Theoretical and Applied Genetics 45, t 70-- 175 (1974) | by Springer-Verlag 1974
Combining Ability Type of Analysis for Triallel Crosses in Maize (Zea mays L.) K. N . PONNUSWAMY, M. N . DAS and M. I. HANDOO Indian Society of Agricultural Statistics, institute of Agricultural Research Statistics, New Delhi, D e p a r t m e n t of Agriculture, Srinagar (India) Summary. Data on 30 three-way maize (Zea mays L.) hybrids formed from 5 inbred lines were subjected to the combining ability type of analysis. The relative importance of general and specific line effects in maize three-way hybrids was studied by the Ponnuswamy and Das (1973) method. It was also shown that this analysis provides the breeder with the basic information necessary to choosing proper breeding materials and in deciding the order in which they should be combined to get desirable three-way hybrids.
Introduction
A three-way cross symbolised b y (AB)C has been defined as a cross between the line C and the unrelated F 1 hybrid (AB), lines A and B being called grand-parental or half-parental lines and line C being known as the (Full) parental line. The triallel cross (T. C.) has been defined b y Rawlings and Cockerham (1962a) as a set of all possible three-way hybrids among a group of (Inbred) lines. Thus, given a set of v lines, the triaUel will consist of a set of [ v ( v - 1)2_(v~ 2)] three-way crosses. The well-known combining ability analysis of diallel cross (De) has helped in understanding the nature of gene action in single cross hybrids, on which depends the breeding policy and selection procedure. It has helped in the evaluation of lines for their suitability in the development of hybrids and synthetic varieties. Methods such as the 'Triallel Analysis' and 'Double Cross Analysis' of Rawlings and Cockerham (1962 a, b), and Analysis of Partial Triallel Cross (P. T. C.) of Hinkelmann (1965), have made it possible to subject the double-crosses and three-way crosses to appropriate statistical and genetical analysis. However, not much attention has been paid to the development of methods for the evaluation of lines based on the results of the triallel experiments. Evaluation of lines is as i m p o r t a n t as the q u a n t i t a t i v e genetic analysis of a mating design to the breeder whose aim is to develop 'strains' or 'hybrids' which are in some way superior to those already in commercial use. Recently some a t t e m p t s have been made b y P o n n u s w a m y (t971) and Ponnusw a m y and Das (t973) to develop suitable models and methods of analysis to s t u d y both the evaluation of lines and quantitative genetics aspects of Triallel. The parameterization of the model in P o n n u s w a m y (197t) and P o n n u s w a m y and Das (1973) is different
from that of Rawlings and Cockerham (1962a). In the former case the model has been developed on the lines of the combining ability model of diallel and is easily understood. The relationships between the parameters of the models in Rawlings and Cockerham (1962a) and Ponnuswamy and Das (1973) have also been established. In the case of Rawlings and Cockerh a m (t962a), both the average two-life specific effects and the two-line order effects involve dominance, whereas in Ponnuswamy and Das (1973) it is only the two-line specific effects of the second kind which involve dominance. This finding has resulted in the development of certain partial triallels which provide complete information on dominance. The parameterization in Hinkelmann (t965) and P o n n u s w a m y and Das (1973) is similar but Hinkelmann considers the reduced model consisting of only general line effects for the analysis of partial triallel cross. As pointed out earlier, b o t h Rawlings et al. (1962a) and Hinkelmann (1965) consider only the quantitative genetic aspects of the triallel, whereas P o n n u s w a m y et al. (1973) consider both aspects. The present investigation is an a t t e m p t to compare the relative importance of general and specific line effects in three-way hybrids involving five inbred lines of maize and show t h a t the triallel experimental d a t a can be used for the evaluation of lines b y subjecting the data to the Analysis proposed b y Ponnusw a m y and Das. The concepts and formulae which are relevant to the present s t u d y are presented in the next Section. Material and Methods Methods f o r the Evaluation o f L i n e s
The model, appropriate for the triallel experiment carried out in a randomized block design layout, is Yiikl :~ /~ + hi + h i + gk + &i + sik + sjk + tiik + rz + eijkz
(1)
K. N. P o n n u s w a m y et al. : Combining Abi'lity T y p e of Analysis for Triallel Crosses in Maize where yiikl is the yield of the cross (ixj)k in t h e lth replicate, rt is t h e lth replicate effect, # is the general mean, hi is the general line effect (g.l,e.) of the first kind, ge is t h e g.l.e, of t h e second kind, di] is t h e two-line specific effect (t.l.s.e.) of t h e first kind (both i and j being grand-parents), si~ is t h e two-line specific effect of the second kind (i being half-parent, k being parent), tii~ is t h e three-line specific effect and ei]~'s are i d e n t i c a l l y i n d e p e n d e n t l y d i s t r i b u t e d n o r m a l v a r i a t e s w i t h m e a n zero and v a r i a n c e a S. The least square e s t i m a t e s of t h e general and specific line effects are given b y = ~ ....
