e t a i l s of results  i n the Appendix shows both  this  i n which  i s e n t i r e l y beyond the scope of the  case  study.  from stepwise r e g r e s s i o n  are t a b u l a t e d and shown i n Table XIV XVI  3sX5  p a r a m e t r i c methods which have  problem i s to apply non-parametric  The  +  data u s i n g l o g a r i t h m i c or square-root  transformations, etc.  the technique  + 3wX 4  assumption. . I f , however,  t i v e ways to deal with t h i s s i t u a t i o n . t r a n s f o r m the raw  3  hundred and n i n e t e e n  are l a r g e enough to v a l i d a t e t h i s the assumption  2  i n the Appendix.  observed  values and  Table  the  c o r r e s p o n d i n g values of apple-crop y i e l d on the b a s i s of the above equation i n v o l v i n g the s e l e c t e d independent 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 .  variables  63  Discussion Quadratic  o f the Results Regression  g r o u n d s , t h e r e s u l t s o f t h e Cobb-  f u n c t i o n show e x p e c t e d s i g n s  efficients Only t h a t  o f the seven r e g r e s s i o n  age  f o r t h e r e g r e s s i o n co-  coefficients  function,only  of t r e e ; cost  were s i g n i f i c a n t  of probability.  the three  of spray;  basic  and p r u n i n g  at the 5 p e r cent  signifi-  Using the  independent v a r i a b l e s , and t h i n n i n g  hours  level.  W h i l e t h e Cobb-Douglas f u n c t i o n not  estimated.  o f t h e t r e e - s i z e v a r i a b l e s was n o t f o u n d  cant at the 5 p e r cent l e v e l quadratic  C o b b - D o u g l a s and  Analyses  On a p r i o r i Douglas  from A p p l y i n g  i n this  case  does  produce a m u l t i c o l l i n e a r i t y problem, the q u a d r a t i c  f u n c t i o n h a s 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 t e d by i n s p e c t i o n o f t h e c o r r e l a t i o n m a t r i x . collinearity very  yield  I f the multi-  c o n d i t i o n e x i s t s b e t w e e n two v a r i a b l e s , i t i s  difficult  nificance  as i n d i c a -  to e s t a b l i s h the level  of coefficients.  Therefore  of s t a t i s t i c a l  sig-  the influence  on c r o p  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  other. I t w o u l d seem r e a s o n a b l e t o s a y t h a t provided two  insufficient  models.  evidence  Preference  work.  grounds t h a t  However, t h i s  function  study  f o r c h o o s i n g between the  f o rthe quadratic  C o b b - D o u g l a s may b e s t a t e d on t h e p u r e l y theoretical  this  f u n c t i o n over the deductive  or  i n t e r a c t i o n o f f a c t o r s c a n be a t  kind of s e l e c t i o n of the quadratic  i s made on t h e same g r o u n d s as e x p l a i n e d  philosophical insights:  by Hume's  64  "When we give the preference to one set of arguments above another, we do nothing but decide from our f e e l i n g concerning the s u p e r i o r i t y  of t h e i r  influence."  CHAPTER VI  TESTING FOR  THE  DIFFERENCE BETWEEN TWO  MEANS  Introduction In c o n d u c t i n g the  t e s t s i t was  necessary to  use  o v e r a l l u n w e i g h t e d 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 of apple p l o t s . enterprises of record total  For  instance,  since  the  sizes of  w e r e i n i t i a l l y d e t e r m i n e d more by  k e e p i n g t h a n by  acres  the  of e n t e r p r i s e s  apple  convenience  r e l a t i o n s h i p they have to  on  the  farms  (not  readily defin-  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 for  e a c h v a r i a b l e on  could for  then allow  an  each p l o t  o v e r a l l p e r - a c r e mean t o be any  permissible,  is  narrow context  derived.  to the  Hence, the  t e s t s may  e n c e s among means, b u t relate  to  to r e a l i z e 1969  butions cost  that  and  o r no  remembered t h a t  with  and  a w e i g h t o f one.  a difference  is with  t h e s e s a m p l e s may regard  of spray,  index,  i n w h i c h they were  the  differmeans  i n d i v i d u a l farmer performance i n apple y i e l d s  w h e r e e a c h f a r m e r has  for  While  their interpretation  show d i f f e r e n c e s  i t must be  These  calculated  sample group.  t h e s e types o f average are limited strictly  average  ( t h e u n w e i g h t e d mean).  a p a r t i c u l a r v a r i a b l e across  the  to  pruning  geographical  age, and  regard  show q u i t e  density, thinning  location.  or l a c k o f d i f f e r e n c e i n the  Also  cost labour  i t is  important  to the  samples  different distriof  fertilizer,  hours, t r e e - s i z e  Therefore,  the  difference  t y p e o f means u s e d can  easily  66  be  s e e n as a r e s u l t  In  conclusion,  careful  of influences  i t c a n be s a i d  interpretation  extending our i n s i g h t s already  studied  two  that  into  the r e s u l t s  picture  the n u l l hypothesis  regard.  of significance  that  lation  variances It  that  a r e t h e same.  The o r d i n a r y m e t h o d o f between  assumes t h a t t h e two p o p u -  1  h a s b e e n assumed a b o u t t h e a p p l e c r o p y i e l d Y,  a s a m p l e mean Y  1 }  i s normally  p o p u l a t i o n mean, u , as f o l l o w s : represents  are " s i g n i f i -  f o r the difference  means o f two i n d e p e n d e n t s a m p l e s  distributed  around the  Y j ^ N ( u j , o / n j ) , where a 2  2  t h e v a r i a n c e o f t h e p o p u l a t i o n , and n t h e s i z e  of  t h e sample drawn.  of  t h e two s a m p l i n g p r o c e d u r e s w i l l  random v a r i a b l e s  Y  Similarly, Y ^N(u 2  and Y  1  2  ,o /n ).  Independence  2  2  2  e n s u r e t h a t t h e two (Y!-Y )^N(ui-  are independent:  2  ,a /n +a /n ). 2  2  1  2  When p o p u l a t i o n v a r i a n c e a  i s u n k n o w n , i t must  2  estimated:  Sf, = (  )  P n  i  + n  2  - 2  ( x 1  -  1  where S , = p o o l e d v a r i a n c e . 2  1  as s l i g h t l y  chapters.  w h e t h e r two s a m p l e s  different" i n this  making a t e s t  be  very  s a m p l e s come f r o m p o p u l a t i o n s w i t h t h e same mean:  cantly  2  bear  the y i e l d r e l a t i o n s h i p  i s used to t e s t  consequently, this tests  y  variables.  a n d a r e t o be s e e n o n l y  i n the previous  A t-test  o f t h e above  G . W. S n e d e c o r and W. pp. 1 1 4 - 1 1 5 , 1969.  M  - XO  2  + E?  1  1  1  The f o r m u l a  -  2  (X  1  - X ) )], 2  2  2  1  f o r the t - t e s t i s :  G. C o c h r a n , S t a t i s t i c a l  Method, 6 t h Ed.,  67  t = Y i - Y / \l S ( l / n i  calculated  2  T h e r e may t h a t o\  = o\  exist  2  n a m e l y , ol^  =  ni  I f so, the formula f o r the  c /n  +  o\/ri\  2  .  2  = n .  holds,  2  The  ordinary  Yy/\j S1 /n 1  t' = Yi -  sample  size  are not of equal s i z e , only  approximate degrees o f  the o t h e r hand,  be c a l c u l a t e d by t h e f o l l o w i n g + Sl/n ) / [(S /n ) /n -l 2  n i  2  2  It  1  1  1  p e r f o r m e d on t h i s  i f the  freedom  + ( S / n ) / n - 2) ] .  3  2  2  2  basis.  R a t h e r , sub-samples  e n t e r p r i s e s were o b t a i n e d  i n t h e t h e s i s , s a m p l i n g was  of tree-size  from the sample  drawn f r o m a s i n g l e p o p u l a t i o n . sizes  t-test  group  and o t h e r  popugroup  of apple producers  In the analyses which  o f groups are u n e q u a l but t h i s  p e r m i t t e d by t h e c o m p u t e r The  are  throughout are  not conducted from the three s e p a r a t e t r e e - s i z e  p.  samples  formula:  and t - t e s t s  As m e n t i o n e d e a r l i e r  low, sample  2  s h o u l d be n o t e d t h a t t h e y i e l d m e a s u r e m e n t s  o b t a i n e d on a p e r - a c r e b a s i s ,  lations.  2  o f each group i s e q u a l ,  On  (S!/  + S /n .  2  t and t ' become i d e n t i c a l .  will  t value  does n o t 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  I f , however,  2  2  still  vari-  When a i s unknown, t h e  is substituted.  2  r e p l a c e d by t h e s t a t i s t i c :  This quantity  i n which the assumption  i n independent samples  unbiased estimator S is  2  situations  i s suspect.  ance o f (Y"i - Y )  + l/n ) .  p  fol-  feature i s  program.  