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Estimation of browse biomass production of Salix SPP. and Betula blandulosa using multiple linear regression Habgood, Helen Leslie 1985

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ESTIMATION OF BROWSE BIOMASS PRODUCTION OF SAL IX SPP. AND BETULA 6LAMM LOSA USING MULTIPLE LINEAR REGRESSION By HELEN LESLIE HABGOOD B . S c , Western Washington U n i v e r s i t y , 1980 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQIUREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department o-f F o r e s t r y We accept t h i s t h e s i s as con-forming -to th.e r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1985 @ Helen L e s l i e Habgood In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 ABSTRACT Browse b i omasa p r o d u c t i o n o-f Saiix spp. and Betula glandulosa on a wetland i n c e n t r a l B r i t i s h Columbia i s e s t i m a t e d . Based on an e x t e n s i v e review o-f much o-f the l i t e r a t u r e p e r t a i n i n g to shrub biomasa and shrub d e n s i t y e s t i m a t i o n , a technique combining r e g r e s s i o n e s t i m a t e s o-f average stem biomass with a d e n s i t y e s t i m a t e o b t a i n e d u s i n g the c o r r e c t e d p o i n t d i s t a n c e method was a p p l i e d . I t was -found t h a t the best r e g r e s s i o n r e l a t i o n s h i p s were o b t a i n e d u s i n g n a t u r a l l o g a r i t h m i c trans-formations o-f the dimension and biomass v a r i a b l e s . I t was p o s s i b l e to o b t a i n a c c e p t a b l e biomass e q u a t i o n s -for the f o u r Salix s p e c i e s encountered without d i f f e r e n t i a t i n g between the s p e c i e s . More a c c u r a t e p r e d i c t i o n s of biomass were achieved u s i n g s i t e - s p e c i f i c e q u a t i o n s and e q u a t i o n s based on pooled s i t e d a t a than with general e q u a t i o n s . I t was concluded th a t the v a l u e of the approach taken i s l i m i t e d i f s i t e - s p e c i f i c e q u a t i o n s are r e q u i r e d because of the c o n s i d e r a b l e time r e q u i r e d f o r sample c o l l e c t i o n and p r e p a r a t i o n . - ii -TABLE OF CONTENTS A b s t r a c t i i Table o-f Contents i i i L i s t o-f T a b l e s v L i s t o-f F i g u r e s v i i Ac know lodgement a v i i i 1 INTRODUCTION AND OBJECTIVES 1 2 LITERATURE REVIEW 4 2 . 1 I n t r o d u c t i o n 4 2 . 2 Review o-f shrub biomass e s t i m a t i o n methods ...... S 2 . 3 R e g r e s s i o n e s t i m a t i o n 8 2 . 3 . 1 D e v e l o p i n g the r e g r e s s i o n 9 2 . 3 . 2 R e g r e s s i o n models •• 1 3 2 . 3 . 3 Sampling the p o p u l a t i o n 1 8 2 . 4 D e n s i t y sampling • 1 9 2 . 4 . 1 P l o t sampling 1 9 2 . 4 . 2 P l o t l e s s sampling 2 0 2 . 5 Biomass per u n i t a r e a 2 5 2 . 6 C o n c l u s i o n 2 8 3 STUDY AREA 2 9 3 . 1 General d e s c r i p t i o n 2 9 3 . 2 S i t e d e s c r i p t i o n s 3 0 3 . 2 . 1 S i t e 1 3 3 3 . 2 . 2 S i t e 4 3 4 3 . 2 . 3 S i t e 6 3 5 3 . 2 . 5 S i t e 7 3 6 4 FIELD METHODS 3 7 4 . 1 R e g r e s s i o n e q u a t i o n s . . 3 7 4 . 2 Stem d e n s i t y 4 0 4 . 3 Biomass per u n i t a r e a 4 1 5 STATISTICAL METHODS 4 3 5 . 1 R e g r e s s i o n e q u a t i o n s 4 3 5 . 1 . 1 Hypothesis 1 4 3 5 . 1 . 2 Hypothesis 2 4 5 5 . 1 . 3 Hypothesis 3 4 7 5 . 1 . 4 Hypothesis 4 4 8 5 . 1 . 5 Hypothesis 5 4 9 5 . 1 . 6 Hypothesis 6 4 9 5 . 2 D e n s i t y 4 9 5 . 3 Biomass per u n i t a r e a • 5 0 6 RESULTS 5 2 6 . 1 Hypothesis 1 5 2 6 . 1 . 1 Sal ix 5 2 6 . 1 . 2 Betula 5 7 6 . 2 Hypothesis 2 6 0 6 . 3 Hypothesis 3 ••• 6 1 6 . 3 . 1 Sal ix 6 1 6 . 3 . 2 Betula 6 3 6 . 4 Hypothesis 4 • • 6 5 6 . 4 . 1 Sal ix 6 5 6 . 4 . 2 Betula 6 7 6 . 5 Hypothesis 5 6 9 6 . 6 H y p o t h e s i s 6 7 1 6 . 7 D e n s i t y 7 2 6 . 8 Biomass per u n i t a r e a ••• 7 3 7 SYNOPSIS AND ANALYSIS 7 8 7 . 1 S y n a p s i s 7 8 7 . 2 C r i t i c a l a n a l y s i s • 8 0 LITERATURE CITED 8 6 APPENDICES 93 A: S c a t t e r p l o t s o-f Sali'x and Betula d a t a 93 B : ANOVA t a b l e s 1 0 6 / V -LIST OF TABLES labia Ease. 4.1 Summary o-f S a l i x common d a t a s e t 3? 4.2 Summary o-f Betula common d a t a s e t 39 4.3 S a m p l e s i z e s f o r d e n s i t y , i n d e p e n d e n t v a r i a b l e s and b i o m a s s on s i t e s 42 5.1 I n d e p e n d e n t v a r i a b l e s t e s t e d i n d e v e l o p m e n t o f m u l t i p l e r e g r e s s i o n e q u a t i o n s 44 6.1 T e s t s o f a s s u m p t i o n s o f r e g r e s s i o n on S a l i x common r e g r e s s i o n 55 6.2 Some e x a m p l e s o f p r e d i c t i o n s and c o n f i d e n c e i n t e r v a l s f o r S a l i x b a s e d on t h e common r e g r e s s i o n e q u a t i o n 56 6.3 T e s t s o f a s s u m p t i o n s of r e g r e s s i o n on Betula common r e g r e s s i o n 58 6.4 Some e x a m p l e s o f p r e d i c t i o n s and c o n f i d e n c e i n t e r v a l s f o r Betula b a s e d on t h e common r e g r e s s i o n e q u a t i o n . . . . . . . . . . 59 6.5 T e s t i n g f o r one e q u a t i o n t o d e s c r i b e f o u r S a l i x s p e c i e s u s i n g r e g r e s s i o n w i t h dummy v a r i a b l e s 60 6.6 T e s t i n g f o r one e q u a t i o n t o d e s c r i b e S a l i x an a l l s i t e s u s i n g r e g r e s s i o n w i t h dummy v a r i a b l e s 62 6.7 T e s t i n g f o r one e q u a t i o n t o d e s c r i b e Betula on a l l s i t e s u s i n g r e g r e s s i o n w i t h dummy v a r i a b l e s 64 6.3 Mean p r e d i c t e d b r o w s e b i o m a s s (grams/stem) and p a i r e d t t e s t s f o r d i f f e r e n c e b etween p r e d i c t e d and o b s e r v e d b i o m a s s o f S a l i x 66 6.9 C o m p a r i s o n o f mean p r e d i c t e d b r o w s e b i o m a s s (grams/stem) and 9 5 % c o n f i d e n c e i n t e r v a l f o r S a l i x , u s i n g t h r e e e q u a t i o n s - 66 6.10 Mean p r e d i c t e d b r o w s e b i o m a s s (grams/stem) and p a i r e d t t e s t s f o r d i f f e r e n c e b e t w e e n p r e d i c t e d and o b s e r v e d b i o m a s s o f Betula .- 68 - v -l a b i a Eaas 6.11 C o m p a r i s o n o-f mean p r e d i c t e d b r o w s e b i o m a s s (grams/stem) and 9 5 % c o n f i d e n c e i n t e r v a l s -for Setula, u s i n g t h r e e e q u a t i o n s . 68 6.12 P a i r e d t t e s t s f o r d i f f e r e n c e b e t w e e n p r e d i c t e d b i o m a s s (grams/stem) of f i r s t s t e m and s e c o n d s t e m o f Setuls on s i t e 4 71 6.13 D e n s i t y e s t i m a t e s , p r o b a b l e l i m i t o f e r r o r and 9 5 % c o n f i d e n c e i n t e r v a l s f o r Salix and Bstuia .... 72 6.14 Salix b r o w s e b i o m a s s and 9 5 % c o n f i d e n c e l i m i t s i n grams p e r s q u a r e m e t r e 74 6.15 Betula b r o w s e b i o m a s s and 95% c o n f i d e n c e l i m i t s i n grams p e r s q u a r e m e t r e 74 vi -LIST OF FIGURES 3.1 Map o-f B r i t i s h C o l u m b i a s h o w i n g l o c a t i o n of s t u d y a r e a 31 3.2 L o c a t i o n s o-f s i t e s w i t h i n s t u d y a r e a 32 - v i i -ACKNOWLEDGEMENTS I would l i k e t o thank my s u p e r v i s o r , D r . P e t e r M a r s h a l l , f o r h i s encouragement d u r i n g t h e d a t a a n a l y s i s and w r i t i n g o-f my t h v s i a , and my -former s u p e r v i s o r , D r . L a r r y L a r s o n , -for h i s s u g g e s t i o n s and encouragement d u r i n g my - f i r s t y e a r a-f g r a d u a t e s t u d i e s . I a l s o a p p r e c i a t e t h e c o n t r i b u t i o n s and a s s i s t a n c e o-f my o t h e r c o m m i t t e e members, D r . J u l i a n D e m a e r s c h a l k and D r . M i c h a e l P i t t . M a r t y B e e t s , J i m Young and Gord W o l f e of t h e F i s h and W i l d l i f e B r a n c h i n W i l l i a m s L a k e , B . C . , v e r y k i n d l y p r o v i d e d t h e l o g i s t i c and f i e l d a s s i s t a n c e n e c e s s a r y f o r c o m p l e t i o n of t h e f i e l d work. Thanks a r e a l s o due t o Anna R o b e r t s f o r a s s i s t a n c e w i t h v e g e t a t i o n i d e n t i f i c a t i o n , A r t Yee f o r s o i l a n a l y s i s , John S m i t h f o r a s s i s t a n c e w i t h d e n s i t y e s t i m a t i o n , B a r r y Wong f o r c o m p u t i n g a s s i s t a n c e , and F r a n c i s Schwab f o r h i s a d v i c e t h r o u g h o u t t h e s t u d y . Most o f a l l I am g r a t e f u l t o my f i a n c e , J i m S i b l e y , f o r p r o v i d i n g d i s t r a c t i o n s when n e e d e d , and w i t h o u t whose s u p p o r t t h i s would n e v e r have been c o m p l e t e d . - v i t t -1. INTRODUCTION AND OBJECTIVES Browse, t h e t w i g s and l e a v e s o-f woody p l a n t s u s e d by a n i m a l s a s -forage, i s an i m p o r t a n t d i e t a r y component -for many a n i m a l s p e c i e s . T h i s i s e s p e c i a l l y s o i n s h r u b d e s e r t c o m m u n i t i e s and d u r i n g t h e w i n t e r i n r e g i o n s o-f h e a v y , l o n g - l a s t i n g snow c o v e r ( T e l f e r 1 9 8 1 ) . In B r i t i s h C o l u m b i a , b r o w s e i s an i m p o r t a n t s o u r c e o-f -forage -for b o t h c a t t l e and w i l d l i - f e (McLean 1 9 7 9 ) . R a n g e l a n d u s e c o n - f l i c t s b e tween d o m e s t i c s t o c k and w i l d u n g u l a t e s may a r i s e , b u t c a n be r e d u c e d o r e l i m i n a t e d by good r a n g e management and u n d e r s t a n d i n g o-f t h e r e s o u r c e . T h i s I n c l u d e s q u a n t i t a t i v e i n v e n t o r y o f b r o w s e b i o m a s s . Browse i n v e n t o r y i s a c o m p l e x p r o b l e m b e c a u s e b r o w s e p l a n t s h a v e p h y s i o g n o m i e s and l i f e h i s t o r i e s d i f f e r e n t f r o m b o t h h e r b a c e o u s r a n g e p l a n t s and t r e e s . The o f t e n h i g h l y v a r i a b l e o c c u r r e n c e o f b r o w s e a l s o makes measurement d i f f i c u l t . A number o f b r o w s e s a m p l i n g m ethods have been d e v e l o p e d , w i t h i n p u t f r o m r a n g e s c i e n t i s t s , f o r e s t e r s and w i l d l i f e b i o l o g i s t s , r a n g i n g f r o m s i m p l e o b s e r v a t i o n o f p l a n t p r e s e n c e t o a c c u r a t e d e l i n e a t i o n o f b i o m a s s ( T e l f e r 1 9 8 1 ) , e a c h w i t h i n h e r e n t s t r e n g t h s and w e a k n e s s e s r e l a t i v e t o s p e c i f i c o b j e c t i v e s ( P i t t and Schwab 1 9 8 5 ) . In t h e C a r i b o o r e g i o n o f B r i t i s h C o l u m b i a some w e t l a n d a r e a s a r e u s e d f o r w i n t e r c a t t l e r a n g e . T h e s e w e t l a n d s a r e a l s o i m p o r t a n t s o u r c e s o f w i n t e r f o r a g e f o r d e e r and moose. W i l d l i f e m a n a g e r s have e x p r e s s e d c o n c e r n t h a t c a t t l e and w i l d l i f e a r e c o m p e t i n g f o r t h e same b r o w s e r e s o u r c e ( B e e t s 1984, p e r s . comm.). The p u r p o s e o f t h i s s t u d y was t o q u a n t i f y b r o w s e p r o d u c t i o n i n one w e t l a n d a r e a t h a t i s u s e d f o r w i n t e r c a t t l e g r a z i n g . T h i s w o u l d a s s i s t i n d e f i n i n g a c a r r y i n g c a p a c i t y f o r t h e a r e a . - 1 -The o b j e c t i v e s o f t h i s s t u d y a r e t o : 1. C r i t i c a l l y r e v i e w l i t e r a t u r e p e r t a i n i n g t o s h r u b b i o m a s s and d e n s i t y e s t i m a t i o n , a s a b a s i s f r o m w h i c h t o s e l e c t a p p r o p r i a t e t e c h n i q u e s f o r e s t i m a t i o n o f c u r r e n t a n n u a l woody t w i g (browse) p r o d u c t i o n o f s h r u b s w i t h i n t h e s t u d y a r e a . 2. D e v e l o p s i m p l e o r m u l t i p l e l i n e a r r e g r e s s i o n e q u a t i o n s t o p r e d i c t b r o w s e b i o m a s s p r o d u c t i o n o f s t e m s o f Saiix s p p . and Setula gZancfulosa w i t h i n t h e s t u d y a r e a . 3. E s t i m a t e stem d e n s i t y by t h e c o r r e c t e d p o i n t d i s t a n c e method. 4. E s t i m a t e b r o w s e p r o d u c t i o n p e r s q u a r e m e t r e and d e f i n e 9 5 % c o n f i d e n c e l i m i t s o f t h e e s t i m a t e . 3. P r o v i d e a c r i t i c a l a n a l y s i s o f t h e t e c h n i q u e s e m p l o y e d i n t h e s t u d y . A f t e r r e v i e w i n g t h e l i t e r a t u r e , l i n e a r r e g r e s s i o n was s e l e c t e d a s t h e b r o w s e e s t i m a t i o n t e c h n i q u e b e c a u s e , a t l e a s t i n t h e o r y , o n c e t h e r e g r e s s i o n e q u a t i o n h a s been d e v e l o p e d i t i s n o n - d e s t r u c t i v e and e f f i c i e n t , and c o n f i d e n c e l i m i t s c a n be e a s i l y a t t a c h e d t o t h e p r e d i c t i o n . Browse was e s t i m a t e d on a p e r s t e m b a s i s a s s t e m s were more c l e a r l y d e f i n e d t h a n i n d i v i d u a l p l a n t s . The c o r r e c t e d p o i n t d i s t a n c e method, a p l o t l e s s t e c h n i q u e u s i n g d i s t a n c e m e a s u r e s , was d e t e r m i n e d t o be t h e most a p p r o p r i a t e d e n s i t y e s t i m a t i o n method a s i t i s more e f f i c i e n t t h a n p l o t m e t h o d s , c o m p e n s a t e s f o r non-random d i s t r i b u t i o n s o f i n d i v i d u a l s , and p r o v i d e s an a p p r o x i m a t i o n o f v a r i a n c e . The f i n a l e s t i m a t e o f b r o w s e p r o d u c t i o n p e r s q u a r e m e t r e was o b t a i n e d by m u l t i p l y i n g t h e mean p r o d u c t i o n p e r stem ( f r o m t h e r e g r e s s i o n ) by t h e mean s t e m d e n s i t y ( f r o m t h e c o r r e c t e d p o i n t d i s t a n c e m e t h o d ) . The r e s p e c t i v e v a r i a n c e s a r e c o m b i n e d f a r t h e c o n f i d e n c e i n t e r v a l o f t h i s e s t i m a t e . Pursuant to the second a b j e c t i v e ( i . e . to develop r e g r e s s i o n equations) the f o l l o w i n g hypotheses were proposed and t e s t e d : 1. R e l a t i o n s h i p s between shrub dimensions and browse biomass p r o d u c t i o n e x i s t and simple or m u l t i p l e l i n e a r r e g r e s s i o n e q u a t i o n s may be developed to p r e d i c t browse biomass p r o d u c t i o n . 2. A s i n g l e , common r e g r e s s i o n e q u a t i o n w i l l d e s c r i b e the f o u r Salix s p e c i e s i n the ar e a . 3. The common r e g r e s s i o n e q u a t i o n s w i l l a dequately p r e d i c t p r o d u c t i o n on s p e c i f i c s i t e s w i t h i n the wetland. 4. The a c t u a l browse biomass on s i t e s (measured from a double sample) i s not s i g n i f i c a n t l y d i f f e r e n t (at the 0.05 s i g n i f i c a n c e l e v e l ) from the p r e d i c t e d biomass. 5. There i s no s i g n i f i c a n t d i f f e r e n c e (at the 0.05 s i g n i f i c a n c e l e v e l ) i n the a c t u a l (measured) biomass between the f i r s t stem and the second stem encountered. 6. There i s no s i g n i f i c a n t d i f f e r e n c e (at the 0.05 s i g n i f i c a n c e l e v e l ) i n the p r e d i c t e d biomass between the f i r s t and second stem encountered. No hypotheses about stem d e n s i t y or the t o t a l browse p r o d u c t i o n per u n i t a r e a were t e s t e d because t h e r e was no check on the accuracy of the stem d e n s i t y e s t i m a t e . 2. LITERATURE REVIEW 2.1 I n t r o d u c t i o n The u s u a l purpose f o r a s s e s s i n g shrub biomass p r o d u c t i o n i s to q u a n t i f y browse a v a i l a b l e to w i l d l i f e and/or l i v e s t o c k (eg. B a s i l e and Hutchings 1966, T e l f e r 1969a, Bobek and Bergstrom 1978, Bartolome and Kosco 1982), or to q u a n t i f y the u t i l i z a t i o n of t h a t browse (eg. T e l f e r 1969a, Ferguson and Marsden 1977, Provenza and Urness 1981). Other purposes i n c l u d e assessment of f u e l accumulation (Dean e t a l . 1981) or q u a n t i f i c a t i o n of biomass components f o r e c o l o g i c a l s t u d i e s ( O r i g a l and Ohmann 1977, H a r r i n g t o n 1979). D e f i n i t i o n s of browse v a r y l g e n e r a l l y i t i s d e f i n e d as the c u r r e n t annual growth of leaves and twigs (eg. B a s i l e and Hutchings 1966, B l a i r 1971, Bobek and Bergstrom 1978). Other d e f i n i t i o n s r e l a t e i t to the l a r g e s t ( T e l f e r 1969a) or average (Shafer 1963) diameter of a browsed twig, which v a r i e s with both shrub s p e c i e s (Shafer 1963) and animal s p e c i e s (Peek et a l . 1971). Browse must a l s o be d e f i n e d v e r t i c a l l y as browsing animals can o n l y reach up a l i m i t e d d i s t a n c e ) t h i s h e i g h t v a r i e s with animal s p e c i e s and snow depth (snow cover may a l s o r a i s e the lower l i m i t of a v a i l a b i l i t y ) ( T e l f e r 1974). By b r e a k i n g stems, moose and deer can browse twigs t h a t would o t h e r w i s e be beyond t h e i r reach ( K r e f t i n g et a l . 1966, T e l f e r and C a i r n s 1978). Not a l l browse s p e c i e s a r e e q u a l l y p a l a t a b l e or p r e f e r r e d , and p r e f e r e n c e v a r i e s among animal s p e c i e s (Coady 1974). Sampling f o r browse p r o d u c t i o n and/or u t i l i z a t i o n i s d i f f i c u l t because of the o f t e n h i g h l y v a r i a b l e s p a t i a l d i s t r i b u t i o n of browse on the p l a n t and of the shrubs themselves. Browse o f t e n i s - 4 -not d i s t i n g u i s h a b l e i n a uniform way i n the - f i e l d , and i t c o n s i s t s o-f very many sma l l d i s c r e t e p a r t s ( R u t h e r f o r d 1979). Some of the methods used f o r shrub biomass e s t i m a t i o n are m o d i f i c a t i o n s of tec h n i q u e s developed f o r herbaceous v e g e t a t i o n . T h i s review w i l l b r i e f l y d e s c r i b e a number of methods which have been used to e s t i m a t e shrub p r o d u c t i o n , and w i l l c o n c e n t r a t e on r e g r e s s i o n e s t i m a t i o n . The reader i s a l s o r e f e r r e d to R u t h e r f o r d ' s (1979) and P i t t and Schwab's (1985) reviews of p l a n t based t e c h n i q u e s f o r e s t i m a t i o n of a v a i l a b l e and u t i l i z e d browse. 2.2 Review of shrub biomass e s t i m a t i o n methods In the weight e s t i m a t i o n method developed by Pechanec and P i c k f o r d (1937), o c u l a r e s t i m a t e s of f o r a g e biomass by s p e c i e s are made on p l o t s by p r a c t i s e d o b s e r v e r s . The e s t i m a t e s are c o r r e c t e d f o r p e r s o n a l b i a s by use of a r e g r e s s i o n developed from e s t i m a t e s on se p a r a t e p l o t s which are c l i p p e d and weighed. A m o d i f i c a t i o n i s to use c l u s t e r s of f i v e p l o t s , where p r o d u c t i o n i n the c o r n e r p l o t s i s esti m a t e d as a percentage of t h a t i n the c e n t r a l p l o t , which i s c l i p p e d and weighed (Hutchings and Schmautz 1969). Problems with the method Include the e x t e n s i v e t r a i n i n g p e r i o d , i n e v i t a b l e p e r s o n a l b i a s , and d i f f i c u l t y i n m a i n t a i n i n g o b s e r v e r c o n s i s t e n c y (Hutchings and Schmautz 1969). The method i s d i f f i c u l t to apply to shrubs (Shafer 1963) and l o s e s e f f i c i e n c y (Shepherd 1962). Wilm e t a l . (1944) m o d i f i e d weight e s t i m a t i o n to i n c o r p o r a t e double sampling. The c o r r e c t i o n f a c t o r was d e r i v e d by c l i p p i n g and weighing a p o r t i o n of the same p l o t s f o r which e s t i m a t e s were made and r e g r e s s i n g a c t u a l weight on es t i m a t e d weight. The r e l a t i o n s h i p may a l s o be expressed by a r a t i o e s t i m a t o r ( B l a i r 1959, F r a n c i s et - 5 -a l . 1979, Ahmed et a l . 