2) ( v -
]
3).1 2
requires t h a t all possible r e l e v a n t comparisons be made, which is a v e r y c u m b e r s o m e and tedious procedure. Moreover, it is also desirable to k n o w the r e l a t i v e imp o r t a n c e of general and specific line effects. To facilitate comparison and also to d e t e r m i n e t h e r e l a t i v e i m p o r t a n c e of t h e general and specific effects, t h e c o m p u t a t i o n of certain statistics as given below is necessary. a.j--
( K - 2' 5 ) _,r ~
+
tr(~----l] (?:- ~)1
(s)
k~i#j
(2)
x
ati"= (v -- 2) (v -- 3) j ir
~i = [ v ~ 2 3 ) r ]
17t
[y..i. + ( v t ~ 2 ) y i . . . -
*~
((v--~-2i) y.... ] ,
(4)
2
k
-- L r ( v - 2) ( v - 3)J (10)
k~i~i
,~ ~_, (~ki)2
at..i ~ (v-- 2) (v-- 3) i k-.i
[ (v2--6v+ 7)~ \r(v -- 2) (v -- 3)/~2,
-~..
(v -
3?
adi'--=- (v t~--2) ~] (dq)'-- [r(v -- l) ~v : 2)) (v -- 4) ] ~ '
(v~ -
2
v(v-
ti y ' ' ' -
4v + 2) r ( h i + h~) -(v - 3) . ~2
s'~i = (~--~) [yi.]. 4-(D)yi.~. 4- ((v D 3 ) ) y i i . . _
Gs. i
1) (v Z 3)-~vZ.--]-)J
i i ",'-'-'; (v I 2 ) ~ _
(;j'd ~ _
(v-
J
i,i 2(VDv--g) y .... -- r ( v - - 2 ) ~ - - ( ( v
D3) r)~'--
[Y~i~ -
y... -
L -
;j -
~e -
Zj -
~ik -
~i4
(7)
where v is the n u m b e r of lines., 5V 4- 5), D 1 = (va -- 7v ~ + 14v -- 7) D~-~ r ( v - - 1) ( v - - 3) (v -- 4) Yiik. is t h e m e a n of the cross (ixj)k, y.... is the t o t a l of all the crosses, yi... is t h e t o t a l of all the crosses which involve line i as g r a n d - p a r e n t , y..i. is the sum of all the crosses which i n v o i v e line i as parent, yq.. is t h e sum of aU t h e D
=
(V ~ - -
crosses which i n v o l v e lines i and j as grand-parents,
yid. is t h e t o t a l of all t h e crosses which i n v o l v e line i as g r a n d - p a r e n t and line j as p a r e n t ; as usual, bar (--) a b o v e y's stands for t h e mean, average being t a k e n over all t h e subscripts which are replaced b y dots(.), and (A) o v e r t h e p a r a m e t e r s stands for t h e estimates. The v a r i a n c e s and covariances of these e s t i m a t e s and o t h e r related statistics for t e s t i n g the hypothesis i n v o l v i n g these p a r a m e t e r s h a v e also been w o r k e d out [c. f. Ponnusw a m y and Das 1973]. F o r t h e sake of r e a d y reference, t h e s t a n d a r d errors (SE) of the e s t i m a t e s are p r e s e n t e d in the Appendix. Comparisons of the p e r f o r m a n c e of t h e i n d i v i d u a l lines as well as t h e c o m b i n e d p e r f o r m a n c e of pairs or t r i p l e t s of lines are of considerable interest. No doubt, comparisons, i n v o l v i n g i n d i v i d u a l effects of general and specific effects of a set of lines w i t h those of s a m e o t h e r set, or t h e same set in different order, will p r o v i d e sufficient i n f o r m a t i o n in this direction; and t tests can be easily devised to m e e t the situation. H o w e v e r , in t h e e v a l u a t i o n of lines, n o t only t h e general line effects of the p a r t i c u l a r line b u t also all t h e two-line and three-line specific effects i n v o l v i n g t h a t line should be considered together. This
Theoret. Appl. Genetics, Vol. 45, No. 4
4v 4- l)
~, (14)
~,, '
(15)
( v - - a)z ]o" (t6) r~(v :- i) (~ - 3) where ~.z is t h e e s t i m a t e of error v a r i a n c e (cf. Ponnusw a m y and Das 1973). The r e l a t i v e m a g n i t u d e of these statistics will indicate t h e r e l a t i v e i m p o r t a n c e of the general and specific effects and will also show w h e t h e r t h e r e is m u c h v a r i a b i l i t y in the specific effects i n v o l v i n g a p a r t i c u l a r line, a pair or triplet of lines. F o r example, the presence of r e l a t i v e l y large v a r i a t i o n s in the specific effects i n v o l v i n g a p a r t i c u l a r line would indicate t h a t t h e r e are specific c o m b i n a t i o n s of t h e iine in question w i t h certain inbreds which yield c o n s i d e r a b l y more t h a n w o u l d be e x p e c t e d on t h e basis of t h e a v e r a g e p e r f o r m a n c e of t h e lines involved, and o t h e r c o m b i n a t i o n s w h i c h yield m u c h less t h a n expected. On the o t h e r hand, if t h e r e is no v a r i a b i l i t y in t h e specific effects associated w i t h a line, say i, t h e n it would a p p e a r t h a t line i u n i