routine  o f t h e TRIP p r o g r a m has  three  2  Ibid.  3  115.  R. E. W a l p o l 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 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. 2 3 0 - 2 3 1 , 1968.  Co.,  68 different Formula  formulae at i t s d i s p o s a l :  (1):  t h e o n l y a s s u m p t i o n made a b o u t t h e p a r e n t lations  i n the d e r i v a t i o n o f t h i s  popu-  formula i s  normality. Formula  (2):  this  i s a special  differences  i n the data p a i r e d  concern i n t h i s Formula  (3):  this  f o r m u l a u s e d when t h e r e a r e  study).  i s a more s e n s i t i v e  Formula  (3) i s v a l i d  variances  s c o r e s ( o f no  v e r s i o n o f Formula ( 1 ) .  o n l y when t h e p o p u l a t i o n  are equal.  In f a c t , users can r e q u e s t the t - t e s t Formula  (1) i f i t f i n d s t h e s a m p l e  different,  and t o u s e F o r m u l a  variances  (3) when t h a t  t o use  significantly i s not the case.  Outcome o f T - t e s t Before showing it  i s desirable  yields  of differences  categories  studied.  The  differences, among  Group  details  and f o l l o w -  o f the relevant  Average Apple Y i e l d by T r e e - S i z e Group  Sample  Standard  23  Semi - S t a n d a r d  62  Semi - D w a r f  |/S,362.«i  10,000 20,000 30,000 Y i e l d (pounds) p e r a c r e  average  respective  are p r e s e n t e d t o serve t h i s purpose  i n g each has a t a b l e showing Tree-Size  f o r average y i e l d  t o have a p i c t u r e  f o r the s p e c i f i c  f i g u r e s below  a test  test.  Size  34 40,000  Figure 4 Differences  i n average apple y i e l d s  among t r e e - s i z e  groups  69  The  results  Table  from t - t e s t s  c o n c e r n i n g F i g u r e 4 a r e shown i n  XVII.  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. Tree-Size  Calculated D.F. T-value  Group  T-Prob.  F-Prob.  Formula Used  Standard vs. Semi - S t a n d a r d  2.132  83  0 .034  0 .163  (3)  Standard vs. Semi - D w a r f  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 usually  t h a n 0.05, i t i s  c o n c l u d e d t h a t t h e s a m p l e means a r e  different. usually  i s less  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  c o n c l u d e d t h a t sample v a r i a n c e s are  different  and t h e r e f o r e f o r m u l a  significantly  (3) i s i n a p p r o p r i a t e f o r  calculating  t.  apple y i e l d  d i f f e r e n c e b e t w e e n S t a n d a r d and  tree-size  significantly  According to these c r i t e r i a ,  groups i s s i g n i f i c a n t l y  the  Semi-Standard  different.  The same i s  t r u e b e t w e e n S t a n d a r d and S e m i - D w a r f t r e e - s i z e t h i s was n o t t h e c a s e w i t h S e m i - S t a n d a r d tree-size  groups where  o f sample v a r i a n c e s were  differences  and t h u s  groups.  But  Semi-Dwarf  t h e d i f f e r e n c e b e t w e e n means  f o u n d n o t t o be s i g n i f i c a n t . pairs  and  average  was  The t h r e e c o r r e s p o n d i n g f o u n d n o t t o show  f o r m u l a (3) was  used  significant  throughout  the t -  70  test Region  Average  A p p l e Y i e l d by  (pounds)  Difference regions  40, 000  5  i n average apple y i e l d s between (across a l l t r e e - s i z e groups)  outcome o f t h e t - t e s t  Table  65  2o,ooo per acre  Figure  Size  54  20, Stf-o. o  South Yield  The  Sample  23. 68V. </  Okanagan N o r t h  Okanagan  Region  concerning Figure  5 a r e shown i n  XVIII.  TABLE X V I I I Results  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 Calculated T-value D.F.  Regions Okanagan N o r t h v s . Okanagan  There the  i s no  South  0 . 581  significant  73  difference  T-Prob.  F-Prob.  0 .570  0.0  v a r i a n c e s are s i g n i f i c a n t l y  CD  i n average y i e l d between  O k a n a g a n N o r t h and Okanagan S o u t h  sample  Formula Used  regions.  different.  But  their  71 Grade Extra  A v e r a g e A p p l e Y i e l d b y Grade Fancy  S amp1e  Size  10, 000.0  Fancy 2,  Cee Cull  iio.il 2, 87/.OS 5,006  Yield  iS.OQO  10,000  (pounds) p e r acre  Figure 6 D i f f e r e n c e s i n a v e r a g e a p p l e y i e l d s among apple grades (across 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 a r e shown i n  Table XIX. TABLE X I X Results  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 Calculated T-value  Grade  D.F,  T-Prob  F-Prob  Formula Used  E x t r a Fancy v s . Fancy  3. 353  159  0. 001  0. 0  CD  Extra Cee  6 .472  134  0.0  0.0  CD  E x t r a Fancy v s . Cull  5. 367  184  0. 0  0.0  (i)  F a n c y v s . Cee  5. 448  191  0.0  0.1  (i)  Fancy v s . C u l l  3. 354  210  0 . 001  0 .006  CD  170  0. 444  0.0  (i)  Fancy v s .  •0.777  Cee  vs. Cull  The  d i f f e r e n c e s between  are  significant with  sample  variances  average y i e l d s f o r p a i r s o f grades  the exception  o f Cee v e r s u s C u l l .  are also s i g n i f i c a n t l y  The  different for a l l  72  pairs  o f grade  Variety  A v e r a g e A p p l e Y i e l d by V a r i e t y  Golden Red  couplings.  Delicious  2l, 3 7 7 . 7  36  21, 7 3 B. 0  42  Delicious  Spartan  31  /£>, 6/0.3  10,000  20,000 Yield Figure  30,000  40,000  (pounds) p e r acre 7  D i f f e r e n c e s i n average apple y i e l d s v a r i e t i e s (across a l l t r e e - s i z e  As  among a p p l e groups)  has been m e n t i o n e d i n t h e i n t r o d u c t o r y  b r e a k d o w n 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  a r b i t r a r y manipulation of data. carried  volve d i f f e r e n t kinds The contributed  rigorous  this  enterprise  a t t e m p t t o g r o u p d a t a by v a r i e t y has i n some i n s t a n c e  partitioning o r i g i n a l enterprises.  procedure the underlying  assumptions o f ensuring seriously,  some c a u t i o n i n a c c e p t a n c e o f t h e r e s u l t s The r e s u l t s  g i v e n i n T a b l e XX.  the tree-size  c o u l d sometimes i n -  y s i s w e r e n o t t h o u g h t t o be i n f r i n g e d  sary.  i n an  o f apple v a r i e t i e s .  t o e n l a r g e d sample s i z e  because i t i n v o l v e d  chapter,  When s u b - s a m p l i n g ' w a s  o u t , i t was done i n a c c o r d a n c e w i t h  g r o u p s , and hence a s i n g l e  Size  40  26,?2*. 7  Mcintosh  the  Sample  of the t - t e s t s  In anal-  although  i s thought  on v a r i e t y  simply  yield  neces-  data are  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 differences r e l a t i n g to v a r i e t y  Variety  Calculated 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 vs. Spartan  -0.747  56  0 . 465  0 .0  (1)  Golden D e l i c i o u s vs. 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 vs. Spartan  -0.719  52  0 .482  0 .0  (1)  Red D e l i c i o u s vs. Mcintosh  1. 336  71  0 .182  0 .129  (3)  Spartan vs. Mcintosh  1.451  49  0.149  0.0  (1)  Varietal  i n mean v a l u e s o f c r o p y i e l d  differences  significant. Delicious as was all  F u r t h e r m o r e , the sample  and Red D e l i c i o u s  the case a l s o  other pairs  variances  are not  f o r Golden  are not 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 Red D e l i c i o u s  and M c i n t o s h .  o f v a r i e t i e s , the v a r i a n c e s were  For  significantly  different.  D i s c u s s i o n o f the t - t e s t The between  significant differences  i n average y i e l d s  t h e S t a n d a r d and S e m i - S t a n d a r d g r o u p s  and  between  surprising  because  the  S t a n d a r d and S e m i - D w a r f g r o u p s was  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  statistically  74  significant  as was d r o p p e d f r o m t h e r e g r e s s i o n  The tation  reason f o r t h i s  may be f o u n d i n t h e i n t e r p r e -  of the characteristics  regression  analysis.  regression  on dummy v a r i a b l e s  different  tree-size  of the t-test  The t - t e s t  groups.  equations.  