Estimation of Her It Ability From Varietal Trials Data

8/2/2019 Estimation of Her It Ability From Varietal Trials Data

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Theor A pp l G ene t ( 1993) 86 :437-441

9 Springer-Verlag 1993

E s t i m a t i o n o f h e r i t a b i l i t y f r o m v a r i e t a l t r i a l s d a t aM . S i n g h , S . C e c c a r e l li , J . H a m b l i nInternat ion al Center for Agr icul tura l Research in the Dry Areas ( IC AR DA ), P.O . Box 5466, Aleppo , SyriaReceived: 20 Oc tober 1992 / Accepted : 11 No vem ber 1992

Abstract. W e p r e s e n t t h e e s t i m a t i o n o f h e r i t a b i li t i e s o f a no b s e r v e d t r a it i n s i t u a ti o n s w h e r e e v a l u a t i o n o f s e v er a lp u r e b r e e d i n g l i n e s i s p e r f o r m e d i n a t r i a l a t a s i n g l el o c a t i o n a n d i n t r i a l s f r o m s e v e r a l l o c a t i o n s . F o r t h es i n g l e l o c a t i o n s i t u a t i o n , w e e v a l u a t e e x a c t c o n f i d e n c ei n t e r v a l s , t h e p r o b a b i l i t y o f i n v a l i d e s ti m a t e s , a n d t h ep e r c e n t a g e p o i n t s o f t h e d i s t r ib u t i o n o f h e ri t ab i l it y . S i m -u l a t i o n s w e r e p e r f o r m e d t o n u m e r i c a l l y v e r if y th e r e s u lt s .A d d i t i o n a l l y , a p p r o x i m a t i o n s t o t h e b i a s a n d s t a n d a r de r r o r o f th e e s t im a t e w e r e o b t a i n e d a n d a r e p r e s e n t e da l o n g w i t h t h e i r s i m u l a t e d v a l u e s a n d c o e f f i c i e n t s o fs k e w n e s s a n d k u r t o s i s . F o r t r i a l s i n s e v e r a l l o c a t i o n s ,e x p l i c it e x p r e s s i o n s f o r e x a c t v a l u e s o f c o n f i d e n c e l i m i t sa r e n o t a v a il a b le . F u r t h e r , o n e w o u l d r e q u i r e k n o w l e d g eo f o n e m o r e p a r a m e t e r , r e p r e s e n t e d b y t h e r a ti o o f g e n o -t y p e x e n v i r o n m e n t ( G x E ) in t e r a c t i o n v a r i a n c e t o e r-r o r v a r i a n c e , i n a d d i t i o n t o t h e n u m b e r o f g e n o t y p e s ,r e p l i c a t i o n a n d t r u e h e r i t a b i l i t y v a l u e . A p p r o x i m a t i o n sw e r e m a d e f o r b i a s a n d t h e s t a n d a r d e r r o r o f e s t im a t e s o fh e r i ta b i li t y . T h e e v a l u a t i o n o f t h e d i s t ri b u t i o n o f h e r it a -b i l i t y a n d i t s m o m e n t s w a s r e c o g n i z e d a s a p r o b l e m o ft h e l in e a r f u n c t i o n o f a n i n d e p e n d e n t c h i - s q u a re . T h em e t h o d s h a v e b e e n i ll u s t ra t e d b y d a t a f r o m e x p e r i m e n t so n g r a i n a n d s t r a w y i e l d o f 6 4 b a r le y g e n o t y p e s e v a l u a t e da t t h r e e l o c a t i o n s .K e y w o r d s : H e r i t a b i l i ty - S t a n d a r d e r r o r - G e n o -t y p e x e n v i r o n m e n t i n t e r a c t i o n - C o n f i d e n c e i n t e r v a l -I n v a l i d e s t i m a t e s

IntroductionA n a s s e s s m e n t o f t h e h e r i t a b i l i t y o f v a r i o u s t r a i t s i s o fc o n s id e r a b le i m p o r t a n c e i n c r o p i m p r o v e m e n t p r o g r a m s ;C om m unica t ed by A . R . H a l l aue rCorrespondence to: M. Singh