1983), but t h i s tends to i n c r e a s e the v a r i a n c e . F r a n c i s et a l . (1979) noted an o v e r a l l tendency toward u n d e r e s t i m a t i o n , and a d e c l i n e i n p r e c i s i o n as the v a r i a b i l i t y i n p l a n t water content I n c r e a s e s . The c l i p and weigh method I n v o l v e s c l i p p i n g a l l -forage m a t e r i a l w i t h i n p r e - d e f i n e d h e i g h t l i m i t s on random (Bobek and Bergstrom 1978, K r e f t i n g and P h i l l i p s 1970) or s y s t e m a t i c a l l y random quadrats ( B l a i r 1971, H a l l s and A l c a n i z 1971, B l a i r and F e d u c c i a 1977, Harlow 1977). The method i s l a b o r i o u s , c o s t l y and d e s t r u c t i v e , and i s not s u i t a b l e f o r permanent p l o t s (Pechanec and P i c k f o r d 1937, Shafer 1963, R u t h e r f o r d 1979). An enormous sampling e f f o r t i s r e q u i r e d f o r r e a s o n a b l e p r e c i s i o n when the method i s a p p l i e d to shrubs (Bobek and D z i e c i o l o w s k l 1972, Harlow 1977). The method may be more u s e f u l in range s i t u a t i o n s where f o r a g e growth Is more homogeneous than shrub growth. Ranked s e t sampling ( M c l n t y r e 1932) may be c o n s i d e r a b l y more e f f i c i e n t than random c l i p and weigh, and b e t t e r s u i t e d f o r s i t u a t i o n s with high l o c a l v a r i a t i o n . A number of s e t s of quadrats are randomly l o c a t e d , and w i t h i n each s e t the quadrats a r e v i s u a l l y ranked by biomass. From the f i r s t s e t , the h i g h e s t r a n k i n g quadrat i s c l i p p e d and weighed, from the second s e t , the second h i g h e s t r a n k i n g quadrat, and so on. In the end, the number of c l i p p e d quadrats e q u a l s the number of s e t s , with equal r e p r e s e n t a t i o n of each rank. The method may be thought of as s t r a t i f i e d random sampling with the ranks as s t r a t a ( D e l l and C l u t t e r 1972). I t w i l l g i v e an unbiased e s t i m a t e of the mean which i s more p r e c i s e than c l i p p i n g the same number of completely randomly l o c a t e d quadrats. Ranking e r r o r s reduce e f f i c i e n c y but do not b i a s the e s t i m a t e ( M c l n t y r e 19S2, D e l l - 6 -and C l u t t e r 1972). The advantage o-f t h i s method i s l o s t i f l o c a l v a r i a b i l i t y i s low r e l a t i v e to l a r g e s c a l e v a r i a b i l i t y ( M c lntyre 1932). S h a f e r ' s (1963) twig count method i s based on c o n v e r t i n g a count of twigs to a weight of browse u s i n g an average twig weight f o r each s p e c i e s . The average browsing diameter i s determined f o r twigs of a g i v e n shrub s p e c i e s , then a sample of unbrowsed twigs are c l i p p e d at t h a t exact diameter t o determine the average twig weight. On random quadrats, a l l twigs are counted and the counts c o n v e r t e d to dry weight u s i n g the average twig weight. S h a f e r ' 3 c l a i m s to the speed and p r e c i s i o n of t h i s method have been qu e s t i o n e d by o t h e r workers who found i t too l a b o r i o u s f o r dense browse (Bobek and Bergstrom 1978, Parker and Morton 1978) and i n s e n s i t i v e to very heavy browse u t i l i z a t i o n (Jensen and S c o t t e r 1977). The a b i l i t y t o a c c u r a t e l y count a l l twigs w i t h i n a quadrat was c o n s i d e r e d q u e s t i o n a b l e by Shafer h i m s e l f . Herbaceous s t a n d i n g crop cover may be e s t i m a t e d , by s p e c i e s , in terms of predetermined cover u n i t s (Anderson and Kothmann 1982). Then mass per cover u n i t i s determined by c l i p p i n g and weighing a p o r t i o n of the samples, with mass per u n i t c a l c u l a t e d by l i n e a r r e g r e s s i o n . S t a n d i n g crop f o r any s p e c i e s i s cover u n i t s times mass per u n i t . The method was developed f o r herbaceous v e g e t a t i o n and l i k e l y i s very d i f f i c u l t to apply to shrubs. Browse u t i l i z a t i o n has been q u a n t i f i e d by the twig length method, a procedure i n v o l v i n g " b e f o r e " and " a f t e r " surveys (Aldous 1944, Smith and Urness 1962). In the f a l l , twigs are tagged and t h e i r l e n g ths measured. In the s p r i n g , f o l l o w i n g winter browsing, t h e i r r e s i d u a l l e n g t h s are measured and percent u t i l i z a t i o n i s - 7 -computed. The method i s more a c c u r a t e than twig counts or percentage e s t i m a t e s o-f browsing in s i t u a t i o n s o-f heavy u t i l i z a t i o n (Jensen and S c o t t e r 1977), but i t i s more l a b o r i o u s and c o s t l y because two surveys are needed ( B a s i l e and Hutchings 1966). Jensen and Urness (1981) developed a method to c a l c u l a t e p ercent u t i l i z a t i o n from measurements of twig diameter at the p o i n t of browsing, the twig b a s a l diameter and the twig t i p diameter. The advantage of t h i s method i s t h a t o n l y one survey i s needed a f t e r browsing animals have l e f t the range. 2.3 R e g r e s s i o n e s t i m a t i o n R e g r e s s i o n e s t i m a t i o n assumes that t h e r e i s a mathematical r e l a t i o n s h i p between shrub biomass and e a s i l y measured dimensions on the shrub. The technique has been used to p r e d i c t t o t a l woody and l e a f y biomass (eg. T e l f e r 1969b), c u r r e n t annual twig biomass (eg. Ohmann et a l . 1976), twig and l e a f biomass (eg. Peek et a l . 1971), f o l i a g e biomass (eg. H a r n i s s and Murray 1976), or f i n e f u e l biomass (eg. Dean et a l . 1981) from measurements such as stem diameter, canopy dimensions and p l a n t h e i g h t . I t has been a p p l i e d i n u t i l i z a t i o n s t u d i e s where the amount of browse consumed, in terms of twig length and/or twig weight, i s p r e d i c t e d from measuring twig diameter at the browsed t i p (eg. T e l f e r 1969a, Lyon 1970, Ferguson and Marsden 1977, Provenza and Urness 1981). The r e g r e s s i o n i s developed by s e l e c t i n g a sample of i n d i v i d u a l s of the p o p u l a t i o n under study on which are measured both the independent (dimensions) and dependent (biomass) v a r i a b l e s . U sing these data, a r e g r e s s i o n e q u a t i o n i s f i t t e d which d e s c r i b e s the r e l a t i o n s h i p between dimensions and biomass. The p o p u l a t i o n i s then - 8 -sampled -Far the s e l e c t e d independent v a r i a b l e ( s ) and the v a l u e s are input i n t o the r e g r e s s i o n e q u a t i o n to p r e d i c t the mean shrub p r o d u c t i o n . No d e s t r u c t i v e sampling i s r e q u i r e d a f t e r the e q u a t i o n has been developed. Shrub d e n s i t y must a l s o be determined to e s t i m a t e biomass on a u n i t a r e a b a s i s . 2.3.1 Developing the r e g r e s s i o n The i n i t i a l sampling -for dimensions and biomass may be random (Ohmann et a l . 1976), s y s t e m a t i c ( R i t t e n h o u s e and Sneva 1977, Dean et a l . 1981) or s e l e c t i v e so as to cover a s i z e or volume range (Tel-fer 1969b, Bryant and Kothmann 1979, Murray and Jacobson 1982) . Brown (1976) s e l e c t e d only undamaged stems. Lyon (1970) recommended s t r a t i f i c a t i o n on each shrub based on l o c a t i o n on the p l a n t . The s e l e c t e d shrubs must be r e p r e s e n t a t i v e of the p o p u l a t i o n i f an unbiased e s t i m a t e i s to be a c h i e v e d . Whittaker and Marks (1973) noted a tendency to s e l e c t v i g o r o u s t r e e s of good form over poorer ones, r e s u l t i n g in an upward b i a s of the e s t i m a t e . Damaged, d i s e a s e d or o t h e r w i s e imperfect twigs or p l a n t s should be i n c l u d e d i n order to d e r i v e an unbiased r e l a t i o n s h i p . U n f o r t u n a t e l y , t h i s i n e v i t a b l y r e s u l t s in reduced goodness of f i t and lower c o r r e l a t i o n c o e f f i c i e n t s ( T e l f e r and C a i r n s 1978, R u t h e r f o r d 1979), but the r e l a t i o n s h i p i s a more v a l i d one (Rutherford 1979). The p r e d i c t i o n s are o n l y r e l i a b l e w i t h i n the range of dimensions o c c u r r i n g in the sample ( T e l f e r 1969a); upward e x t r a p o l a t i o n , p a r t i c u l a r l y with 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 , may produce m i s l e a d i n g d a t a (Rutherford 1979). A b s o l u t e v a l u e s of dimensions w i l l y i e l d b e t t e r p r e d i c t i o n s than s i z e c l a s s e s (Ohmann et a l . 1976). Some commonly used independent v a r i a b l e s i n c l u d e stem - 9 -diameter measured e i t h e r at ground l e v e l (eg. T e l f e r 1969b) or at some a r b i t a r y h e i g h t (Ohmann et a l . 1976, Bobek and Bergstrom 1978) and t o t a l p l a n t h e i g h t (eg. R i t t e n h o u s e and Sneva 1977, Murray and Jacobson 1982). Often diameter and heigh t are combined as DxH (Bobek and Berstrom 1978) or D ZH (Crow 1978). Canopy dimensions may be u s e f u l p r e d i c t o r v a r i a b l e s . H a r n i s s and Murray (1976) used p l a n t h e i g h t times c i r c u m f e r e n c e . Dean et a l . (1981) used maximum and minimum crown diameter, crown depth and crown denseness. Peek (1970) measured the maximum crown diameter and the diameter at r i g h t a n g l e s to t h a t , and c a l c u l a t e d canopy a r e a f o r both an e l l i p s e and a c i r c l e and c a l c u l a t e d p l a n t volume as a r e a times h e i g h t . Bryant and Kothmann (1979) c a l c u l a t e d crown volume as an i n v e r t e d cone f o r f o u r d e s e r t s p e c i e s . Murray and Jacobson (1982) developed 15 independent v a r i a b l e s ( c i r c u m f e r e n c e , s u r f a c e a r e a and volume f o r v a r i o u s shapes) from diameter and h e i g h t measurements to p r e d i c t l e a f and twig biomass. Other l e s s common p r e d i c t o r v a r i a b l e s i n c l u d e shrub r i n g widths (averaged from two r a d i a l measurements) (Davis e t a l . 1972), b a s a l diameter of second order stems (stems b r a n c h i n g o f f of the primary s u p p o r t i n g stem) (Bartolome and Kosco 1982) and number of a e r i a l stems (Tappeiner and John 1973). R e g r e s s i o n e s t i m a t i o n has a l s o been a p p l i e d to u t i l i z a t i o n s t u d i e s . B a s i l e and Hutchings (1966) and Ferguson and Marsden (1977) were a b l e to p r e d i c t t o t a l twig length and weight f o r b i t t e r b r u s h iPurshia tridentata) b e f o r e browsing from the diameter at the base of the browsed twig. The remaining p o r t i o n c o u l d be c l i p p e d and weighed or i t s length measured, and the percentage u t i l i z e d e a s i l y c a l c u l a t e d by s u b t r a c t i o n . B a s i l e and Hutchings suggested t h a t s e p a r a t e e q u a t i o n s should be developed f o r each s i t e , but Ferguson - 10 -and Marsden -f e l t t h a t a more ge n e r a l e q u a t i o n would u s u a l l y be adequate. Provenza and Urness (1981) used a s i m i l a r method with b l a c k b r u s h (Coleogyne ramosissima), but r e l a t e d branch diameter to branch l e n g t h and weight (where a branch c o n t a i n s a number o-f t w i g s ) . They a l s o p r e d i c t e d the amount o-f browsed m a t e r i a l -from diameter measurements made at the browsed t i p . T h i s avoided the need to a c t u a l l y measure any l e n g t h s , s i n c e u t i l i z a t i o n was c a l c u l a t e d as p r e d i c t e d browse length or weight as a p r o p o r t i o n o-f p r e d i c t e d t o t a l l e n gth or weight. Tel-fer (1969a) developed r e g r e s s i o n e q u a t i o n s r e l a t i n g a i r dry twig diameter to oven dry twig weight -for 22 s p e c i e s o-f t r e e s and shrubs. The e q u a t i o n s c o u l d be used i n s u r v e y s where counts o-f twigs are combined with an average diameter a t p a i n t o-f browsing (dpb) . The weight c o r r e s p o n d i n g to each dpb i s e s t a b l i s h e d -from the s p e c i f i c curve f o r th a t s p e c i e s and the mean weight of the twig i s computed. U t i l i z e d twig weights, t a l l i e d by diameter c l a s s e s , c o u l d a l s o be e s t imated. A p o t e n t i a l problem with u s i n g browsed twig diameters to e s t i m a t e u t i l i z a t i o n i s the d i s c r e p a n c y between the f r e s h , a i r dry and oven dry diameters. F r e s h l y c o l l e c t e d twigs w i l l not always c l o s e l y approximate the dpb of twigs i n the f i e l d d u r i n g a browse survey. S p e c i e s d i f f e r e n c e s (such as amount of sap in twigs and whether or not the browsed end d i e s back) and the time of browsing (twigs browsed e a r l y i n winter have longer to dry) w i l l a f f e c t the dpb's measured d u r i n g browse s u r v e y s . For example P o t v i n (1981) found an average diameter decrease ( f o r a l l s p e c i e s ) of 8% f o l l o w i n g one week's a i r d r y i n g . - 11 -U n i f o r m l y measuring twig diameters may p r e s e n t d i f f i c u l t i e s i n s p e c i e s t h a t have f l a t t e n e d twigs. Twig diameters may be determined by a v e r a g i n g maximum and minimum diameters ( B a s i l e and Hutching 1966) or two diameters at 90 degrees from each other (Ferguson and Marsden 1977, Jensen and Urness 1981). B a s i l e and Hutchings (1966) noted s m a l l but s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s i n r e g r e s s i o n c o e f f i c i e n t s f o r measurements taken at d i f f e r e n t p a i n t s an the p l a n t . The p o i n t of measurement f o r twig b a s a l b a s a l diameter should be s p e c i f i c a l l y d e f i n e d . I t may be a g i v e n d i s t a n c e above the b a s a l bud s c a r (Lyon 1968, Ferguson and Marsden 1977) or immediately above the bud s c a r (Jensen and Urness 1981). Browsing i n t r o d u c e s an unknown amount of v a r i a b i l i t y i n t o the r e g r e s s i o n by a f f e c t i n g a shrub's growth form and amount of biomass p r o d u c t i o n . The o n l y author encountered who used browse c o n d i t i o n as an independent v a r i a b l e was Schwab (1985). He used t h r e e s u b j e c t i v e l y determined browse c o n d i t i o n c l a s s e s of unbrowsed, browsed and h e a v i l y browsed, d e s i g n a t e d as 0, 1 and 2 r e s p e c t i v e l y . Schwab found the v a r i a b l e browse c o n d i t i o n squared was very important i n a c c o u n t i n g f o r v a r i a t i o n . He suggested u s i n g at l e a s t f i v e browse c o n d i t i o n c l a s s e s r a t h e r than t h r e e , and s t r a t i f i c a t i o n based on browse c o n d i t i o n . Browsing e f f e c t s vary c o n s i d e r a b l y with both p l a n t s p e c i e s and the s e v e r i t y of browsing. C l i p p i n g to s i m u l a t e browse has been used to study the e f f e c t of continued browsing an p r o d u c t i o n . White b i r c h (Betula alba v a r . p a p y r i f e r a ) withstood c l i p p i n g of 50% of the c u r r e n t annual growth f o r s i x y e a r s without reduced p r o d u c t i o n , and w i l l o w (Salfx spp.) i n c r e a s e d p r o d u c t i o n in - 12 -response to 100% c l i p p i n g o-f c u r r e n t annual growth (Aldous 1952). Mountain maple \Acsr spicatun) which had 100% of i t s c u r r e n t annual growth c l i p p e d -for ten y e a r s maintained g r e a t e r regrowth than other clumps s u b j e c t e d to l e s s or no c l i p p i n g (Kre-fting et a l . 1966). Rabbitbrush (Cfirysatftannus i / i s c i d i f lorus) and snowberry \Synphoricarpos usee in iaides) were s e n s i t i v e to t i m i n g and i n t e n s i t y o-f d e f o 1 i a t i o n , with adverse e f f e c t s caused by removal o-f g r e a t e r than 30% o-f c u r r e n t growth ( W i l l a r d and McKell 1978). Browse p r o d u c t i o n was g r e a t e s t on Salix scouleriana shrubs in A l a s k a t h a t had been most h e a v i l y browsed the p r e v i o u s winters t o t a l c l i p p i n g -for two y e a r s a l s o s t i m u l a t e d browse p r o d u c t i o n i n t h i s s p e c i e s (Wol-f-f 1978). Animal p r e f e r e n c e i s another c o n s i d e r a t i o n , as some shrub s p e c i e s are p r e f e r r e d over o t h e r s and degree of u t i l i z a t i o n may vary c o n s i d e r a b l y (Coady 1974, Ponto 1983). S t r a t i f i c a t i o n by s p e c i e s groups based on browse p r e s s u r e or apparent animal p r e f e r e n c e may be u s e f u l . I t makes l i t t l e sense to i n c l u d e an u n p a l a t a b l e s p e c i e s in the same e s t i m a t e as d e s i r a b l e browse s p e c i e s . 2.3.2 R e g r e s s i o n models Forms of r e g r e s s i o n models t h a t have been used f o r e s t i m a t i o n of shrub biomass i n c l u d e simple and m u l t i p l e l i n e a r r e g r e s s i o n , q u a d r a t i c , s e m i - l o g , a l l o m e t r i c (and i t s l i n e a r farm of l o g - l o g ) , e x p o n e n t i a l and weighted l i n e a r . U n l e ss o n l y a small range of shrub dimension s i z e o c c u r s , b i o l o g i c a l d a t a r a r e l y f i t a simple l i n e a r r e l a t i o n s h i p , and a c u r v i l i n e a r r e g r e s s i o n e q u a t i o n w i l l p r o v i d e a b e t t e r f i t to the d a t a ( T e l f e r 1969a, R u t h e r f o r d 1979). For example, B a s i l e and Hutchings - 13 -(1966) were a b l e to - f i t a l i n e a r r e l a t i o n s h i p between twig diameter and twig weight o-f b i t t e r b r u s h s i n c e most o-f t h e i r twig diameters were l e s s than 2.4 mm. Tel-fer (1969a) encountered a diameter range o-f almost 40 mm and -found a c u r v i l i n e a r e q u a t i o n more a p p r o p r i a t e . L i n e a r r e g r e s s i o n theory r e q u i r e s t h a t the data - f i t a s t r a i g h t l i n e r e l a t i o n s h i p , and t h a t the v a r i a n c e or spread o-f o b s e r v a t i o n s i s equal at a l l p o i n t s a l o n g the r e g r e s s i o n l i n e . The advantage o-f a l i n e a r r e g r e s s i o n i s t h a t s o l u t i o n i s staight-forward u s i n g the standard l e a s t squares technique. However these assumptions a r e o f t e n v i o l a t e d by biomass data; the r e l a t i o n s h i p o f t e n appears c u r v i l i n e a r when p l o t t e d on normal graph paper with the v a r i a n c e i n c r e a s i n g as the magnitude of the independent v a r i a b l e i n c r e a s e s . These problems may be overcome by some type of t r a n s f o r m a t i o n of the data, so t h a t the transformed v a l u e s meet the assumptions of l i n e a r r e g r e s s i o n and the l e a s t squares technique may then be used. The a l l o m e t r i c e q u a t i o n , of the form Y = B X a, i s widely used as a b i o l o g i c a l model (Zar 1968) and o f t e n i s a p p l i e d to shrub biomass e s t i m a t i o n i n i t s l i n e a r i z e d form, achieved by 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 . The l o g - l o g model, lnY » InB + a lnX, i s c o m p a t i b l e to o r d i n a r y l e a s t squares f i t t i n g . Although m a t h e m a t i c a l l y e q u i v a l e n t to the a l l o m e t r i c model, i t i s not s t a t i s t i c a l l y e q uvalent f o r l e a s t squares s o l u t i o n , and can lead to b i a s e d e s t i m a t e s (Zar 1968). The l e a s t squares curve f i t t i n g t echnique seeks t o f i n d the unique s e t of parameter e s t i m a t e s which minimizes the r e s i d u a l sum of squares ( i . e . , £ (Yj - Y . ) 2 i s minimized). Using l e a s t squares to f i t the l o g a r i t h m i c form £ (lnYj - lnYj )* i s minimized, which - 14 -Is c l e a r l y not the same as the p r e v i o u s q u a n t i t y (Zar 1968). Zar does suggest c o n s i d e r i n g the l o g - l o g trans-formation when the homogeneity o-f v a r i a n c e assumption i s v i o l a t e d and the v a r i a n c e o-f the r e s i d u a l s (Yj - Y ; > i s r e l a t e d to the s i z e o-f X, but he c a u t i o n s about the in h e r e n t b i a s o-f the e s t i m a t e . C o n v e r t i n g the l o g a r i t h m i c e s t i m a t e o-f the dependent v a r i a b l e <lnY| ) and i t s v a r i a n c e back to o r i g i n a l u n i t s i s not as simple as merely t a k i n g the antil-ogs. I-f the d i s t r i b u t i o n o-f InYj at X; i s normal, the d i s t r i b u t i o n o-f Yf w i l l c e r t a i n l y be skewed (Zar 1968, B a s k e r v i l l e 1972), and in -fact, i-f the lo g a r i t h m s of the Y; 's are e x a c t l y normally d i s t r i b u t e d the a n t i l o g s of the lnY; 's w i l l be the medians, r a t h e r than the means of the skewed a r i t h m e t i c d i s t r i b u t i o n s ( B a s k e r v i l l e 1972). (Otherwise, i f not p e r f e c t l y lognormal, they w i l l be the geometric means (Munro 1974).) The b i a s e d e s t i m a t i o n of Yj , by not a c c o u n t i n g f o r the skewness of the d i s t r i b u t i o n , s y s t e m a t i c a l l y g i v e s an u n d e r e s t i m a t i o n of the t r u e v a l u e of Yf ( B a s k e r v i l l e 1972). The e s t i m a t e w i l l be b i a s e d by a f a c t o r of e^ and s e r i o u s u n d e r e s t i m a t i o n w i l l r e s u l t when the v a r i a n c e i s l a r g e (Mount-ford and Bunce 1973) . S e v e r a l authors have recommended adjustments to c o r r e c t t h i s b i a s ( B a s k e r v i l l e 1972, Beauchamp and Olson 1973, Mountford and Bunce 1973, Madgwick 1976). H a f l e y (1969) d i s c u s s e d the e f f e c t of the random element (£ ) on the model. When the e f f e c t of E, on the dependent v a r i a b l e (Y) v a r i e s with the v a l u e of the independent v a r i a b l e (X), E, has a m u l t i p l i c a t i v e e f f e c t : Y = E, B X a and the transformed l i n e a r form i s lnY = InB + a lnX +1n£ which may be s o l v e d by l e a s t squares. When the random e f f e c t i s independent of X, E, has an a d d i t i v e e f f e c t : Y = B X a + E, > and an i t e r a t i v e procedure r a t h e r than the - 15 -normal l e a s t squares must be used to e s t i m a t e the parameters (Hafley 1969). Zar <1968) recommended u s i n g an i t e r a t i v e procedure -for s o l v i n g a l l o m e t r i c r e g r e s s i o n models, r a t h e r than r e s o r t i n g to the l o g - l o g l i n e a r approximation, i n order to a v o i d the b i a s -from 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 . I t e r a t i v e procedures l o c a t e the minimum o-f the sum o-f the squared r e s i d u a l s by repeated computation u s i n g d i f f e r e n t v a l u e s f o r the parameters on each i t e r a t i o n u n t i l the sum of squares has ceased to change. Zar demonstrated an example i n which the i t e r a t i v e s o l u t i o n to the a l l o m e t r i c f u n c t i o n y i e l d e d s i g n i f i c a n t l y d i f f e r e n t parameter e s t i m a t e s and s m a l l e r s t a n d a r d e r r o r e s t i m a t e s than those from a standard l e a s t squares s o l u t i o n of the l o g - l o g e q u a t i o n . However one disadvantage of the i t e r a t i v e s o l u t i o n method i s the l a r g e amount of computation time i n v o l v e d ( H afley 1969). A l s o , the d i s t r i b u t i o n p r o p e r t i e s of the parameter e s t i m a t e s are not e a