i nrelation to  may be s e e n as a s i m p l e  representing the three The f o l l o w i n g  situation  c a n be  depicted: Dummy 0  0  0  Semi - S t a n d a r d g r o u p  1  0  0  Semi-Dwarf  0  1  0  Standard  group  group  A dummy v a r i a b l e needed because differentials situation  variables  f o r t h e S t a n d a r d group  S e m i - S t a n d a r d and Semi-Dwarf measured  Yi  0  1  0  Y2  0  0  1  i nmatrix  0  1  0  0  0  1  Dummy v a r i a b l e s data.  as h a s b e e n of  reflect This  notation:  3i  B  ing  groups  from t h e S t a n d a r d group b a s e .  c a n e a s i l y be v i s u a l i z e d  Y 1 1 9  i s not  3  .  a r e n o t t h e o n l y means o f a d j u s t -  A n o t h e r method i s t o d e v i s e done e a r l i e r i n t h i s  crop y i e l d t o the t r e e - s i z e  study.  variable  a scale  for  The d e s i r e d  tree-size relation  c a n now be e s t i m a t e d  75  by  a simple  r e g r e s s i o n of apple  yield  on  t r e e - s i z e index.  This  l a t t e r method i s advantageous b e c a u s e i t i s n o t  neces-  sary  t o assume d i s c r e t e s h i f t s .  any  observation  v a r i e s f r o m one  same i d e a was Slopes  t-test the  t r e e - s i z e groups.  production  are  observation  and  The tion  i n the  i m p l i e d w i t h i n the  o f the  context  type  of the  u s e d has study.  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 t i s p r e c i s e l y our  ences c o n s t a n t  when t e s t i n g  Nevertheless vide  remember  framework  the of vari-  the  t e s t s are  limited applica-  I t i s a meaningful c o n d i t i o n of s i n g l e  inability  obtained  held  to hold other  results  from the  t-test  from m u l t i p l e r e g r e s s i o n .  of e i t h e r a f f o r d i n g a  a t t a c k or even a d d i n g to the  d i f f e r e n c e s among g r a d e s .  influ-  differences for pairs  of i n t e r e s t because they  f u r t h e r i n s i g h t s by way  different e.g.,  than those the  of  by  i n f l u e n c e s are  the y i e l d  of c a t e g o r i e s , which renders less powerful  to  seven independent  method d e r i v i n g i n f o r m a t i o n under the  constant.  Equality  regressors.  t-test  data  This  r e l a t i o n s h i p s under examination  theoretically  twenty  the  It is crucial  m u l t i p l e r e g r e s s i o n model u s i n g  ables  for  to another.  u t i l i z e d when h y p o t h e s i z i n g  among t h e  t h a t the  Thus a d j u s t m e n t  do  pro-  slightly  overall analysis,  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 sion  objective of this  s t u d y was t o e s t i m a t e  r e l a t i o n s h i p s between apple y i e l d s  encing  factors  and c e r t a i n  f o r t h e Okanagan a r e a o f B r i t i s h  regresinflu-  Columbia i n  1969 . Two t y p e s yield  relationships.  Quadratic  follows:  explanatory  v a r i a b l e s u s e d i n r e g r e s s i o n were  (1) d e n s i t y p e r a c r e ,  cost o f f e r t i l i z e r applied per acre, acre,  T h e s e w e r e t h e C o b b - D o u g l a s and  forms. The  as  o f e q u a t i o n were employed t o r e p r e s e n t  (2) age o f t r e e ,  applied per acre, (5) p r u n i n g  (3) t h e  (4) t h e c o s t o f s p r a y  and t h i n n i n g l a b o u r hours p e r  (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  index. When a C o b b - D o u g l a s f u n c t i o n was f i t t e d sample o b s e r v a t i o n s v a r i a b l e s were found  (across t r e e - s i z e significant  groups),  at 5 p e r cent  p r o b a b i l i t y with the exception o f t r e e - s i z e other hand, a Quadratic  function involving  to a l l  t h e independent level of  index.  On t h e  twenty-eight  77  independent terms, nificant  at the  included only eleven  5 per  cent  level  s e l e c t e d v a r i a b l e s w e r e as cost of spray hours per (5)  per  acre,  squared  acre,  (4)  pruning  squared  pruning  and  of p r o b a b i l i t y .  follows:  (3)  t e r m s as b e i n g  (1) and  fertilizer  age  spray  (8)  pruning  and  tree-size  spray and  acre,  index,  costs per  (9)  (10)  and  the  the  resulted  (11)  to the  on  cost per  acre  and  acre,  and t h i n n i n g  tree-size  of tree  But  index.  along w i t h i t s squared  cause a m u l t i c o l l i n e a r i t y  I f two  term  has  equal i n f l u e n c e .  repre funct  use  of  may  independent  problem i t i s  i n f l u e n c e o f one  The  vari-  extremely  of the v a r i a b l e s  t h e d e p e n d e n t v a r i a b l e b e c a u s e i t m i g h t w e l l be  other variable  made i n  squaring  u s e d as r e g r e s s o r s .  some c o l l i n e a r i t y .  t o deduce the  function  the Q u a d r a t i c from  and  pruning  Cobb-Douglas f u n c t i o n to  i n a complicated problem a r i s i n g  have g e n e r a t e d  difficult  acre,  c r o s s - t e r m between  apple-yield relationship.  independent v a r i a b l e  ables  (6)  o f p r o d u c t i o n , a c h o i c e was  i n d e p e n d e n t v a r i a b l e s t o be an  term,  view of the p r o p e r t i e s of a Q u a d r a t i c  favour of i t r e l a t i v e sent  acre  c r o s s - t e r m between f e r t i l i z e r  a c r e , and  economic theory  (2)  term,  c r o s s - t e r m b e t w e e n age  t h i n n i n g labour hours per In  the  acre  cost per  (7) c r o s s - t e r m b e t w e e n d e n s i t y and  labour hours per  of t r e e ,  cost per  fertilizer  c r o s s - t e r m b e t w e e n d e n s i t y and  Here  thinning labour  t h i n n i n g hours per  c r o s s - t e r m b e t w e e n d e n s i t y and  sig-  that  the  78  Conclus i o n L o o k i n g back  to the regression  a s s u m p t i o n s w e r e made s o t h a t regression  equations could Suppose t h a t  inferences  model,  from e s t i m a t e d  be made.  the following  functional  exists  i n regard t o apple y i e l d Y = B i + B X  +  + B&XG  2  B5X5  about  e:  +  + e .  B7X7  namely,  A strong  e^N(0,a ). 2  that  a l l independent v a r i a b l e s  only  random v a r i a b l e  the  fact that  several  relationship  +  2  B3X3  + BnX  H  a s s u m p t i o n must be made  I n t h i s model i t i s i m p l i e d are t r e a t e d  as f i x e d .  i n t h e model i s Y w h i c h  The  i s deduced  from  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 a s s u m p t i o n made i n t h e m o d e l i s that  a l l independent v a r i a b l e s  a n o t h e r , and o f t h e e r r o r  a r e i n d e p e n d e n t o f one  t e r m e.  dom v a r i a b l e " a n d " i n d e p e n d e n c e "  A l t h o u g h t h e terms  "ran-  t e s t o u r p o w e r s 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 features various  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 c o n s e q u e n c e rules  employed  Treatment  to  In f a c t , e r r o r  variables  the only  error  involved  a l l o w e d was an input  The e r r o r may be due  f o rinstance,  i n t h e c o l l e c t i o n and r e c o r d i n g The  as t a k i n g  i n t h e measurement o f t h e i n c l u d e d  i s extremely l i k e l y .  "human e l e m e n t "  occur  that  i n t h e e q u a t i o n due t o t h e o m i s s i o n o f some  factors. input  statisticians.  o f a l l independent v a r i a b l e s  fixed values implies error  by  of  m i s t a k e s may  of data.  observed values o f the v a r i a b l e s  arenot  79  strictly the  comparable because of l a c k of homogeneity which i s  case w i t h  thinning  fertilizer  labour  hours.  n e c e s s a r i l y present  i n the  u s e d , but  c o s t , and  Therefore observation  method o f a d j u s t i n g the m i g h t be  c o s t , spray  data.  data  pruning errors  and are  I t i s possible that  to take  account of  some  heterogeneity  e v e n so  i t is difficult  to c o n t r i v e  most c e r t a i n l y w o u l d l e a v e  s o m e t h i n g t o be  desired.  case of the not  As  long  as  the  independent v a r i a b l e s are  Xi•& 2  cannot y i e l d  a satisfactory result.  =  O  f o r example, stepwise  statistical  i t i s t o be  sumptions r e q u i r e d  are  c o n t r o l l e d e x p e r i m e n t w o u l d be retical  framework, p r e f e r e n c e  form of y i e l d  apple y i e l d  be  r e l a t i o n s h i p over that  Douglas f u n c t i o n .  