f o r e x a m p l e , t o p r e d i c t r e s p o n s e t o s e l e c t io n . E s t i m a t e s o fh e r i t a b i l i t y a r e a v a i l a b l e i n s e v e r a l e x p e r i m e n t a l s i t u a -t i o n s , b u t t h e s t a n d a r d e r r o r s o f t h e s e e s t im a t e s , o r t h ec o n f i d e n c e in t e r v a l s o f h e r i t a b i l it y , h a v e b e e n r e p o r t e dm o s t l y f o r p a r e n t - o f f s p r i n g d a t a ( G r a y b i l l e t a l. 1 9 56 ;B o g y o a n d B e c k e r 1 9 6 3; B r o e m e l i n g 1 9 69 ). F a l c o n e r( 1 98 2 , p p 1 6 5 - 1 6 7 ) g a v e a l a r g e s a m p l e s t a n d a r d e r r o rw h e r e t h e h e r i t a b i li t y w a s o b t a i n e d f r o m t h e r e g r e s s io n o fo f f s p r i n g o n p a r e n t s . E x a c t c o n f i d e n c e i n t e r v a l s f o r h e r i -t a b i l i t y w e r e o b t a i n e d b y K n a p p e t a l . ( 1 9 8 5 ) w h e n t h ed a t a w e r e c o ll e c t ed o n a p r o g e n y - m e a n b a s is f r o m s e v e r a le n v i r o n m e n t s . T h e s t a n d a r d e r r o r s a n d c o n f i d e n c e i n t e r-v a l s o f r e s p o n s e t o s e l e c t i o n w e r e g i v e n b y B r i d g e s e t a l.( 19 91 ). G e n e r a l f o r m u l a e f o r t h e r a t i o o f v a r i a n c e s w e r ep r o v i d e d b y G r a y b i l l e t a l . ( 1 9 5 6 ) f o r p a r e n t - o f f s p r i n gd a t a a n d G r a y b i l l a n d W a n g ( 1 9 79 ) f o r a t w o - f a c t o r n e s t -e d m o d e l .

T h e p u r p o s e o f t h e p r e s e n t p a p e r i s t o p r o v i d e e x p r e s-s i o n s f o r t h e s t a n d a r d e r r o r s o f t h e e s t i m a t e o f h e r i t a b i l i t yf r o m t h e a n a l y s i s o f v a r i a n c e o n d a t a g e n e r a t e d i n ar a n d o m i z e d c o m p l e t e b l o c k d e s i g n c o n d u c t e d i n o n e e n -v i r o n m e n t ( o r s in g l e t r ia l ) a n d i n s e v e r a l e n v i r o n m e n t s ( o rm u l t i - l o c a t i o n a l t ri a ls ) . T h i s p a p e r a l s o s t u d i e s t h e b e h a v -i o r o f t h e d i s t r i b u t i o n o f h e r i t a b i l it y , u s i n g s i m u l a t i o nt e c h n i q u e s . B a r l e y d a t a h a v e b e e n u s e d t o i l l u s t r a t e t h em e t h o d o l o g y .Mat er i a l s and m et hodsEst imat ion o f her i tab i l i ty & a s ing le t r ia lCon sider es timating the heritabili ty h 2 of a trait Y from theresponses of a se t of v inbred l ines, chosen ra ndom ly to representa popu l a t i on o f li ne s , w hen g row n in b r andom ized com ple teblocks in a single trial. L et Yij be the respon se o f the i-th g eno typegro wn in the j-th bloc k (i = 1, 2 . . . v, j = t , 2, . . . , b). A mod el forYij and the parameters involved isYij = la + gi + flj + ~ij (1)


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w h e r e # i s g e n e r a l m e a n , g i r e p r e s e n t s t h e e f f ec t o f i -t h g e n o t y p ea n d i s a ss u m e d t o b e i n d e p e n d e n t a n d n o r m a l l y d i s t r ib u t e d w i t hz e r o m e a n a n d v a r i a n c e rrg f lj i s t h e u n k n o w n f i x e d e f fe c t s o f th ej - t h b l o c k , a n d e i~ v a l u e s a r e e x p e r i m e n t a l e r r o r s a s s u m e d t o b ei n d e p e n d e n t l y n o r m a l l y d i s t r i b u t e d w i t h z e r o m e a n a n d v a r i -ance c r~ ( i = 1 . . . v ; j = 1 , 2 , . . . b ) . B ro ad s ens e h er i t ab i l i ty (h 2 ) o fa t r a i t i s t h e r a t i o o f g e n e t i c v a r i a b i l i t y (G ~ ) t o p h e n o t y p i c v a r i -abi l i ty ( r r~ + ~r~) an d is gi ven byh 2 = a ~ (ag + a~) . (2 )G e n e r a l l y , e s t i m a t i o n o f v a r i a n c e c o m p o n e n t s i s b a s e d o n a n a l -y s i s - o f - v a ri a n c e (A N O V A ) e s t im a t e s o b t a i n e d a s o u t l i n e d i nTable 1 . Es t im ates o f ~ and rr~ a r e^ 2 = M e a n d ~ g 2 = ( M g - M e ) / b ."