s i l y determined and cannot be r e l a t e d to the d i s t r i b u t i o n of the random element, so p r o b a b i l i t y statements cannot be made about them. Furthermore, because the i t e r a t i v e procedure i s a s o r t of "search" of an e r r o r s u r f a c e f o r a minimum, the e s t i m a t e s may converge on the s u r f a c e to a l o c a l minimum, not the a b s o l u t e minimum. The problem becomes more s e r i o u s as the number of parameters i n c r e a s e s (Hafley 1969). The problems of b i a s i n t r o d u c e d by u s i n g a 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 , and of s o l u t i o n by i t e r a t i v e r a t h e r than l e a s t squares, may be avoided by the use of weighted r e g r e s s i o n s (Schreuder and Swank 1971, 1973, Crow and L a i d l y 1980). The weights should be i n v e r s e l y p r o p o r t i o n a l to the v a r i a n c e of the r e s i d u a l s ( F u r n i v a l 1961). - 16 -Crow and L a i d l y (1980) -found t h a t the v a r i a n c e o-f Y (aboveground shrub biomass) was p r o p o r t i o n a l to the s i z e of the c o r r e s p o n d i n g X (basal d i a m e t e r ) , and so w e i g h t i n g f a c t o r s c o u l d l o g i c a l l y be expressed as some f u n c t i o n of the independent v a r i a b l e . The v a r i a n c e was d i r e c t l y e s t i m a t e d from the data. They t e s t e d weighted and unweighted l i n e a r and n o n - l i n e a r models, and found t h a t both n o n - l i n e a r and weighted l i n e a r models were a c c e p t a b l e a l t e r n a t i v e s to the l o g - l o g model, and recommended t h a t more c o n s i d e r a t i o n be given to a l t e r n a t i v e models. Schreuder and Swank (1973) a l s o recommended a weighted l i n e a r r e g r e s s i o n t echnique f o r e s t i m a t i o n of t r e e biomass and s u r f a c e area. Schreuder and Swank (1971, 1973) d e r i v e d t h e i r w e i g h t i n g f a c t o r s by d i v i d i n g sampled t r e e s i n t o t h r e e e q u a l l y spaced diameter c l a s s e s and c a l c u l a t e d v a r i a n c e w i t h i n each c l a s s . Choosing the "best" model can be a p e r p l e x i n g problem, e s p e c i a l l y when t r y i n g to e v a l u a t e d i f f e r e n t types of models, d i f f e r e n t s c a l e s , or d i f f e r e n t numbers of independent v a r i a b l e s . C o n v e n t i o n a l l y , the c o e f f i c i e n t of d e t e r m i n a t i o n , R z, has been used to measure goodness of f i t . I t i s v a l i d p r o v i d i n g the c o n d i t i o n s under which i t i s meant to be a p p l i e d are met (Schreuder and Swank 1971): 1) The independent v a r i a b l e s must be normally d i s t r i b u t e d (when lnX or X* i s used i t i s immediately known th a t these v a r i a b l e s are not normally d i s t r i b u t e d ) } and 2) i s never decreased by the i n c l u s i o n of other independent v a r i a b l e s , so comparison of R 2 i s meaningful o n l y when models have the same number of c o e f f i c i e n t s . F u r n i v a l (1961) suggested an approach which e v a l u a t e s models by comparing the product of the l i k e l i h o o d s of the v a l u e s of the - 17 -dependent v a r i a b l e under the d i f f e r e n t models. Schreuder and Swank <1971, 1973) and Crow and L a i d l y (1980) s u c c e s s f u l l y used i t to e v a l u a t e weighted and unweighted, l i n e a r and n o n - l i n e a r models. The l i k e l i h o o d s i n d i c a t e the p r o b a b i l i t y t h a t the d a t a came from the s p e c i f i e d model. The more c l o s e l y the s p e c i f i e d model resembles the t r u e model, the l a r g e r the l i k e l i h o o d . The approach assumes each model has normally and independently d i s t r i b u t e d e r r o r s with zero means and c o n s t a n t v a r i a n c e . The computed l i k e l i h o o d s not o n l y r e f l e c t the magnitude of the r e s i d u a l s but a l s o p a s s i b l e d e p a r t u r e s from the assumptions of n o r m a l i t y and c o n s t a n t v a r i a n c e , making t h i s a very u s e f u l s t a t i s t i c a l t o o l f o r e v a l u a t i n g a l l types of models (Schreuder and Swank 1971, Crow and L a i d l y 1980). Models must be d e f i n e d b e f o r e data a n a l y s i s , e n s u r i n g t h a t some f o r e t h o u g h t i s g i v e n to d e f i n i n g meaningful models. A disadvantage i s t h a t the number of c o e f f i c i e n t s to be determined a f f e c t s the s i z e of the l i k e l i h o o d (Schreuder and Swank 1971). Formulae f o r c a l c u l a t i o n of l i k e l i h o o d f u n c t i o n s f o r s p e c i f i c models are given in Schreuder and Swank (1971) and Crow and L a i d l y (1980). 2.3.3 Sampling the p o p u l a t i o n Once the shrub dimensions most important i n a c c o u n t i n g f o r v a r i a b i l i t y have been determined and a r e g r e s s i o n model has been developed, the next s t e p s are to sample the p o p u l a t i o n f o r the dimension measurements and f o r d e n s i t y in order to a r r i v e at a biomass e s t i m a t e on a per u n i t a r e a b a s i s . D e n s i t y e s t i m a t i o n w i l l be on the b a s i s of stems or p l a n t s , depending on whether biomass i s e s t i m a t e d f o r stems or p l a n t s . There are a number of d e n s i t y e s t i m a t i o n methods a v a i l a b l e - 18 -which are d i s c u s s e d in the next s e c t i o n . I-f one has decided to s t r a t i f y the p a p u l a t i o n , such as i n t o t a l l shrubs and law shrubs,, one should e s t i m a t e d e n s i t y f o r each s t r a t a s e p a r a t e l y . I t i s p r o b a b l y most convenient to make the p l a n t dimension measurements at the same time as one makes the d e n s i t y counts or measurements to maximize f i e l d e f f i c i e n c y . Although i t may be mare con v e n i e n t to a s s i g n p l a n t s to dimension c l a s s e s (eg. stem diameter c l a s s e s ) , more p r e c i s e measurements w i l l g e n e r a l l y lead to a more p r e c i s e e s t i m a t e (Ohmann et a l . 1976), so the a d d i t i o n a l e f f o r t i s worthwhile. 2.4 D e n s i t y sampling 2.4.1 P l o t sampling P l o t t e c h n i q u e s are probably the most commonly used technique to e s t i m a t e p l a n t d e n s i t y and the e a s i e s t to understand (Lyon 1968). A l l p l a n t s w i t h i n two dimensional sampling u n i t s are counted. Quadrats may be c i r c u l a r , square or r e c t a n g u l a r , of a g i v e n s i z e and shape and may be l o c a t e d e i t h e r randomly or s y s t e m a t i c a l l y w i t h i n the study a r e a . Quadrat s i z e should be r e l a t e d to the s i z e and B p a c i n g of the i n d i v i d u a l s i n order to m a i n t a i n a c c u r a c y of counts (Mueller-Dombois and E l l e n b u r g 1974). Three d i f f i c u l t i e s i n the a p p l i c a t i o n of quadrat counts are 1) the r e c o g n i t i o n of i n d i v i d u a l s , 2) boundary e f f e c t s ( i n c l u s i o n or - e x c l u s i o n of i n d i v i d u a l s on the boundary l i n e ) and 3) time r e q u i r e d to count i n d i v i d u a l s . The f i r s t two problems r e q u i r e a r b i t r a r y d e c i s i o n s to be made, wh i l e the t h i r d r e q u i r e s a c l e a r e v a l u a t i o n of the purpose of the study (Mueller-Dombois and E l l e n b u r g 1974). Quadrat methods have o f t e n been used as a check or i n - 19 -comparison with one or more p l o t l e s s methods (Cottam et a l . 1953, P i e l o u 1959, R i s s e r and Z e d l e r 1968, Boyd 1980), sometimes with s e v e r a l quadrat s i z e s b e i n g t e s t e d (Lyon 1968, Oldemeyer and R e g e l i n 1980). A c h i e v i n g a p r e c i s e d e n s i t y e s t i m a t e by p l o t sampling in clumped d i s t r i b u t i o n s may be very d i f f i c u l t . Cottam and C u r t i s (1956) s t a t e d "Quadrat d a t a in aggregated stands are so v a r i a b l e t h a t the number of quadrats r e q u i r e d i s almost i m p o s s i b l e to a t t a i n . " Lyon (1968) found t h a t no quadrat t e c h n i q u e c o u l d e s t i m a t e shrub d e n s i t y to h i s s p e c i f i c a t i o n s of 95% c o n f i d e n c e l e v e l + 10%, without needing a p r o h i b i t i v e l y l a r g e sample s i z e . Using 20 f t by 20 f t quadrats r e q u i r e d a s e a r c h of n e a r l y 2 a c r e s , and s m a l l e r more e f f i c i e n t quadrats r e q u i r e d from 400 to s e v e r a l thousand samples c o v e r i n g c l o s e to one a c r e . Oldemeyer and R e g e l i n (1980) found a c o n s i d e r a b l e i n c r e a s e in p r e c i s i a n in going from a 1 ma quadrat to a 5 m 1 quadrat, but l i t t l e or no improvement i n going to a 10 m z quadrat, and they recommended the 5 m* s i z e f o r shrub d e n s i t y sampling. 2.4.2 P l o t l e s s sampling P l o t l e s s or d i s t a n c e sampling i n v o l v e s the measurement of d i s t a n c e from random or s y s t e m a t i c p o i n t s to shrubs, or d i s t a n c e s between shrubs. Cottam et a l . (1953) and Cottam and C u r t i s (1956) developed f o u r d i s t a n c e measures which are based on the i d e a of the mean are a per p l a n t (M) which i s the r e c i p r o c a l of d e n s i t y , r a t h e r than on the number of p l a n t s per u n i t a rea. N/M i s a d i r e c t i n d i c a t i o n of p l a n t s p a c i n g , and can be c a l c u l a t e d by each of the d i f f e r e n t methods. In - 20 -the c l o s e s t i n d i v i d u a l method, the mean d i s t a n c e from sampling p o i n t to c l o s e s t p l a n t i s determined, which e q u a l s h a l f of the square r o o t of the mean area, so that mean d i s t a n c e times 2 . 0 0 e q u a l s \/M. In the nearest neighbour method an e s t i m a t e of the mean d i s t a n c e from each i n d i v i d u a l to i t s c l o s e s t neighbour i s made, and t h i s d i s t a n c e times 1 . 6 7 e q u a l s vfT. The random p a i r s method uses an angle of e x c l u s i o n c e n t r e d on the sampling p o i n t and b i s e c t i n g the c l o s e s t p l a n t ; the i n t e r p l a n t d i s t a n c e to the c l o s e s t p l a n t o u t s i d e the angle i s measured, then m u l t i p l i e d by a c o r r e c t i o n f a c t o r to o b t a i n the c o r r e c t d i s t a n c e . The p o i n t c e n t r e d q u a r t e r method uses f o u r quadrants c e n t r e d on the sampling p o i n t , i n each of which the p o i n t to n e a r e s t p l a n t d i s t a n c e i s measured; the average of the fo u r d i s t a n c e s e q u a l s vTf. These f o u r methods a l l assume a random d i s t r i b u t i o n of the i n d i v i d u a l s i n the p o p u l a t i o n . That i s , each o b j e c t i s l o c a t e d independently of a l l o t h e r s , and any o b j e c t has an equal and independent chance of o c c u r r i n g at any l o c u s . If a number of e q u a l - s i z e d sampling u n i t s were l a i d down randomly, the frequency of I n d i v i d u a l s per quadrat would be a Po i s s o n d i s t r i b u t i o n (Cattam et a l . 1 9 3 3 ) . Shrub p o p u l a t i o n s are r a r e l y random; i n d i v i d u a l s tend to be clumped or aggregated. When t e s t e d with nan-random shrub, bunchgrass and a r t i f i c i a l dot p o p u l a t i o n s the above f o u r methods y i e l d e d b i a s e d d e n s i t y e s t i m a t e s because they do not compensate f o r non-random d i s t r i b u t i o n (Lyon 1 9 6 8 , R i s s e r and Z e d l e r 1 9 6 8 , B a t c h e l e r 1 9 7 1 , Laycock and B a t c h e l e r 1 9 7 5 , Boyd 1 9 8 0 , Oldemeyer and R e g e l i n 1 9 8 0 ) . S e v e r a l methods have been developed which attempt to compensate f o r non-random d i s t r i b u t i o n s . In the wandering quarter - 2 1 -method Catana (1963) measured s e q u e n t i a l p l a n t to p l a n t d i s t a n c e s a l o n g a meandering t r a n s e c t determined by a c o n s t a n t compass b e a r i n g and a 90 degree angle o-f i n c l u s i o n c e n t r e d on s u c c e s s i v e p l a n t s . Catana ordered a l l d i s t a n c e measurements i n a -frequency d i s t r i b u t i o n , s e p a r a t i n g w i t h i n clump and between clump d i s t a n c e s . He c a l c u l a t e d d e n s i t y and s i z e o-f clumps and between clump d e n s i t i e s , and recombined the i n f o r m a t i o n -for an o v e r a l l d e n s i t y e s t i m a t e . U n f o r t u n a t e l y Catana gave no c a l c u l a t i o n f o r v a r i a n c e of the d e n s i t y e s t imate. M o r i s i t a (1937) developed the angle order method to g i v e unbiased d e n s i t y e s t i m a t e s of non-random p o p u l a t i o n s . The a r e a around each sampling p o i n t i s d i v i d e d i n t o f o u r equal quadrants c e n t r e d on the p o i n t s in each quadrant the d i s t a n c e to the t h i r d c l o s e s t p l a n t i s measured. T h i s y i e l d s two d e n s i t y e s t i m a t e s : f o r i n d i v i d u a l quadrants ( M A ) and quadrants combined (M 3), The d e v i a t i o n of these e s t i m a t e s from one another i s i n f l u e n c e d by the degree of non-randomness p r e s e n t . If M i i s g r e a t e r than M a , the M i e s t i m a t e i s accepted; i f s m a l l e r , then the two e s t i m a t e s are averaged (Boyd 1979). No v a r i a n c e c a l c u l a t i o n f o r m u l a i s g i v e n . Lyon (1968) assumed the same v a r i a n c e as t h a t of an e a r l i e r v e r s i o n of the method which d i d not c o r r e c t f o r non-randomness, and Laycock and B a t c h e l e r (1975) suggested u s i n g p o i n t e s t i m a t e s of d e n s i t y to c a l c u l a t e v a r i a n c e . B a t c h e l e r (1971, 1973) developed the c o r r e c t e d p o i n t d i s t a n c e method f o r o b t a i n i n g unbiased d e n s i t y e s t i m a t e s . At each sampling p o i n t one measures the d i s t a n c e from the p o i n t to the c l o s e s t i n d i v i d u a l ( r p ) , the d i s t a n c e from th a t p l a n t to i t s nearest neighbour ( r n ) and from t h a t i n d i v i d u a l to i t s n e a r e s t neighbour <<" ). - 22 -The b a s i c premise of the method i s t h a t the p o i n t to n e a r e s t p l a n t measurement <f*p) g i v e s the t r u e mean i n t e r p l a n t d i s t a n c e -for a random p o p u l a t i o n , and the two a d d i t i o n a l d i s t a n c e s c o r r e c t -for d e p a r t u r e s -from non-randomness. The r n and r m d i s t a n c e s c o r r e c t f o r f i r s t and second degree of a g g r e g a t i o n , r e s p e c t i v e l y ( i . e . clumping, and clumping w i t h i n clumps) ( B a t c h e l e r and B e l l 1970). B a t c h e l e r (1975) developed an e m p i r i c a l f o r m u l a f o r the "probable l i m i t of e r r o r " or PLE, which takes the farm PLE = t A D / \fr\ (where t i s Student's t, A i s a measure of non-randomness, D i s the d e n s i t y e s t i m a t e and n i s the number of sample p o i n t s ) . A D / V^n can be c o n s i d e r e d the " e m p i r i c a l analogue of the s t a n d a r d e r r o r " ( B a t c h e l e r 1975) and PLE i s the p r o b a b l e l i m i t of e r r o r of the d e n s i t y e s t i m a t e at a s p e c i f i e d c o n f i d e n c e l e v e l . The PLE broadens as the degree of d e p a r t u r e from u n i f o r m i t y i n c r e a s e s . B a t c h e l e r (1973) concluded t h a t PLE i s a r e a s o n a b l e e s t i m a t e of the e r r o r , p r o v i d e d t h a t e s t i m a t e s of t o t a l a g g r e g a t i o n are made, and t h a t few samples are of repeated d i s t a n c e measures to the same members. The c o r r e c t e d p o i n t d i s t a n c e method may be used with or without a maximum sea r c h d i s t a n c e (R) ( B a t c h e l e r 1973, 1975, Boyd 1979). If used, one would not s e a r c h beyond a p r e s p e c i f i e d d i s t a n c e R to f i n d an i n d i v i d u a l . A l a r g e s e a r c h d i s t a n c e may cause problems with having a few l a r g e " t a i l end" d i s t a n c e s in the r p d i s t r i b u t i o n , which i n turn a f f e c t s the d e n s i t y e s t i m a t e (D). G e n e r a l l y , p r e c i s i o n (as d e f i n e d by the r a t i o of PLE/D) i s improved as R i n c r e a s e s . But in a very aggregated p o p u l a t i o n , the PLE/D r a t i o becomes l e s s p r e c i s e a f t e r a p o i n t ( u s u a l l y a f t e r about 80% of the sample), and B a t c h e l e r - 23 -(1975) c o n s i s t e n t l y -found t h a t the most p r e c i s e e s t i m a t e ( i . e . minimum PLE/D) was a l s o most a c c u r a t e . Boyd (1979) kept running c a l c u l a t i o n s o-f PLE and D, made over a range o-f i n c r e a s i n g R (usi n g the r p , r n and r m d i s t a n c e s a s s o c i a t e d with the changing search l i m i t ) , the d e n s i t y e s t i m a t e with the s m a l l e s t PLE/D becoming the - f i n a l e s t i m a t e . However the method d i d not s i g n i f i c a n t l y Improve r e s u l t s , and in some cases more a c c u r a t e e s t i m a t e s were o b t a i n e d f o r se a r c h d i s t a n c e s t h a t were not optimum by the PLE/D c r i t e r i o n . PLE/D o v e r e s t i m a t e d the a c t u a l e r r o r , with the amount of o v e r e s t i m a t i o n i n c r e a s i n g as non-randomness i n c r e a s e d i n the p a p u l a t i o n (Boyd 1979). A disadvantage of any d i s t a n c e measure t e c h n i q u e i s t h a t i f one d e s i r e s d e n s i t y e s t i m a t e s f o r each s p e c i e s , d i s t a n c e s must be measured s e p a r a t e l y f o r each s p e c i e s (Laycock and B a t c h e l e r 1975). For t h i s reason Oldemeyer and R e g e l i n (1980) advocated the use of quadrats over d i s t a n c e measures. However f o r s i n g l e s p e c i e s i n f o r m a t i o n they d i d recommend the angle order method over c o r r e c t e d p o i n t d i s t a n c e , s i n c e they found i t to be s l i g h t l y more p r e c i s e than B a t c h e l e r ' s method (using the p o i n t e s t i m a t e method to e s t i m a t e v a r i a n c e ) . T h e i r e s t i m a t e s u s i n g c o r r e c t e d p o i n t d i s t a n c e a l s o encompassed the t r u e d e n s i t y w e l l w i t h i n 95% c o n f i d e n c e l i m i t s . Average e r r o r s were 14.3% f o r angle order and 16.7% f o r c o r r e c t e d p o i n t d i s t a n c e (Oldemeyer and R e g e l i n 1980). Although the c o r r e c t e d p o i n t d i s t a n c e method may be a c l o s e second to angle order in accura c y and p r e c i s i a n , i t i s much mare e f f i c i e n t . T e s t i n g the two methods with a r t i f i c i a l dot p o p u l a t i o n s , in t h r e e out of f o u r cases Boyd (1980) got more a c c u r a t e e s t i m a t e s with angle order, but both methods had l e s s than 20% e r r o r i n t h r e e - 24 -out of -four cases. In - f i e l d time t r i a l s , c o r r e c t e d p o i n t d i s t a n c e r e q u i r e d 60% o-f the time needed by angle o r d e r . Laycock and B a t c h e l e r (1975) o b t a i n e d esimates w i t h i n 20% o-f t r u e d e n s i t y with both methods i n t e s t s with n a t u r a l bunchgrass p o p u l a t i o n s t h a t were clumped to v a r y i n g degrees. C o r r e c t e d p o i n t d i s t a n c e r e q u i r e d h a l f the time of angle o r d e r . In summary, the c o r r e c t e d p o i n t d i s t a n c e method appears to be the most p r o m i s i n g t e c h n i q u e f o r e s t i m a t i n g d e n s i t y of low d i v e r s i t y shrub p o p u l a t i o n s . Because i t i s f a s t e r than angle o r d e r , i t may be p a s s i b l e to get a more p r e c i s e e s t i m a t e by sampling a g r e a t e r number of p o i n t s in the same time. To d e s c r i b e a comp1 ex'community (where i t i s not d e s i r a b l e to group shrub s p e c i e s ) , i t may be best to use a quadrat method f o r d e n s i t y e s t i m a t i o n . 