However, the  serious  statistical  problem  already  been d i s c u s s e d  under s i m i l a r circumstances  that  In  this  given  as-  equation,  Within  the  t o the  a  theo-  Quadratic  s u m m a r i z e d by  f o r m e r f u n c t i o n has  a Cobbposed  ( m u l t i c o l l i n e a r i t y ) , which  a t some  Needless to say,  fore-  t o meet a l l t h e  necessary. may  t o say  inconclusive.  suggested that  f o r d e r i v i n g an  regression  issues' i n the  analysis, i t i s appropriate  empirical r e s u l t s obtained  not  1  In view of the m e t h o d o l o g i c a l  connection  the  proportion.  orthogonal,  the  In  s t u d y i t i s assumed t h a t m e a s u r e m e n t e r r o r i s  of s e r i o u s  going  and  a  has  length.  i t i s hoped t h a t to the  one  Mv'onnacott, e t a l . , E c o n o m e t r i c s , pp.  any  w h i c h has 309-  312.  future been  study con-  80  ducted 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 t h e Cobb-Douglas  f u n c t i o n over s e l e c t e d r e l e v a n t ranges avoid the m u l t i c o l l i n e a r i t y  problem  o f data and, thereby,  as met i n t h e Q u a d r a t i c  analysis. Since the r e g r e s s i o n theory most d i f f i c u l t  In  real  the case  tions. in  of the t e s t s  although  T h e s e as w e l l  a preceding  t o remind  o b s t a c l e s to the type  brief  the reader of  of analysis  of significance  among means, t h e c o n c e p t u a l understand,  to the  conceptual part of the t h e s i s , t h i s  summary and c o n c l u s i o n s h a s h e l p e d the very  gives r i s e  undertaken.  for differences  f r a m e w o r k i s somewhat e a s i e r t o  the a n a l y s i s  rests  as t h e r e s u l t s  on d e f i n i t e  of that analysis  c h a p t e r , and t h e r e f o r e no a t t e m p t  r e p e a t t h e summary a l r e a d y g i v e n .  assumpoccur  i s made t o  81  BIBLIOGRAPHY  82 BIBLIOGRAPHY  B j e r r i n g , J . H. a n d S e a g r a v e s , P. UBC T R I P ( 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., C o m p u t i n g C e n t r e , November 1 9 7 0 . B r a s e , K. D. a n d Way, R. 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" I m p l i c a t i o n s o f E c o n o m i c s on O r c h a r d Management," The 1969 A p p l e Forum, P u b l i s h e d Proceedings 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 Conference. November 1 9 6 9 . Van  R o e c h o u d t , L. L. Some F a c t o r s w h i c h I n f l u e n c e t h e Use o f D w a r f and S e m i - d w a r f A p p l e T r e e s f o r Commerc i a l O r c h a r d s i n t h e 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 , L o n d o n , 1 9 6 8 . Ware, D. W., Woodward, E. D. a n d T r e v o r , H. W. 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V I I , M.IV  Semi - s t a n d a r d  M.II,  M.M.Ill,  Standard  Seedling,  Source:  1/5 t o 1/4  M.M.104  Anchorage  Poor  1/3  Poor t o F a i r  t o 3/4  F a i r t o Good  2/3  M.XVI, M.XXV, M.M.109  Tree  Full  size  Good  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 e s e a r c h S t a t i o n , S u m m e r l a n d , B.C., R e s e a r c h B r a n c h , Canada D e p a r t m e n t  of Agriculture.  March, 1966.  87  TABLE I I VARIABLES USED IN MODELS  Variable  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 8  GEOGRAPHICAL DUMMY TREE SIZE INDEX  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28  SQUARE OF VARIABLE 2 SQUARE OF VARIABLE 3 SQUARE OF VARIABLE 4 SQUARE OF VARIABLE 5 SQUARE OF VARIABLE 6 CROSS TERM BETWEEN VARIABLE 2 AND VARIABLE 3 CROSS TERM BETWEEN VARIABLE 2 AND VARIABLE 4 CROSS TERM BETWEEN VARIABLE 2 AND VARIABLE 5 CROSS TERM BETWEEN VARIABLE 2 AND VARIABLE 6 CROSS TERM BETWEEN VARIABLE 2 AND VARIABLE 8 CROSS TERM BETWEEN VARIABLE 3 AND VARIABLE 4 , CROSS TERM BETWEEN VARIABLE 3 AND VARIABLE 5 CROSS TERM BETWEEN VARIABLE 3 AND VARIABLE 6 CROSS TERM BETWEEN VARIABLE 3 AND VARIABLE 8 CROSS TERM BETWEEN VARIABLE 4 AND VARIABLE 5 CROSS TERM BETWEEN VARIABLE 4 AND VARIABLE 6 CROSS TERM BETWEEN VARIABLE 4 AND VARIABLE 8 CROSS TERM BETWEEN VARIABLE 5 AND VARIABLE 6 CROSS TERM BETWEEN VARIABLE 5 AND VARIABLE 8 CROSS TERM BETWEEN VARIABLE 6 AND VARIABLE 8  88 TABLE  III  ESTIMATED SIMPLE LINEAR REGRESSION EQUATION E q u a t i o n f o r the Standard-Tree Group 1.  VAR 1 =  2.  VAR 1 =  3.  VAR 1 =  4.  VAR 1 =  5  VAR :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)  1  S i g n i f i c a n c e of b Estimate at .05 L e v e l NON. SIG.  0.0017  NON. SIG.  0.0296  NON. SIG.  0.0078  NON. SIG.  0.0054  NON. SIG.  0.0914  R  Equation f o r the Semi-Standard-Tree Group 1.  VAR 1 =  2.  VAR 1 =  3.  VAR 1 =  4.  VAR 1 =  5.  VAR 1 =  15240 (3957) - 3239 (4434) 14810 (2289) 14010 (3651) 106.1 (168.9)  -  15. 21 VAR 2 (21. 2 3) + 2114 VAR .3 (413.5) + 149 . 1 VAR 4 (61 .55) + 67. 60 VAR 5 (51. 18) + 69. 40 VAR 8 (219. 4)  2  NON. SIG.  0..0085  SIG.  0.,3034  SIG.  0..0891  SIG.  0. 0236  NON. SIG.  0. 0017  r  Equation f o r the Semi -Dwarf-Tree Group 1.  VAR 1 =  2.  VAR 1 =  3.  VAR 1 =  4.  VAR 1 =  5.  VAR 1 =  6.  VAR 1 =  7688 (5664) 12520 (7226) 13110 (2709) 12330 (2993) 13620 (3056) 15260 (9345)  +  41. 7 3 VAR (19. 6S) + 616. 1 VAR (702 .2) + 171. 1 VAR (46. 07) + ' 60.21 VAR (18. 07) + 59.62 VAR (22. 11) + 9189 VAR (26200)  2  SIG.  0. 1235  3  NON. SIG.  0..0235  4  SIG.  0..3012  5  SIG.  0..2575  6  SIG.  0..1851  8  NON. SIG.  0..0038  Equation T o t a l Data 1.  VAR 1 =  2.  VAR 1 =  3.  VAR 1 =  4.  VAR 1 =  5.  VAR 1 =  6.  VAR1  =  18800 + 7.220 VAR (2639) (12.67) 13800 + 479. 2 VAR (2431) (150 . 16920 + 145. 2 VAR (1650) (39. 26) 16410 + 51. 79 VAR (1921) (18. 03) 15530 + 68. 50 VAR (1852) (18. 13) 341.0 - 245. 5 VAR (27.74) (37. 81)  2  R  2  2  NON. SIG.  0,.0028  3  SIG.  0..0795  4  SIG.  0,.1047  5  SIG.  0..0659  6  SIG.  0,.1087  8  SIG.  0..2649  'Data i n b r a c k e t s r e f e r t o r e g r e s s i o n c o e f f i c i e n t standard e r r o r s .  89 TABLE I V EVALUATION OF SIZE-CONTROLLING EFFECTS OF ROOTSTOCK, INTERMEDIATE FRAMEWORK STOCK AND STRAIN OF SCION VARIETY ON TOTAL TREE S I Z E IN TERMS OF AN INDEX VALUE Index A. B. C.  Standard Standard Standard  Reduction  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 v i g o u r framework v a r i e t y vigour scion variety  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  We w o u l d r a t e s e m i - s t a n d a r d  stocks  We w o u l d r a t e s e m i - d w a r f s t o c k s  s u c h as  s u c h as  We w o u l d r a t e d w a r f r o o t s t o c k s s u c h as  Reduction  i n t r e e s i z e by framework s t o c k  Value  1. 0 1 .0 1 .0 to A  M.11 M.M.Ill A. 2  a t 0 . 75  M.M.104  a t 0 . 85  M.M.106 M. I V M.VII M. 26  at at at at  M. I X M.VIII  — at a t 0 . 20  in relation  0 . 50 0 .40 0 . 33 0 .25  to B  We w o u l d 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 s u c h as H a r a l s o n ( o n l y one on o u r 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 o f t h e u s e o f a s p 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 w o u l d r a t e r e d u c t i o n o f t r e e s i z e by u s e o f s p u r type s t r a i n s  a t 0.75  Application 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 u n d e r A, B, and C, a t r e e - s i z e i n d e x v a l u e c a n be e s t a b l i s h e d . F o r e x a m p l e - S p u r 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 f r a m e w o r k s t o c k on M.IV = 0.74 x 1.0 x 0.40 = 0.30 Mcintosh standard vigour intermediate f r a m e w o r k s t o c k on M . V I I I = 1.0 x 1.0 x 0.33 = 0. 33  11.K : VI A I N IPUT RATA (,ri:BTAnRA 'I1 'I1 ' SI; PLOTS) •'l -l11_ > ' PPr.,l l!\ TU 1  CY  TrocSiie ("roup: 8 O.:::JI: SS 4 ..0JO 4510.500.0 670..6:o0 2 .S.091: O O 46 80 4950 1 4 . r.mi 2 .mi to.li..51" 5 . .il :4 0 .3>-5>1rt':.| nOsS 444HS0..H30M 0.o.1 . s 0 . 0 JO 5 :o5? t> .:i u 5 *0.1 0(1.0 14ti859...00*>00: 0 ^ . 8 3 1 O S l i ) 4 3 1.4 9250 18.1M ..'tf-1OS 5574.33:o .1:::433o.0..o00.o0 :;i.o58.P....O0*SOI0".i 00.o.6: 0 1 : :tf:1 -oss0 i 7508..0'00 ::.270.oo0 2274..602.0 i-.41o o :s1sE n 002 3 ...2 S 9 OS S 1 10 08 80 0.. I11S25.000..000 33868...550660 S 32SH LO 0 1 . 7 1 O S 1 0 8 0 . 190.0 2235..4304 o.SJj; sS 1 10 08 80 0 4 ..40o10321ii:;: oO O Q S 7 0 0CC....0OO 11:::s9.o7o.0.0o.o00 110376,..81.9900 0 4 .948E 0s5 4 8 o.4ci' : i " : o 7 3 0 .113S7 74 35 90 .0 0 5 .,8 0.1 tE05OS 3 468.0 .0i"0 4 .0 19 .1 00 T r e e S i : e C r o u p : S e m S i t a n d a r d ft 4444. 