A n e s t i m a t eT h u s a n e s t i m a t e f i 2 o f h e r i t a b i l i t y h 2 isf~2 = 4gz/(~2 + 4gZ)= (V, -- 1)/(V~+ b - 1) (3)w h e r e V ~ = M g / M , i s t h e v a r i a n c e r a t i o . N o t e t h a t t h e e x a c td i s t r i b u t i o n o f

2 2W = V ~ ar + b f ig 2) = V r / [ J + b h 2 (1 - h 2 ) - 11i s t h e F d is t ri b u ti o n w it h q = v - 1 a n d f = ( v - 1 ) ( b - 1 ) d f. T h u st h e e x a c t d i s t r i b u t i o n o f f i2 c a n e a s il y b e t r a c e d . W e k n o w t h a tt h e e s t i m a t e f i 2 i s b i a s e d f o r h 2 a n d m a y a s s u m e n e g a t i v e( i n v a li d ) v a l u e s. T h e a m o u n t o f b ia s i s o f i n t e r e s t t o p l a n t b r e e d -e r s. W e g i v e e x p r e s si o n s f o r b i a s, s t a n d a r d e r r o r , p r o b a b i l i t y o fi n v a l i d e s t i m a t e s , a n d c o n f i d e n c e i n t e r v a l b e l o w . A p p e n d i x Ig i v e s t h e m e t h o d o f o b t a i n i n g t h e m a n d t h e e x p r e s s io n s f o r t h ec o e f f ic i e n t s i n v o l v e d b u t n o t r e p r o d u c e d i n t h i s s e c t i o n .Bia s b (~2) = 2 (1 - h 2) [1 + (b - 1) h 2] [1 + (b - 2) h2]/(h 2 b f).S tan dard e r ro r SE( fa 2)= ( l - h 2 ) [1 + (b - l ) h 2 ] [2/ (b f111 /2 .P ro bab i l i ty o f inva l id es t im ate H [q , f , ( 1 + b 2 2 ) - 1 ] .P r o b a b i l i t y d i s t r i b u t i o n o f fa2 H [q, f, y] .1 0 0 ( 1 - o 0 % c o n f i d e n c e i n t e r v a l ( L , h ~ ) w h e r e

f i2 = ( V , - Fu) / [V~+ (b - 1) Fu]~ l2 = ( V - - F L ) / [ V r + (b - 1) FL].

A n e s t i m a t e o f b ia s i n fi 2 c a n b e o b t a i n e d b y s u b s t i t u t i n g t h ee s t i m a t e o f h 2 . T h i s e s t i m a t e c a n b e s u b t r a c t e d f r o m [ a2 to c o r r e c tf o r t h e b i a s. S e v e r a l s ta t i s ti c a l p a c k a g e s a n d l i b r a ri e s o f m a t h e -m a t i c a l s u b r o u t in e s c o n t a i n t h e e v a l u a t i o n o f th e p r o b a b i l i t y o fl o w e r t a i l o r p e r c e n t a g e p o i n t s o f t h e F d i s t r i b u t i o n a n d m a yf a c i l i t a t e t h e c o m p u t a t i o n s i n t h i s s e c t i o n . I n o r d e r t o e v a l u a t et h e u p p e r a p r o b a b i l i t y p o i n t o f t h e d i s t r i b u t i o n o f fi 2 , l e t y , b et h e u p p e r ~ p r o b a b i l i t y p o i n t o f t h e F d i s t r i b u t i o n w i t h q a n d fd r, i. e. P r o b ( F , f > y , ) = a t h e n t h e c o r r e s p o n d i n g p o i n t la= f o r. . . . 2 "~' 2 2 1 2 2he n ta bd l ty h i s h~ - - [(1 + b 2 ) y ~ - ] / [ (1 + b ) y~ + b - 1 ]. Fu r -ther , i n o rd er to t es t a g iven va lue o f her i t ab i l i ty , s ay h 2 , onem a y c o m p u t e t h e e x a c t p r o b a b i l i t y l e v e l = P r o b (fa2 > ho ) = 1- n ( q , f , u ), w h e r e u = [ 1 + b h 2 /( 1 - h 2 ) ] / [ 1 + b h 2 / ( l - h 2 )] .