f o r m l y t r a n s m i t s its high-yielding (or lowyielding) ability to all of t h e crosses which involve line i. Suppose the v a r i a n c e s a,h, as~ and aq are s u b s t a n t i a l l y larger t h a n as.i, m..i, then, in order to get a high-yielding c o m b i n a t i o n , it m a y be b e t t e r to use line i as a g r a n d p a r e n t t h a n as a parent. Of course these decisions should t a k e into a c c o u n t the general and specific effects of t h e o t h e r lines too. In o r d e r to d e t e r m i n e e x a c t l y t h e lines which should be considered, and the order in which t h e y should be c o m b i n e d w i t h the given line, the general and specific effects m a y be tested t h r o u g h the use of approp r i a t e l test. T h u s the r e l a t i v e m a g n i t u d e of the r e l e v a n t statistics, for example, ~ , and 3~ as c o m p a r e d w i t h those of ~s, ~2 ~2 ~ ~g . . . ad~ and at~ (as, & o~..i), will p r o v i d e sufflcmnt k n o w l e d g e to decide w h e t h e r t h e general line effects alone are
~,~hi= (~i) ~-
~~ ~
( 2(v - 1) ~7~(~7-Z!
-~~ ~'ri = ( g i P -
2)(v 2 -
' (13)
[
t 72
K. N. P o n n u s w a m y et al. : Combining Ability T y p e of Analysis~ or Triallel Crosses in Maize
sufficient, or w h e t h e r two-line and three line specific effects should also be t a k e n into consideration in the e v a l u a t i o n of t h e b r e e d i n g w o r t h of lines. The inform.ation p r o v i d e d by these variances, s u p p l e m e n t e d b y knowledge gained f r o m t h e p a r t i c u l a r comparisons, should be a d e q u a t e to d e t e r m i n e w h e t h e r it is advisable to go for s y n t h e t i c s by using specific lines, or to go for the devel o p m e n t of hybrids, and if deciding on hybridization, which are the lines t h a t should be considered and the order in which t h e y should be c o m b i n e d in a t h r e e - w a y cross. M e t h o d s to s t u d y the Q u a n t i t a t i v e G e n e t i c A s p e c t o f T r i a l l e l
Considering the set of lines as a r a n d o m sample from an infinite population, and assuming the p a r t i c u l a r v a r i a n c e c o v a r i a n c e s t r u c t u r e for t h e p a r a m e t e r s in m o d e l (1) [viz., E ( h ~ ) = a~, E ( h i g , ) = agh E (gi)2 = a~, E(d~) = % E(d~js~i) = E(d~jsi~) = ,,d,, ~(s~j) = ,,'~, E ( s q s i i ) = ass, E(t~j~) = a~, E(tijktikj) = E(tijktiki ) = art,
E(e~.m) = a s and all the o t h e r covariances being zero@ P o n n u s w a m y and Das (1973) h a v e shown t h a t the variances and covariances c o m p o n e n t s of the general effects (viz. a~, a~ and agh) are functions of a d d i t i v e and a d d i t i v e • a d d i t i v e t y p e of epistasis only. The c o m p o n e n t s a~ and ads are functions of a d d i t i v e • a d d i t i v e t y p e of epistasis only. as2 and ass are the o n l y two c o m p o n e n t s which involve d o m i n a n c e component. The c o m p o n e n t s a~ and art are functions of epistatic c o m p o n e n t s other t h a n a d d i t i v e • additive. Thus, t h e non-significance of the three-line specific effects will p r o v i d e positive evidence of the absence of all epistasis o t h e r t h a n a d d i t i v e • additive. The pooled m e a n square of d effects and t effects when tested against error will show t h e absence of all t y p e s of epistatic gene action. Similarly, t h e pooled m e a n square of s and t effects when tested against error will provide an indirect t e s t for t h e absence of all gene action other t h a n additive, and a d d i t i v e • a d d i t i v e t y p e of epistasis. Likewise, the non-significance of the pooled m e a n square of s, d and t Table 1. A n a l y s i s
effects when tested against error will show t h e absence of all t y p e s of n o n - a d d i t i v e gene action. However, for the e s t i m a t i o n of all the nine design components, the n u m b e r of lines should be at least 6. M a t e r i a l f o r the S t u d y
H a n d o o ' s (1964) d a t a p e r t a i n i n g to 30 t h r e e - w a y h y b r i d s formed f r o m 5 inbred lines of maize c o n s t i t u t e d t h e m a t e r i a l for the present study. The d a t a were subjected to the analysis discussed above and the salient features of the 'results (presented in Tables I and 2) are now discussed.