118.0 6.00 6 .30 . 0 3 9 3 0 . 1 1 8 5 0 . 0 1 1 ..8 00 2992. 0 5..0 9 ..0 0 1 3 ,8 172031 ..214E 05 6 2 9 0 0 4 0 0 7 1 . 7 0 2 10880 .0 8 .0 0 1 .6 7 1 .6S4637E F. O S 1 . 1 2 0 .0 0 I1S8 .2 04 0 00 .03 2 O S 1 0 8 . 0 1 1 . 0 7 , 7O 9.0.1 . 01E OS 1 19 94 4 0....000 114631..000.0.0000 47145787....32139260 S2 SO 1 0 8 6 3 . 4 S 6 E 0 5 1 0 8 002 .3 6S 2S 1E E0 05 5 1 641..0 14 40 .0 0 1 8..5 87 3 .1 13 00 1 ..0 135 O 1 . 2 1 0 E O S 8 6 . 0 1 1 0 0 7 . , 2 0 8163 66 9 4 ..0 65.0 0 177 ,,4 .5 0 22 0..777E OS 1 1 3 2 0 1 0 .0 0 1 5 0 . 1 3 2 . 0 1 5 . 0 1 9 . 4 0 4OS 0088..0 .0 0 9 5 .,4 70 0 021 .0..S61E 05 1 19 00 16 7 .0 0 866 4 2 0 7.0 .0 0.0 0 .0 0 1 .,4 6 1 0 7 1 . 2 1 8 5 1 4 3 0 128054 .S97E 05 29 18 ..0 00 252.0 0 1 4,4 7S .1 0 0 .0 0 193 ,86 Q . 2 2 4 E O S 2 0 3 . 0 1 3 . 0 5 , . 005 ..S 3SO E0 O5S 2 18 8..0 40.0.00 31396,,4..460 2 B 83E 2 1 0 1 1 1 0 . 1 6 1 1 O S 1 0 8 . 0 1 2 0 57 ,..78200 002 .2 S 6750K 0 51 10 08 80 ..0 1 3 0 ...0 0 9 . 3 E O S 1 9 0 0 1 3 ..212 •002 .00434 2C EO OS 08 00 2 12 30 31 .2 S 129 52..0 0 ...0 0DO. 17 ..627080 3 4 6 8 : 7 . 0 5 5 93406.1.I33E 05 12 99 7..0 0 1 2 0 . 0 6 .01 0 0 13 00 .0 0 14 ..3 0S 7 .6 7 4E E0 O5S 6 20 1S 6..0 0 1 .0 427 5 470 .01 0 2 2 8 7 0 . 0 . 2 . 0 .5 2 E O5S 1 8.0 .00 0 18 2..1 64 0 1 SS 31 0E E0 10 04 8...0 0 9 0004 ..3 6 7 O S 1 0 8 0 I S . 0 0 9 1 . 5 8 6 150.00 .0 0 25 .0.93030 62 S342 3...763E 05 1 1 08 8...0 0 71 4 1 0 0 8 . 0 0 6 ..57 0 04 4 .1 4S 75 9E E0 5 1 30 5..0 1 2 0 .0 0 1 2 0 .02 0 0 5 1 5 0 1 2 . 0 1 3 . 9 0 ..7 E0 055 2 17 09 7..0 0 14 0 ..0 313...83260 001 1 09 1S 3E E 4 10 3 ..2 1 6 0 O S1 9 4 .0 0 1 16 40 0 .. 0 0 7 001 B 0 E 0 5 1 0 8 . 0 0 1 7 ..1 2 ..3 5 40E 0055 19 07 80 61.0 0 1S71 1 ,18 70 . 0 2 7 1 1 . 0 0 1 0 . 0 . 75 65 49 2.. 21188.00. 7.0 0 .0 0 0 6 ..8 0 9 2 6 0 6 8 00 0 002 .1 1 3 9 E O S 1 0 8 . 0 1 2 0 . 0 • 2 0 . 2 5 .0.I311962E E0 05 5 1 4 2 . 0 S 0 . 0 3 0 . 1 8 1 0 .0 8 0 0 9 .6 0 08 1 .01 1 0 08 8..0 0 .0 0 96 ..8 80 00U9 9 . 76E 05 1 1S 08 81...0 6 0 6 0 7 3 4 7 . 0 6 . 0 0 1 0 . 5 , 6 . 0 5 82 95 4.. 1S 0. 0 0 1504 .,55 60 2 4 701.00 6 7 .0 S 99 5S 1 5 0 9 4 ..,4 40 0 21 8.. 13344..0 0 50 ..00 94 Tree-Sic Group:SemD -i warf 01 .630E 05 .151,0 100 .0 18,43 001 .4 4 8 6 1 : O S 4 1 8 . 0 5 0 . 0 2 0,.0 6 ..172E OS 2 16 18,.0 0, 1820 0 .0 0 2 18 3 95 0 7 4 9 S 1 . , 4 . 0 59 88 83 3.. 2 1 8 8 .0 00 0 •1830 ,.6,04 . .0.. 4 2 1 8 7 . 8 3 4 S . 0 9 . 0 0 2 7 1 . 0 O . 5 0 9 O I i O S 004 ..3 4 6 F OS S3 81 8..0 920 5 0 .39 71 00 2P 1:.. O 11 7 0 2 0.00 9 15 6.0 1 01 2 .4 7 0055 2 8 ..0 184 0 ..0 0 ,..6 9S 0 . 0 7 7 » 3 3 9 0 . 0 5 7 7 199051 ..376F. 0 5 49S730 .,0 0 1 3 0 . 0 7 . 1 6 0 . S.00 .0 0 70 ,.3 0 0 2 . 6 S 9 E 0 5 4 5 3 , 0 . 8 6 8 . 2 0 BOO . 3E OS 29071..0 00 .0 0 104.1,40300C01 ..S2 00 1 17 3 .0 O 6 39 30E 05 6 0 5 . 0 0 . 0 97 28 S . 1 3 0 . 0 1 088.00. 130.0 1223334...22765 7 7 0 . 1 0 993042 .384E 05 1 0 10 0 .0 0 52 4..0 40 0 1 08 8...0 0 1 8 1 3..6 . 2 9 0 0 14 8.0 .0.00 22 ..0 80 70 ]34411872 3 8 8 . 0 3 . 1 4 2 . 0 7 0 . 0 7 . 2 60 838062 . 11; 05 2 2 .0 (J S .0 0 247.4 20 l'1J7 I.0 7.0 0 .0 . 330 0G 1.,,1126 1 1 ! 0 5 0 8 . 9 0 1 2 . 7 289011 051451.0 9.00 12.61 Tree - SI in HrM ow nn > jr: 0.1077J! 05 3 39 0. 80 .0 S7.75 0 . 2 2 2 7 ) : 0 5 1 3 4 . 1 6 .0 0 7.3 20Ofl!0 97O 60 .7 d 3. 0 50 0 .0 lO .30'Jh'.« :! O05i 3 3 3 8 0 . 9 .00 0.!Mi 0 . 2 I . 3 K 7 . 0 7 0 . ! > fll 3 « a . 7 0 . 0 J J ' O 0 . 2 ' J ' G I . 0. , ' 0 119 <il,.-«-rtf»i i,.ni I«] JI1 1< J .(!'«.<• - of I rri'Ji'iti :  . :.o  KTJ  P..  tl;  14 1422.3 9 .8 6 . 7J S4 1O 080 0 .,8I 5 2 . 75 52 .0 0 7 ..2 1 9 6 5 .0 14 16 44 .41 5 5 . 7 6 0 6 25 4.6 .8 8 4 .6 22 4 0 0 . 77 4.5.515 5 2 5 5 ..9 2 9 17 00 42 6 .4 0 . 18 .03  8.00.00 4 12 0 .8.00 580 4 3o.0n0 io. 2 1 0 , 2 1 0 ..0 0 1 7 0 10 06 80 0 .0 1 . 13 1S 60 ..0 0 1 6 6 . 1 6 6 0 ..0 15 06 00 0 1045.0.0 .00 33 44 S0 .0 .0 5 7 0 . 0 96.00  1 0 .0 0 0 1 .0 00 0 1 0 . 1 O .0HO 1 0 . 0 1 .000 10 0 .0 00 1 . 10 .0 00 0 1 ..0 0 1 0 0 0 1 0 .00 000 1 10 0 .0 0.0 0 1 . 1 0 . 0 0 10 0 .0 00 0 1 1 0 ...0 0 1 0 0 0 1 0 ..0 0 1 10 0 .0 00 0  1 7 ,1 4 1 88 9 3 4 8 ..6 1 3 6 . 6 3 4 2 . 2 600 .0 14050.7 .17 5 3 . 0 3 5 5 ..9 0 2 8 7 0 14 76 27 ..56 7 6 4 3 . 14202.0 2 .2 12 55 30 2 .0 ..8 1 48 1 7 1 . 8 4 0 . 7 9 4 1 6 6.2 4 ..5 6 4 5 34 39 1 .2 1410 65 ..7 4 5 4 0 ..0 5 8S 0 5 .5 O 778 8 .71 4 8 .5 4 4 9 . 9 4 3 17 00 ..60 19495.3 0 .8 2 2 .5 5 84 .6 8.93 3 1 1 2 1 . 0 53 19 .3 2 .4 8 2 61 .6 3 3 0 . 0 17 057O .. S 9 1 2 .. 140 035 54 2 .0 7 3 3 . 77 3 3 . 0 3 0 6 . 6 4 96 1 4 6 ..9O 0 4 6 S .0 4 0 . 0 7 3 9 .9 8 7 33 .6 8 1 6 . 19 9.8 .81 1 1  Z 30 0 .0 0 12 8 0 ..0 0 4 '5 6 . 0 13 30 6 0 ...0 0 13 30 5 0 ..0 0 1 7 0 0 1 7 0 . 0 15 02 70 .0 ' .0 4 3 0 . 264700 .00 . 0 11 09 7.0 0 .0 3 4 0 . 0 8 60 .0 0 157 ..00 0 62.00 7 ..0 0 31 10 0 0 ..00 2 0 52 4.0 0 .0 0 9 186700 ..0 0 25530 .0 00 2 3 0 ...0 2 8 0 5 0 ..0 0 510 70 0 0 . 0 190 .0 18 74 60 .0 ..0 3 2 0 3 0 0 ..0 0 7 8 0 0 8 7 0 .0 0 6 3 . 0 1 9 S 0 . 14 34 60 .0 ..0 41 4.0 0 0 4 1 9 0 ..0 0 1 9 0 0 8 1 .0 36 0.0 .0 0 9 0 0 98 6.0 .00 0 1 2 0 . 0 1 2 0 . 0 S44.00 .0 0 4.00  0 8 ..5 00 0 8 5 0 0 7 .S 50 0 0 0 7 . 00 0 0 8 . 5 0 0 7 . 5 0 0 0 7 ..5 0 0 0 7 0 0 0 8 ..5 5 0 0 0 7 5 0 0 0 8 . 5 0 0 0 7 . S 0 0 07 7 .5500O 0 0 . 0 8 .7 00 0 ..S 4 0 0 0 7 0 0 0 0 7 . 5 0 0 0 7 . 5 0 0 0 7 . 5 0 0 0 8 . 5 0 0 0 8 ..5 0 0 7 50 0 07 7 .S S 00 0 0 . 0 0 0 7 . 5 0 0 0 7 .8 S 0 0 5O 00 0 07 8 ..5 8 0 0 . 0 0 007 .7 550000 .7 O .5 0 0 8 . 3O 00 0 0 7 . 5 0 0 8 50 00 0 07 ..5 5 0 ..7 0 0 0 7 5 0 0 0 7 ..5 0 0 0 8 1 0 0 0 7 . 5 0 0 0 6 . 7 0 0 0 6 .8 10 00 0 0 7 . 0 7 . S 0 O 0 7 . 5 0 0 0 8 ..4 0 0 8 40 0 0 0 7 . S 0 0 0 6 .6 S7O 0 0 . 0 0 0 7 . 5 0 0 0 7 . 5 0 0 0 7 SO OO 0 0 7 ..7 S 0 . 5 0 0 0 ..5 08 6 40 00 0 .0.8.45000 .07 07 .50 00  9 9.3 9 .5 0 0 3 72 86 6 .0 6 6 . 6 6 0 . 8 7 9 .4 1 7 8 0 0 . 27 50 73 .50 . 0 .2 16 81 26 0 .61 2 . 7 51 7 .5 72 6 8 ..57 1 2 7 3 . 231970 .6 .4 2 35 95 ..741 8 122070 . 0? 1433 .2..9 14 0 5 .7 2 1723 30 .2 8 . 97 . 75*  51 6.0 .0 0 6 5 0490 0 ..0 36 66 2.0 0 .0 0 9 3 ..0 26 50455.00 0 10 .0 1 .00 1 5 0 ..0 1 9 0 0 392.00 .0 28 40 0 .0 0 3 . 0 3 7 0 . 0 3 7 0 .0 0 3 7 . 8 90 .0 0 334 .0 0 . 0 0 2 2 0 . 0 1 4 0 ..0 1 2 0 3 0 .0 0 810 .0 0  0 4 00 00 0 0 2 ...5 0 3 3 0 0 0 3 . 3 0 0 0 303000 05 ...03 0 1 0 . 4 000 00 0 0 3 ..3 0 0 4 0 0 0 0 3 . 3 0 0 0 2 .3 0 0 0 ..7 3 0 0 0 3 8 0 0 0 3 . 6 0 0 0 3 . 3 0 0 0 5 . 0 0 0 0 3 . 3 0 0 0 .3 0 03 .3 3.0 05 0900 0 0 . S C O JO 0 2 . 5 0 0 03 .3509000 0 . 0 3 ..3 00 0 03 3 .3 30 00 0  14820 .10 .0 5 4 . 319.7.01^40 t.00  100 .0 3 .0 0 3 2 0 . 0 3 3 .00 1 1 T.I). 50.00  0 ..0 0 02 2 02 000 00 0 . 0 2 ..120000 0 02 .0000(1  TABLE V I I I CORRELATION MATRIX FOR SIMPLE LINEAR REGRESSION WITH ONE HUNDRED AND NINETEEN PAIRS OF VAR 5  VAR4  VAR1  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  Table I I f o r d e f i n i t i o n of v a r i a b l e s .  VAR 5  1  Variable  'See  VAR 2  OBSERVATIONS  VAR6  VAR 8  1.0000  92  TABLE XI RESULTS OF EQUALITY OF SLOPE TEST FOR THREE T R E E - S I Z E GROUPS D e p e n d e n t V a r i a b l e (VAR1) i s A p p l e Y i e l d P e r A c r e (Quadratic Equation) 1  Slope 2  SCP(1)  Coefficients 90 . 190  SCP(2)  =  3430. 676  =  -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  SCP(ll)  - 1 . 511  SCP(12)  =  0 . 633  SCP(13)  7. 174  SCP(14)  =  . 1.463  SCP(15)  - 1 . 393  SCP(16)  •-  1. 050  -225. 679  SCP(18)  =  -33. 096  SCP(19)  -4. 074  SCP(20)  =  - 1 . 760  SCP(21)  = - 2 2 5 5 . 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  Test  =  f o r Hypothesis F  J  Data  SCP(3)  SCP(17)  2  f o r Pooled  of Slope =  Coefficients  0.29  D.F.  N2  =  54  D.F.  Ni  =  35  PROB.  =  1.00  (Therefore H  See Table I I f o r d e f i n i t i o n of v a r i a b l e s . SCP stands f o r Slope C o e f f i c i e n t s f o r pooled data tree-size  0  is  accepted)  (across groups).  TABLE X I I SIGNIFICANT COEFFICIENTS AT .05 LEVEL FOR COBB-DOUGLAS MODEL Dependent V a r i a b l e  ST.  Error  'F' V a l u e  'See  (VAR1) i s A p p l e Y i e l d  1  Per Acre  Constant  VAR2  VAR3  VAR4  VAR5  VAR6  VAR 7  1.8530  0.4932  1.2182  0.2240  0.2533  0.1978  0.1062  (1.3990)  (0.2040)  (0.2218)  (0.0962)  (0.1049)  (0.0746)  (0.0422)  0.0165  0.0000  0.0206  0.0166  0.0089  0.0128  Table I I f o rd e f i n i t i o n  of variables  TABLE  XIII  CORRELATION MATRIX FOR COBB-DOUGLAS MODEL  VAR1  VAR2  VAR3  VAR4  VAR5  1  VAR6  VAR7  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  :  See  Table I I f o r d e f i n i t i o n of v a r i a b l e s .  