T h e s t a n d a r d e r r o r ( S E ) g i v e s a m e a s u r e o f th e p r e c i s i o n o ft h e e s t i m a t e . I n c a s e s w h e r e c o n f i d e n c e i n t e r v a l s a r e r e p o r t e d ,t h e S E o f ~ 2 m a y n o t b e n e c e s s a r y . F o r s i t u a t io n s , w h e r e t h e2d i s t r i b u t i o n o f h a p p r o x i m a t e s t o t h e n o r m a l d i s tr i b u t i o n , t h e n2e v a l u a t i o n o f S E ( h ) i s r e q u i r e d t o o b t a i n t h e s a m p l i n g d i s t r ib u -t i o n o f t h e e s ti m a t e . I f s e v e r a l i n d e p e n d e n t e s t i m a t e s o f h e r i ta b i l -i t y a r e t o b e c o m b i n e d , t h e s t a n d a r d e r r o r s d e t e r m i n e t h ew e i g h t s u s e d i n t h e p o o l e d e s t i m a t e o f h e r it a b i l i ty . F u r t h e r ,S E ( ~ 2 ) d o e s n o t r e q u i r e a n y a d d i t i o n a l p a r a m e t e r , s u c h a s ac o n f i d e n c e c o e f f i c i e n t i n t h e c o n f i d e n c e i n t e r v a l .

T a b l e 1. A n a l y s i s o f v a r i a n c e f o r s i n g le l o c a t i o n d a t aS o u r c e d f M e a n s q u a r e E x p e c t e d m e a n

s q u a r eB l o c k b - 1 N o t r e l e v a n t N o t r e l e v a n tG e n o t y p e s q = V - - 1 M g 0 2 + b a 2E r r o r f = ( v - 1 ) ( b - 1 ) M e a 2

T a b l e 2 . A n a l y s i s o f v a r i a n c e f o r m u l t i l o c a t i o n d a t aS o u r c e d f M e a n s q u a r e E x p e c t e d m e a ns q u a r eE n v i r o n - L - 1 N o t r e l e v a n tm e n t ( g )B l o c k s / ( r - 1 ) L N o t r e l e v a n te n v .G e n o t y p e q = v - 1 M g(G)G x E ( v - 1 ) ( L - 1 ) M ~E r r o r f = ( v - 1 ) M ~

9 ( b - l ) L

2a ~ + b ~ r ~+ b L ~g.~+b.~

Estim ation o f heritabili ty fr om multi- location trialsC o n s i d e r a s e r ie s o f t r i a ls c o n d u c t e d i n r a n d o m i s e d c o m p l e t eb l o c k d e s i g n w i t h v g e n o t y p e s a n d b r e p l i c a t i o n s , o v e r L e n v i -r o n m e n t s . L e t h 2 b e t h e h e r i t a b i l i ty o f t r a i t Y o n w h i c h w eobs erved Yl jka s t h e r e s p o n s e f r o m i - t h g e n o t y p e ( i = 1 . . . . v ), j - t hb l o c k ( j = 1 . . . , b ) o v e r t h e k - t h e n v i r o n m e n t ( k = 1 , . . . , L ). T h em ode l fo r Yl jk i sYijk = ~ + g i + ~k + filk+ fljk + ~ijk (4)wh ere # i s the gene ra l mea n , g l i s the e f f ec t o f i - th geno ty pe , ~ki s the e f f ec t o f the k - th env i ronm ent , 6~k i s the in t e r a c t ion e f f ec tb e t w e e n t h e i - t h g e n o t y p e a n d t h e k - t h e n v i r o n m e n t , a n d / 3 j ~ i st h e e f f e ct o f th e j - t h b l o c k w i t h i n t h e k - t h e n v i r o n m e n t . T h eef f ec ts g i s , 3 ik S , and e s a r e a s s um ed to b e inde pen den t ly and2r a n d o m l y d i s t r i b u t e d w i t h z e r o m e a n s a n d v a r i a n c e s a g , a r a n da~ . Th e her i t ab i l i ty o f t r a i t Y, deno ted by h 2 , i s def ined ash 2= ~/(~ + o ? + ~ / . ( 5 1E s t i m a t i o nT o e s t i m a t e h z w e c o n s t r u c t a n A N O Y A t a b l e ( T a b l e 2 ).E x p e c t e d v a l u e s o f m e a n t h e s q u a r e a p p e a r s i m i l a r t o t h o s eg i v e n i n T a b l e 2 , f o r a n y o f th e f o l l o w i n g t h r e e m o d e l s : ( 1 ) g e n o -typ ic e f f ec ts f ixed an d e n v i r o n m e n t r a n d o m w h e r e a ~ = v a r i a n c eo f t h e f ix e d g e n o t y p e s , (2 ) g e n o t y p e r a n d o m a n d e n v i r o n m e n tf i xe d , a n d ( 3 ) b o t h g e n o t y p e a n d e n v i r o n m e n t r a n d o m . T h e v a r i -a n c e c o m p o n e n t s a r e e s t i m a t e d a s^ 2 2G = M e, ch = (MI - - Mr and ag = (Mg - - Ml ) / (b L ) .L e t u s d e f i n e t h e t w o v a r i a n c e r a t i o s a sV g = M g /M ~ a n d V t = M J M o .T h u s a n e s t i m a t e o f h 2 isfi2 = &g2/(3.2+ 02 + 0~) (6)

= (M g-- M 1)/[Mg + (L - - 1) M I + L (b - 1) M~]= ( V g - - 1)/[Vg+ L - - 1 + L ( b - - 1 )/ V~ ].