R e s u l t s and D i s c u s s i o n s W e s h a l l c o n s i d e r t h e r e s u l t s of t h e c h a r a c t e r ' m e a n g r a i n y i e l d p e r p l a n t ' for d e t a i l e d d i s c u s s i o n a n d b r i e f l y i n d i c a t e t h e s a l i e n t f e a t u r e s of t h e r e s u l t s for the other characters. Mean
Grain
Yield per Plant (in gins/at)
!5%
moisture)
T h e a n a l y s i s of v a r i a n c e is p r e s e n t e d in t a b l e 1, a n d t h e e s t i m a t e s of t h e g e n e r a l a n d s p e c i f i c e f f e c t s a n d o t h e r r e l a t e d s t a t i s t i c s for m e a n y i e l d p e r p l a n t are p r e s e n t e d in t a b l e 2. T h e a n a l y s i s s h o w s t h a t t h e g e n e r a l line e f f e c t s ( b o t h first a n d s e c o n d kinds) as w e l l as t h e t w o - l i n e s p e c i f i c e f f e c t s of t h e first k i n d are h i g h l y s i g n i f i c a n t , t h e t w o - l i n e s p e c i f i c e f f e c t s of t h e s e c o n d k i n d (s~is) are s i g n i f i c a n t , w h e r e a s t h e t h r e e - l i n e s p e c i f i c e f f e c t s are n o t s i g n i f i c a n t . T h i s s h o w s t h e a b s e n c e of all t y p e s of e p i s t a s i s o t h e r t h a n a d d i t i v e • a d d i t i v e . I t is also e v i d e n t t h a t t h e n a t u r e of g e n e a c t i o n is predominantly additive. However, because the numb e r of lines w a s less t h a n six, t h e m a g n i t u d e s of
o f v a r i a n c e f o r c h a r a c t e r s 1 to 7 ( m e a n s q u a r e s )
Characters Source 1) General line effect of t h e first k i n d ( h ' s e l i m . g's) General line effect of t h e 2nd k i n d ( g ' s e l i m . h's) T w o line specific effects of t h e first kind ( d ' s e l i m . s's) T w o line specific effects of t h e second k i n d (s's elim. d's) Three-line specific effects Crosses Error
d.f.
Mean yield
Mean number of days
Mean ear length (cm.) (7)
girth (cm.) (8)
Mean plant height (cm.) (9)
gms./plant
kg./per ha.
to tassel
to silk
(2)
(3)
(4)
(5)
(6)
v1 = 4
1471.41"*
2632220**
3.631"*
3.417"*
3.798**
2.569**
109.18"*
v1 = 4
1865.71"*
2859570**
56.322**
55.5430**
3.366**
4.181"*
3084.37**
v2 = 5
552.05**
457134*
2.135"
3.290**
2.t44"*
0.122NS
219.70"
va = II
270.73*
474425*
0.705NS
1.227NS
0.972*
0.279**
130.55NS
155821NS
0.496NS
0.670NS
0.603NS 0.125NS
139.63NS
797579** 161 057
8.836** 0.907
9.133"*
1.490"* 0.521
554.61"* 89.62
v4 = 5 v5 = 29 v6 = 177
58.46NS 506.49** 89.34
0.947
0.863** 0.096
where v1 ~ (v -- 1), v 2 ~ v(v -- 3)/2, v 3 = (v 2 - - 3v + 1), v 4 = (v ~ -- 6v + 7)/2, v 5 ~ [v(v -- ~) (v -- 2)/2 -- 11 and v 6 ~ vs(r--~ ) r being the number of replications and v the number of lines. ** denotes significance at t % level, * denotes significance at 5% level and NS stands for not significant. T h e o r e t . A p p l . Genetics, Vol. 45, N o . 4
K. N. P o n n u s w a m y et al. ; C o m b i n i n g A b i l i t y T y p e of Analysis for Triallel Crosses in Maize
~73
Table 2. The estimates of general and specific effects of lines and other related statistics
(mean yield per plant)
General line effects Two line specific effect [dq, sq] Lines
of first kind
of 2nd kind
PI
8.719
t3.78o
P~
--5.733
-9.646
P~
6.499
6.020
P~
--4.619
--6.173
P~
--4.866
--3.979
Lines 1
2
--
3 5.625
(6.650)
5.625 (--1.859) -- 9599 (1.444) 3.o58 (6.694) --8.083 (--6.280)
-- 5.975 (3.420) --4.742 (--4.642) 5.092 (--5.429)
4
-- .599
3.058
(-2.395) --5.975 (-- .129) -2.633 9 (--1.088) 3.942 (3.612)
5
(2.964) --4.742 (--5.432) 2.633 (-- 5.628) --- .950 (8.096)
--8.083
(-7.220) 5.092 (7.420) 3.942 (0.762) --.950 (--.963) --
Standard Errors:
SE(h,) = 1.544, SE(ffi) = t.894, SE('hi-- "hi~~ 2.4t7, SE(hi~-- g i ) ~ 1.890, SE(gi -- g"/) = 2.989 SE(dq) = 2,363, SE(~'q) = 2.745, SE(dii -- dik) = 3.858, SE(dij -- det) = 2,728 SE(dij -- ~ii) = 2.745, SE(dii--('si j + "sii)/2) = 2.046, SE('~ii -- 2ii) = 3.66t, SE(2ij -- ~i/c) = 4.484 SE(~q -- ~'ei) = 3.661, SE(~'ii -- "2kl) = 3.504. Figures in the bracket stand for (sq) effects. Three-line specific effects. tx~3 = 0.629, tlz4 = 0.866, ta~5= -- t.495, tla~ = 2.291, t x s ~ = - 2.145, tx.~5= -- 0.t45, ~14,= -2.083 t'x,. = .44t, t~,,5= 1.64t, t'152= -- 0.208, t;,, = -- 1.070, txs, = 1.279, t'2,1= -- 2.920, t;41 = 1.216, f2sl = t .704
SE(~ie) = 1.930 Note 1: Estimates for only 15 out of 30 parameters are given here, since when the number of lines is 5, the parameter tqe = time for all i, j, k, m, i #/" ~ k 9a t ~ m. Also for the estimation of a~ and ate, v should be greater than 5 and hence genetic components could not be estimated in this case. Note 2: Also when the number of lines v = 5, SE('gi) and SE(h| -- "gi) are equal, so also SE(~q -- "did and SE('~q -- "d~i). Lines
ahi
agi
~di
~si.
~Ts.i
~ti..
fft..i
Px P~ P3 P4 -P5
73.638 30.485 39.855 18.953 21.295
t86.3Ol 89,472 32.667 34.532 12.259
25.767 30.800 tl.725 5.439 28.116
27.866 t9.294 5.295 t2.772 39.117
19.879 25.596 --3.389 35.120 26.180
--3.562 --3.125 --4.090 --4.346 --5.666
1.16t --1.025 --6.288 --2.787 --4.146
different genetic components could not be worked out. F r o m t h e r e s u l t s p r e s e n t e d i n t a b l e 2, i t is e v i d e n t t h a t lines P1 a n d P8 a r e s i g n i f i c a n t l y s u p e r i o r to t h e r e s t of t h e lines i n t h e i r a v e r a g e p e r f o r m a n c e , Also, there are no marked differences in the average perf o r m a n c e of lines P1 a n d P3 as g r a n d - p a r e n t s or a m o n g t h e r e s t of t h e t h r e e l i n e s b o t h as p a r e n t s a n d as g r a n d - p a t e n t s . L i n e Pa is b e t t e r t h a n l i n e P3 i n p a r e n t a l p e r f o r m a n c e . L i n e Pa p e r f o r m s b e t t e r as p a r e n t t h a n as g r a n d - p a r e n t , w h i l e t h e r e v e r s e is t r u e for l i n e P z ; i n t h e case of t h e o t h e r s t h e r e are n o m a r k e d differences. T h e c o m p a r a t i v e l y large m a g n i t u d e of { ~ } a n d {~h,) c o m p a r e d w i t h { ~ i , ~ i , a n d ~.i} i n all t h e lines
Theoret. Appl. Genetics, Vol. 45, No. 4
e x c e p t P4 a n d P~ s h o w s t h a t t h e g e n e r a l l i n e effects are m o r e i m p o r t a n t t h a n t h e specific effects i n lines P1, P2 a n d Pz, w h e r e a s for l i n e s P~ a n d P5 t h e specific effects are m o r e i m p o r t a n t . A l t h o u g h l i n e s P1 a n d P3 are s i m i l a r i n t h a t b o t h e x h i b i t h i g h g e n e r a l line effects, t h e y a t t a i n t h e i r h i g h average performance by entirely different means. T h e r e l a t i v e l y low v a r i a t i o n s i n t h e specific effects [ d3., a~3. a n d a~.3~ a s s o c i a t e d w i t h l i n e P8 i n d i c a t e t h a t line P3 u n i f o r m l y t r a n s m i t s i t s h i g h - y i e l d i n g a b i l i t y to all i t s t h r e e - w a y crosses. O n t h e o t h e r h a n d , t h e h i g h v a r i a t i o n s a s s o c i a t e d w i t h l i n e P1 s h o w t h a t t h e r e are specific c o m b i n a t i o n s of l i n e P1 w i t h c e r t a i n inbreds which yield considerably more than would be expected and certain other combinations which
t74
K. N. Ponnuswamy et al. : Combining Ability Type of Analysis for Triallel Crosses in Maize
yield much less than expected on the basis of the average performance. For this reason line Pz is probably superior to line P1 for inclusion in the production of a synthetic variety, line P1 is probably superior to line P3 if specific high-yielding combinations are desired. Also, since there are no specific effects involving lines P1 and P3, probably a cross involving both the lines might be used for the development of a synthetic variety. If one is interested in developing superior threeway hybrids from the given material, Px and P3 cannot be ignored as they are the only two lines which have high general effects, which means