VAR8  1. 0000  95  TABLE  XIV  S I G N I F I C A N T COEFFICIENTS AT .05 LEVEL FOR QUADRATIC MODEL Dependent V a r i a b l e  (VAR1) i s A p p l e Y i e l d P e r A c r e  ST.  Error  F' V a l u e  Constant  5733.2578  (7318.3136)  VAR 3  2739.7249  (1162.8074)  VAR 5  398.6005  (85.1116)  0 .0000  VAR 6  1096.3665  (192.8729)  0.0000  (2.8635)  0.0033  0.0193  VAR11  -8 . 6445  VAR13  0.9406  (0.2208  0.0001  VAR 15  3. 5579  (1.0176)  0. 0008  VAR16  -2.2731  (0.4338)  0.0000  VAR17  1.9925  (0.4324)  0.0000  VAR2 2  2671.2339  (1094.0824)  VAR2 3  2.8387  (0.9108)  0.0025  VAR2 8  866.9939  (143.6294)  0.0000  'See T a b l e  II fordefinition  of variables.  0 .0156  1  TABLE XV CORRELATION MATRIX FOR QUADRATIC MODEL INVOLVING ONLY SIGNIFICANT VARIABLES 1  Variable VAR1 VAR2 VAR 5 VAR4 VAR5 VAR 6 VAR 7 VAR8 VAR 9 VAR 10 VAR 11 VAR12 VAR13 VAR14 VAR15 VAR 16 VAR 17 VAR18 VAR 19 VAR 20 VAR 21 VAR 2 2 VAR2 3 VAR 2 4 VAR2 5 VAR2 6 VAR2 7 VAR2 8 Variable VAR15 VAR 16 VAR17 VAR18 VAR 19 VAR20 VAR21 VAR2 2 VAR 2 3 VAR24 VAR2 5 VAR 2 6 VAR2 7 VAR28  VAR4  VAR6  VAR7  VAR2  1,.0000 0..0526 0..2820 0.. 3235 0,.2566 0., 3298 -0..0255 0..1370 0,.1169 0 .1631 . 0 .2945 , 0.. 2098 0..2410 0,.3697 0.. 3277• 0 .2754 , 0,.2729 0..0952 0..3658 0.. 3264 0..3801 .0..2501 0..2368 0,.2446 0 .3274 , 0.. 2490 0..2884 0..3500  1..0000 -0.,4374 0..4166 0,.2495 0..0072 0..0627 -0,.5147 0..9559 -0 .3535 , 0.. 3853 0., 1882 0..0242 0..6630 0.. 5443 0,.5247 0,.4303 0.. 7061 0,.2117 -0.,0980 -0.,2155 -0..4666 0..2507 0., 2141 0..3410 0..1630 0,.0245 -0.,2026  VAR15  VAR16  VAR17  VAR18  VAR19  VAR 20  VAR21  1., 0000 0,.7750 0..4254 0,.5597 0.. 7556 0.. 2985 0,.0311 -0,. 1333 0 .6532 0..6159 0,.8788 0..4986 0.,4401 0..0490  1..0000 0., 5727 0.. 3268 0,,6666 0., 5130 0..0673 -0.,1888 0.. 8843 0,.8238 0..5187 0,. 8290 0., 5922 0.,0376  1,.0000 0,.2004 0,.4265 0..2488 0..4787 -0,.1927 0,. 5218 0,.6662 0,.3192 0 .6793 0,.2543 0,.4281  1,.0000 0,, 2320 -0 ,.1139 -0..1902 -0,,2343 0 .1296 0,.1196 0..5422 0,.0236 0,.1833 -0..0596  1..0000 0..5352 0..2520 0.. 1451 0..6970 0,. 7480 0,. 8180 0,.6115 0,.4918 0..2338  1.,0000 0. 2899 0., 5609 0,.5412 0,.5149 0..2269 0..5791 0,.7S76 0., 1508  1,,0000 0,, 3943 0 .1404 0 .2934 0 .1085 0 .3215 0 .1756 0,.9011  'See Table  VAR 3  VAR 5  VAR1  1,.0000 -0 .0920 1..0000 -0 .0094 1,.0000 0,.6882 0,.0481 0,.2444 0 .2745 , 1,. 0000 -0,. 2136 -0,.0539 -0,.1389 -0,.1480 0 .4314 -0,.1310 -0..1726 -0..0669 -0..2964 ' 0,.4453 0.,2347 0.,0036 0..9545 -0,.1004 -0.,0072 -0..0242 -0.,0787 0., 7473 0.,9321 0..2362 -0..0409 0.,6780 0..8997 0., 2959 0 .0026 0 .2385 . 0,.9122 0.. 1890 0,. 1975 0,.2733 0,.1194 0,.4039 -0,.1236 0,.5881 0,.1536 0.,9258 -0..1565 0..9104 . 0,.2475 0..8092 -0,.1411 0.,4974 0,. 7927 0..4837 -0,. 3124 0,.1047 0.,4063 -0..0807 0,.1840 0.,6274 0,.2789 0,,9153 0.,7189 0,.5966 0., 3901 0..1916 0..1407 0,.4123 0..1095 0., 8489 0.,9654 -0.,0416 0.,0178 -0. 1158 -0..0579 0.. 8677 0.,2872 0., 775 7 -0,.0468 0. 8169 0,.4706 0., 7816 -0..0313 0,,8813 0..3845 0,.1681 -0..0306 0,,6509 0.,8737 0,.5046 0..2160 0,,4741 0.. 7931 0..1704 1317 ,0954 0., 2055 0. 0. 0., 8251  VAR9  VAR10  VAR11  1..0000 -0.,2238 0.4218 0.,1595 0..0042 0..6756 0,.6131 . 0..5203. 0., 3872 0..7370 0.,2514 -0.,0325 -0,,1514 -0..3114 0.,2386 0..1882 0.,4166 0.,1297 0..0786 -0,,1439  1.,0000 -0.,0690 -0,,0355 -0,,0369 0,.1542 -0,.1037 -0,.1294 -0,.1456 -0,.2613 0,. 1321 0., 59 77 0..3058 0,.9399 -0.,0512 -0,.0581 -0..0522 -0,.0428 0,.1949 0,.0923  1,.0000 0., 8080 0 .1951 0 .3413 0,.9012 0,. 8445 0 .4895 0,. 3815 0,.8078 0,.4393 0,.1019 -0.,0930 0,.8894 0,.8562 0 .7753 , 0 .7648 0 .5103 , 0 .0972 ,  VAR22  VAR2 3  VAR24  VAR 2 5  VAR 2 6  VAR 2 7  VAR2 8  1.,0000 -0,.0844 -0..0728 -0,.0063 -0.,0641 0..2594 0..2432  1 .0000 0,.9527 0..4600 0 .9371 0,. 5280 0,.0965  1,,0000 0.,5144 0.,9527 0..4980 0,.2710  1,.0000 0..3272 0,.3914 0..1943  0000 5610 2645  1.0000 0.1892  1.0000  VAR8  1,.0000 -0,.0594 1 .0000 0,.0687 -0..3863 -0,.1942 0., 3867 -0,.0275 -0..1100 -0,.1120 -0..1594 -0 .1793 -0 .1409 -0,. 1019 -0,.2557 0,.0111 -0..1300 -0 .0896 -0,.2855 -0,.0982 -0,.3255 0,.0303 0,.1226 -0,.1048 0.,0015 -0,.2376 0., 1537 -0,.1912 . 0.1959 , -0., 1814 0,,6151 -0,.0909 -0,.1612 .1343 -0,.0862 . -0, 0..0220 0 .1148 -0,,1433 -0.,1589 -0,.0825 0,,2728 -0 ,.0897 0., 3093  VAR12  VAR13  1.,0000 0 .2579 1..0000 0..1875 0..0782 0,.1239 0.. 5355 0.. 8621 0.,2241 0,. 7150 0..5151 -0 ,.0887 0..0425 0..6170 0.,1948 0..5903 • 0,.1455 0.,7488 0.. 1506 -0..0679 -0 .,0375 0.,2504 0..9827 0..9342 0,.3980 0,.0884 0,.3319 0,.9606 • 0. , 4554 0.,5747 0..0955 0..1003 0.,6411  II 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 VARIABLES  VAR14  1,.0000 0,.4471 0,.4188 0 .4018 , 0,. 5165 0,.4274 0.,2876 0., 1068 0.,0854 0..2342 0,. 2441 0,. 3548 0,.1952 0,.1S65 0..0043  97  TABLE OBSERVED AND  C A L C U L A T E D V A L U E S OF I N V O L V I N G ONLY  No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. SS. 56. 57. 58. 59 . 60.  Observed  Calculated  22220 . 25090. 4951.0 17350. 36760. 20760. 23830. 14800. 9251. 0 28100. 28140. 62220 . 22550 . 35910. 28350. 17120. 54470. 40130. 46020. 49480. 46020. 18740 . 11370. 44 44 .-0 393.0 2992.0 1723.0 12140. 15670. 26430. 31010 . 279.0 " 5500.0 34560. 26210. 31550. 12100 . 8666.0 132.0 27770. 4050.0 15610. 4202.0 1071.0 1285.0 45970. 12240. 55300. 28580 . 16190 . 25650. 23700. 20320. 20442. 3468. 9346 .0 11330. 76740. 52280 . 15520.  Autocorrelation Durbin-Watson  25870. 14013. 34297. 31585. 25195. 19789. 13026. 13026. 21225. 31564. 35365. 33262. 32531. 26331. 22101. 27863. 31001. 29709. 24938. 46341. 30649 . 28916. 29489. 690 3. 5 5115.1 26570. 6211.3 14213. 20162. 17032. 26450. 4730.9 5677.7 17805. 26874. 22932. 31349. 15275. 23189. 28988. 7442.4 12657. 14711. 6545.6 6640 . 8 31403. 22657. 24749 . 24561. 18802. 23354. 34929. 22872. 28198. 9224.4 16222 . 15743. 23925. 60396 . 7730.5  Coefficient  D Statistic  - -3650.0 . 11077. -29346. -14235. 11565. 971.32 10804. 1774.4 -11974 . -3464.2 -7225.2 28958. -9980.8 9578.7 6249.2 -10743. 23469 . 10421. 21082. 3139.2 15371. -10176. -18119. -2459 .5 -4722 .1 -23578. -4488.3 -2072.7 -4492.3 9398.0 4560.5 -4451.9 -177.65 16755. -664.44 8617 . 7 -19249. -6608.6 -23057. -1217.9 -3392.4 2952.9 -10509. - 54 74 .6 -5355 . 8 14567. -10417. 30551. 4018.6 -2611 . 8 2296 . S -11229 . -2551.9 -7755.9 -5756.4 -6876.1 -4412 .6 52815. -8115.6 7789.5  1.948  APPLE  YIELDS  SIGNIFICANT  Residual  0.024  XVI BASED ON  QUADRATIC  MODEL  VARIABLES  No.  Observed  61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. -84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100 . 101. 102. 103. 104 . 105. 106. 107. 108. 109. 110. 111. 112. 113. 114 . 115. 116. 117. 118. 119.  15304 . 46710. 37630. 6543.0 . 4232.0 44790. 41550. 27950. 10130. 11600. 12800 . 23540. 27100. 7642.0 9559.0 21390 . 11920 . 13160. 11760. 998.0 7347.0 5894.0 2425.0 595 5.0 2198.0 16000. 14880. 41720. 7495.0 5883.0 4983.0 50900. 43460. 47020. 27100. . 10770. 1995.0 13760. 26590. 8005.0 12930. 63300. 9285.0 777.0 9934 .0 23840. 1436.0 1417.0 3182.0 8386.0 21610 . 11280. 28900. 10770. 22270. 976.0 20880. 23930 . 29660.  Calculated 12966. 31555. 19907. 5165.8 10588. 21210. 22056. 22411. 18988 . 25000 . 10261. 10259 . 15886. 9317.1 . 7146.7 20684. 14588. 15346. 15346. 4169.1 7592 . 3 7115.3 9676.1 3866.3 3866.3 17459. 11282. 45607. 20675. 14922. 15094 . 50890. 23871. 34947. 34165. 25954. 16300. 8904.6 18478. 13991. 17348. 47204. 18777. 18779. 17388 15579. 34242. 3261 .1 9171.4 7242.5 8799.6 12355. 16522. 24339. 23062. 8091.6 19102. 22415. 22435.  Res i d u a l 2338. 4 15155. 17723. 1377. 2 - 6 3 5 6 .,2 23580. 19494. 5538. 9 - 8 8 5 8 .,4 -13400. 2539. 1 13281. 11214. -1675. 1 2412. 3 7 0 5 .,68 -2667. 8 -2185. 6 -3585. 6 - 3 1 7 1 .,1 - 2 4 5 ..27 - 1 2 2 1 .,3 - 7 2 5 1 .,1 2 0 8 8 .,7 - 1 6 6 8 .,3 - 1 4 5 8 .,8 3 5 9 7 .,9 - 3 8 8 7 ..1 -13180. - 9 0 3 9 .,S -10111. 9.! 5859 19589. 12073. - 7 0 6 5 ., 5 -1S184, -14305. 4 8 5 5 ,.4 8 1 1 2 .,4 - 5 9 8 6 ,.5 - 4 4 1 7 .,9 16096, - 9 4 9 1 ,!s -18002, - 7 4 5 3 ,. 7 8261 .2 -32806, - 1 8 4 4 ,. 