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Table 3. Analysis of variance on grain and straw yields from three trials together with estimates o f heritabilities439

Item Gra in yield Straw yieldTrials Trials1 2 3 1 2 3

Mean squaresSource df Mean squaresBlocks 2 633,095 143,392 61,707 51 ,6 91 1, 130,316Genotype 63 509,155 249,945 51,668 263,573 94,976Error (4~ ) 126 58,801 19,268 7,010 82,794 45,866Variance component estimates~z 58,801 19,268 7,010 82,794 45,866~ 150,118 76,892 14,886 60,260 16,370gEstimates of heritabili ty~2 0.72 0.80 0.68 0.42 0.2695% confidence intervalsLower 0.6112 0.7160 0.5630 0.2685 0.1090Upper 0.8068 0.8653 0.7781 0.5698 0.427099% confidence intervalsLower 0.5726 0.6850 0.5219 0.2201 0.0628Upper 0.8298 0.8820 0.8040 0.6127 0.4771SE, bias, skewness and kurtosis

194,00889,14832,124

32,12419,008

0.37

0.21710.5265

0.16890.5720

Approx. s~ (la2) 0.0499 0.0379 0.0550 0.0776 0.0818 0.0797Simul. s~ (f~2) 0.0524 0.0402 0.0575 0.0796 0.0832 0.0861Approx. bias 0.0087 0.0062 0.0099 0.0191 0.0286 0.0214Simul. bias -0.005 2 -0.0045 -0.00 54 -0.0045 -0.0027 -0.0041Simul. skewness -0. 72 -0. 83 -0. 67 -0. 35 -0. 14 -0. 29Simul. kurtosis 0.27 0.43 0.20 -0 .1 2 -0 .23 -0 .1 5Approx.: approximated values. Simul.: simulated values based on 2,000 runs

The dist ribution of ~2 is based on three independent quadraticforms (Mg, MI, M~) or on two dependent var iance ratios Vg andV~ (see Appendix II). If h z is based on progeny means (Knapp etal. 1985), then the her itabi lity estimate is expressed only in termsof V~; therefore its distribution is determined as in the previoussection. The probability distribution of fi2 can be expressed asthat of a linear combination of independen t chi-square variables.Its distribution, however, depends on four parameters (v, b, Land fl~) in addi tion to h 2 where 2 2 2ll = al/ae the ratio of G x Einteraction variance to error variance. A brief derivation of theresults presented in the following is in Appendix II.Bias h 2[Cl l/A 1-C12 /(A 1 A2)Standard error h 2 C l l / A 2 + C 2 2 / A 2 - 2 C 12 /( A 1 A 2) ]1/2Probabili ty of invalid estimateH[q, p, (1 +b 22)/(1 +b )o2+b L 2~z)].Probabili ty distribut ion Prob (Z~ > Z)100 (i - c 0 % asymptotic confidence in terva l[fa2 -- Z=/2S E ( f 1 2 ) , fl2 1 - Z = / 2 SE (f12)]

o2 and )2eritability in terms of/~g2 2 2 2 2 2 2h =2J(2g +-~i + 1), where 2g= a g / ~ eObtaining exact confidence limits for h 2 appears to be cumber-some, since it involves two ratios ()2 and fi~) whose estimates arenot independent, if we take the ratios of mean squares. Anasymptotic confidence interval is given in the above using anasymptot ic standard error and percentage points Z~/2 of thestandard normal distribution.

R e s u l t s a n d d i s c u s s io n

Sixty four genotypes from three trials were evaluated inrandomized complete block designs with three replica-tions at three locations (Tel Hadya, Bouider and Breda)in Syria. These trials were condu cte d by the Cereals Pro-gram, ICARDA. Data on straw and grain yields werecollected during May, 1990. The analyses of these data


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44 0Table 4 . Com bined analys is of var iance of the three t r ia ls an destimation of heritabili tyM ean squa resSource df Gra in yie ld St raw yie ldLoc at ion (L) 2 1 .314 x l0 s 1 .153 x l0 sBloc ks/L oc 6 2.794 x 105 4.587 x 105Gen otyp e (G) 63 5 .127 x 10 ~ 2 .078 x 10 sG x L 126 1.490 x 105 1.200 x 105Err or (6~) 378 2.836 x t0 ~ 5.359 x 104Variance c om pon ent es t imates~i 4"0411 x 104 0"9756 x 1044.0213 x 1 0 4 2.2137 x 10 ~d~ 2.8360 X 1 0 4 5.3590 x 104Est imates of her i tabil ity and s tandard er rorsla2 0.3710 0.1140SE 0.0680 0.0490Bias 0.0160 0.0220

f o r a n e s t i m a t i o n o f h e ri t a b il i ty a r e g i v e n i n T a -b l e s 3 a n d 4 .