that a third line has to be selected from the remaining three lines which will have high positive interactions with lines P1 and Pa- The significantly high negative values associated with the specific effects of line P5 with line Pa, as well as the negative general effects of line Ps, indicate that line P5 is not a desirable one. Lines P2 and P4 have both negative and positive interactions with lines P1 and Pz. By considering the two-line specific effects, it can be shown that the triplets of lines (P,, P2 and Pa) or (P,, Pa and P4) m a y be chosen and combined as (13)2 and (34)1, respectively, to produce superior three-way hybrids. Comparison of the yields of the different inbred lines shows line P 3 i s the highest yielder and line P1 is the lowest yielder. This observation, and information on the average performance of lines P1 and Pa in the three-way hybrid Combinations, indicate that inbred line P1 probably contains a large number of unfavourable alleles whereas line P3 contains a larger number of favourable alleles. This aspect needs to be examined further.
Other Characters The results for the character mean yield per hectare broadly agree with the conclusions drawn from character mean yield per plant. The results indicate that eloss (t3)2 is probably superior to other crosses. The results pertaining to m a t u r i t y characters, such as number of days to tassel, number of days to silk etc., show that lines t92 and Pa are always associated with early m a t u r i t y and P1 and Pa with late maturity. They also indicate that it is possible to combine earliness of m a t u r i t y and high-yielding potential in the three-way hybrid b y forming crosses which include lines P2 or P4 with P1 and Pa. It is evident from the results of the ear characteristics, such as mean ear length and mean ear girth, that the highyielding potential associated with lines -Pl and Pa can probably be ascribed to the large and lengthy ears associated with lines ,ol and Pa. The poor potential of line -P2 m a y be attributed largely to shorter ear length rather than smaller ear girth, the reverse being true for line P~. An interesting point
here is the absence of all epistasis other than additive • additive for all the characters considered. Conclusions
As pointed out earlier, the analysis carried out indicates that it m a y be possible to develop synthetic varieties using the materials at hand. It also shows that it may be possible to exploit the highyielding potential, early maturity and other desired characteristics which are present in different lines, to obtain superior three-way hybrids by selecting the suitable lines and combining them in appropriate order. AeknGwlcdgements
The authors are grateful to the referee for his valuable suggestions and comments. Literature
Griffing, B.: Concept of general and specific combining ability in relation to diallel systems. Aust. J. Biol. Sc. 9, 463--493 0956). Handoo, M. Iqbal: Studies on the combining ability, gene-action and performance of inbred lines in single, three-way and double cross combinations in Maize (Zea mays L.). Unpublished M. Sc. thesis I. A. R. I., N. Delhi (1964). Hinkelmann, K. : Partial triallel crosses. Sankhya, Series A. 27 , 173--195 (1965). Ponnuswamy, K. N. : Some contributions to design and analysis for diallel and triallel crosses. Ph. D. thesis, Indian Agricultural Research Institute, New Delhi (1971). Ponnuswamy, K. N., Das, M. N. : Design and analysis for triallel cross. Communicated to Biometrics (1973). Rawlings, J. O., Cockerham, C. C. : Triallel analysis. Crop Sci. 2, 228--23t 0962a). Rawlings, J. O., Cockerham, C. C.: Analysis of double cross hybrid populations. Biometrics 18, 229--244 (1962 b). Sprague, G. F., Tatum, L. A. : General vs. Specific combining ability in single cross of corn. Jour. Amer. Soc. Agron. 34, 923--932 0942). Appendix
The standard errors of the estimates are given below : --
=[
.....