2.5 Biomass per u n i t a r e a The e s t i m a t e of shrub biomass p r o d u c t i o n per u n i t a r e a w i l l be the e s t i m a t e of mean shrub biomass p r o d u c t i o n per stem (or p l a n t ) times the e s t i m a t e of d e n s i t y of stems (or p l a n t s ) per u n i t a r e a . It i s d e s i r a b l e to be a b l e to make a p r e d i c t i o n with a given amount of c o n f i d e n c e and p r e c i s i o n . L i n e a r r e g r e s s i o n p e r m i t s t h i s , but t h e r e are problems with e x p r e s s i n g the mean and v a r i a n c e a s s o c i a t e d with transformed d a t a and n o n - l i n e a r r e g r e s s i o n s . The problem i s compounded by the f a c t t h a t shrubs are seldom randomly d i s t r i b u t e d and a d e n s i t y method that accomodates non-randomness i s needed ( i . e . angle order or c o r r e c t e d p o i n t d i s t a n c e ) . Such d e n s i t y measures are not conducive to exact c a l c u l a t i o n of v a r i a n c e . M o r i s i t a (1957) d i d not g i v e a f o r m u l a f o r c a l c u l a t i n g v a r i a n c e f o r the angle order method, but a few authors have attempted to e s t i m a t e - 25 -i t (Lyon 1968, Laycock and B a t c h e l e r 1975). B a t c h e l e r ' s (1975) "probable l i m i t o-f e r r o r " (PLE) -formula i s an e m p i r i c a l approximation o-f the c o n f i d e n c e l i m i t s -for the c o r r e c t e d p o i n t d i s t a n c e d e n s i t y e s t imate. R u t h e r f o r d (1979) a l l u d e s to problems with e x p r e s s i n g f i n a l c o n f i d e n c e l i m i t s of biomass da t a per u n i t area. The o n l y d i s c u s s i o n encountered i n the r e l e v a n t l i t e r a t u r e was by Schwab (1985) who used l o g - l o g r e g r e s s i o n f o r browse biomass and the c o r r e c t e d p o i n t d i s t a n c e method f o r d e n s i t y . Because of the l o g - l o g t r a n s f o r m a t i o n , the c o n f i d e n c e i n t e r v a l around the e s t i m a t e i s assymmetrlc when the log-based upper and lower c o n f i d e n c e l i m i t s are back-transformed, i n d i c a t i n g t h a t t h e r e i s no simple r e l a t i o n s h i p between transformed and back-transformed v a r i a n c e . Schwab compromised by u n d e r e s t i m a t i n g the upper l i m i t and o v e r e s t i m a t i n g the lower l i m i t . For the v a r i a n c e of the d e n s i t y e s t i m a t e he used the P L E 3 . (Schwab noted t h a t the PLE might be reduced by r e a l l o c a t i n g d e n s i t y sampling e f f o r t to c o n c e n t r a t e more e f f o r t on heterogeneous p l o t s . ) For the e s t i m a t e of biomass per u n i t a r e a , the e r r o r s of the r e g r e s s i o n and d e n s i t y e s t i m a t e s must be combined, but t h e r e appears to be no g e n e r a l l y accepted way. Schwab used Goodman's (1960) fo r m u l a f o r the exact v a r i a n c e of p r o d u c t s , but questioned the s t a t i s t i c a l v a l i d i t y of the r e s u l t i n g v a r i a n c e , e s p e c i a l l y because the e m p i r i c a l PLE e s t i m a t e was used f o r the d e n s i t y v a r i a n c e . The o n l y other example found i n v o l v i n g e s t i m a t i o n of shrub p r o d u c t i o n on an a r e a b a s i s u s i n g r e g r e s s i o n e s t i m a t i o n was by Bobek and Bergstrom (1978), who combined r e g r e s s i o n e s t i m a t i o n of browse biomass of a l l s p e c i e s with quadrat d e n s i t i e s . (Twig counts were used f o r r a r e r s p e c i e s . ) They claimed t h e i r method was 11 times more - 26 -e f f i c i e n t than t r a d i t i o n a l c l i p and weigh! however they d i d not i n c l u d e time spent d e v e l o p i n g the r e g r e s s i o n e q u a t i o n . To a c h i e v e a 93% c o n f i d e n c e i n t e r v a l + 20%, they c a l c u l a t e d t h a t 17 3 m x 5 m p l o t s needed to be c l i p p e d and weighed, w h i l e the r e g r e s s i o n method r e q u i r e d 21 p l o t s with much l e s s t o t a l sampling time. However the d i s t r i b u t i o n of browse of i n d i v i d u a l s p e c i e s was h i g h l y v a r i a b l e and would have r e q u i r e d an enormous amount of sampling e f f o r t to q u a n t i f y to the same degree of p r e c i s i o n . For example, p i n e (Pinus silx/estrus) would have r e q u i r e d 104 p l o t s , spruce (Picas abiea) 533, b i r c h (Betula spp.) 77 and rowan (Sorbus aucuparia) 326. Schwab (1985) used r e g r e s s i o n with p l o t l e s s d e n s i t y sampling f o r browse biomass of t h r e e shrub groups: c o n i f e r o u s , t a l l deciduous and low deciduous, on 31 p l o t s (each 70 m x 120 m>. In g e n e r a l he achieved an 80% c o n f i d e n c e i n t e r v a l . + 30%, and on over h a l f of the p l o t s a c h i e v e d + 20%. Sampling each p l o t f o r dimensions and d i s t a n c e s took, on average, e i g h t man-hours. Four hundred and f o u r shrubs were d e s t r u c t i v e l y sampled to develop the r e g r e s s i o n e q u a t i o n s , t a k i n g 70 man-days to c o l l e c t m a t e r i a l and 90 man-days to a n a l y s e data. Although he c o u l d have improved e f f i c i e n c y by sampling fewer shrubs and r e a l l o c a t i n g sampling e f f o r t , t h i s i s p r o b a b l y a r e a s o n a b l e example of the time r e q u i r e d to develop the r e g r e s s i o n s . If an i n t e n s i v e i n v e n t o r y of an a r e a i s planned, i t may be worthwhile to spend the time to develop a r e g r e s s i o n t h a t can be used l o c a l l y over a p e r i o d of time. For a s i n g l e time study of a small area, the u t i l i t y and e f f i c i e n c y i s q u e s t i o n a b l e , and other methods may be more s u i t a b l e . The optimum sampling method should minimize complexity of f i e l d measurements, minimize c a l c u l a t i o n time, and maximize accura c y - 27 -of e s t i m a t i o n (Rut her-ford 1979). Although i t i s u s u a l l y p o s s i b l e to a t t a i n a c c e p t a b l e p r e c i s i o n through r e g r e s s i o n e s t i m a t i o n , o-f ten much e-f-fort i s r e q u i r e d . Because q u i c k e r and cheaper methods w i l l l i k e l y not p r o v i d e a c c e p t a b l e i n f o r m a t i o n , i n a r e s t r i c t e d r e s e a r c h program the q u e s t i o n of whether browse biomass e s t i m a t i o n should be done at a l l i s a very v a l i d one. 2.6 C o n c l u s i o n For t h i s study, a sampling method was d e s i r e d t h a t would be e f f i c i e n t in the f i e l d and o f f i c e , would p r o v i d e an a c c u r a t e and p r e c i s e e s t i m a t e with a known l e v e l of c o n f i d e n c e , and would be a p p r o p r i a t e f o r a p p l i c a t i o n on an o p e r a t i o n a l s c a l e . I t was f e l t t h a t r e g r e s s i o n e s t i m a t i o n of browse biomass based on shrub dimensions, combined with the c o r r e c t e d p o i n t d i s t a n c e method of d e n s i t y e s t i m a t i o n , would be most a p p r o p r i a t e f o r these purposes. R e g r e s s i o n e s t i m a t i o n p r o v i d e s an e f f i c i e n t and s t a t i s t i c a l l y a c c e p t a b l e e s t i m a t e . Once the r e g r e s s i o n r e l a t i o n s h i p has been e s t a b l i s h e d no a c t u a l c o l l e c t i o n of biomass i s needed. The c o r r e c t e d p o i n t d i s t a n c e method appeared t o be the most a p p r o p r i a t e d e n s i t y technique because i t i s designed f o r aggregated p o p u l a t i o n s , p r o v i d e s an e s t i m a t i o n of the l e v e l of e r r o r , and i s e f f i c i e n t i n terms of f i e l d time. The combination of the r e g r e s s i o n and d e n s i t y e s t i m a t e s would y i e l d an e s t i m a t i o n of browse biomass per u n i t area. - 28 -3. STUDY AREA 3.1 General d e s c r i p t i o n The study a r e a was l o c a t e d at Dick Meadow ( e l e v a t i o n 1400 m), an e x t e n s i v e and complex wetland system i n the F r a s e r P l a t e a u of c e n t r a l i n t e r i o r B r i t i s h Columbia ( F i g u r e 3.1). Dick Meadow i s in the B i g Creek d r a i n a g e , a t r i b u t a r y o-f the C h i l c o t i n R i v e r , and i s s i t u a t e d a p p r o x i m a t e l y 30 km south o-f B.C. Highway 20. I t i s i n the dry Sub-Boreal Spruce (SBSa) b i o g e o c l i m a t i c subzone, a r e g i o n c h a r a c t e r i z e d by r e l a t i v e dryness, extreme c o l d and a s h o r t growing season (Annas and Coupe 1979). The s o i l of the a r e a i s very rocky, d e r i v e d from morainal d e p o s i t s of unsorted g r a v e l s , c o b b l e s and stones l a r g e l y of v o l c a n i c o r i g i n . In d e p r e s s i o n a l areas t h i s may be o v e r l a i n by a s i I t y g l a c i o f l u v i a l or g 1 a c i o 1 a c u s t r i n e veneer. In most areas of the wetland, m i n e r a l s o i l i s o v e r l a i n by f i b r i c or mesic o r g a n i c d e p o s i t s ; these are t h i n ( l e s s than 15 cm) in meadows and s h r u b - c a r r s , and t h i c k e r i n f e n s . The main wetland e c o l o g i c a l a s s o c i a t i o n s are sedge f e n , shrub f e n , s h r u b - c a r r and meadow, d e f i n e d mainly by v e g e t a t i o n and depth of the o r g a n i c l a y e r (Runka and Lewis 1981). There a r e a l s o s e v e r a l s h a l low lakes and ponds i n the ar e a . The s u r r o u n d i n g f o r e s t cover i s lodgepo l e p i n e <Pinus contorts) of poor s i t e q u a l i t y , with l i t t l e f o r a g e or browse i n the u n d e r s t o r y . Within the wetland " i s l a n d s " of t r e e s occur on s l i g h t l y r a i s e d mounds. The meadows and s h r u b - c a r r s tend to occur i n bands al o n g the f o r e s t edge and around t r e e i s l a n d s , with the shrub and sedge fe n s occupying the lower and more c e n t r a l a reas. - 29 -A t r a c k -from the east -follows higher ground a l o n g the -forest edge and meadows to a c o r r a l and c a b i n , -from which a c a t t l e t r a i l l eads north to a shallow lake and another wetland system. From the abundance o-f droppings, c a t t l e use appears to be c o n c e n t r a t e d a l o n g the road and t r a i l , i n the -forest edges and m i n e r a l meadows b o r d e r i n g the -forest. Dick Meadow i s crown land a p a r t -from a s m a l l p r i v a t e h o l d i n g which i n c l u d e s the c a b i n and p a r t o-f the road. I t i s w i t h i n the Winter Range range management u n i t a d m i n i s t e r e d by the Range Branch o-f the B.C. F o r e s t S e r v i c e i n W i l l i a m s Lake. The Winter Range u n i t ( t o t a l a r e a a p p r o x i m a t e l y 230 km58 or 23,000 ha) i s used -for l a t e - f a l l and e a r l y w i n t e r c a t t l e g r a z i n g . In 1984, f o u r permitees held p e r m i t s -for a t o t a l o-f 765 head -from November 1 to December 31. C a t t l e tend to c o n c e n t r a t e i n the open meadows o-f the wetland systems which are common i n t h i s u n i t , where -forage and browse are a v a i l a b l e . No data are a v a i l a b l e an w i l d l i - f e use o-f Dick Meadow but - f i e l d o b s e r v a t i o n i n d i c a t e d u t i l i z a t i o n by moose d u r i n g both summer and wi n t e r . 3.2 S i t e d e s c r i p t i o n s Four s i t e s were s e l e c t e d to r e p r e s e n t wetland shrub a s s o c i a t i o n s w i t h i n the wetland ( F i g u r e 3.2). ( O r i g i n a l l y seven s i t e s were i d e n t i f i e d . Due to time c o n s t r a i n t s the number o-f s i t e s sampled was reduced to f o u r , but the o r i g i n a l s i t e numbers were r e t a i n e d . ) S i t e d e s c r i p t i o n methodology and ter m i n o l o g y f o l l o w Walmsley et a l . (1980); wetland a s s o c i a t i o n c l a s s i f i c a t i o n i s based on Runka and Lewis (1981) and Roberts (1984), the l a t t e r of which i s s p e c i f i c to the SBSa subzone. - 30 -Figure 3.1. Map of B r i t i s h Columbia showing location of study area. F i g u r e 3,2. L o c a t i o n s o-f s i t e s w i t h i n the study area. 3.2.1 S i t e 1 T h i s s i t e i s predominantly an o r g a n i c s h r u b - c a r r (Runka and Lewis 1981) or "Grey-leaved w i l l o w - moss - s h r u b - c a r r a s s o c i a t i o n " (Roberts 1984). I t has at l e a s t 10 cm o-f moderately to w e l l decomposed o r g a n i c s o i l o v e r l y i n g m i n e r a l s o i l . In p l a c e s , d e p o s i t s o-f up to 60 cm o-f o r g a n i c m a t e r i a l would make the c l a s s i - f i c a t i o n shrub -fen, however the v e g e t a t i o n i s mast s i m i l a r to Roberts' wet s h r u b - c a r r a s s o c i a t i o n . The s i t e i s l o c a t e d i n what pro b a b l y was once a small d r a i n a g e channel. Parent m a t e r i a l s a r e a s i l t y g 1 a c i o 1 a c u s t r i n e or glacio-f l u v i a l veneer o v e r l y i n g eroded a b l a t i o n moraine with a high content o-f v o l c a n i c o r i g i n c o b b l e s and stones. These c o b b l e s and ston e s o c c a s i o n a l l y appear at the s u r f a c e i n s o r t e d rock " p o o l s " . The s i t e i s l e v e l with a s t r o n g l y mounded s u r f a c e . The e c o l o g i c a l m o i s t u r e regime i s s u b h y g r i c to h y g r i c and the n u t r i e n t regime i s permesotrophic to e u t r o p h i c . The s i t e i s l o c a t e d w i t h i n an e x t e n s i v e shrubby a r e a . To the north and south a l o n g the o l d dr a i n a g e channel are shrub fen a s s o c i a t i o n s and to the e a s t and west on s l i g h t l y higher ground are dry s h r u b - c a r r s , s l o p i n g up to i s l a n d s of lo d g e p o l e p i n e f o r e s t . Shrub cover i s patchy with some dense areas and o t h e r s with s p a r s e cover. Dominant v e g e t a t i o n i s as f o l l o w s ( o c c u l a r l y e s t i m a t e d , in percent c o v e r ) : - 33 -shrub Iayer: Salix glauca 20% Betula glandulosa 13% dwarf shrub l a y e r : Salix myrtillifolia 20% Arctostaphylos u\/a-ursi 2% herb l a y e r : Carex aguatillis 5% Calanagrostis strict a 3% Valeriana dioica 2% Fragaria virginiana 1% Achillea millefolium 1% moss l a y e r : 20% Maxlmium shrub h e i g h t i s about 1.3 m; most shrubs a r e l e s s than 1 m t a l 1 . 3.2.2 S i t e 4 T h i s s i t e i s a f r e s h m i n e r a l s h r u b - c a r r (Runka and Lewis 1981) or a "Scrub b i r c h - k i n n i k i n n i c k s h r u b - c a r r a s s o c i a t i o n " (Roberts 1984). I t has o n l y a very t h i n (0 to 3 cm) l a y e r of moderately decomposed o r g a n i c s o i l o v e r l y i n g m i n e r a l s o i l . The mi n e r a l s o i l i s c o a r s e t e x t u r e d (SL t o S) and has a high c o a r s e fragment content (up to 40%) of mainly g r a v e l s d e r i v e d from a b l a t i o n moraine. The e f f e c t i v e r o o t i n g zone i s shallow (20 cm) due to the shallow depth to parent m a t e r i a l . The s i t e i s n e a r l y l e v e l with a moderately mounded s u r f a c e . The e c o l o g i c a l m o i s t u r e regime i s mesic and the n u t r i e n t regime i s mesotrophlc. V e g e t a t i o n i s more homogeneous than at S i t e 1 with more uniform shrub cover. B i r c h comprises the shrub l a y e r j w i l l o w i s v i r t u a l l y absent at t h i s s i t e . Dominant v e g e t a t i o n ( i n perc e n t c o v e r ) . i s : shrub l a y e r : Betula glandulosa 15% dwarf shrub l a y e r : Salix bracftycarpa 8% Arctostaphylos uva-urs i 2% herb l a y e r : /(atresia myosuro ides 50% fluf) lenbergia richardsonis 3% Carex praegraciI is 2% moss l a y e r ! 4% - 34 -Maximum shrub h e i g h t i s about 1.5 mj most shrubs are about .75 m or 1 ess. The s i t e grades i n t o a m i n e r a l meadow then i n t o l o d g e p o l e p i n e -forest. I t i s b i s e c t e d by the road to the c o r r a l and c a b i n . 3.2.3 S i t e 6 T h i s s i t e i s a -fresh m i n e r a l s h r u b - c a r r (Runka and Lewis 1981) or a "Scrub b i r c h - k i n n i k i n n i c k s h r u b - c a r r a s s o c i a t i o n " (Roberts 1984). There i s a t h i n (1 to 7 cm) l a y e r o-f p o o r l y decomposed o r g a n i c s a i l over m i n e r a l s o i l . Parent m a t e r i a l s a r e a s i l t y glacio-f l u v i a l or g 1 ac i o l a c u s t r i ne veneer with a high c o n t e n t (50%) o-f c o b b l e s and stones, over a g r a v e l l y c l a y - l o a m morainal d e p o s i t . The e-f-fective r o o t i n g zone i s shallow (30 cm) due to the high c o a r s e -fragment c o n t e n t . The s i t e i s n e a r l y l e v e l with a moderately mounded s u r f a c e . Rock " p o o l s " are common. The e c o l o g i c a l m o isture regime i s mesic and the n u t r i e n t regime i s permesotrophic. T h i s s i t e appears s l i g h t l y m o i s t e r and r i c h e r , with g r e a t e r v e g e t a t i o n a l d i v e r s i t y than S i t e 4 which i s a l s o a dry s h r u b - c a r r . S i t e 6 has g r e a t e r shrub cover with a much g r e a t e r w i l l o w component than s i t e 4. V e g e t a t i o n ( i n pe r c e n t caver) i s : shrub l a y e r : Betula glandulosa 25% Salix glauca 20% dwarf shrub l a y e r : Salix brachycarpa 10% Arctostaphylas uva-urs i 8% Salix myrtillifolia 2% herb l a y e r : /(atresia myosuroides 20% Antennari a pulcherrima 4% F ragar i a virginiana 2% Achillea millefolium 1% moss l a y e r : 8% Maximum shrub h e i g h t i s 1.5 mi most shrubs are 1 m t a l l or l e s s . T h i s s i t e s l o p e s very s l i g h t l y down to the south and grades - 35 -i n t o a narrow band o-f o r g a n i c s h r u b - c a r r where i t bo r d e r s on a sedge -fen and small pond. In the o p p o s i t e d i r e c t i o n i t s l o p e s s l i g h t l y upward, b o r d e r i n g a m i n e r a l meadow at the edge o-f lodgepole p i n e •forest. 3.2.4 S i t e 7 T h i s s i t e Is i n a deep mesic shrub -fen (Runka and Lewis 1981) or " M a c c a l l ' s w i l l o w - t a l l shrub -fen a s s o c i a t i o n " (Roberts 1984), with over 60 cm o-f moderately decomposed o r g a n i c s o i l . The s i t e i s l e v e l with s e v e r e l y mounded micratopography. The e c o l o g i c a l m o i s t u r e regime i s s u b h y d r i c and the n u t r i e n t regime i s meso- to permesotrophic. There was no s u r f a c e water i n August 1984, but an i n t e r m i t t e n t -frozen l a y e r was encountered at 35 cm depth. The s i t e i s l o c a t e d w i t h i n the c e n t r a l d r a i n a g e channel o-f the wetland system and i s surrounded by sedge -fen a s s o c i a t i o n s . Dominant p l a n t cover i s : shrub l a y e r : Salix arbusculo ides 20% Betula glandulosa 15% Salix mace a. 11 i ana 10% Salix glauca 5% dwarf shrub l a y e r : Salix m y r t i l l i i o l i a 10% herb l a y e r : Carex aquatill i s 25% moss l a y e r : 65% D e s p i t e the i m p l i c a t i o n of the name " t a l l shrub f e n " , the o v e r a l l appearance i s of dwarf shrubs, g e n e r a l l y l e s s than 50 cm and with many shrubs b a r e l y r e a c h i n g 30 cm. There are a few t a l l e r clumps (up to 2 m> alo n g the edges of the shrub f e n . - 36 -4. FIELD METHODS 4.1 R e g r e s s i o n e q u a t i o n s A d e c i s i o n was made to e s t i m a t e browse biomass on a per stem b a s i s , r a t h e r than on a per clump b a s i s , because of the d i f f i c u 1 t y of d e f i n i n g a clump. For d e v e l o p i n g r e g r e s s i o n e q u a t i o n s , S a l i x spp. and B e t u l a g l a n d u l o s a stems t a l l e r than 30 cm were c o l l e c t e d s e l e c t i v e l y from throughout the study area, with the i n t e n t i o n of r e p r e s e n t i n g the range of s i z e v a r i a t i o n p r e s e n t . These data are r e f e r r e d to as the "common data s e t " (as opposed to data from s p e c i f i c s i t e s ) and were used to develop "common" r e g r e s s i o n e q u a t i o n s . (Data from t r a n s e c t s on s p e c i f i c s i t e s were used to develop " s i t e - s p e c i f i c e q u a t i o n s " and " p o o l e d - s i t e e q u a t i o n s " . The f o l l o w i n g data were recorded from each stem c o l l e c t e d ( a b b r e v i a t i o n s used in r e g r e s s i o n e q u a t i o n s are shown in p a r e n t h e s e s ) : 1. b a s a l stem diameter (DIAM) - measured with d i a l c a l i p e r s to the nearest 0.1 mm at 1 cm above the ground. 2. stem length (LENGTH) - the length from the base of the stem (at the p o i n t where diameter was measured) to the stem t i p , measured to the nearest 1 cm. 3. canopy depth (DEPTH) - the d i s t a n c e from stem t i p to the base of the lowest p r o d u c t i v e branch ( i . e . branch with c u r r e n t annual twig growth), measured to the n e a r e s t 1 cm. 4. canopy width 1 (WID1) - the diameter of the canopy at the widest p o i n t , measured to the n e a r e s t 1 cm. 5. canopy width 2 (WID2) - the diameter of the canopy at r i g h t a n g l e s to canopy width 1, measured to the n e a r e s t 1 cm. 6. browse c o n d i t i o n (BRS) - c l a s s i f i e d i n t o c a t e g o r i e s based on a c c u i a r e s t i m a t e s of the degree of browsing the p r e v i o u s year. (Because the study was done i n J u l y and August, t h e r e was no browsing of the c u r r e n t y e a r ' s twig p r o d u c t i o n . ) - 37 -0 1 no browsing 1-25% o-f twigs browsed 2 25-50% o-f twigs browsed or s l i g h t hedging 3 50-75% of twigs browsed or moderate hedging 4 75-100% o-f twigs browsed or severe hedging (Hedging i s browsing o-f twigs o l d e r than the p r e v i o u s y e a r ' s growth.) 7. browse biomass (BIOMASS) - the oven dry weight o-f the c u r r e n t y e a r ' s woody twig p r o d u c t i o n . C u r r e n t annual twigs were c l i p p e d , the leaves removed, and the woody m a t e r i a l oven d r i e d at 50° C to a cons t a n t weight (at l e a s t 72 hours) and weighed an a d i g i t a l b a l ance to 0.01 g. Items 1 - 6 were used as independent v a r i a b l e s i n the development o-f r e g r e s s i o n e q u a t i o n s ; item 7, biomass, was the dependent v a r i a b l e . T a b l e s 4.1 and 4.2 summarize the raw stem dimension, browse and biomass d a t a t h a t were used i n e q u a t i o n development. Salix s p e c i e s were i d e n t i f i e d and r e c o r d e d . The number o-f o b s e r v a t i o n s -for each s p e c i e s was: Salix glauca Salix arbuscuIoides Salix maccalIi ana Salix myrti11iioIia n = 73 n » 26 n = 36 n » 25 - 38 -T a b l e 4.1. Summary o-f Salix common dat a s e t . V a r i a b l e N Mini mum Max imum Mean Std . Dev 1. Diameter (mm) 160 3.4 23. 1 8.4125 3.9765 2. Length (cm) 160 27 182 57.537 29.822 3. Depth (cm) 160 10 147 39.787 24.309 4. Width 1 (cm) 160 3 69 21.037 13.641 5. Width 2 (cm) 160 3 66 13.837 10.791 6. Browse 160 0 4 1.2562 1.5057 7. Biomass (g) 160 0.01 7.07 0.95544 1.2170 Table 4.2. Summary of Betula 3landulasa common d a t a s e t . Var i ab1e N Mi n imum Max imum Mean St d . Dev. 1. D i ameter (mm > 112 3. 5 14.3 7.7937 2. 7462 2. Length (cm) 112 27 153 65.723 29. 301 3. Depth (cm) 70 12 130 48.057 25. 993 4. Width 1 (cm) 112 3 50 18.214 9. 5610 5. Width 2 (cm) 112 2 35 11.839 6. 7831 6. Browse 112 0 4 .83929 1. 3190 7. Biomass (g) 112 0.01 4.51 .71804 . 