T h e s t a n d a r d e r r o rs a p p r o x i m a t e d f r o m t h e f o rm u l ai n (4 ) a r e r e a s o n a b l e c l o se t o t h e c o r r e s p o n d i n g s i m u l a t e dv a lu e s9 R e l a t i v e y l o w m a g n i t u d e s o f s k ew n e s s a n d k u r t o -s is v a l u e s i n d i c a t e t h a t t h e d i s t r i b u t i o n o f e s t i m a t e s o fh e r i ta b i l it y c a n b e r e a s o n a b l y a p p r o x i m a t e d b y t h e n o r -m a l d i s t r i b u t i o n . H o w e v e r , t h e a p p r o x i m a t e d a n d s i m u -l a t e d b i a se s a r e s i m i la r a n d l o w m a g n i t u d e f o r h i g h v a l-ues o f he r i t ab i l i ti e s .

A p p e n d i x IBias and standard error of ]1 2A n a p p r o x i m a t i o n t o t h e b i a s i s o b t a i n e d u s i n g T a y l o r ' ss e r ie s e x p a n s i o n ( K e n d a l l a n d S t u a r t 1 96 9 ).B ias (~2) = E ( fi z) - h 2 = E ( N /D ) - h 2w h e r e th e n u m e r a t o r N = M g - M ~ a n d t h e d e n o m i n a t o rD = M g + ( b - 1 ) M e . A n a p p ro x i m a t i o n o f t he bi as isB ( fi 2 ) = [ E ( N ) / E ( D ) ]

9 { v a r ( N ) / [E ( N )] 2 _ c o v ( N , D ) / [ E ( N ) ~E D ) ] } .T h e e x p e c t e d v a l u e o f r a n d o m v a r i a b le Z2 i s v w i th av a r i a n c e 2 v ; f u r t h e r , q M g ~ ( a ~ 2 + b o . g ) 2 Zq2 a n d f M o~ o ' 2 Z } , w h e r e q = v - 1 , f = q ( b - 1 ) . W e o b t a i n t h e e x -p e c t a t i o n s , v a r ia n c e s a n d c o v a r i a n c e s o f N a n d D a sE (N) = b o 2 = b 2 o .~ , wh ere

-- 2 22 - Crg/o.e= h2 / (1 - - h 2 ) exp re s sed i n t e rm s o f h 2 ,E (D ) = b (o.~ + o-~)= b o-2/(1 - h / ) ,

v a r ( N ) = v a t (Mg ) -}- va r ( M e) = 2 (a 2 + b o-gz)2/q+ 2 o.~/f= 2 o.e [(1 + b 22)2/q + 1 / 0 ,

va r (D ) = 2 o ,4 [ (1 + b 22)2/q + ( b - 1)2/f],co v (N , D ) = 2 o -~ [ (1 + b L 2 ) / / q - ( b - 1 ) / t ] .A f t e r a l g e b r a i c s i m p l i f i c a t i o n , w e g e tB ( f i2 ) = 2 (1 -h 2 ) [ 1 + ( b - 1 ) h21 [1 + (b - 2 ) hZ] / (h 2 b 0 .T h e m e a n s q u a r e e r r o r ( o r v a r i a n c e i g n o r i n g b i as ) o f ta 2i s g i v e n b yva r ( f i 2 ) = [E (N ) /E(D ) ] 2 {v a r (N ) / [E (N ) 2

+ v a r (D ) / [E (D )]2 _ 2 co v (N , D ) / [E (N ) E (D )]}= 2 (1 - h 2 ) 2 [ 1 + ( b - 1 ) h 2 1 2 / ( b f ) .