-]
Ll/rv2(v-2) (v - 3)J a,
SU(g~)
SE(dq)
= l/2(v -- 1) V ~Tff~( --
= []/r(v
sE(&)-
~) •
o,
(_v ~ 3 )
(~-2)
] (Y,
1/!~-4~+')
1/
(v~ - 6v + 7) SE'(i,k) = V~(~-- 'Y(~ = ~ i x ~, S E ( h i - hi)
1/
= ~rv(V
SE(~i -- gi) --
2(v -- 1/ "
2) (V ~ 3)
X a,
2a
Vrv(v - 3)
TheoreL Appl. Genetics, Vol. 45, No. 4
K. N. P o n n u s w a m y et aI. : Combining A b i i i t y T y p e of Analysis for Triallel Crosses in Maize = l/3(~ SE(s
SE(~j
SE(~ i
- - '2~)
-
Vr~(~
^
1) -
V
2) ~ '
2
2(v - 3) ~
- - dik) = _ _ r ( v - - 1) (v -- 2) (v -- 4)
-
=
2(v - - 3) /~(~ - 1)(~-
• (7
SE(h~k -
~#.) = V
~
1/2(v
= g2r(v
~kj) = V r ( v
-
1) •
- - 1) 2 •
a,
~'
1/2(~ ~ 6~ +_ 71 x (7,
2) •
S E ( 2 q ~ - - hik) = g r ( v
~'
SE(hjk-
sE(~,.j -
175
BZm)
- - 1) (v - - 3)
1/2(v~ -- I iv ~ + 3Sv -- 39)
x (7, -
2) (v = -
4v + 1) 4i • (7,
= V r(v -
S E ( ~ I j - - s'ie) = Vr~(v 2_~) ( v - ~ j ) ( v ~
l ) (v -
3) (v -
4)
(7,
] / 2 ( v ~ - - t l v ~ - 39v ~ 4J)_ SE(hik SE(sq
- - $ki) = V r v ( v _
1) (v - 4) • a ,
1/ 2(~_ _- 3~ + 1)
S E ( ~ i i - - "s,~l) = g r v ( v
- - 1) (v -- 3) X
Combining Ability Type of Analysis for Triallel Crosses in Maize (Zea mays L.) K. N . PONNUSWAMY, M. N . DAS and M. I. HANDOO Indian Society of Agricultural Statistics, institute of Agricultural Research Statistics, New Delhi, D e p a r t m e n t of Agriculture, Srinagar (India) Summary. Data on 30 three-way maize (Zea mays L.) hybrids formed from 5 inbred lines were subjected to the combining ability type of analysis. The relative importance of general and specific line effects in maize three-way hybrids was studied by the Ponnuswamy and Das (1973) method. It was also shown that this analysis provides the breeder with the basic information necessary to choosing proper breeding materials and in deciding the order in which they should be combined to get desirable three-way hybrids.
Introduction
A three-way cross symbolised b y (AB)C has been defined as a cross between the line C and the unrelated F 1 hybrid (AB), lines A and B being called grand-parental or half-parental lines and line C being known as the (Full) parental line. The triallel cross (T. C.) has been defined b y Rawlings and Cockerham (1962a) as a set of all possible three-way hybrids among a group of (Inbred) lines. Thus, given a set of v lines, the triaUel will consist of a set of [ v ( v - 1)2_(v~ 2)] three-way crosses. The well-known combining ability analysis of diallel cross (De) has helped in understanding the nature of gene action in single cross hybrids, on which depends the breeding policy and selection procedure. It has helped in the evaluation of lines for their suitability in the development of hybrids and synthetic varieties. Methods such as the 'Triallel Analysis' and 'Double Cross Analysis' of Rawlings and Cockerham (1962 a, b), and Analysis of Partial Triallel Cross (P. T. C.) of Hinkelmann (1965), have made it possible to subject the double-crosses and three-way crosses to appropriate statistical and genetical analysis. However, not much attention has been paid to the development of methods for the evaluation of lines based on the results of the triallel experiments. Evaluation of lines is as i m p o r t a n t as the q u a n t i t a t i v e genetic analysis of a mating design to the breeder whose aim is to develop 'strains' or 'hybrids' which are in some way superior to those already in commercial use. Recently some a t t e m p t s have been made b y P o n n u s w a m y (t971) and Ponnusw a m y and Das (t973) to develop suitable models and methods of analysis to s t u d y both the evaluation of lines and quantitative genetics aspects of Triallel. The parameterization of the model in P o n n u s w a m y (197t) and P o n n u s w a m y and Das (1973) is different