1 - 5 9 8 9 ,.4 1143 . 5 12810, -1075 . 3 12378 -13569 - 7 9 2 ,.12 -7115 .6 1 7 7 8 ,.1 1515 .1 7224 .7  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 t h e E c o n o m i c s B r a n c h - V a n c o u v e r , C a n a d a D e p a r t m e n t 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:  ,  P.O. Address:  Record No. .  Date Taken:  District:  Taken by:  Marketing Point:  Check by; Land E escripti on and Value j Orel-lard Other Improved Unimproved 1 Acres Value/Acre Acres Value/Acre Acres :Value/Acre  Owned  Rented  Additional Acres Owned Land Suitable f o r Orchard  Waste Acres  Total Acres  Clearing  Page - 2  LAND PURCHASES & SALES  LAND IMPROVEMENTS  Disc and Harrow  Breaking  Picking Roots Stones  1  Purchases  Acres  Sales  Price Paid  Cover description  Received  Hours - Farm tractor  Owing  - Unpaid Labor  Yes  ' Cost - Hired tractor  Purchased  Land  S o l d Land  Cropped  Cropped  Date o f Purchase  - Hired labor  '  !  i  o r Sale  Orchard  i  Other costs  ; No  j  Other im-i Unproved .! improved  Acres Purchased  Total Costs  Acres Sold  E x p e n d i t u r e s on New C o n s t r u c t i o n , Improvements and R e p a i r s A s s o c i a t e d w i t h C A S H H i r e d Mach. Hrs,  Mater'1  Costs Cost,?  C O S T S  Land  Value of Farm C o n t r i b u t i o n s  H i r e d Labor j T o t a l  Farm T r a c t o r I'taber !! Unpaid  Urr,  Hrs., C o s t  Cost s  Ce.sh  1  Labor  Total  Cach Cost  Value  Repairs  Hr:;... \ Value  Value  !  rrrg„ D i t c h e s Drainage Farm Reads  i  Fences Other  j i ! :  1  .  i  1 1  i i  1  Page - 3  Inventory of Buildings  Year Built  Replacement Cost Beg. of Year  •  Typ- of Cone t T  Cash Ccct New Bldg , Imp,. Equip & I'-Iaterial  Hir?d Labor  Value of 37arm Cont r i b u t i o n s Fcrm Tractor Value  Total ReUnpaid Labor Costr p a i r s  I-iatr'l | Value  •Hrs.!Value i i  . t  Machine Shed  _  •__  Tool Shed  i •1 i  • i  i! ;j 1!  !  Garage  :  Pump House  i  i  i  J i i i  !  |  Pickers Cabins  " "f 1  i  i  1  i  !  i 1 j  i  :  1 ! i } 1  Road Side Stand  \ 1 I  i  -  l  ! 1!  1  =  j  j  ! *  ;  j  i  t  I  i  1  1  !  !  !  i  . iI  TREE FRUIT INVENTORY  STANDARD APPLES Variety  Red  Delicious  Golden D e l i c i o u s  Mcintosh  Winesap  Newton  Rome Beauty  Spartxi  Spacing  Trees/ Acre  Planted 1969  1 to $ Yrs.  Over 6 to 11 t o 10 Y r s . 20 Yrs, 20 Yrs,  Page - k Total Trees  Est. Acres  } I n t e r - j W i n t e r 69 j Plants  Killed  1  Dam.  TREE FRUIT INVENTORY Planted  Planted Planted! Planted Planted 1966 1968 | 1967 1969  Spacing T r e e s / Acre  Variety  Page - $ j Total  ! 1 i  1  Serai-Standard A p p l e s  1  1  i 1  t  1  J  \  i  * j  i  ! i 1  i !  ! !  1  i :  !j  i  -i  :  \ t  ! 1  !  i  1 • 1  !  i  :  l  l <  i  •.  •  •  :  I i i  1  ii  \  1  .  '•  1  ! -  ! f  !  I  :  i  !  1  t  '  '  !  i  •  1  j  r  ;  !  I  •  *  1  i  1  \- \  1  .  i  1  1  \  t  !  1  i1 !;i • 1  !  \ i !  ; i 1:  !  .  f i  1  | •  !  i,  ]  i  1  !  i  !  } i  !  1  j  .  i  ! 1 |  I ; i  ;.  1 ! i  !  :  !  i ! i  i ! ! •: !  i 1 !  \ i  j •  •  !  i i i  !  i  i !  1 i  i 1  !  i  ! '  j  1 i 'i  Semi-Dwarf A p p l e s  ! W i n t e r I969 1 Killed ] Dam,,  |Est. jAcres '  :  !  i  Dwarf Apples  Over  1965-61 5 Y r s . ? Trees  i [  *  i 1  Page - 6  TREE FRUIT INVENTORY  Variety  Spaaing  1 Trees/ Acre  Planted I969  11 t o 1 to j 6 to £ Yrs.i 10 Yrs. 20 Yrs.  Winter I969 Est. Acres IKilled . Damaged „  Total Trees  Over 20 Yrs.  Fsars  i ....  1-  i  .  -~™  Peaches  1  1  i  i 1 * j  1  1 . . . .  j  ;  i  !  ]  1  i  !  .  1  i  1  ..  !  !  ...  p.  i 1  !  " 1 •  '  — _  1  1  1  Page - 7  TREE FRUIT INVENTORY Variety-  Trees/j Planted Acre .; I 9 6 9  Spacing  11 to 1 to 6 to $ Yrs. 10 Yrs. 20 Yrs.  Over  20 Yrs ,  Total Trees  Est. Acres  I  Winter I 9 6 9  i Damaeed i  Killed  Apricots  \  !  1 j  | i  1  1i  1  i  j  1  i  '  1i  ;  1  :  !  i  1  1  1 5 i  I  Cherries  j  ! >  ! i  . ! | j I  1 j j  i j  •  1  i  •  Plums & Prunes  1  _________  i  i 1 1  !  !  i  i  i  !  !  1  !} 1 j  1  t  1  J  1  *l 1  t  1  1  PHYSICAL-DATA - RE SPECIFIC APPLE PLOTS  - -  j! PRODUCTION AND RECEIPTS FROM SPECIFIC APPLE PLOTS Intensive  Plot of Standard!  P l o t of Intensive p l a n t i n g  Planting  H  r  V a r i e t y  U Type of Pack  Variety-  Standard  ;  \- E x t r a Fancy: Large :  Main Root Stock S i z e o f Block Ac. i j  Spacing  Small  T o t a l No. of Trees | Trees per A r e  j. Fancy:  :  C  •  !  j  ;  i  ;  1  Small  !  Winter Damage 69  |  Trees K i l l e d No. . Trees Damaged No.  Cee !: C u i i s  \  j; i;  •  !  .A  \  r Receipts  \  .  Extra Fancy  i  Fair  Fancy  j  t  Poor  Cee  i1  Rebates  1 j  C u l l Returns  I  1  D i r e c t i o n o f Slope Degree of Slope  i ! ;  i I i  Good  A i r Drainage  Cover Crop  !  Large Medium  Year Planted  S o i l Tvoe  f  I! 1 1  Medium  Level  1  Slight  }l Total Receipts  Moderate  ji !!, Farm Sales:  Steep  u  1  j Yes i No  :  i  Lbs. I  Receipts Home Use: Lbs. Valu°  1  1  ! j  1  Page - 8~  •  GRADE AND SIZE OF FRUIT DELIVERED TO THE PACKim HOUSE  Page - 9  Name of Packing House APPLES Type of Pack  Variety  Fancy  Extra Fancy Large  Medium  Large  Small  Small  Medium  j Culls  Cees  !  1 •-  -  - t-  .  j I  •  \\  -j  !  I i i  Apple Receipts Variety  1  variety  j  T  y  p  e  p  a  c  °  Extra Fancy  f  Fancy  Cee  Rebates  Cull Return  Total [ Cull Receipts ' Charge j  k  i  I f  ;  ;  1 j  I 1  ii  i i  1  l  1  ;  GRADE AND SIZE OF FRUIT DELIVERED TO THE PACICING HOUSE P E A R S Variety-  Type of  I  Extra Fancy  Fancy  Culls  CGGS  rack  Small  Mediiim  Small  Lar^e  Pear Receipts Variety  t T Type  Pack  of  Extra Fancy  Fancy  Cee  Rebates  Total Receipts  Cull Charge  H  -(  I f  GRADE AND SIZE  RECEIPTS  Page - 11  i  Plums and Prunes Variety-  ( 1 1  No.  Select  1  2  No.  No.  Select  Culls  1;  2  No.  Cull Charge Receipts  Total  Rebates  i ;  Peaches Variety  i  Domestic Grade  Domestic Grade No.  1  No..  uUXIS  2  i  1  No.  i I i  No.  Kebaties  2  Total  Cull  Receipts  Charge j 1  l i  I  i  •  Apricots Variety  Domestic Grade No.  1  No.  2  Domestic No.  ouxis  3  r  No.  1  Grade  No.  2  No.  1U_JU ct Uco  3  Total  Cull  Receipts  Charge  i  I  1 ii  1  i i  i  I 1  t Cherries Variety  No.  1  i Orchard Run  Culls  No.  i i  1  Orchard Run  debates  Total Receipts  ! !  1  j  • •  r i :  Cull Charge  1  1  i1  FRUIT SOLD AND USED ON FARM Farm S a l e s  Used on Farm  Receipts  Pounds  Pounds  k'axm S a l e s  |. Pounds  Value  Apples  ;  Receipts  Used on l''ar,n Pounds  Value  Peaches  !; A p r i c o t s Pears )'• C h e r r i e s Plums and Prunes  •  XXX  f: T o t a l s  XXX  FERTILIZER USED Kind o f Fertilizer  Total ; Quantity  Orchard Cost  \  I n t e n s i v e Apple  i  Quantity  ILbs/Tree  , , , . , S t a n d a r d Apples  Other S p e c i f i e d Fruit  ;  c  Plot Costs  jibs/Tree Q u a n t i t y !;  |  Costs  : Quan.  !|  1! j;  i;j  |  j;  ii ii  j!  ii  ii  I;  i•  t'  \  ||  |i  :  N  !  V:  1: j• •  \  ll  Costs  SPRAY MATERIAL Kind of Sprav  Total Orchard Quantity  Boron  Page - 13 Standard Apple P l o t  ! Intensive Apple P l o t  Costs  Quantity  Costs  Quantity  Costs  Costs  i i 1 ! I  1  it t I i  Magnesium Manganese  Quantity  ''  i  \  Zinc  Other S p e c i f i c F r u i t  i  t  !  Iron Urea  I  Dinitrocresol  1  ii  Naphthalene acetmide (Amid Thin)  !!  1 1  1  Sevin  !  1I  Triethanolamine S a l t  1 i  of 2 , U, 5 - T.P. Naphthalene A c e t i c  •i  J  ;l  Acid (N.A.A.) Alar Dormant SprayMoras tan Keithane  i  Tedion l  Ethion  II  jj ii  i  Kava thane Movocide  i  D Lmethoate (Cygon, Roger)  i  Total.  i  1  ;  i  i  X X X  |  i  1  i  I1 1  i  i X X X  X X X 1  ,  •  !  i  T o t a l Orchard Kind o f  Spray  I  I n t e n s i v e Apple  Costs  Quantity  S t a n d a r d Apple  Plot  Costs  Quantity  Other S p e c i f i c F r u i t  Plot  Quantity  Costs  Quantity  Costs  Guthion P e r t h ane  |  Supreme and S u p e r i o r i I  Type O i l s  1  |  D.D.T.  j  Parathion  Ii  l  Diazinion  i  Thiodan Para dichlorobenzene  !  Lime S u l p h u r  fl 1  !  Glyodin-Dodine  \  !  (Glyodax) Dodine  (Cyprex)  Dichlone  (Phygon)  !  J  i  !  !  ]  i  Ferbam Maneb  &  |  <  i  Zirim  !  ! i  1  Captan Bordeaux i  Bot r a n  1  I  i  Malathion Wettable  i  Sulphur  Paste  Sulphur  Fixed  Copper  l  1 1  I i i i i!  '  iii I!  !i !  1  i i i  | !  I1  i  i  1  «  !  r i!  1  n  1  1  1  I i i  1  • —  i 1 ! ii |  1 j  i i 1  1 1  1  i  j I >  COST OF SEEDS AND Total  Trees  Cost  j  Page - Ih  I n t e n s i v e Apple P l o t j Standard  Orchard  j QuantityFruit  PLANTS  J  -  -  i  Quantity  Cost  Apple P l o t  H '  Cost  Quantity  jOther S p e c i f i c Fruit j Quantity  Cost  ] !  !  i  G r a s s and P l a n t s  Total CUSTOM WORK T o t a l Oi-chard  Rate  Received  Paid  Paid Intensive  {Paid  Apple P l o t  Apple  Standard  P a i d Other Specific Fruit  Plot  Plowing Discing i  Mowing  i  \  \j  Raking  \  :  i j  Ditching Spraying  |  Hauling Other t r u c k i n g  i i  |  Total  XXX  i j  1 1  Ii i  !• ! ;  1  ! i  - *S  GENERAL MACHINERY AND EQUIPMENT Total Orcha?-! Begin of Yr. Value  No.  Sales  Purchases  Cost of Repairs  / Share t o j Intensive j Apple P l o t  • i  cnare t o i ftlic.re to ' Star/., arc' 'Other SpeApple Plot c i f i c F r u i t  !  Irrigation -  | It ! I  I  i  Orchard  1  1i  & Mask Spray Costumes  i  i  i  j  i  !  •  !  1  Pruning Equipment  j1 1  Props  j  i  Ladders P i c k i n g Bags Orchard Boxes Equipment P i c k e r s Cabin  1  i '  j  Sub T o t a l  1  j  i  i  i  I i  -  Ditcher  I  i1 1  j  i  i  j  1  !  |  t  i;  1  i  | j  i XXX  i  :  11  i  1  j  •i  j  1 !  1  I  GENERAL MACHINERY AND EQUIPMENT - Continued T o t a l Orchard N o  -  'Beg. of Y e a r jValue  Plow  Share t o j Share t o Share t o '. Other SpeI n t e n s i v e j Standard ' 'Repairs ° !i Apple P l o t Apple P l o t i c i f i c Fruit i ii : i ii !| (  S i aa il ee ss  Purchases  l  C  0  S  t  f  !  !  i  Disk  i  IJ  i  i! II  Row Crop o r F i e l d Cultivator  i  t  ii it ,1 i•  i  Harrows  * •  1  Mower  i  i  Rake Hand S p r a y e r Trailer  j  ' <  : !  !  !  I ! j  i  Wagon C h a i n Saw  | i j !  Motors  i  ir  ! 1 t |  i  !  ii I: 1 ii '!  I  I  !  I  !  i I  i!j  j  i i i  Electric  Ea.g«.- 1 6 -  ;  i ;, ; 1  I  \  l  '•  ' .t 1 1i  !  i  !  S m a l l T o o l s and Garden T o o l s  j  j  I  i  i i  1 i <  Total  i  !  i  ' ! ',  i .  ' ; 1  !  •  !  ;  :  ;  i  •>  •i  i 1  i i  .  !  i  !  ;ge Car  Truck  i  •' Garden • T r a c t o r Sprayer  Tractor  |  •  : i  i  Year  !  i j  ii  !  Hours Used Value Beg. o f Y r .  !  i  Purchase P r i c e  Operating Cost  ii  i  .  ;  1  :  !  !  ;  ;  '  I  :  i  Costs  i  i  of Fuel  "  O i l and Grease Repairs  ;  ;  Tires  • j  !  i  • :  1  •'  Licence Insurance  1  :  !  1  i  !  '•  ! •  1  i  M i l e s t o Farm  Price  i  1  i  l i l i e s f o r Year  I  1  ,  :  i  Size  Sales  :  j  Make  ( Roto j j Mower f  Giraffe Squirrel etc.  |  |  i T o t a l Operating \ Costs Proportion of Totals ! I n t e n s i v e Apple P l o t  j  S t a n d a r d Apple P l o t  f  ! !  f i  j  :  •  :  ;  1 1  ' t  !  i j  ! 1  i  •  i  '.  '  i  I  '  i  '  !  \  j  \  Other S p e c i f i c F r u i t N. B. I f g r o s s f i g u r e s only a v a i l a b l e on c o s t s e s t i m a t e  !  1 1  !  t  !  !  !  <  •:  1  ,  • j  *:'  i  i  i i  c o s t o f c a r o p e r a t i o n o r t r u c k i f used i n s t e a d o f c a r .  Page - 1 8  LABOR RECORD Rate of Wages Hired:  Total Orchard Total Hours  Total Wages  jj Intensive Apple Plot ! Total l Hours  Board  Total  Total Hon rs  Board  |  Month  j Standard Apple Plot Tot"!  Board  j Other S p e c i f i c F r u i t Total Hours  1  i i  I  Total; Board Wao-e i  1  1I  i Day  i  i !  Piece Work  1  1  i  i|  I  1  ;  i  |  1  Family - Daughl:er Son (Age  )  .Son (Age Wife  )  i  Operator  it  t  I  I  I i  1i  i  1  i  1 i  i  i  i  !  •  j 1  ii i  1  LABOR INPUTS RE PLOTS Intensive Apple Plot Hired Pruning, Grafting, Cultivating  i  Operator  Family  11  Repairing and Removing trees  1  I; i1  Fertilization  i  Standard Apple Plot i i Hired Family j Operator  j 1  1i 1  1  3  i  \  i i  i  j  i  |  [  |  Picking  j  | 1 i  ;| ;  1  \ \ J '  <• .!  i 1  — — — — — — — — — — — — — — — —  »——  !  i ,  i  Irrigating  DC io sl tl re ic bt ui tn ig n gand & hauling j s t o r i n g boxes boxes t o packing house !  !  i  i  1  : i  Spraying Thinning, propping  !  !  !  Mowing  !  _  _  !  i  i  !  i i  ! j  FARM LIABILITIES Borrowed During the Year  Amt. Owing Deg. o f Yr.  Purpose  Borrowed From  •  Term  Amt.  ;  Paid During Year  Rate f  |  P r o v i n c i a l Gov't  .  Int.  Princ.  Page - 19 , Owing End of Year  i  Farm C r e d i t  1  i  Farm Improvement V.L.A. Bank  !  1  i  i  C r e d i t Union Mortgage Co.  1 t  Finance Co.  !! i  i  1  Machinery Co.  i  Other  | t>  i  j  1 i ti 1  Current Borrowing Bank  .  !  1 j  C r e d i t Union Other  I t  1  ! i  i  1 1  !  1  I  I  t  1  I  1  1  .....  1  OF FAMILY "  " RECEIPTS' Intensive Apple Plob  Total Orchard  Oth  oir.i'.'liircl  Soocifio Fruit  Apple Plot  Current Receipts Fruit:  Sox Operator  ii |  Wife  |  •  Apples  1 i  l i i  Peaches  Children: 1  ji  2  !  Cherries Farm Sales Custom Work i  xxx  i i  ! 1 1 1  u  !  1  $  i i  I I i!  6  !  ;  8  1  Others  ,  1i  i  i  1  1  7  ?  i  1  i  1  iI  !  i  !  i  i I  j Monthsct  j  3  i  Total  A-c  Pago' - 20  t  Apricots •  Other  v  1  1 !  Non-Farm Earnings  |  i  Pears Plums and Prunes  "  j  I  C a p i t a l Receipts Real Estate S a l e s Power Equipment Sale? General Equipment  —1  1 1  j  SSIP-  Gthcr  1  I  i  Year Operator Started on t h i s Farm  i 1 I1  Acres Orchard  i  Acres Improved  i  Acres Unimproved  1  Total I  i i  i 1 |  i  I  I  EXPENSES T o t a l jj Intensive Orchard i; Apple Plot  Spaniard Anple F nt  |  Current Expenses  Other  Total Orchard  Fruit Current Expenses ,,  Cash Rent  E l e c t r i c ( T J t o farm)  Land Taxes.  Phoneto  Irrigation- x|£  er  farm)  F r e i g h t & Express  Water T o l l  Accounting  Electricity  Interest(current)  Gas and O i l  Membership Fees  F i r e Insurance  I  1  1  Orchard Box Rental  H a i l Insurance  i  Repairs - Land  j  Other  Buildings j C u l l Charges i P l a n t s &. Seeds , ! Purchased \ Fertilizer } T  Spray M a t e r i a l j Operating Costs o f j a l l Equipment Labor:  Mages Un«  T o t a l Current  j  j  C a p i t a l Expenses  !  i  New Cons't Lr.-d Building  Ir.v.n  1  C.P.P.  Liab,. In3, ___________ p • Custom Wcrk j Meed Spr-.vs 1  Land Ir^prcveir. ?rvb r,  , !  Small Haivlware  Power Equip, Fur,. Geu-val Equip, P r ^ . Ofcbr.r  !  Mien. O i l & Greaco|  j  I  1 . 1  1  Total C a p i t a l  j  Page - 2 1 Intensive Standard Other Specific Apple Apple Fruit Plot Plot  LABOR TIME SHEET  Semi-Standard o r Standard Apple P l o t  Intensive Apple P l o t Date  D e s c r i p t i o n of Work  Full-time Employee and Operator  F u l l - t i m e Family Labor Employee Adult j Under Hired and j 15 Yrs. Operator Hours  I  Family Labor Adult I Under ;15 Yrs  ' Casual Hired  Hours  ! I  1  1  ' !r 1! ji  1  L.  ;..... ,  1  1  1  i 1 i i  """"  i  i <  .  j  i  I  ! 5 i i i Ii i  i  II .  f  i  1 i 1 1  CASfl EXPENSES  Date  Description (including custom work)  Intensive Apple Plot Quantity or Hrs  Cost  Standard o? Semi-Standard Apple Plot Quantity or Hrs".  •  *  Y  Cost  i CASH RECEIPTS  Dace  Kind of Apple  Semi-Standard o r Standard Apple P l o t  Intensive Apple Plot Pounds  Grade  "CuTls  Pounds  i  Total  Cull  Receipts  Charges  Ui  Pounds  Grade  Culls  Total  Pounds  Receipts  Cull  Charges  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 Spacing  Value of i r r i g a t i o n system  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 Est. of taxes  Cost of trees  Total Orchard Size - Acres Date  .. _  1/  Quantify  Hours Description of 1/ Item  Custom Work  1  Weed Spray- Tree ReplaceConing ment t r o l  FerI r r i g a - Prunt i l i z e r tion Thitnning  JJoliars  1  I f operator or family labor - Do not put i n value but indicate item applicable with "U-".  Mowing  Picking & Sundry Hauling  NET ESTABLISHMENT COST IN I969 OF AN INTENSIVE APPLE PLOT PLANTED IN  Date  Quantity Hours 1/  1/  Description of Item  Cont'd  Weed I r r i g a - Pruning Spray- Tree Custom FerConing ReplaceThint i l i z e r tion Work trol ing ment i Dollars  I f operator or family labor - Do not put i n value but indicate item applicable with  "K",  Mowing  Picking & Sundry Hauline  

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