Probab i l i ty of a negat ive est imate o f h 2S i n c e fi 2 h a s a d i f fe r e n c e e x p r e s s i o n , M g - M o , a s it s n u -m e r a t o r , w h i c h m a y s o m e t i m e s b e n e g a t i v e, t h is r e s u lt s i na n i n v a l i d e s t i m a t e o f h 2 . T h e p r o b a b i l i t y o f s u c h c a s e s( G i l l a n d J e n s e n 1 9 6 8 ) i s g i v e n b yP r o b (fx2 < 0 ) = P r o b ( M g < M , ) = P r o b ( V < 1 )

= P r o b [ W < o~/(a~ + b o-~)]= P r o b [ W < (1 + b ~ 2 ) - 1 ]= H f q , f , ( 1 + b 2 2) - 1 ]

w h e r e H [ n l , n 2 , x ] i s t h e l o w e r - t a i l p r o b a b i l i t y a t p o i n t xo f t h e F d i s t r i b u t io n w i t h n 1 a n d n 2 d f a n d 2 2 = h 2 /( 1 - h 2 ) . T h e i n t e g r a l f o r m o f H [ n 1 , n 2 , x ] is

xH [n l , n 2, x] = S [B (nl / 2 , n2/2)] - 1 (nl /n2 )nl 1 /209y(nl /2)- 1 ( ] + nl y /n2 ) - (nl +n2) /2 dy

a n d B ( n, m ) i s t h e b e t a f u n c t i o n .Pro bab ili ty distribution of h 2T h e e x a c t p r o b a b i l i t y d i s t r i b u t i o n o f fl 2 i sP r o b ( fi 2 < X) = P ro b [(V r - l )/ (V~ + b - 1) < x]

= P r o b ( w < y ) = H [ q , f, y ]w he re y = [1 + x b/ (1 - x) ] /(1 + b 22) .Confidence intervalT o c o m p u t e t h e 1 0 0 ( 1 - ~ ) % c o n f id e n c e in t e rv a l f o r h 2,l e t F L a n d F u b e r e s p e c t i v e l y t h e c ~/ 2 a n d 1 - c ~ / 2 l o w e rp r o b a b i l i t y p o i n t s o f t h e F d i s t r i b u t i o n w i t h ( q , f ) d e g r e e so f f r e e d o m 9 U s i n g t h e d i s t r i b u t i o n a l b e h a v i o r o f W a s t h eF d i s t r i b u t i o n , i t i s e a s y t o t r a n s l a t eP r o b ( F L < W < F v ) = l - c~i n t oP ro b [ (V - Fu) / (V + (b - 1) Fu) _< h 2

_


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T h u s a n e x a c t 1 0 0 (1 - ~ ) % c o n f i d e n c e i n t e r v a l i s t1 12 ~ 2 ]k L~ U I,w h e r e1 ] ~ = ( V ~ - V v ) / [ V , + ( b - 1 ) Fu]a n d1 12 = ( V r _ F L ) / [ V r + ( b - 1 ) V L ] .

A p p e n d i x I IE v a l u a t i o n o f b i a s, s t a n d a r d e r r o r , a n d t h e p r o b a b i l i t y o fa n e g a t i v e e s t i m a t e o f h e ri t a b il i ty f r o m m u l t i - l o c a t i o nt r ia ls .D e p e n d e n c e o f V g a n d V IC o m p u t e t h e c o v a r i a n c e c o y ( Vg , V r)C O V ( V g , V i )

= E (v ~ v , ) - E ( v 0 E (V e)

= g ( m g ) g ( M ~ ) - g ( m g ) g ( ~ ) g ( m I ) E ( M - -~ )

B i a s a n d m e a n s q u a r e e r r o r o f 1~W e s h a ll f o ll o w t h e T a y l o r s er ie s e x p a n s i o n a p p r o a c h o n112 i n te r m s o f t h e n u m e r a t o r N = M g - M ~ a n d d e n o m i n a -t or D = M g + ( L - 1 ) M ~ + L ( b - 1 ) M e. N o t ic e t ha t2 a n d E ( D ) = b L 2 " 2 2(N ) = b L 2~ a~ , ( r e ( ,t g + 2 , + 1 ).B ( ~ 2 ) = E(~2)-h 2 = h 2 [ C 1 f f A 1 - C 1 2 / ( A 1 A 2 ) ] ,va r (~2) = h a [C 1 fA2 + C 22/A 2 - 2 C 12/ (A1 A2)]w h ere 2 2 2 2O'g/0"e ,1 = b L 2g, 2 g = a n d t h e r a t i o o f g e n e t i c v a ri -a n c e t o e r r o r v a r i a n c e c o m p o n e n t s , A 2 = b L ( 2 2 + 2 2 + 1 ),C , ~ = 2 [ ( 1 + b 2 ~ + b LC 2 2 = 2 [ ( 1 + b 2 1 2 + b L

+ L 2 ( b - 1)2 / I ] ,C 1 2 = 2 [ ( 1 + b 2 1 2 +b L

) 2 ) 2 / q q _ ( 1 q _ b 2 2 ) 2 / b ] ,292)2/q+ (L - 1) 2 (1 +b ,~,i2)2/b

2 ~ ) 2 / q - 2 ( L - 1 ) ( 1 + b 2 12 )2 /b 1.T h e h e r i t a b i l i t y i n t e r m s o f 2 2 a n d ).2 i s