82277 - 39 -4.2 Stem d e n s i t y To e s t i m a t e shrub p r o d u c t i o n on a biomass per u n i t a r e a b a s i s , an e s t i m a t e o-f stem d e n s i t y i s needed as w e l l as an e s t i m a t e o-f biomass per stem. D e n s i t y was e stimated u s i n g the c o r r e c t e d p o i n t d i s t a n c e method ( B a t c h e l e r 1973, 1975), d i s c u s s e d in d e t a i l in the l i t e r a t u r e review. B r i e - f l y , the method uses d i s t a n c e measures r a t h e r than p l o t s and c o r r e c t s -for the non-random d i s t r i b u t i o n o-f i n d i v i d u a l s . In t h i s study area, i n d i v i d u a l s (stems) were s t r o n g l y clumped, so the c o r r e c t e d p o i n t d i s t a n c e method was c o n s i d e r e d to be very s u i t a b l e . On each s i t e , at e q u a l l y spaced p o i n t s on t r a n s e c t s , d i s t a n c e measurements were c o l l e c t e d as -follows (to the n e a r e s t 1 cm): the d i s t a n c e -from the p o i n t to the c l o s e s t Salix stem (over 30 cm high) was recorded, then the d i s t a n c e s -from tha t stem to i t s c l o s e s t Salix neighbour and -from tha t neighbour to i t s c l o s e s t Salix neighbour were recorded. These measurements were repeated -for Betula, thus g i v i n g two s e t s o-f d i s t a n c e measurements, one -for each s p e c i e s . T r a n s e c t l i n e s were e s t a b l i s h e d to t r a v e r s e a given type o-f wetland a s s o c i a t i o n , a v o i d i n g edges and heterogeneous v e g e t a t i o n . The number o-f t r a n s e c t l i n e s v a r i e d with the s i t e , depending on the s i z e and shape o-f the wetland a s s o c i a t i o n . P o i n t s on the t r a n s e c t l i n e s were 2 - 4 m a p a r t , depending on the s i z e and d e n s i t y o-f the s i t e . The d i s t a n c e between p o i n t s i s not c r i t i c a l , p r o v i d i n g i t i s great enough that the same stem i s not measured -from two c o n s e c u t i v e p o i n t s . The search l i m i t , or the maximum d i s t a n c e w i t h i n which to s e a r c h -for a stem of the a p p r o p r i a t e s p e c i e s , was equal to the d i s t a n c e between p o i n t s . As i t was, stems were so dense t h a t the problem of remeasuring d i d not occur, and a stem was always - 40 -encountered w i t h i n the search l i m i t . 4.3 Biomass per u n i t a r e a Biomass data c o l l e c t e d from the s i t e s was combined with the d e n s i t y data to y i e l d an e s t i m a t e of browse biomass p r o d u c t i o n per u n i t a r e a (see Chapter 6, " S t a t i s t i c a l methods"). The t r a n s e c t s d e s c r i b e d above were a l s o used to c o l l e c t biomass data. At every second p o i n t on the t r a n s e c t s , stem dimensions and browse c o n d i t i o n ( v a r i a b l e s 1 - 6 d e s c r i b e d i n S e c t i o n 4.1) were c o l l e c t e d on the c l o s e s t stem of each of Salix and Betula. At every f o u r t h p a i n t , the c u r r e n t browse biomass of the measured stem was c o l l e c t e d , d r i e d and weighed. Thus the s i t e - s p e c i f i c data s e t i n c l u d e d a double sample f o r which both independent and dependent v a r i a b l e s were known. On s i t e 4, which c o n s i s t e d almost e n t i r e l y of Betula, data were c o l l e c t e d on both the f i r s t and second stems encountered. T h i s was done due to a suspected b i a s in the sampling d e s i g n , in tha t f i r s t stems tended to be ones on the o u t s i d e s of clumps and may have been s m a l l e r than i n s i d e stems. Table 4.3 shows the number of o b s e r v a t i o n s f o r d e n s i t y , indepependent v a r i a b l e s and biomass on each s i t e , f o r Salix and Be tula. - 41 -Table 4.3. Sample s i z e s -for d e n s i t y , independent v a r i a b l e s and biomass on s i t e s . S i t e Spec i e s Dens i t y Independent V a r i a b l e s Biomass Sa I i x BetuIs n = 200 n = 200 n n 100 100 n n 50 50 BetuIs Betu Is 146 n n 72 (1) 72 <2> n n 35 (1) 35 (2) Sal i x BetuIs n n 200 200 n n 150 150 n n 38 37 SaI ix Betula n n 195 195 n n 150 149 n n 42 42 <1) F i r s t stems (2) Second stems - 42 -5. STATISTICAL METHODS 5.1 R e g r e s s i o n e q u a t i o n s An i n i t i a l d e c i s i o n was made to keep the Salix and Betula d a t a s e p a r a t e , r a t h e r than develop one eq u a t i o n to d e s c r i b e both genera, because o-f suspected d i-f-f e rences i n animal p r e f e r e n c e . Separate p r o d u c t i o n e s t i m a t e s may be more u s e f u l to a w i l d l f e or range manager than a combined e s t i m a t e . A l l s t a t i s t i c a l t e s t s employed throughout the study used a 0.05 s i g n i f i c a n c e l e v e l . 5.1.1 H y p o t h e s i s 1: R e l a t i o n s h i p s between shrub dimensions and browse biomass p r o d u c t i o n e x i s t and simple or m u l t i p l e l i n e a r r e g r e s s i o n e q u a t i o n s may be developed to p r e d i c t browse biomass product i on. Least squares m u l t i p l e l i n e a r r e g r e s s i o n s f o r Salix and Betula were developed with the a i d of the MIDAS s t a t i s t i c a l package (Fox and G u i r e 1976), u s i n g the common d a t a s e t s ( i . e . those d a t a c o l l e c t e d from throughout the study a r e a s p e c i f i c a l l y f o r the r e g r e s s i o n e q u a t i o n s , not from s p e c i f i c s i t e s ) . Biomass was p l o t t e d a g a i n s t each stem dimension to examine the r e l a t i o n s h i p s , which were t y p i c a l l y c u r v i l i n e a r with i n c r e a s i n g v a r i a n c e (see Appendix A). T r a n s f o r m a t i o n s of the independent v a r i a b l e s were examined in an attempt to remove the c u r v i 1 i n e a r i t y and o b t a i n a homogeneous v a r i a n c e . T a b l e 5.1 l i s t s the v a r i a b l e s t e s t e d . - 43 -T a b l e 5.1. Independent v a r i a b l e s t e s t e d i n development o-f m u l t i p l e r e g r e s s i o n e q u a t i o n s . diameter (DIAM) length (LENGTH) * canopy depth (DEPTH) * canopy width 1 (WID1) * canopy width 2 (WID2) * browse (BRS) browse squared (BRS 2) browse cubed (BRS 3) diameter squared (D a) # diameter squared X length (D ZL) # (diameter squared X length) squared ( ( D 2 ) L ) 2 c i r c u l a r canopy a r e a (AREA C) * e l l i p t i c a l canopy a r e a (AREA E) # c i r c u l a r p l a n t volume (VOL C) * e l l i p t i c a l p l a n t volume (VOL E> * # n a t u r a l l o g a r i t h m s o-f these v a r i a b l e s were a l s o t e s t e d The c i r c u l a r canopy a r e a (AREA C) was c a l c u l a t e d -from the •formula -for the a r e a o-f a c i r c l e , TT r 2 , where the r a d i u s was h a l f of the average of the two width measurements! thus The e l l i p t i c a l canopy a r e a (AREA E) was computed as f o l l o w s : The c i r c u l a r and e l l i p t i c a l p l a n t volumes (VOL C and VOL E, r e s p e c t i v e l y ) were computed by m u l t i p l y i n g the r e s p e c t i v e areas by stem l e n g t h . The untransformed dependent v a r i a b l e , browse biomass, and i t s n a t u r a l l o g a r i t h m were r e g r e s s e d on the above v a r i a b l e s , u s i n g the MIDAS "forward" and "backward" procedure o p t i o n s . V a r i a b l e s were excluded from the model i f they were not s i g n i f i c a n t at the 0.05 AREA C - 77 AREA E =( - 2 - ) (WID 1 X WID 2) 4 l e v e l . - 44 -Once e q u a t i o n s were developed, p l o t s o-f the r e s i d u a l s a g a i n s t the p r e d i c t i o n s were examined -for l i n e a r i t y and homoskedast i c i t y . T e s t s -for the assumptions o-f l i n e a r i t y , equal v a r i a n c e and n o r m a l i t y were per-formed on the r e s i d u a l s as -follows: 1. L i n e a r i t y : R e s i d u a l s were ordered by the s i z e of the p r e d i c t i o n and d i v i d e d i n t o groups (four f o r Salix and s i x f o r Betula). A t t e s t f o r zero mean was made on each group. A l l t t e s t s had to be n o n - s i g n i f i c a n t at the 0.0S l e v e l f o r the model to be c o n s i d e r e d l i n e a r . A standard lack of f i t t e s t was not done because of lack of repeated o b s e r v a t i o n s f o r any gi v e n v a l u e of the independent v a r i a b l e s . 2. Equal v a r i a n c e . R e s i d u a l s were grouped as d e s c r i b e d above and a B a r t l e t t s t e s t f o r equal v a r i a n c e performed. 3. N o r m a l i t y of d i s t r i b u t i o n . A Chi sqare goodness of f i t t e s t f o r n o r m a l i t y was performed on r e s i d u a l s . 5.1.2 H y p o t h e s i s 2: A s i n g l e common r e g r e s s i o n e q u a t i o n w i l l d e s c r i b e the f o u r Salix s p e c i e s i n the area. T h i s h y p o t h e s i s was t e s t e d with a r e g r e s s i o n u s i n g dummy v a r i a b l e s f o r the f o u r s p e c i e s (Cunia 1973). Dummy v a r i b l e s were a s s i g n e d as f o l l o w s : i i f s p e c i e s 1 D l a 0 i f spec i es 2, 3, or 4 D 2 a 1 i f spec i e s 2 D 2 a 0 i f s p e c i e s 1, 3 or 4 D 4 : 0 i f spec i es 1,2 or 3 - 45 -New independent v a r i a b l e s were c r e a t e d by m u l t i p l y i n g each independent v a r i a b l e , , by the c o r r e s p o n d i n g dummy v a r i a b l e , D , such t h a t D l X 1 ' *1 i-f s p e c i e s 1 D l X 1 = o i f s p e c i e s 2, 3 or 4 D l X 2 - *2 i-f s p e c i e s 2 D l X 2 - 0 i-f s p e c i e s 1, 3 or 4 D 4 * 4 ™ * 4 i + s P e c i e s X ^  = 0 i-f s p e c i e s 1,2 or 3 A r e g r e s s i o n without i n t e r c e p t was - f i t t e d , u s i n g the dummy v a r i a b l e s and the new independent v a r i a b l e s . E s s e n t i a l l y such an eq u a t i o n i s e q u i v a l e n t to -four s e p a r a t e e q u a t i o n s . The dummy v a r i a b l e can be on l y 0 or 1, so the c o e f f i c i e n t f o r each dummy v a r i a b l e i s the i n t e r c e p t f o r t h a t s p e c i e s , and the c o e f f i c i e n t f o r each new v a r i a b l e i s the s l o p e f o r t h a t s p e c i e s . To t e s t i f i n t e r c e p t s were not s i g n i f i c a n t l y d i f f e r e n t , another r e g r e s s i o n was f i t t e d with dummy v a r i a b l e s but u s i n g o n l y one i n t e r c e p t . The d i f f e r e n c e i n the r e s i d u a l sum of squares was determined and d i v i d e d by the d i f f e r e n c e i n the r e s i d u a l degrees of freedom to o b t a i n the d i f f e r e n c e mean square, MS ^ . An F t e s t was performed as f o l l o w s : MS res where MS i s from the f i r s t ( l a r g e s t ) e q u a t i o n . res To t e s t i f s l o p e s were s i g n i f i c a n t l y d i f f e r e n t , a r e g r e s s i o n - 46 -with dummy v a r i a b l e s was - f i t t e d , u s i n g -four i n t e r c e p t s but o n l y one s l o p e coe-f-f i c i e n t -for each independent v a r i a b l e , and a s i m i l a r F t e s t was performed. To t e s t i f both s l o p e s and i n t e r c e p t s t o g e t h e r were s i g n i f i c a n t l y d i f f e r e n t , an eq u a t i o n was f i t t e d with one i n t e r c e p t and one s l o p e c o e f f i c i e n t f o r each independent v a r i a b l e ( i . e . the common e q u a t i o n ) , M S ^ f c a l c u l a t e d and a s i m i l a r F t e s t performed. 3.1.3 Hyp o t h e s i s 3: The common r e g r e s s i o n e q u a t i o n s w i l l a d e q uately p r e d i c t p r o d u c t i o n on s p e c i f i c s i t e s w i t h i n the wetland. In a manner analagous t o that d e s c r i b e d above, a r e g r e s s i o n with dummy v a r i a b l e s r e p r e s e n t i n g s i t e s was used to t e s t i f one common r e g r e s s i o n e q u a t i o n would p r e d i c t f o r s p e c i f i c s i t e s as well as s e p a r a t e e q u a t i o n s . The common dat a s e t was combined with the s i t e - s p e c i f i c data and dummy v a r i a b l e s a s s i g n e d t o r e p r e s e n t s i t e s 1, 4, 6, 7, and the common dat a s e t . T e s t s f o r s i g n i f i c a n t d i f f e r e n c e s i n i n t e r c e p t s , s l o p e s , and i n t e r c e p t s and s l o p e s t o g e t h e r were performed as p r e v i o u s l y d e s c r i b e d . The outcome of these t e s t s , f o r both Salix and Betula, i n d i c a t e d t h a t a s i g n i f i c a n t l o s s i n p r e c i s i o n r e s u l t e d from u s i n g the common eq u a t i o n s on s p e c i f i c s i t e s (see S e c t i o n 6.3). T h e r e f o r e , a d d i t i o n a l e q u a t i o n s were developed f o r both genera. T r a n s e c t data from a l l s i t e s were combined i n t o a s i n g l e data s e t and used to devlop " p o o l e d - s i t e e q u a t i o n s " f o r Salix and Betula (the common dat a s e t was not i n c l u d e d ) . " S i t e - s p e c i f i c e q u a t i o n s " were developed u s i n g only s i t e d a t a from the r e s p e c t i v e s i t e . In a l l cases the eq u a t i o n s were of the l o g - l o g form. The r e g r e s s i o n e s t i m a t e s of mean biomass per stem on each s i t e were c o r r e c t e d f o r the b i a s i n h e r e n t i n l o g a r i t h m i c - 47 -t r a n s f o r m a t i o n . Each p r e d i c t i o n o-f logY^ was back-transformed by a 2? t a k i n g the a n t i l o g , then c o r r e c t e d -for b i a s by the f a c t o r e (where a 2 was the r e g r e s s i o n mean square r e s i d u a l ) (Mountford and Bunce 1973), then the average of the back-transformed and c o r r e c t e d e s t i m a t e was c a l c u l a t e d . The v a r i a n c e of the es t i m a t e d mean p r o d u c t i o n per stem was c a l c u l a t e d i n a s i m i l a r f a s h i o n . The v a r i a n c e . S'2 , was logY c a l c u l a t e d f o r each p r e d i c t i o n , logY^ . Each v a r i a n c e was back-transformed by t a k i n g the a n t i l o g , c o r r e c t e d by the f a c t o r & 1 and the mean of these back-transformed and c o r r e c t e d v a r i a n c e s was c a l c u l a t e d to y i e l d the v a r i a n c e of the mean biomass per stem, S 2 | , in o r i g i n a l u n i t s . C o n f i dence l i m i t s were a t t a c h e d to the e s t i m a t e of mean p r o d u c t i o n per stem in the normal way: Y ± t • S£ These c a l c u l a t i o n s were made f o r Salix and Betula on a l l s i t e s u s i n g the common, p o o l e d - s i t e and s i t e - s p e c i f i c e q u a t i o n s . 5.1.4 Hy p o t h e s i s 4: The a c t u a l browse biomass on s i t e s i s not s i g n i f i c a n t l y d i f f e r e n t from the p r e d i c t e d biomass. T h i s h y p o t h e s i s was t e s t e d with a p a i r e d t t e s t , where each o b s e r v a t i o n of a c t u a l (measured) biomass was p a i r e d with i t s r e s p e c t i v e p r e d i c t i o n , and the d i f f e r e n c e ( e q u i v a l e n t to the r e s i d u a l ) found. The average of these d i f f e r e n c e s or r e s i d u a l s was compared with zero i n the t t e s t : where d = Y^ - Y^ . T h i s was done f o r Salix and Betula, on each s i t e , u s i n g p r e d i c t i o n s from the common, p o o l e d - s i t e and - 48 -s i t e - s p e c i-f i c e q u a t i o n s . 5.1.5 H y p o t h e s i s 5: There i s no s i g n i f i c a n t d i f f e r e n c e i n the a c t u a l browse biomass between the f i r s t stem and the second stem encountered of Betula. Using s i t e 4 data, t h i s h y p o t h e s i s was t e s t e d with a p a i r e d t t e s t , where o b s e r v a t i o n s of a c t u a l (measured) biomass from f i r s t and second stems were p a i r e d and the d i f f e r e n c e computed: d = Y i ( l ) - Y K 2 ) and the average of these d i f f e r e n c e s compared with zero. 5.1.6 H y p o t h e s i s 6: There i s no s i g n i f i c a n t d i f f e r e n c e i n the p r e d i c t e d browse biomass between the f i r s t stem and the second stem encountered of Betula. Again a p a i r e d t t e s t was used with s i t e 4 data, where p r e d i c t i o n s of biomass from f i r s t and second stems were p a i r e d and the d i f f e r e n c e computed: d • Y i ( l ) " Y i ( 2 ) The average of these d i f f e r e n c e s was compared with zero. T h i s was done f o r p r e d i c t i o n s from the common, p o o l e d - s i t e and s i t e - s p e c i f i c e q u a t i o n s . 5.2 D e n s i t y A computer program f o r a m o d i f i c a t i o n of B a t c h e l e r ' s (1971, 1973) c o r r e c t e d p o i n t d i s t a n c e method, o b t a i n e d from the Canadian W i l d l i f e S e r v i c e , was used to e s t i m a t e Salix and Betula stem d e n s i t y on the s i t e s . In t h i s program s u c c e s s i v e i t e r a t i o n s are made, u s i n g s u b s e t s of the e n t i r e d a t a s et with p r o g r e s s i v e l y - 49 -i n c r e a s i n g s e a r c h l i m i t s (the maximium d i s t a n c e from the t r a n s e c t p o i n t w i t h i n which to encounter a stem). The program then s e l e c t s the e s t i m a t e which has the s m a l l e s t v a r i a b i l i t y as the best d e n s i t y e s t imate. At each t r a n s e c t p o i n t , d i s t a n c e measurements were made s e p a r a t e l y f o r Salix and Betula, making i t p o s s i b l e to run the program on each data s e t Independently. T h i s r e s u l t s i n a b i t t t e r e s t i m a t e than when d i s t a n c e measures t o d i f f e r e n t s p e c i e s groups are combined (Laycock and B a t c h e l e r 1975). The p r o b a b l e l i m i t of e r r o r (PLE) gi v e n by t h i s program i s t r e a t e d as the standard d e v i a t i o n of the d e n s i t y e s t i m a t e (Smith 1985, p e r s . comm.), t h e r e f o r e i n the v a r i a n c e c a l c u l a t i o n s (see next s e c t i o n ) the v a r i a n c e of the d e n s i t y e s t i m a t e i s r e p r e s e n t e d by P L E 2 . 5.3 Browse biomass per u n i t a r e a Browse biomass i n grams per square metre was c a l c u l a t e d as mean p r o d u c t i o n per stem times stem d e n s i t y . Because t h i s f i n a l e s t i m a t e i s a product of two e s t i m a t e s which each have v a r i a n c e s , the v a r i a n c e of the f i n a l e s t i m a t e i s determined by combining the two v a r i a n c e s u s i n g the f o l l o w i n g f o r m u l a (Goodman I960): 2 2 2 2 s s s s o o y ? x x y s = x + Y -X y n ( Y ) n(X) n(X) n ( Y ) where X = estimated browse p r o d u c t i o n per stem, and Y =° estimated stem d e n s i t y (The per stem biomass e s t i m a t e and v a r i a n c e are back-transformed and c o r r e c t e d f o r b i a s . ) The c o n f i d e n c e l i m i t s of the est i m a t e d biomass per square - 50 -metre are, there-fore, XY + t • These c a l c u l a t i o n s were made -for Salix and Betula, on a l l s i t e s , u s i n g the common, p o o l e d - s i t e and s i t e - s p e c i-f i c e q u a t i o n s . - 31 -6. RESULTS 6.1 Hypo t h e s i s 1: R e l a t i o n s h i p s between shrub dimensions and browse biomass p r o d u c t i o n e x i s t and simple l i n e a r or m u l t i p l e r e g r e s s i o n e q u a t i o n s may be developed to p r e d i c t browse biomass p r o d u c t i o n . P l o t s o-f Salix and Betula browse biomass a g a i n s t the independent dimension v a r i a b l e s (diameter, l e n g t h , depth, width 1 and width 2) r e v e a l e d c u r v i l i n e a r r e l a t i o n s h i p s with the v a r i a n c e i n c r e a s i n g p r o p o r t i o n a l l y to the independent v a r i a b l e . No r e l a t i o n s h i p was apparent i n the p l o t of browse biomass on browse c o n d i t i o n . (See Appendix A.) Using the MIDAS s t a t i s t i c a l package (Fox and G u i r e 1976), v a r i o u s trans-formations, i n c l u d i n g n a t u r a l l o g a r i t h m s , of the independent and dependent v a r i a b l e s ( l i s t e d i n Table 5.1) were t e s t e d , and l e a s t squares m u l t i p l e l i n e a r r e g r e s s i o n s were developed. For both the Salix and Betula common d a t a s e t s , the r e l a t i o n s h i p s were best d e s c r i b e d by l o g - l o g models. 6.1.1 Salix The best e q u a t i o n developed f o r Salix was log BIOMASS - -6.1708 + .46478(log D») + .68509(log DEPTH) + .44336(log WID2) + .10892(BRS) R 2 » .81697 SE »• .48720 n » 160 The a n a l y s i s of v a r i a n c e i s g i v e n i n Appendix B. S l i g h t l y higher R* v a l u e s were o b t a i n e d with models u s i n g the f o l l o w i n g v a r i a b l e s : l o g - D z L , l o g LENGTH, log WID2 and BRS (R 2 = .81904) - 52 -l o g D a, log VOL E, log AREA E and BRS (R* • .81837) However the - f i r s t model was s e l e c t e d because i t r e q u i r e s -fewer measurements i n the f i e l d ( i . e . f o u r v e r s u s f i v e and f i v e , r e s p e c t i v e l y ) . In l i n e a r r e g r e s s i o n c e r t a i n assumptions are made about the e r r o r s or r e s i d u a l s : t h a t they a r e independent, have a mean of zero, have a c o n s t a n t v a r i a n c e , and f o l l o w a normal d i s t r i b u t i o n . The model must be c o r r e c t and meet these assumptions i f the mean square r e s i d u a l i s to p r o v i d e a v a l i d e s t i m a t e of a 2 - If t h e r e i s lack of f i t , t h a t i s , the d a t a are not l i n e a r , the model w i l l p r e d i c t i n a d e q u a t e l y . C o n fidence i n t e r v a l s w i l l not be v a l i d i f the model 1 i n c o r r e c t (Draper and Smith 1966). The f o l l o w i n g t e s t s were performed on the r e s i d u a l s c a l c u l a t e d by the Salix r e g r e s s i o n e q u a t i o n to t e s t the assumptions of r e g r e s s i o n : 1) t t e s t s on f o u r groups of r e s i d u a l s to t e s t f o r l i n e a r i t y 2) B a r t l e t t s t e s t s f o r equal v a r i a n c e , u s i n g f o u r groups of r e s i d u a l 3) Goodness of f i t t e s t f o r normally d i s t r i b u t e d r e s i d u a l s . R e s u l t s of the t e s t s are summarized i n T a b l e 6.1. The model d i d meet the assumption of l i n e a r i t y but f a i l e d the t e s t s f o r equal v a r i a n c e and normal d i s t r i b u t i o n of r e s i d u a l s . In the t e s t f o r v a r i a n c e , the f i r s t group of r e s i d u a l s (n»40) had a v a r i a n c e two to th r e e times t h a t of the other groups. P r e d i c t i o n s from t h i s model should be a c c u r a t e because the model passed the t e s t f o r l i n e a r i t y . A p l o t of the r e s i d u a l s of the transformed model showed a marked r e d u c t i o n i n h e t e r o s c a d a s t i c i t y from the untransformed data, even though the assumption was not met. - 53 -A weighted r e g r e s s i o n may have b e t t e r handled the unequal v a r i a n c e , but i t i s more complex to compute. I t i s r e a s o n a b l e to assume that the non-transformed dependent v a r i a b l e s are normally d i s t r i b u t e d at every v a l u e of X. If t h i s i s t r u e , then in a l o g - t r a n s f o r m e d model the r e s i d u a l s would not be normally d i s t r i b u t e d but would be skewed by the t r a n s f o r m a t i o n . T r a n s f o r m a t i o n may s o l v e some of the problems of v i o l a t i n g assumptions, but w i l l i n t u r n c r e a t e new problems by v i o l a t i n g other assumptions. In t h i s case, the l i n e a r i t y was a c h i e v e d by t r a n s f o r m a t i o n , homogeneity of v a r i a n c e was improved by t r a n s f o r m a t i o n (although the assumption was not met), but the d i s t r i b u t i o n of r e s i d u a l s was skewed by the t r a n s f o r m a t i o n . The user should be aware of these l i m i t a t i o n s of the p r e d i c t i o n s d e r i v e d from the e q u a t i o n . Because of the d i f f i c u l t y i n g r a p h i c a l l y d e p i c t i n g c o n f i d e n c e i n t e r v a l s f o r r e l a t i o n s h i p s of more than two dimensions, s e v e r a l examples r e p r e s e n t i n g v a r i o u s s i z e s of shrub stems with d i f f e r e n t degrees of browsing a r e g i v e n , with the p r e d i c t e d browse biomass and a s s o c i a t e d c o n f i d e n c e i n t e r v a l (Table 6.2) In a l l cases the p r e d i c t i o n s and c o n f i d e n c e l i m i t s shown are backtransformed to o r i g i n a l u n i t s and c o r r e c t e d f o r b i a s by the f a c t o r e (Mountford and Bunce 1973). The c o r r e c t i o n f a c t o r f o r Salix was 1.12602. In oth e r words, simply t a k i n g the a n t i l o g of l o g y ± would underestimate the c o r r e c t v a l u e by 12.6%. The Salix common r e g r e s s i o n model suggests t h a t browsing s t i m u l a t e s browse biomass p r o d u c t i o n . A h e a v i l y browsed stem should produce more browse biomass than a l i g h t l y browsed or unbrowsed stem of the same dimensions. T h i s s t i m u l a t i o n by browsing i s supported by - 54 -browse s i m u l a t i o n s t u d i e s on v a r i o u s s p e c i e s i n c l u d i n g Salix (Aldous 1952, K r e f t i n g et a l . 1966). The c o n f i d e n c e i n t e r v a l around the p r e d i c t i o n i s skewed because o-f the back-trans-format ion from l o g a r i t h m i c v a l u e s . The width of the c o n f i d e n c e i n t e r v a l as a percentage of the mean ranges from a p p r o x i m a t e l y + 7.5% at the means of the l o g a r i t h m s of the independent v a r i a b l e s , to a p p r o x i m a t e l y + 30% at the maximum observed v a l u e s f o r the independent v a r i a b l e s . T a b l e 6.1. T e s t s of assumptions of r e g r e s s i o n on Salix common r e g r e s s i o n . C a l c u l a t e d C r i t i c a l S i g n i f i c a n c e T e s t s t a t i s t i c v a l u e at 0.05 l e v e l 1. L i near i ty t, = 0.1707 t = 2.021 N. S. t t =-0.6001 t = 2.021 N.S. = 0.4649 t = 2.021 N.S. =-0.2172 t = 2.021 N.S. 2. Equal v a r i a n c e -x2 = 16.9293 x2 = 7.815 S i g . 3. Normali ty x2 = 102.781 x2 = 16.919 S i g . - 55 -Table 6.2. Some examples o-f p r e d i c t i o n s and confidence i n t e r v a l s -for Satix based on the common regression equation. Predicted DIAM DEPTH WID 2 BRS Browse 93% Confidence Interval Bi amass (mm) (cm) (cm) (g) 3. 4 10. 0 3.0 0 + 0.05797 0.04687 0.07172 3.4 10.0 3.0 1 0.06463 0.03344 - 0.07820 3.4 10.0 3.0 2 0.07208 0.05977 - 0.08693 3.4 10.0 3.0 3 0.08038 0.06333 - 0.09839 3.4 10.0 3. 0 4 0.08963 0.07072 - 0.11339 6.4 23. 0 9.0 0 0.31893 0.28297 - 0.35945 6.4 23.0 9. 0 2 0.39635 0.35909 - 0.43792 6. 4 23.0 9.0 4 0.49307 0.40717 - 0.39709 7.66 34.0 11.0 0 0.30946 0.43689 - 0.56809 7. 66 34.0 11.0 1. 256 * 0.38416 0.34169 - 0.62996 7.66 34. 0 11.0 4 0.78763 0.63303 - 0.94995 9.3 39. 8 13.9 0 0.73246 0.66232 - 0.83460 9.3 39.8 13.9 1. 236 ** 0.36279 0.79133 - 0.94044 9.3 39.8 13.9 4 1.16331 0.97314 - 1.38779 16. 2 93.0 40.0 0 3.61409 3.01554 - 4.33144 16.2 93. 0 40. 0 2 4.49371 3.74661 3.38979 16.2 93.0 40.0 4 3.38743 4.33346 - 7.20425 23. 1 147.0 66.0 0 8.39662 6.77647 •- 10.90367 23. 1 147.0 66.0 1 9.38586 7.62335 - 12.03011 23. 1 147.0 66. 0 2 10.68894 3.43833 - 13.33978 23. 1 147.0 66. 0 3 11.91895 9.19610 - 13.44800 23. 1 147. 0 66. 0 4 ++ 13.29050 9.90493 - 17.33318 + Minimum observed values of untransformed independent v a r i a b l e s ++ Maximum observed values of untransfarmed v a r i a b l e s # Averages of log-transformed independent v a r i a b l e s #* Averages of untransformed Independent v a r i a b l e s - 56 -6.1.2 Betula The best e q u a t i o n developed f o r Betula was: log BIOMASS = -6.8506 + .66984(log VOL E) - .0068697(BRS 3) R z = .67404 SE = .67759 n = 112 The a n a l y s i s of v a r i a n c e i s g i v e n i n Appendix B. Higher R z v a l u e s were o b t a i n e d f o r models without the l o g - l o g t r a n s f o r m a t i o n . For example a model u s i n g <D ZL) Z and WID1, and no t r a n s f o r m a t i o n of the dependent v a r i a b l e , had an R a = .82516. However p l o t t i n g the r e s i d u a l s showed seve r e h e t e r o s c a d a s t i c i t y and suggested n o n - 1 i n e a r i t y ; t h i s was confirmed by B a r t l e t t s t e s t and t t e s t s f o r l i n e a r i t y . The non-transformed model was t h e r e f o r e r e j e c t e d in favour of the l o g - l o g model, as the l a t t e r met more of the assumptions of r e g r e s s i o n . R e s u l t s of t e s t s of the assumptions are shown in Table 6.3. The l o g - l o g model met the assumptions of l i n e a r i t y and equal v a r i a n c e , but d i d not pass the t e s t f o r normal d i s t r i b u t i o n of r e s i d u a l s . Examples r e p r e s e n t i n g v a r i o u s s i z e s of shrub stems with d i f f e r e n t browse i n t e n s i t i e s are given in T a b l e 6.4 with the p r e d i c t e d browse biomass p r o d u c t i o n and a s s o c i a t e d c o n f i d e n c e i n t e r v a l . As with Salix, the p r e d i c t i o n s and c o n f i d e n c e l i m i t s have been backtransfarmed and c o r r e c t e d . The c o r r e c t i o n f a c t o r f o r Betula was 1.25805; i . e . the b i a s a s s o c i a t e d with the 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 would r e s u l t in an underestimate of 25.8% when t a k i n g the ant i1ag. U n l i k e the Salix model, the Betula model p r e d i c t s t h a t browsing i n h i b i t s the p r o d u c t i o n of browse biomass. L i k e the Salix model, the c o n f i d e n c e i n t e r v a l s are skewed i n o r i g i n a l - 57 -u n i t s . T h i s r e g r e s s i o n i s l e s s p r e c i s e than the Salix r e g r e s s i o n . The width o-f the c o n f i d e n c e i n t e r v a l ranges -from approximately + 12.5% at the means o-f the trans-formed independent v a r i a b l e s , to appro x i m a t e l y + 53% at the maximum observed v a l u e s . T a b l e 6.3. T e s t s o-f assumptions o-f r e g r e s s i o n on Betula common r e g r e s s i o n . Test C a l c u l a t e d s t a t i s t i c Cr i t i c a l v a l ue Signi-f i c a n c e at 0.05 l e v e l 1. L i n e a r i t y =-0.0257 =-0.0440 = 1.2413 = 1.1180 =-1.5267 = 1.0420 t = t = t = t = t = t = 2. 093 2. 093 2.093 2.093 2.093 2.201 N.S. N.S. N.S. N.S. N.S. N.S. 2. Equal v a r i a n c e = 10.2104 11.070 N.S. 3. Normali ty = 36.87 16.919 S i g . - 58 -Table 6.4. Same examples o-f p r e d i c t i o n s and confidence i n t e r v a l s -for Betula based an the common regression equation. VOL E (cm3) BRS Pred i c t e d Browse Biamass (g> 93% Confidence In t e r v a l 179. 0 179. 0 179.0 179.0 179.0 0 + 1 2 3 4 0.04302 0.04273 0.04072 0.03374 0.02772 0.02933 0.0293S 0.02S27 0.02474 0.01720 0.06268 0.06214 0.03866 0.05162 0.04468 3899.3 3899.3 3899.3 0 2 4 0.33878 0.32067 0.21826 0.29068 0.27885 0.14897 0.39484 0.36876 0.31978 7619.8 7619.8 7619.8 7619.8 18,967.8 18,967.8 18,967.8 18,967.8 80,781.2 80,781.2 80,781.2 133,937.2 133,937.2 133,937.2 133,937.2 133,937.2 0 1 2.88 * 4 0 0. 769 2 4 0 2 4 0 1 2 3 4 ++ 0.33064 0.32701 0.30128 0.34187 0.97748 0.97332 0.92321 0.62975 2.38009 2.44211 1.66223 3.97384 3.94663 3.76133 3.30109 2.36017 0.46283 0.46080 0.44216 0.23240 0.83634 0.83398 0.79598 0.41942 2.01315 1.90332 1.04046 2.94673 2.92713 2.78238 2.34699 1.34326 0.60838 0.60274 0.36830 0.30289 1.14216 1.13641 1.07342 0.94333 3.30668 3.13309 2.63358 5.33896 5.32122 3.08471 4.64305 4.24166 + Minimum observed values of Independent v a r i a b l e s ++ Maximlum observed values of independent v a r i a b l e s * Averages of transformed v a r i a b l e s Averages of non-transformed v a r i a b l e s - 59 -6.2 Hypo t h e s i s 2: A s i n g l e common r e g r e s s i o n e q u a t i o n w i l l d e s c r i b e the -four Salix s p e c i e s i n the area. Using the common Salix d a t a s e t (n=160>, a r e g r e s s i o n with dummy v a r i a b l e s r e p r e s e n t i n g the -four Salix s p e c i e s was used t o t e s t the -following hypotheses: 1) i n t e r c e p t s are not signi-f l e a n t l y di-f-ferent 2) s l o p e s a r e not signi-f i c a n t l y di-f-ferent 3) i n t e r c e p t s and s l o p e s t o g e t h e r are not signi-f l e a n t l y d i f f e r e n t . In a l l cases the c a l c u l a t e d F v a l u e was not s i g n i f i c a n t at the 0.05 l e v e l (Table 6.5) There-fore, one equa t i o n may be used to d e s c r i b e the -four s p e c i e s i n the study and t h e r e would be no s i g n i f i c a n t g a i n i n p r e c i s i o n by d e v e l o p i n g s p e c i e s - s p e c i f i c e q u a t i o n s . T a b l e 6.5. T e s t i n g f o r one equa t i o n to d e s c r i b e f o u r Salix s p e c i e s u s i n g r e g r e s s i o n with dummy v a r i a b l e s . c a l c u l a t e d c r i t i c a l H y p o t h e s i s DF F F 1. I n t e r c e p t s are not (3,140) 0.288 2.67 s i g n i f i c a n t l y d i f f e r e n t 2. S l o p e s are not (12,140) 0.427 1.82 s i g n i f i c a n t l y d i f f e r e n t 3. I n t e r c e p t s and s l o p e s are not (16,140) 0.884 1.74 s i g n i f i c a n t l y d i f f e r e n t - 60 -6.3 Hypo t h e s i s 3: The common r e g r e s s i o n e q u a t i o n s w i l l adequately p r e d i c t p r o d u c t i o n on s p e c i f i c s i t e s w i t h i n the wetland. A r e g r e s s i o n with dummy v a r i a b l e s r e p r e s e n t i n g s i t e s was used, but t h i s time the d a t a s e t c o n s i s t e d of the common data s e t p l u s those s i t e d a t a f o r which biomass had been measured. The common data s e t i s r e f e r r e d to as " s i t e 99" and as s i g n e d a dummy v a r i a b l e a c c o r d i n g l y . 6.3.1 S a l i x For Salix, the r e g r e s s i o n s with dummy v a r i a b l e s i n d i c a t e d 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 i n t e r c e p t s , but t h e r e were s i g n i f i c a n t d i f f e r e n c e s i n s l o p e s , and in s l o p e s and i n t e r c e p t s t o g e t h e r , as shown i n Table 6.6. Due t o the l o s s of p r e c i s i o n a r i s i n g from p o o l i n g s i t e - s p e c i f i c Salix data with the common data, s i t e - s p e c i f i c r e g r e s s i o n e q u a t i o n s were developed. The e q u a t i o n s are given below; a l l a n a l y s i s of v a r i a n c e t a b l e s a r e i n Appendix B: SITE l : l og BIOMASS = -6.9076 + 1.7391<log DEPTH) R* = .59908 SE » .52053 n = 50 SITE 6: log BIOMASS = -4.8459 + .52938 (LOG D=» DEPTH) R" = .48251 SE = .46465 n = 38 SITE 7: log BIOMASS = -7.0869 + .71605(log DIAM) + 1.4038<log DEPTH) R* • .77815 SE = .46334 n = 42 In a d d i t i o n to equat i o n was developed the i n d i v i d u a l s i t e e q u a t i o n s a " p o o l e d - s i t e " u s i n g those d a t a from s i t e s 1, 6 and 7 f o r - 61 -which biomass had been measured <n = 130). The common dat a set was excluded. The p o o l e d - s i t e e q u a t i o n i s given below and the a n a l y s i s o-f v a r i a n c e i s in Appendix B. l o g BIOMASS = -6.3720 +.24038(log D») + 1.0731(log DEPTH) + .29389(log WID2) + .081898(BRS) R* - .63201 SE = .50130 n = 130 I n t e r e s t i n g l y , t h i s e q u a t i o n uses the same independent v a r i a b l e s as the common eq u a t i o n but with d i f f e r e n t c o e f f i c i e n t s . The s i t e - s p e c i f i c e q u a t i o n s use fewer and i n some cases d i f f e r e n t v a r i a b l e s . T a b l e 6.6. T e s t i n g f o r one e q u a t i o n to d e s c r i b e Salix on a l l s i t e s u s i n g r e g r e s s i o n with dummy v a r i a b l e s . c a l c u l a t e d c r i t i c a l H y p o t h e s i s DF F F 1. I n t e r c e p t s are not (3,270) 2.050 2.60 s i g n i f i c a n t l y d i f f e r e n t 2. S l o p e s are not (12,270) 1.854 * 1.75 s i g n i f i c a n t l y d i f f e r e n t 3. I n t e r c e p t s and s l o p e s a r e not (16,270) 1.905 # 1.67 s i g n i f i c a n t l y d i f f e r e n t * S i g n i f i c a n t at 0.05 l e v e l - 62 -6.3.2 Betula There was no s i g n i f i c a n t d i f f e r e n c e i n i n t e r c e p t s , nor i n s l o p e s , i n d i c a t e d by t e s t s on the r e g r e s s i o n with dummy v a r i a b l e s f o r s i t e s , but the F s t a t i s t i c was s i g n i f i c a n t at the 0.05 l e v e l i n the t e s t combining i n t e r c e p t s and s l o p e s (Table 6.7). T h e r e f o r e s i t e - s p e c i f i c e q u a t i o n s were developed f o r Betula and are shown below. ( A n a l y s i s of v a r i a n c e t a b l e s are given i n Appendix B.) SITE l : l o g BIOMASS = -6.3201 + .36705(log VOL E) R* » .43897 SE =» .52652 n = 50 SITE 4: l o g BIOMASS = -3.4574 + 1.3084(log WID2) - .2329(BRS) R a = .70266 SE = .62091 n =» 35 (The s i t e 4 e q u a t i o n was based on f i r s t stems only.) SITE 6: log BIOMASS = -8.1241 + .65842(log D ZL) + .65218(log WID2) R a a .53050 SE = .76266 n = 37 SITE 7: log BIOMASS = -6.6450 + .61017(log VOL E) R a = .60114 SE = .59433 n = 42 A p o o l e d - s i t e equation was a l s o developed f o r Betula, u s i n g o n l y those d a t a from s i t e s 1, 4 ( f i r s t stems), 6 and 7 f o r which biomass had been measured, and e x c l u d i n g the common data s e t . The e q u a t i o n i s shown below and the a n a l y s i s of v a r i a n c e i s in Appendix B. - 63 -l o g BIOMASS = -7.2008 + .68993<log VOL E) R a « .56112 SE = .67188 n » 164 Table 6.7. T e s t i n g -for one eq u a t i o n to d e s c r i b e Betula. on a l l s i t e s u s i n g r e g r e s s i o n with dummy v a r i a b l e s . c a l c u l a t e d c r i t i c a l H y p othesis DF F F 1. I n t e r c e p t s are not (4,261) 0.260 2.37 s i g n i f i c a n t l y d i f f e r e n t 2. S l o p e s are not (8,261) 1.405 1.94 s i g n i f i c a n t l y d i f f e r e n t 3. I n t e r c e p t s and s l o p e s are not (13,261) 2.184 # 1.74 s i g n i f i c a n t l y d i f f e r e n t # S i g n i f i c a n t a t 0.05 l e v e l - 64 -6.4 H y p o t h e s i s 4: The a c t u a l browse biomass on s i t e s i s not s i g n i f i c a n t l y d i f f e r e n t from the p r e d i c t e d biomass. 6.4.1 Salix The r e s u l t s of the p a i r e d t t e s t s i n d i c a t e t h a t the s i t e - s p e c i f i c and p o o l e d - s i t e e q u a t i o n s performed much b e t t e r than d i d the common e q u a t i o n . T h i s i s not s u r p r i s i n g c o n s i d e r i n g the r e s u l t s of the F t e s t s f o r s i t e d i f f e r e n c e s u s i n g the r e g r e s s i o n with dummy v a r i a b l e s . T a b l e 6.8 shows the mean p r e d i c t e d biomass and the r e s u l t s of the p a i r e d t t e s t s , u s i n g the common, s i t e - s p e c i f i c and p o o l e d - s i t e e q u a t i o n s . Table 6.9 compares mean p r e d i c t e d browse biomass per stem, 93% c o n f i d e n c e i n t e r v a l s and the c o n f i d e n c e i n t e r v a l as a p e r c e n t of the mean p r e d i c t e d by the t h r e e e q u a t i o n s . These p r e d i c t i o n s are made from the f u l l d a t a s e t , r a t h e r than the subset used f o r the p a i r e d t t e s t , t h e r e f o r e the p r e d i c t i o n s d i f f e r from those shown in Table 6.8. On a l l s i t e s the common eq u a t i o n i s l e s s a c c u r a t e than the s i t e - s p e c i f i c and p o o l e d - s i t e e q u a t i o n s , as i n d i c a t e d by the p a i r e d t t e s t s (Table 6.8), yet y i e l d s narrower c o n f i d e n c e I n t e r v a l s than the other e q u a t i o n s (Table 6.9). The common equa t i o n used d a t a t h a t were p u r p o s e l y s e l e c t e d t o r e p r e s e n t the f u l l s i z e range of stems (which would i n c r e a s e SSX and reduce the v a r i a n c e ) w h i l e the other two e q u a t i o n s used s y s t e m a t i c a l l y c o l l e c t e d data, r e s u l t i n g i n c o l l e c t i o n of many of the more common smal1 stems and few of the r a r e r l a r g e stems. The s i t e - s p e c i f i c e q u a t i o n i s not the most a c c u r a t e in a l l c a s e s ! i t was outperformed by the p o o l e d - s i t e e q u a t i o n on s i t e 1. - 65 -Table 6.8. Mean predicted browse biomass (grama/stem) and paired t t e s t s •for d i f f e r e n c e between predicted and observed biomaas f o r Salix. Actual b i omasa Common equat ion Site S i t e - s p e c i f ic equat i on Pooled-si te equat i on 1 30 .32440 6 38 .43447 7 42 .39000 .60686 2.310* .38009 2.897* .79009 3.973* .54373 0.603 .43382 0.017 .60199 0.261 .51396 -0.235 .49262 0.903 .60424 0.337 * S i g n i f i c a n t at 0.03 level Table 6.9. Comparison of mean predicted browse biomass (grams/stem) and 93% confidence i n t e r v a l s f o r Salix, using three equations. S i t e n Common equation Y + 93% C I (% of mean) S i t e - s p e c i f i c equat i on + 93% C I (% of mean) Poole d - s i t e equat i on + 93% C I (% of mean) 1 100 .38076 6 130 .53107 7 130 .37204 ± .20876 ( + 36%) + .17046 ( ± 31%) + .17067 ( t 30%) ,53340 ,43234 ,44687 + .21630 ( ± 40%) ± .17549 ( ± 40%) + .17332 ( + 39%) .49708 .47472 43399 + .21184 ( ± 43%) + .17118 < + 36%) t .17142 < ± 38%) * Confidence Interval - 66 -6.4.2 Betula The s i t e -spec i-f i c e q u a t i o n s performed b e t t e r than e i t h e r the common or p o o l e d - s i t e e q u a t i o n s on t h r e e o-f the -four s i t e s , as shown by the p a i r e d t t e s t s (Table 6.10). T h i s i s not s u r p r i s i n g as the t t e s t i s a p p l i e d to the same data s e t as i s used to develop the s i t e -spec i -f i c e q u a t i o n . S i t e 4 was the e x c e p t i o n , where the common equ a t i o n gave the most a c c u r a t e p r e d i c t i o n s . On s i t e s 1 and 6 t h e r e were s i g n i f i c a n t d i f f e r e n c e s (at the 95% l e v e l ) between p r e d i c t e d and a c t u a l biomass u s i n g both the common and p o o l e d - s i t e e q u a t i o n s . T a b l e 6.11 shows the mean p r e d i c t e d browse biomass p r o d u c t i o n , 95% c o n f i d e n c e i n t e r v a l and c o n f i d e n c e i n t e r v a l as a perc e n t of the mean f o r the t h r e e e q u a t i o n s on f o u r s i t e s . These d a t a are based on the complete s i t e data s e t s . On s i t e 1, 6 and 7 the c o n f i d e n c e i n t e r v a l s as a percent of the mean were s m a l l e s t f o r the common e q u a t i o n . However the a c t u a l width of the c o n f i d e n c e i n t e r v a l ( i n the same u n i t s as the mean) was narrowest with the s i t e - s p e c i f i c e q u a t i o n f o r s i t e s 1, 4 and 7, and with the p o o l e d - s i t e e q u a t i o n f o r s i t e 6. - 67 -Table 6.10. Mean predicted browse biomass (grama/atsm) and paired t tes t -for di-f-ference between predicted and observed biomass o-f Betula. S i t e Actual biomass Common equat i on Si te-speci-f i c equat ion Pooled-sit« equat ion 1 SO .4446 .60213 4.74* .44873 0.136 .31014 2.092* 4<1) 33 1.0306 .93732 -0.838 1.1291 1.004 .86467 -1.834 4(2) 33 .96171 .88903 -0.740 1.0073 0.412 .79961 -1.629 6 37 .33865 .36198 3.139* .38470 1.267 .49187 3.963* 7 42 .70214 .82911 1.795 .70365 0.024 .72802 0.386 • S i g n i f i c a n t at 0.05 level (1) F i r s t stems <2) Second stems Table 6.11. Comparison of mean predicted browse biomass (grams/stem) and 95% confidence i n t e r v a l s f o r Betula, using three equations. S i t e Common Equation 7 * 95% C I (% of mean) S i t e - s p e c i f i c equat ion f ± 93% C I (% af mean) Pooled-si te equation + 95% C I (% of mean) 1 100 .65866 4 72 .98692 6 150 .37198 7 149 .74923 + .22036 < + 33%) + .26031 ( i 26%) + .18029 < ± 31%) + .18043 ( • 24%) .38398 1.2219 .37405 ,49812 + .21223 < + 35%) ± .25921 ( + 21%) + .20900 ( + 36%) ± .17683 < + 35%) .33991 ,87774 49839 .63772 + .21991 < + 39%) + .23938 ( + 30%) + .17967 ( ± 36%) ± .17978 ( + 27%) * Confidence Interval - 68 -6.5 Hypothesis 3: There i s no s i g n i f i c a n t d i f f e r e n c e i n the a c t u a l browse biomass between the f i r s t stem and second stem encountered of Betula. The p a i r e d t t e s t f o r t h i s h y p o t h e s i s gave a t v a l u e of 0.0824 i n d i c a t i n g no s i g n i f i c a n t average d i f f e r e n c e (at the 95% c o n f i d e n c e l e v e l ) i n the a c t u a l biomass of the f i r s t and second stems. The h y p o t h e s i s was t e s t e d because of a s u s p i c i o n t h a t b i a s might be e n t e r i n g i n t o the sampling method. When the stem to be measured i s a r b i t r a r i l y e s t a b l i s h e d as the f i r s t stem encountered ( i . e . the stem c l o s e s t to the t r a n s e c t p o i n t ) i t i s u s u a l l y a stem on the o u t s i d e of a clump. Due to the s t r o n g l y clumped nature of the shrubs i n t h i s study area, the t r a n s e c t l i n e p o i n t u s u a l l y f e l l between clumps. Thus the f i r s t d i s t a n c e measurement was u s u a l l y made to a stem on the o u t s i d e of a clump. However, because stems are very c l o s e t o g e t h e r w i t h i n a clump ( t y p i c a l l y the d i s t a n c e from the f i r s t stem to the second was l e s s than 10 cm) the second stem would a l s o be c l o s e to the o u t s i d e of the clump. T h i s h y p o t h e s i s d i d not r e a l l y address the q u e s t i o n : are o u t s i d e stems d i f f e r e n t from i n s i d e stems in biomass p r o d u c t i o n ? F i e l d o b s e r v a t i o n s were t h a t stems on the i n s i d e of a clump tended to be t a l l e r and have a l a r g e r b a s a l diameter than those on the o u t s i d e of a clump. U n f o r t u n a t e l y the q u e s t i o n of a d i f f e r e n c e i n biomass p r o d u c t i o n has not been answered, nor has the q u e s t i o n of b i a s i n the sampling method. To t e s t f o r biomass d i f f e r e n c e s between o u t s i d e and i n s i d e stems i n the same clump, one c o u l d sample p a i r s of stems: the f i r s t stem encountered and a stem from the c e n t r e of the clump, then t e s t f o r d i f f e r e n c e s with a p a i r e d t t e s t . Some s e t of r u l e s to d e f i n e the c e n t r a l stem - 69 -would be needed to av o i d p e r s o n a l b i a s . I-f indeed t h e r e i s a s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e between outer and inner stems, the p r e s e n t sampling scheme would have to be m o d i f i e d . One needs some method of d e t e r m i n i n g which stem to measure; s e l e c t i n g the f i r s t stem was convenient and seemed a p p r o p r i a t e when the sampling scheme was designed. - 70 -6.6 Hypothesis 6: There i s no s i g n i f i c a n t d i f f e r e n c e i n the p r e d i c t e d browse biomass between the f i r s t stem and second stem encountered of Betula. P a i r e d t t e s t s to t e s t t h i s h y p o t h e s i s were not s i g n i f i c a n t at the 0.05 l e v e l f o r p r e d i c t i o n s u s i n g the common and p o o l e d - s i t e e q u a t i o n s , as shown in Table 6.12. The s i t e - s p e c i f i c e q u a t i o n d i d , however, g i v e s i g n i f i c a n t l y d i f f e r e n t p r e d i c t i o n s f o r the f i r s t and second stems (t=2.025, n=72) with g r e a t e r biomass on the f i r s t stem, on the average. Table 6.12. P a i r e d t t e s t s f o r d i f f e r e n c e between p r e d i c t e d biomass (grams/stem) of f i r s t stem and second stem of Betula on s i t e 4. 1st stem 2nd stem P a i r e d t t e s t : E q uation n Y Y t v a l u e Common 72 .98692 .85700 t = 1.856 S i t e - s p e c i f i c 72 1.2219 1.0319 t = 2.025 * P o o l e d - s i t e 72 .87774 .77037 t =-1.779 * S i g n i f i c a n t at 0.05 l e v e l - 71 -6.7 D e n s i t y Betula had a g r e a t e r d e n s i t y than Salix on the th r e e s i t e s where both o c c u r r e d . On s i t e 1, Betula had about 1.3 times the d e n s i t y of Salix, on s i t e 6 i t was 1.6 times and on s i t e 7, 2.1 times. The g r e a t e s t d e n s i t y o v e r a l l , 42 stems/m 3 5, was on s i t e 4, where v i r t u a l l y no Salix o c c u r r e d . D e n s i t i e s , p r o b a b l e l i m i t s o-f e r r o r (PLE) and 95% c o n f i d e n c e i n t e r v a l s are shown in Table 6.13. Table 6.13. Density estimates, probable l i m i t o-f e r r o r and 95% confidence Intervals f o r Salix and Betula. S A L I X B E T U L A S i t e n Density PLE + 95% CI* n Density PLE + 95% CI (stems/m 2) (% of mean) (stems/m 2) <% of mean) 1 200 12.4232 .4513 . 8843 200 16.3130 .4392 . 8608 ( + 7. 1%) ( + 3.3%) 4 - - - - 146 42.0908 . 7038 1. 383 ( + 3.3%) 6 200 12.4977 . 4363 + . 8831 200 20.0936 . 4343 . 8516 < 6.3%) ( 4.2%) 7 195 6.4230 .3741 1. 123 193 13.4207 .6164 ± 1 . 208 ( + 17.3%) ( 9. 0%) # Confidence i n t e r v a l - 72 -6.8 Biomass p r o d u c t i o n per u n i t a r e a The p o i n t e s t i m a t e o-f browse biomass per u n i t a r e a i s made simply by m u l t i p l y i n g stem d e n s i t y by biomass per stem. The v a r i a n c e o-f t h i s e s t i m a t e i s o b t a i n e d by combining the r e s p e c t i v e v a r i a n c e s u s i n g the -formula given i n S e c t i o n 5.35 -from t h i s the c o n f i d e n c e i n t e r v a l i s determined i n the normal f a s h i o n . The t h r e e forms of eq u a t i o n s , common, s i t e - s p e c i f i c and p o o l e d - s i t e , r e s u l t i n d i f f e r e n t e s t i m a t e s of the mean and v a r i a n c e , and t h e r e f o r e lead to d i f f e r e n t e s t i m a t e s of biomass per u n i t a r e a . E s t i m a t e s u s i n g the d i f f e r e n t e q u a t i o n s are shown in Tab l e s 6.14 and 6.15. The columns headed "Sample" r e f e r to the double sampled biomass data combined with dens i t y . One r e s u l t of combining the v a r i a n c e s of the d e n s i t y e s t i m a t e with the biomass per stem e s t i m a t e i s a r e d u c t i o n i n the s i z e of the c o n f i d e n c e i n t e r v a l r e l a t i v e to the mean. As a perc e n t of the mean, the c o n f i d e n c e i n t e r v a l f o r the r e g r e s s i o n e s t i m a t e of biomass per stem i s roughly between + 25-50%S f o r the d e n s i t y e s t i m a t e s i t mostly ranges from +. 4-10%, and f o r the combined e s t i m a t e the c o n f i d e n c e i n t e r v a l has been reduced to approximately + 3-5% of the mean. T h i s i s l a r g e l y due to the l a r g e sample s i z e s used f o r both the d e n s i t y and biomass e s t i m a t e s . Note t h a t n ( d e n s i t y ) and n(biomass) are in the denominator of Goodman's e q u a t i o n . Comparison of the 95% c o n f i d e n c e l i m i t s of est i m a t e d biomass per square metre f o r the d i f f e r e n t r e g r e s s i o n e q u a t i o n s , shown in Ta b l e s .6.14 and 6.15, i n d i c a t e s t h a t f o r a l l Betula s i t e s and Salix s i t e 6 t h e r e i s no o v e r l a p p i n g at a l l on the same s i t e , and minimal o v e r l a p f o r Salix on s i t e s 1 and 7. - 73 -T a b l e 6 . 1 4 . Smltx broNii blomiii and 93% c o n f i d e n c e l i m i t s i n g rama p e r s q u a r e m e t r e . S i t e Common E q u a t i o n BI omasa 93% C . L . j t <g/m2> ( H o f f l S i t e - s p e c i f i c E q u a t i o n B i o m a s s 93% C . L . ^ <g/m2> I • « o f VI P o o l e d - a l t e E q u a t i o n B i o m a s s 93% C . L . <g/m z ) I i X o f VI S a m p l e B i o m a s s 93% C . L . ^ <g/m2> I i » o f ? l 7 . 2 1 3 6 . 9 3 3 - 7 . 4 7 7 < ± 3.6%) 6 . 6 3 1 6 . 3 8 7 - 6 . 9 1 6 < + 4 .0X1 6 . 173 3 . 9 7 4 - 6 . 3 7 6 ( ± 3 . 3 X ) 6 . 4 6 3 . 1 4 - 7 . 7 7 ( ± 20%) 6 . 6 8 7 6 . 7 1 0 - 7 . 0 6 4 ( 1 2.6%) 3 . 4 0 6 3 . 2 3 1 - 3 . 3 8 1 ( ± 3.2%) 3 . 9 2 9 3 . 7 3 2 - 6 . 1 0 6 C ± 3.0%) 5 . 6 2 4 . 3 1 - 6 . 9 4 ( ± 24%) 3 . 6 7 4 3 . 3 7 4 - 3 . 7 7 3 ( + 2.7%) 2 . 8 7 0 2 . 7 7 4 - 2 . 9 6 6 ( ± 3.3%) 2 . 9 1 6 2 . 8 1 9 - 3 . 0 1 3 < ± 3.3%) 3 . 7 9 2 . 7 7 - 4 . 8 1 < • 27%) # C o n f i d e n c e L l m l t i T a b l e 6 . 1 3 Batula b r o w s e b l o m a a a and 93% c o n f i d e n c e l i m i t s i n g r a m s p e r s q u a r e m e t r e . S i t e Common E q u a t i o n B i o m a s s 93% C . L . * S i t e - s p e c i f i c E q u a t i o n B i o m a s s 93% C . L . ^ ( g / m 2 ) < + % of ?) P o o l e d - s i t e E q u a t i o n B i o m a s s 93% C . L . A (g/ro 2 ) ( • % o f Y l Samp 1e B i o m a s s 93% C . L . ( g / m z ) < ± % o f Y) 1 0 . 7 4 3 1 0 . 3 8 3 - 1 1 . 1 0 7 ( ± 3.3%) 3 . 9 4 9 - 6 . 6 4 3 ( + 3.3%) 9 . 134 8 . 7 7 3 - 9 . 4 9 4 ( * 3.9%) 7 . 18 3 . 6 3 - 8 . 7 1 ( + 21%) 4 1 . 3 4 0 4 0 . 2 4 3 - 4 2 . 8 3 7 ( i 3.1%) 3 1 . 4 3 1 3 0 . 1 3 7 - 3 2 . 7 2 4 ( ± 2.3%) 3 6 . 9 4 3 3 3 . 6 3 4 - 3 8 . 2 3 6 ( ± 3.3%) 4 3 . 3 3 3 1 . 9 3 - 5 4 . ( + 26%) 1 1 . 4 9 3 1 1 . 1 9 7 - 1 1 . 7 9 8 ( ± 2.6%) 7 . 3 1 7 7 . 1 7 3 - 7 . 8 6 1 < 1 4.6%) 1 0 . 0 2 0 9 . 7 2 4 - 1 0 . 3 1 6 ( ± 3.0%) 6 . 83 4 . 9 6 - 8 . 7 1 ( ± 27%) 1 0 . 0 3 3 9 . 8 4 6 - 1 0 . 2 6 4 ( + 2.1%> 6 . 6 8 5 6 . 4 8 6 - 6 . 8 8 4 ( + 3.0%l 8 . 8 2 7 8 . 6 2 1 - 9 . 0 3 3 ( ± 2.3%) 9 . 3 9 6 . 3 3 - 1 2 . < ± 30%) * C o n f i d e n c e L i m i t s T h i s leads to the obvious q u e s t i o n : which i s the best e q u a t i o n to use? One answer c o u l d be to r e t u r n to the p a i r e d t t e s t f o r s i g n i f i c a n t d i f f e r e n c e between p r e d i c t i o n and o b s e r v a t i o n (d = Yj - Yj ), where the e x p e c t a t i o n i s that the average of the d i f f e r e n c e s or r e s i d u a l s i s c l o s e to zero. The c l o s e r d i s to zero, the more a c c u r a t e the e q u a t i o n . In most cases the s m a l l e s t t v a l u e i s f o r the s i t e - s p e c i f i c e q u a t i o n . However, because the s i t e - s p e c i f i c e quation i s o n ly based on the data from t h a t s i t e , which are e x a c t l y the same dat a used in the t t e s t , the comparisons are p o s s i b l y b i a s e d in f a v o u r of the s i t e - s p e c i f i c e q u a t i o n s . In terms of p r e c i s i o n , t h e r e was not a g r e a t d i f f e r e n c e between the t h r e e types of e q u a t i o n s , and t h e r e was no one type that was c o n s i s t e n t l y the most p r e c i s e . D i f f e r e n c e s i n p r e c i s i o n are masked when the d e n s i t y and biomass e s t i m a t e s are combined. T h e r e f o r e , e q u a t i o n accuracy should be the main concern in s e l e c t i n g the best e q u a t i o n . From a p r a c t i c a l p o i n t of view, i t i s much more d e s i r a b l e to be a b l e to develop one general e q u a t i o n t h a t can be widely a p p l i e d , r a t h e r than having to develop s p e c i f i c e q u a t i o n s f o r every s i t e . I d e a l l y , once a s a t i s f a c t o r y g e n e r a l e q u a t i o n has been developed, no biomass sampling on s i t e s should be r e q u i r e d . Indeed, i f t h i s method i s to be used o p e r a t i o n a l l y on a l a r g e s c a l e , g e n e r a l - t y p e e q u a t i o n s would be a n e c e s s i t y . Developing s i t e - s p e c i f i c e q u a t i o n s f o r dozens or hundreds of s i t e s would be a p r o h i b i t v e task. U n f o r t u n a t e l y in t h i s study the accuracy of the general e q u a t i o n s was poor. Another disadvantage to the s i t e - s p e c i f i c and p o o l e d - s i t e e q u a t i o n s , as they were developed in t h i s case, i s that biomass data were c o l l e c t e d s y s t e m a t i c a l l y r a t h e r than s e l e c t i v e l y , with the - 75 -r e s u l t t h a t most o-f the stems c o l l e c t e d were - f a i r l y s m a l l . A b e t t e r and more e-f-ficient way to develop a r e g r e s s i o n e q u a t i o n i s to spread the sampling over the e n t i r e s i z e range and to get good r e p r e s e n t a t i o n o-f i n d i v i d u a l s at both ends o-f the s i z e s c a l e . T h i s w i l l i n c r e a s e SSX, reduce the v a r i a n c e , and de-fine the s l o p e o-f the 1 i ne b e t t e r . An a l t e r n a t i v e method o-f sampling on the s i t e s , which would ensure b e t t e r r e p r e s e n t a t i o n o-f a l l s i z e s o-f stems, would be to e s t a b l i s h a number o-f s i z e c l a s s e s and determine the number o-f i n d i v i d u a l s d e s i r e d in each c l a s s . Then i n d i v i d u a l s c o u l d be sampled as they were encountered to - f i l l the requirements o-f the s i z e c l a s s e s . Many sma l l stems would be passed by because the s m a l l e r s i z e c l a s s e s would - f i l l q u i c k l y , and more o-f the l a r g e r stems, because they are l e s s common, would be c o l l e c t e d . Biomass per square metre was a l s o e s t i m a t e d without the use o-f r e g r e s s i o n e q u a t i o n s ; the mean p r o d u c t i o n per stem was based only on data -from those stems a c t u a l l y c o l l e c t e d -for biomass. The v a r i a n c e o-f t h i s mean was c a l c u l a t e d i n the normal method -for simple random sampling, and the combined v a r i a n c e c a l c u l a t e d with Goodman's -formula. The r e s u l t s are shown on T a b l e s 6.14 and 6.15 under the heading "Sample". The c o n f i d e n c e i n t e r v a l was many times wider with t h i s method than with any of the r e g r e s s i o n s (averaging about +, 25% of the mean) and in most cases encompassed the e s t i m a t e s from the t h r e e r e g r e s s i o n e q u a t i o n s . Using any one of the t h r e e r e g r e s s i o n e q u a t i o n s would t h e r e f o r e be s u p e r i o r to random or s y s t e m a t i c sampling, with the same amount of e f f o r t , because of the gain in prec i s i o n . An approximate breakdown of the e f f o r t spent in c o l l e c t i n g - 76 -data and p r e p a r i n g samples i s : Common r e g r e s s i o n data: c o l l e c t i n g independent v a r i a b l e and biomass d a t a ( i n c l u d i n g sample p r e p a r a t i o n ) 15 man-days S i t e data: c o l l e c t i n g biomass -from -four s i t e s ( i n c l u d i n g sample p r e p a r a t i o n ) 15 man-days S i t e data: c o l l e c t i n g d e n s i t y and independent v a r i a b l e data -from -four s i t e s 12 man-days C o l l e c t i n g biomass data i s by -far the most time consuming o-f the - f i e l d work because o-f the sample p r e p a r a t i o n . C urrent annual twig growth must be removed -from each stem, leaves removed -from the twigs, and the m a t e r i a l bagged, d r i e d and weighed. In t h i s study t h i s consumed 30 man-days, e q u a l l y d i v i d e d between the m a t e r i a l used •for the common r e g r e s s i o n ( t h i s i n c l u d e d time -for measuring the independent v a r i a b l e s on t h i s m a t e r i a l ) and the m a t e r i a l c o l l e c t e d -from the s i t e s . and no s i t e biomass data co 1 1 e c t e d , the t o t a l - f i e l d time requirement would have been 36% l e s s . With the sampling regime used ( c o l l e c t i n g biomass at every -fourth p o i n t ) the biomass sample p r e p a r a t i o n time •for each s i t e was approximately 1.25 times g r e a t e r than the time used to c o l l e c t d e n s i t y and independent v a r i a b l e data. I-f o n l y the common r e g r e s s i o n e q u a t i o n s had been deve1 oped 7. SYNOPSIS AND ANALYSIS 7.1 S y n o p s i s Much o-f the a v a i l a b l e l i t e r a t u r e c o n c e r n i n g shrub biomass and shrub d e n s i t y was reviewed in Chapter 2. I t was determined t h a t the most a p p r o p r i a t e method o-f browse biomass i n v e n t o r y was r e g r e s s i o n e s t i m a t i o n of biomass combined with the c o r r e c t e d p o i n t d i s t a n c e method, a p l o t l e s s d e n s i t y e s t i m a t i o n technique. Stems of Salix spp. and Betula glandulosa were c o l l e c t e d from thoughout the study a r e a and measured f o r the independent v a r i a b l e s ( s e v e r a l stem dimensions and browse c o n d i t i o n ) and the dependent v a r i a b l e (oven dry weight of c u r r e n t annual woody growth), f o r the purpose of d e r i v i n g r e g r e s s i o n e q u a t i o n s . A l s o , independent v a r i a b l e and d e n s i t y data were c o l l e c t e d on f o u r s i t e s r e p r e s e n t i n g d i f f e r e n t wetland shrub a s s o c i a t i o n s . A p o r t i o n of these stems were a l s o sampled f o r browse biomass. Least squares m u l t i p l e l i n e a r r e g r e s s i o n e q u a t i o n s u s i n g n a t u r a l 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 of dependent and independent v a r i a b l e s were developed to d e s c r i b e the r e l a t i o n s h i p s between browse biomass p r o d u c t i o n per stem and shrub c h a r a c t e r i s t i c s ( i . e . dimensions and browse c o n d i t i o n ) , f o r S a l i x and B e t u l a . These e q u a t i o n s were developed from d a t a c o l l e c t e d s p e c i f i c a l l y f o r the purpose of d e r i v i n g r e g r e s s i o n e q u a t i o n s and are r e f e r r e d to as the common e q u a t i o n s . The f o u r S a l i x s p e c i e s p r e s e n t on the study a r e a c o u l d be d e s c r i b e d by a s i n g l e r e g r e s s i o n e q u a t i o n . P r e d i c t i o n s would not be s i g n i f i c a n t l y improved by d e v e l o p i n g s e p a r a t e e q u a t i o n s . However, - 7 3 -•for both S a l i x and B e t u l a , s i g n i f i c a n t l y b e t t e r p r e d i c t i o n s -far s p e c i f i c s i t e s were o b t a i n e d by d e v e l o p i n g s i t e - s p e c i f i c e q u a t ions, or by p o o l i n g a l l s i t e data and d e r i v i n g e q u a t i o n s from t h i s d a t a s e t . On a l l t h r e e s i t e s where S a l i x o c u r r e d , t h e r e were s i g n i f i c a n t d i f f e r e n c e s i n the S a l i x mean biomass p r e d i c t e d by the common eq u a t i o n and a c t u a l measured biomass from a double sample. For B e t u l a , t h i s was the case on two of fo u r s i t e s . Using s i t e - s p e c i f i c e q u a t i o n s , i n no case were t h e r e s i g n i f i c a n t d i f f e r e n c e s between p r e d i c t e d and a c t u a l biomass, w h i l e the p o o l e d - s i t e e q u a t i o n f o r B e t u l a r e s u l t e d i n s i g n i f i c a n t d i f f e r e n c e s on two of fo u r s i t e s , and none f o r S a l i x . In g e n e r a l , the 95% c o n f i d e n c e i n t e r v a l s f o r the common e q u a t i o n s were narrower ( + 30 -36%) than those f o r the s i t e - s p e c i f i c ( + 21 - 56%) and p o o l e d - s i t e e q u a t i o n s < + 27 - 43%), but the common equa t i o n s y i e l d e d l e s s a c c u r a t e p r e d i c t i o n s . The common equa t i o n e s t i m a t e s had a 95% c o n f i d e n c e i n t e r v a l of + 30 - 36%. On one s i t e , in an attempt to determine i f b i a s was e n t e r i n g the sampling procedure by u s i n g the f i r s t stem encountered at each t r a n s e c t p o i n t , data were c o l l e c t e d from the f i r s t and second stems encountered. There were no s i g n i f i c a n t d i f f e r e n c e s in the a c t u a l measured biomass of these stems, nor were there s i g n i f i c a n t d i f f e r e n c e s i n p r e d i c t e d biomass u s i n g the common and p o o l e d - s i t e e q u a t i o n s . The s i t e - s p e c i f i c e q u a t i o n d i d , however, y i e l d s i g n i f i c i a n t d i f f e r e n c e s i n p r e d i c t i o n s between f i r s t and second stems. The c o r r e c t e d p o i n t d i s t a n c e method was employed on the fo u r s i t e s f o r stem d e n s i t y e s t i m a t e s . The method g i v e s an approximation of the c o n f i d e n c e of the e s t i m a t e with the "probable l i m i t of e r r o r " - 79 -(PLE). On a l l s i t e s B e t u l a had a g r e a t e r d e n s i t y than S a l i x , and the s i t e which was v i r t u a l l y a l l B e t u l a was a l s o the most dense o-f the •four s i t e s . S i x o-f the seven e s t i m a t e s had a 95% c o n f i d e n c e i n t e r v a l o-f + 10% or l e s s (one was + 13%). S a l i x and B e t u l a browse biomass p r o d u c t i o n per stem was esti m a t e d on each s i t e u s i n g the t h r e e e q u a t i o n -forms (common, s i t e - s p e c i-f i c and p o o l e d - s i t e ) . Browse biomass i n grams per square metre was estimated by m u l t i p l y i n g the per stem e s t i m a t e by the d e n s i t y e s t i m a t e . A l l e s t i m a t e s o-f biomass and t h e i r accompanying v a r i a n c e s were c o r r e c t e d -for the b i a s i n h e r e n t in l o g a r i t h m i c trans-formation by the -factor e . The mean biomass per stem was the mean o-f the backtrans-formed and c o r r e c t e d p r e d i c t i o n s , and the v a r i a n c e of the mean biomass per stem was the mean of the backtransformed and c o r r e c t e d v a r i a n c e s of the p r e d i c t i o n s . The v a r i a n c e of the f i n a l e s t i m a t e of browse biomass per square metre was ob t a i n e d by combining the v a r i a n c e s of the per stem biomass e s t i m a t e and the d e n s i t y e s t i m a t e . The v a r i a n c e of the stem d e n s i t y was taken to be PLE 3 8. S a l i x browse p r o d u c t i o n was on the order of 3 to 8 g/m3. B e t u l a was more p r o d u c t i v e , on the order of 5 to 12 g/mz except f o r one s i t e where e s t i m a t e s were from 30 to 50 g/ra a. These e s t i m a t e s had 95% c o n f i d e n c e i n t e r v a l s of + 2.3 - 5.5% (Tables 6.14 and 6.15). 