2 2 2 2h = 2g/(2g + 2i + 1).P r o b a b i l i t y o f n e g a t iv e e s t i m a t e s o f h 2T h e q u a n t i t y h 2 w o u l d b e e s t i m a t e d a s n e g a t i v e i fM g < M I. T h u s ,P r o b (1]2 < 0 ) = P r o b ( M g < M I )

= P r o b [ W < ( a 2 + b a 2 ) / ( a 2 + b cr~ + b L ag2)]

44 1w h ere W = [M g/ (% + b a 2 + b L a2)] /[M, / (cr 2 + b cq2)] an dw o u l d f o l l o w t h e F - d i s t r i b u t i o n w i t h q a n d p d f . T h u sP r o b ( h 2 < 0 ) = H [ q , p , (1 + b 2 2)/(1 + b 2 2 + b L 2 2) ].P r o b a b i l i t y d i s t r i b u t io n o f I~W e c a n e x p r e s s t h e p r o b a b i l i t y P r o b (f l 2 > x ) a s a p r o b a -b i li ty o n a l in e a r c o m b i n a t i o n o f i n d e p e n d e n t c h i - s q u a r e sv a r i a b l e s .P r o b ( ~ 2 > x )

= P r o b { ( M g - g i ) / [ g g + ( L - 1 ) g i + L ( b - 1 ) M e ] > x} .2 2 p M i ~ ( l + b ~ . i 2 +in ce q M g ~ ( l + b 2 2 + b L 2 2 ) a e Z q ,

2 2 p M I ~ ( 1 + b 2 2 ) b - 1 2 2 a n d f M eL 2 2 ) a e Z q , a e Z p ,2 Zr2 . W e hav e , a f t e r s im p l i f i ca t i on ,"

Pr ob ( ft 2 > x ) = P ro b (01 Z2 + 02 Zp + 03 )~r > 0 ).F o l l o w i n g a n a p p r o x i m a t i o n g i ve n b y I m h o f f (1 9 61 ), t h ea b o v e p r o b a b i l i t y c a n b e a p p r o x i m a t e d b y P r o b ( Z 2 > Z ) ,a s a p r o b a b i l i t y f u n c t i o n h a v i n g a c h i - s q u a r e d i s t r i b u t io nw h e r e

3 3V = C 2 / C 3 ,Z = - - C 1 ( v / C 2 ) 1 /2 q - v ,C 1 = 0 1 q + O 2 p + 0 3 f ,C 2 = q O 2 + p O 2 + f O 2 ,C 3 = q 0 3 + p 0 3 + f 0 3 a .T h u s t h e p r o b a b i l i t y p o i n t o f th e f t2 d i s t r i b u t io n c a n b eo b t a i n e d b y u s i n g a s u it a b l e n u m e r i c a l a l g o r i t h m .

R e f e r e n c e sBogy o TP, Becker WA (1963) Exac t conf idence in tervals forheritability estimate d from paren tal half-sib correla tions.B iometr ics 19:494-496Bridges J r , Knap p S , Cornel ius SJ (1991) Standard er rors an dconfidence interval estimators for expected selection re-sponse . Crop Sci 31:253-255Broemeling L D (1969) Con fidence intervals for measures o fheritabili ty. Biometrics 25:424-427Falcon er DS (1982) Int rod uct ion to quan t i ta t ive genet ics . Long-m an Inc ., N ew Y orkGil l JL , Jensen EL (1968) Probabi l i ty of obta ining negat iveestimates of heritability. B iometrics 24 :51 7-5 26Graybi l l FA, Chih-M ing Wa ng (1979) Conf idence in tervals forpropo r t ions o f var iabil ity in two-facto r nes ted var iance com-ponen t m odel s. J A m S ta t A ssoc 74 :368-37 4Graybi l l FA, Mart in F, Godfrey G (1956) Conf idence in tervalsfor varianc e ratios specifying genetic heritabili ty. Biometrics12:99 109Kend al l MG , Stuar t A (1969) Ad vanc ed theory o f s tat is ticsvol I . Char les Gri f f in and Co. , Lo nd onK n a p p SJ , St roup WW , Ross WM (1985) Exact conf idence in ter -vals for heri tabi li ty on a pro geny mea n bas is . Cro p Sci2 5 : 1 9 2 - 1 9 4Im h of f JP (1961) C om put ing t he d i s t ri bu t ion o f quad ra t i c f o rm sin normal var iables . B iometr ika 48:419-425

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