7.2 C r i t i c a l a n a l y s i s . The p r e c i s i a n of the r e g r e s s i o n e q u a t i o n s c o u l d have been improved ( i . e . narrower c o n f i d e n c e l i m i t s ) by sampling g r e a t e r - 80 -numbers o-f l a r g e stems. D e s p i t e the attempt to r e p r e s e n t evenly the range o-f s i z e s p r e s e n t , sampling was c o n c e n t r a t e d on the more abundant s m a l l e r stems. Sampling -for the r e g r e s s i o n c o u l d have been improved by de-fining s i z e c l a s s e s -for the independent v a r i a b l e s and sampling to s a t i s f y the number of o b s e r v a t i o n s d e s i r e d f a r each c1 ass. The sample s i z e s were adequate f o r the common r e g r e s s i o n s ( S a l i x n=160, B e t u l a n=112) and p o o l e d - s i t e e q u a t i o n s ( S a l i x n=130, B e t u l a n=164), but c o u l d have been g r e a t e r f o r the s i t e s p e c i f i c e q u a t i o n s (the s m a l l e s t sample s i z e was 35). Rather than s t r a i g h t f o r w a r d s y s t e m a t i c sampling f o r the s i t e - s p e c i f i c e q u a t i o n s , sampling e f f o r t c o u l d have been o p t i m i z e d by sampling so as to s a t i s f y s i z e c l a s s c r i t e r i a , as d i s c u s s e d above and in S e c t i o n 6.8. A p o r t i o n of the s i t e d a t a was double sampled f o r biomass as a check on the accuracy of the r e g r e s s i o n s . In g e n e r a l , the accuracy of the common eq u a t i o n was po o r e s t and that of the s i t e - s p e c i f i c e q u a t i o n s was bes t . T h i s supports the r e s u l t s of the t e s t s u s i n g r e g r e s s i o n with dummy v a r i a b l e s r e p r e s e n t i n g s i t e s , which i n d i c a t e d t h a t b e t t e r p r e d i c t i o n s would r e s u l t from s p e c i f i c e q u a t ions, and c a s t s doubt on the v a l u e of a common r e g r e s s i o n e q u a t i o n and i t s r e s u l t i n g p r e d i c t i o n s . No check was a v a i l a b l e on the accuracy of the d e n s i t y e s t i m a t e . A c c o r d i n g to i n v e s t i g a t o r s who have used the c o r r e c t e d p o i n t d i s t a n c e method on a r t i f i c i a l ( B a t c h e l e r 1973, 1975; Boyd 1.979, 1980) and n a t u r a l p o p u l a t i o n s (Laycock and B a t c h e l e r 1975), the method i s e f f i c i e n t and y i e l d s a c c e p t a b l e e s t i m a t e s . I t c e r t a i n l y seems to be the most a p p r o p r i a t e method to use with a very clumped p o p u l a t i o n (most shrub p o p u l a t i o n s ) , as most other p l o t l e s s methods - 81 -do not account f o r a non-random d i s t r i b u t i o n . P l o t sampling i s very t e d i o u s and would r e q u i r e a l a r g e sample s i z e and/or l a r g e p l o t s due to high v a r i a b i l i t y between p l o t s . The c o r r e c t e d p o i n t d i s t a n c e method c e r t a i n l y has not been p e r f e c t e d . One problem i s d e t e r m i n i n g what a c t u a l l y i s the "best d e n s i t y e s t i m a t e " . D i f f e r e n t d e n s i t y e s t i m a t e s , and d i f f e r e n t PLE's, w i l l r e s u l t from v a r y i n g the s e a r c h l i m i t (Boyd 1979). The "best d e n s i t y e s t i m a t e " as d e f i n e d by B a t c h e l e r (1975) i s t h a t which i s most p r e c i s e ( i . e . has a minimum r a t i o of P L E / d e n s i t y , analagous to the c o e f f i c i e n t of v a r i a t i o n ) , which he found to be the most a c c u r a t e as w e l l . However Boyd (1979) found t h a t in some cases the most a c c u r a t e e s t i m a t e d i d not have the lowest P L E / d e n s i t y r a t i o . The program s u p p l i e d f o r use in t h i s study i t e r a t e s d e n s i t y c a l c u l a t i o n s with v a r y i n g s e a r c h l i m i t s and s e l e c t s the e s t i m a t e with the minimum PL E / d e n s i t y r a t i o . With no check, such as independent quadrat sampling, t h e r e i s no way of knowing i f t h i s was in f a c t the most a c c u r a t e e s t i m a t e . If the d e n s i t y e s t i m a t e s can be accepted as v a l i d , the r e s u l t s were very s a t i s f a c t o r y : a 95% l e v e l of c o n f i d e n c e +, 10% or b e t t e r f o r 6 out of 7 e s t i m a t e s (one e s t i m a t e was + 18%). The c o r r e c t e d p o i n t d i s t a n c e method i s r e l a t i v e l y s imple and f a s t to conduct in the f i e l d , a l l o w i n g a l a r g e sample s i z e to be obtained q u i c k l y . For t h i s study d e n s i t y sample s i z e s were from 150 to 200. The f i n a l e s t i m a t e s of biomass per square metre, ob t a i n e d by m u l t i p l y i n g the r e g r e s s i o n e s t i m a t e of biomass per stem by the stem d e n s i t y e s t i m a t e , had high l e v e l s of p r e c i s i o n : 95% c o n f i d e n c e l e v e l s + 3 - 5%. The r e l a t i v e l y l a r g e sample s i z e s of both the r e g r e s s i o n and d e n s i t y e s t i m a t e s c o n t r i b u t e d to the r e d u c t i o n of the c o n f i d e n c e - 82 -i n t e r v a l s of the f i n a l e s t i m a t e s , as p o i n t e d out i n S e c t i o n 6.3 R e g r e s s i o n e s t i m a t i o n of p l a n t biomass i s a p o t e n t i a l l y very u s e f u l method because, once e q u a t i o n s have been developed, sampling i s quick and n o n - d e s t r u c t i v e and the p r e d i c t i o n s are more p r e c i s e than e s t i m a t e s from many other methods. In the p r e s e n t study, sampling f o r the r e g r e s s i o n v a r i a b l e s c e r t a i n l y was f a s t e r than measuring browse biomass d i r e c t l y through the t r a d i t i o n a l c l i p and weigh method, f o r example. For t h i s study, a p p r o x i m a t e l y 2 hours were needed, on average, to remove a l l c u r r e n t twig growth from each stem. T h e r e f o r e , f o r an average stem d e n s i t y of 28 steras/ra 2, 56 hours would be r e q u i r e d to c l i p and prepare a browse sample from a s i n g l e square metre p l o t u s i n g the c l i p and weigh method. Re g r e s s i o n p r o v i d e s a more p r e c i s e e s t i m a t e than random or s y s t e m a t i c sampling, as shown by the r e g r e s s i o n e s t i m a t e c o n f i d e n c e l i m i t s l i m i t s which are in a l l cases much narrower than those of the e s t i m a t e s d e r i v e d o n l y from the s y s t e m a t i c samples on s i t e s . T h i s i s shown in T a b l e s 6.14 and 6.15 and i s d i s c u s s e d in S e c t i o n 6.8. However, the need f o r s i t e - s p e c i f i c e q u a t i o n s ( d i s c u s s e d in S e c t i o n 6.3) reduces the advantage of r e g r e s s i o n e s t i m a t i o n because of the time r e q u i r e d f o r d e v e l o p i n g e q u a t i o n s ( p r i m a r i l y sample p r e p a r a t i on) . When r e g r e s s i o n i s combined with a s e p a r a t e d e n s i t y e s t i m a t e , the v a r i a n c e of the r e s u l t i n g combined e s t i m a t e i s a combination of the i n d i v i d u a l v a r i a n c e s . However t h e r e does not appear to be a w e l l d e f i n e d , s t a t i s t i c a l l y a c c e p t a b l e way to c a l c u l a t e the v a r i a n c e of the mean r e g r e s s i o n p r e d i c t i o n when a 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 of the dependent v a r i a b l e i s used. The v a r i a n c e s c a l c u l a t e d in t h i s - 83 -study by the method d e s c r i b e d i n S e c t i o n 5.1.2 can only be regarded as approximations. S i m i l a r l y the v a r i a n c e o-f the d e n s i t y e s t i m a t e P L E 3 s h o u l d o n l y be c o n s i d e r e d an approximation. Thus the combined v a r i a n c e o-f the biomass per u n i t a r e a e s t i m a t e i s q u e s t i o n a b l e , but un-fortunate 1 y i t i s the best t h a t can be o b t a i n e d . A p p l i c a t i o n o-f t h i s method on an o p e r a t i o n a l l e v e l would r e q u i r e much r e - f i n i n g o-f the t e c h n i q u e s . I t would o n l y be p r a c t i c a l i-f g e n e r a l biomass e q u a t i o n s are a p p l i c a b l e , as the work r e q u i r e d to develop an equa t i o n -for each s i t e would negate the e-f-ficiency o-f the method. Even so, the method i s t e d i o u s i-f many shrub dimensions must be measured at each p o i n t . Determining biomass over a l a r g e a r e a such as the wetland i n t h i s study c o u l d pose problems due to the very patchy nature o-f v e g e t a t i o n d i s t r i b u t i o n . In wetlands, v e g e t a t i o n o f t e n o c c u r s i n narrow bands or s t r i p s , making the detemination o-f the are a o-f a p a r t i c u l a r a s s o c i a t i o n d i f f i c u l t . An i n a c c u r a t e e s t i m a t e of t o t a l a r e a would devalue an a c c u r a t e and p r e c i s e browse p r o d u c t i o n e s t i m a t e , so c a r e f u l t y p i n g and d e l i n e a t i o n of shrub a s s o c i a t i o n s on a e r i a l photographs would be necessary. P i t t and Schwab (1985) suggested t h a t shrub biomass should not be measured on i n d i v i d u a l p l a n t s , as t h i s n e c e s s i t a t e s a d e n s i t y e s t i m a t e , thus g e n e r a t i n g two sources of v a r i a b i l i t y . In order to l i m i t the sources of v a r i a b i l i t y , they recommended a double sampling procedure o u t l i n e d by Zamora (1981) i n which canopy volume i s esti m a t e d i n a t h r e e dimensional p l o t . Then a p o r t i o n of the p l o t s are c l i p p e d and weighed, so that the e s t i m a t e becomes an independent v a r i a b l e f o r p r e d i c t i n g biomass. A disadvantage of the method i s - 84 -t h a t s e p a r a t e r e g r e s s i o n s would l i k e l y be neeeded to r e - f l e c t v a r i a b i l i t y i n o b s e r v e r , s i t e and season. P i t t and Schwab (1983) a l s o suggested t h a t , i f r e g r e s s i o n e s t i m a t i o n i s used, shrub volume c o r r e l a t e s w e l l with biomass and should p r o v i d e good p r e d i c t i v e r e l a t i o n s h i p s . However, models should "be developed from s i t e / s e a s o n / s p e c i e s s p e c i f i c canopy volume and biomass data c o l l e c t e d d u r i n g the shrub i n v e n t o r y program". C l e a r l y , any shrub biomass e s t i m a t i o n method i n v o l v i n g r e g r e s s i o n , with e i t h e r v i s u a l e s t i m a t e s or shrub dimensions as independent v a r i a b l e s , w i l l i n v o l v e much t e d i o u s work, at l e a s t i n i t i a l l y , and w i l l l i k e l y not be u n i v e r s a l l y a p p l i c a b l e ( i . e . s i t e and s p e c i e s s p e c i f i c e q u a t i o n s w i l l p r o b a b l y be n e c e s s a r y ) . I t should a l s o be c o n s i d e r e d t h a t a change in u t i l i z a t i o n l e v e l (such as from a s i t u a t i o n of no browsing to one of heavy use) c o u l d a l t e r the p l a n t s ' growth form and n e c e s s i t a t e development of new r e g r e s s i o n e q u a t i o n s . 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Min. of For., C a r i b o o For. Region. - °to -Runka, G.Q. and T, Lewis. 1981. P r e l i m i n a r y wetland managers manual; C a r i b o o r e s o u r c e management r e g i o n . B.C. Min o-f E n v i r . APD Tech. Pap. 5. 112 pp. Ruther-ford, M.C. 1979. P l a n t based t e c h n i q u e s -for d e t e r m i n i n g a v a i l a b l e browse and browse u t i l i z a t i o n : a review. Bot Rev. 45(2):203-228. Schreuder, H.T. and W.T. Swank. 1971. A comparison o-f s e v e r a l s t a t i s t i c a l models i n -forest biomass and s u r f a c e a r e a e s t i m a t i o n . In: F o r e s t Biomass S t u d i e s . Univ. o-f Maine P r e s s , Orono. pp. 125-136. Schreuder, H.T. and W.T. Swank. 1973. S t a t i s t i c a l c o n s i d e r a t i o n s i n sampling biomass and s u r f a c e a r e a over time f o r a Pirtus stratus L. f o r e s t . In: IUFR0 Biomass S t u d i e s . Univ. of Maine P r e s s , Orono. pp. 131-141. Schwab, F.E. 1985. Moose h a b i t a t s e l e c t i o n i n r e l a t i o n to f o r e s t c u t t i n g p r a c t i c e s in north c e n t r a l B r i t i s h Columbia. PhD. t h e s i s , Univ. of B r i t i s h Columbia, Vancouver, B.C. (In p r e p a r a t i o n . ) Shafer, E.L. 1963. The twig-count method f o r measuring hardwood deer browse. J . W i l d l . Manage. 27(3):428-437. Shepherd, W.O. 1962. Herbage sampling f o r y i e l d : n a t u r a l p a s t u r e s and range. Pp 102-105 in'. P a s t u r e and range r e s e a r c h t e c h n i q u e s . Comstock Publ. Assoc., I t h a c a , NY. 242 pp. Smith, A.D. and P.J. Urness. 1962. A n a l y s e s of the twig length method of d e t e r m i n i n g the u t i l i z a t i o n of browse. Utah Div. of F i s h and Game B u l l . 62-9. 34 pp. ( c i t e d by Jensen and Urness 1981). Smith, J . 1985. P e r s o n a l communication. B i o m e t r i c i a n , Environment Canada, Canadian W i l d l i f e S e r v i c e , Ladner, B.C. Tappeiner, J.C. II and H.H. John. 1973. Biomass and n u t r i e n t c ontent of h a z e l undergrowth. Ecology 54(6):1342-1348. T e l f e r , E.S. 1969a. Twig weight-diameter r e l a t i o n s h i p s f o r browse s p e c i e s . J . W i l d l . Manage. 33:317-321. T e l f e r , E.S. 1969b. Weight-diameter r e l a t i o n s h i p s f o r 22 woody p l a n t s p e c i e s . Can. J . Bot. 47:1851-1855. T e l f e r , E.S. 1974. V e r t i c a l d i s t r i b u t i o n of c e r v i d and snowshoe hare browsing. J . W i l d l . Manage. 38(4):944-946. T e l f e r , E.S. 1981. Browse i n v e n t o r i e s : t e c h n i q u e s and e v a l u a t i o n . Pp. 67-82 in: M i l l e r , F.L. and A. Gunn ( e d s . ) . Symposium on census and i n v e n t o r y methods f o r p o p u l a t i o n h a b i t a t s . Proc. N.W. W i l d l . Soc. Univ. Idaho. Tel-fer, E.S. and A. C a i r n s . 1978. Stem breakage by moose. J . Range Manage. 42(3):639-642. Walmsley, M., G. U t z i g , T. Void, D. Moon and J . van Barneve Id ( e d s . ) . 1980. D e s c r i b i n g ecosystems in the f i e l d . B.C. Min. of E n v i r . and Min. of For. RAB Tech. Pap. 2, Land Manage. Rep. 7. 224 pp. Whittaker, R.H. and P.L. Marks. 1975. Methods of a s s e s s i n g t e r r e s t r i a l p r o d u c t i v i t y . Pp. 55-118 in'. L i e t h , H. and R.H. Whittaker ( e d s . ) . Primary p r o d u c t i v i t y of the b i o s p h e r e . S p r i n g e r - V e r l a g New York Inc. W i l l a r d , E.E. and CM. M c K e l l . 1978. Response of shrubs to s i m u l a t e d browsing. J . W i l d l . Manage. 42(3):514-519. Wilm, H.G., D.F. C o s t e l l o and G.E. K l i p p l e . 1944. E s t i m a t i n g f o r a g e y i e l d by the double sampling method. J . Am. Soc. Agron. 36:194-203. Wolff, J.O. 1978. Burning and browsing e f f e c t s on w i l l o w growth i n i n t e r i o r A l a s k a . J . W i l d l . Manage. 42(1):135-139. Zamora, B.A. 1981. An approach to p l o t sampling f o r canopy volume in shrub communities. J . Range Manage. 43(2):155-156. Zar, J.H. 1968. C a l c u l a t i o n and m i s c a l c u l a t i o n of the a l l o m e t r i c e q u a t i o n as a model i n b i o l o g i c a l data. B i o s c i e n c e 18(12):1118-1120. - <?2 -APPENDIX A: SCATTER PLOTS OF SALIX AND BETULA DATA - 93 -BIOMASS 7.0700 S.28SS + 5.S011 + 4.71S7 + 3.1478 * 2.3633 + 3 •2 .79444 + • *2 2 2 " • • . . . 3 . . 3 * • "2 3'"2'222 • * "'2346 3" • 2' ' 2'445"332 • 2 • .10000 -1* •• -3-4000 7.7778 12.156 16.533 20.911 DI AM 5.5889 9.9667 14.344 18.722 23.100 F i g u r e A . l . S c a t t e r p l o t o-f Salix browse biomass (g) on stem diameter (mm). - 9* -BIOMASS 7.0700 5.5011 • 4.7167 * 3.9322 * 2.3633 * 1.5789 * .79444 * •2 •2 2 * * * • 2" • 23222 232 • "3222 236*" 2 2 • " •238254" « 2 " .10000 -1* •"• 2 7 0 0 0 '„„,,, S 1' 4 4 4 9 9 8 8 9 ' 3 0 3 3 '64.78 LENGTH 44.222 78.667 ,13.11 147.56 182.00 F i g u r e A.2. S c a t t e r p l o t o-f Salix browse biomass (g) on stem length (cm). - 95 -BIOMASS 7.0700 + 5.23S6 * 5.5011 * 4.7167 * 2.3633 + 1.S789 * .79444 + • « * * 2 2"" "2 2 * "2 442*2 " " • "22243 '222 "• • •2*3»2 33523"' • . 10000 - 1 * " -10.00O 40.444 70.889 101.33 131.78 DEPTH 25.222 55.S67 86.111 116.56 147.OO F i g u r e A.3. S c a t t e r p l o t of Salix browse biomass (g) on canopy depth (cm). - 96 -BIOMASS 7.0700 + 6 . 2 8 5 6 * 3 . 1 4 7 8 + 2 . 3 6 3 3 + . 7 9 4 4 4 * . 2 2 ' * ' 2 - - 2 4 2 2 - 2 * • 2 • 3 3 6 3 4 4 • « « 5 2 - X 3 ' 3 . 1 0 0 0 0 - 1* 2 • 3 0 0 0 0 10 3 3 3 , 7 S S 7 ™ 3 2 " 3 4 7 0 0 0 ^'V'"^ 1 0 3 3 3 2 5 . 0 0 0 3 9 . 6 6 7 5 4 . 3 3 3 =o nr F i g u r e A.4. S c a t t e r -plot o-f Salix browse biomass <g) on canopy width 1 (cm). - 97 -BIOMASS 7.0700 6.2856 * 4.7167 * 3.9322 • 3.1478 *. 2.3633 * 1.5789 + 2 • • 2 •2 *• • .79444 + • 22 • • . . .22 . 2 • . • 3 54 222 -• 65 38 «2* 3 45 66 22-. 10000 - 1*" 3.0000 17.000 31.000 45.000 59.000 WI02 10.000 24.000 38.000 ' 52.000 66.000 F i g u r e A.5. S c a t t e r p l o t o-f Salix browse biomass on canopy width 2 (cm). - 90 -BIOMASS 7.0700 2 * 2 +2 2 + - • 3 2 2 3 2 2 +3 2 * * 4 • " 7 • ' 3 +6 4 5 3 X 4 4 4 3 X 5 4 2 -1+3 BRS 4.0000 F i g u r e A . 6 . S c a t t e r p l o t o-f Salix browse biomass (g) on browse c o n d i t i o n . - 9«? -BIOMASS 4.5100 + 4.0100 +• 3.5100 + 3.0100 + 2.5100 + 1 .5100 + .51000 + « *2 • 2 2 10000 « 3 ' 5 0 ° ° « nrm 5 9 0 0 0 8 : 3 0 0 0 1 0 - 7 0 0 1 3 . 1 0 o " " o i A M 4- 7 0 0 0 7.1000 9.5000 11.900 14 30 F i g u r e A. 7. S c a t t e r p l o t o-f Betula browse biomass (g) on stem diameter (mm). too BIOMASS •4.5100 + 4.0100 + 3.0100 + 2.5100 + 2.0100 • 1.5100 + * • - • 2 2 ... ,,..2 3 • 232 • "« 2 2 . 10000 -1 + « * • " 27.000 55.000 83.000 111.00 139.00 LENGTH 41.000 69.000 97.0O0 125.00 153.00 F i g u r e A.8. S c a t t e r p l o t o-f Betula browse biomass <g) on stem length (cm). - 101 -B I O M A S S 4 . 5 1 0 0 + 3 . 5 1 0 0 + 3 . 0 1 O 0 * 2 . 0 1 0 0 + 1 . 5 1 0 0 * 1 . 0 1 0 0 + . 5 1 0 0 0 + + • 2 2 " * * 2 • • " * » * • * 2 * l O O O O - 1 + » 1 2 0 0 0 3 8 . 2 2 2 6 4 . 4 4 4 9 0 . S 6 7 1 1 6 . 8 9 D E P T H 2 5 . 1 1 1 5 1 . 3 3 3 7 7 . 5 5 6 1 0 3 . 7 8 1 3 0 . 0 0 F i g u r e A.9. S c a t t e r p l o t of Betula browse biomass <g> on canopy depth (cm). - I0Z -BIOMASS 4.5100 + 4.0100 * 3.5100 + 2.51O0 * 2.01O0 * 1.5100 + 1.0100 * 2 • * 2 * 2 * • 2 2 3' .51000 + 2 * 2 * 2 • . . 1O0O0 -1* * - 13-444 23.889 34.333 44.778 WID1 3 2 2 2 2 '8.667 29.111 39.556 SO.00 F i g u r e A. 10. S c a t t e r p l o t o-f Betula browse biomass (g) on canopy width 1 (cm). - 103 -BIOMASS 4.5100 3.5100 + 3.0100 + 2.5100 + 2.0100 + 1.5100 * - 2 • * 3 2 « 2 2 - » " » 3 2 ' 2 • 2 4 « 3 2 . 1000O -1+ 2 2.0000 9.3333 1G.S67 24.000 31.333 WID2 5.S667 13.000 20.333 27.SS7 35.000 F i g u r e A. 1 1 . S c a t t e r p l o t o-f Betula browse biomass (g) on canopy width 2 (cm). - 1 0 4 -BIOMASS 4.S100 • 4.0100 + 3 .0100 * 2.5100 • 2.0100 *• 3 1.5100 * 1.0100 • 2 2 2 +3 3 2 3 4 .51000 +4 3 3 4 +8 . S 4 . 100OO - 1 + 3 44444 8 8 8 8 9 , 3 3 3 3 ,' 7 7 7 8 , „ „ 2 8 6 6 7 3 • 5 5 5 6 8RS 1-3 33 2.2222 3.1111 i n F i g u r e A. 12. S c a t t e r p l o t o-f Betula browse biomass (g) on browse c o n d i t i o n . - 105 -APPENDIX B: ANOVA TABLES - 1 0 S -T a b l e B . l . ANOVA -for S a l i x common r e g r e s s i o n e q u a t i o n . Source DF Sum o-f squares Mean square F Re g r e s s i o n 4 164.72 41.035 172.92 E r r o r 155 36.792 .23737 T o t a l 159 201.01 T a b l e B.2. ANOVA -for B e t u l a common r e g r e s s i o n e q u a t i o n . Source DF Sum o-f squares Mean square F Reg r e s s i o n 2 103.49 51.744 112.70 E r r o r 109 50.045 .45913 T o t a l 112 Ta b l e B.3. ANOVA -for S a l i x s i t e 1 r e g r e s s i o n e q u a t i o n . Source DF Sum a-f squares Mean square F Re g r e s s i o n 1 19.433 19.433 71.723 E r r o r 48 13.005 .27095 T o t a l 49 - 107 -T a b l e B.4. ANOVA f o r S a l i x s i t e 6 r e g r e s s i o n e q u a t i o n . Source DF Sum o-f squares Mean square F Re g r e s s i o n 1 7.2470 7.2470 33.567 E r r o r 36 7.7723 .21590 T o t a l 37 Tab l e B.5. ANOVA -for S a l i x s i t e 7 r e g r e s s i o n e q u a t i o n . Source DF Sum o-f squares Mean square F Re g r e s s i o n 2 29.367 14.684 68.397 E r r o r 39 8.3727 .21468 T o t a l 41 37.740 Ta b l e B.6. ANOVA f o r S a l i x p o o l e d - s i t e r e g r e s s i o n e q u a t i o n . Source DF Sum o-f squares Mean square F Re g r e s s i o n 4 53.950 13.488 53.672 E r r o r 125 31.412 .25130 T o t a l 129 - 108 -T a b l e B.7. ANOVA -for B e t u l a s i t e 1 r e g r e s s i o n e q u a t i o n . Source DF Sum of squares Mean square F Regre s s i o n 1 10.413 10.413 37.557 E r r o r 48 13.308 .27725 T o t a l 49 23.720 Table B.8. ANOVA f o r B e t u l a s i t e 4 r e g r e s s i o n e q u a t i o n . Source DF Sum o-f squares Mean square F Regre s s i o n 2 29.154 14.577 37.811 E r r o r 32 12.337 .38552 T o t a l 34 Table B.9. ANOVA f o r B e t u l a s i t e 6 r e g r e s s i o n e q u a t i o n . Source DF Sum o-f squares Mean square F Re g r e s s i o n 2 22.345 11.173 19.209 E r r o r 34 19,776 .58165 T o t a l 36 42.121 - 10«? -Table B.10. ANOVA f o r B e t u l a s i t e 7 r e g r e s s i o n e q u a t i o n . Source DF Sum o-f squares Mean square F Reg r e s s i o n 1 21.293 21.295 60.286 E r r o r 40 14.129 .35323 T o t a l 41 35.424 Table B . l l . ANOVA -for B e t u l a p o o l e d - s i t e r e g r e s s i o n e q u a t i o n . Source DF Sum o-f squares Mean square F Reg r e s s i o n 1 93.500 93.500 207.12 E r r o r 162 73.130 .45142 T o t a l 163 166.63 - 1IO -

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