from that of Rawlings and Cockerham (1962a). In the former case the model has been developed on the lines of the combining ability model of diallel and is easily understood. The relationships between the parameters of the models in Rawlings and Cockerham (1962a) and Ponnuswamy and Das (1973) have also been established. In the case of Rawlings and Cockerh a m (t962a), both the average two-life specific effects and the two-line order effects involve dominance, whereas in Ponnuswamy and Das (1973) it is only the two-line specific effects of the second kind which involve dominance. This finding has resulted in the development of certain partial triallels which provide complete information on dominance. The parameterization in Hinkelmann (t965) and P o n n u s w a m y and Das (1973) is similar but Hinkelmann considers the reduced model consisting of only general line effects for the analysis of partial triallel cross. As pointed out earlier, b o t h Rawlings et al. (1962a) and Hinkelmann (1965) consider only the quantitative genetic aspects of the triallel, whereas P o n n u s w a m y et al. (1973) consider both aspects. The present investigation is an a t t e m p t to compare the relative importance of general and specific line effects in three-way hybrids involving five inbred lines of maize and show t h a t the triallel experimental d a t a can be used for the evaluation of lines b y subjecting the data to the Analysis proposed b y Ponnusw a m y and Das. The concepts and formulae which are relevant to the present s t u d y are presented in the next Section. Material and Methods Methods f o r the Evaluation o f L i n e s
The model, appropriate for the triallel experiment carried out in a randomized block design layout, is Yiikl :~ /~ + hi + h i + gk + &i + sik + sjk + tiik + rz + eijkz
(1)
K. N. P o n n u s w a m y et al. : Combining Abi'lity T y p e of Analysis for Triallel Crosses in Maize where yiikl is the yield of the cross (ixj)k in t h e lth replicate, rt is t h e lth replicate effect, # is the general mean, hi is the general line effect (g.l,e.) of the first kind, ge is t h e g.l.e, of t h e second kind, di] is t h e two-line specific effect (t.l.s.e.) of t h e first kind (both i and j being grand-parents), si~ is t h e two-line specific effect of the second kind (i being half-parent, k being parent), tii~ is t h e three-line specific effect and ei]~'s are i d e n t i c a l l y i n d e p e n d e n t l y d i s t r i b u t e d n o r m a l v a r i a t e s w i t h m e a n zero and v a r i a n c e a S. The least square e s t i m a t e s of t h e general and specific line effects are given b y = ~ ....
2) ( v -
]
3).1 2
requires t h a t all possible r e l e v a n t comparisons be made, which is a v e r y c u m b e r s o m e and tedious procedure. Moreover, it is also desirable to k n o w the r e l a t i v e imp o r t a n c e of general and specific line effects. To facilitate comparison and also to d e t e r m i n e t h e r e l a t i v e i m p o r t a n c e of t h e general and specific effects, t h e c o m p u t a t i o n of certain statistics as given below is necessary. a.j--
( K - 2' 5 ) _,r ~
+
tr(~----l] (?:- ~)1
(s)
k~i#j
(2)
x
ati"= (v -- 2) (v -- 3) j ir
~i = [ v ~ 2 3 ) r ]
17t
[y..i. + ( v t ~ 2 ) y i . . . -
*~
((v--~-2i) y.... ] ,
(4)
2
k
-- L r ( v - 2) ( v - 3)J (10)
k~i~i
,~ ~_, (~ki)2
at..i ~ (v-- 2) (v-- 3) i k-.i
[ (v2--6v+ 7)~ \r(v -- 2) (v -- 3)/~2,
-~..
(v -
3?
adi'--=- (v t~--2) ~] (dq)'-- [r(v -- l) ~v : 2)) (v -- 4) ] ~ '
(v~ -
2
v(v-
ti y ' ' ' -
4v + 2) r ( h i + h~) -(v - 3) . ~2
s'~i = (~--~) [yi.]. 4-(D)yi.~. 4- ((v D 3 ) ) y i i . . _
Gs. i
1) (v Z 3)-~vZ.--]-)J
i i ",'-'-'; (v I 2 ) ~ _
(;j'd ~ _
(v-
J
i,i 2(VDv--g) y .... -- r ( v - - 2 ) ~ - - ( ( v
D3) r)~'--
[Y~i~ -
y... -
L -
;j -
~e -
Zj -
~ik -
~i4
(7)
where v is the n u m b e r of lines., 5V 4- 5), D 1 = (va -- 7v ~ + 14v -- 7) D~-~ r ( v - - 1) ( v - - 3) (v -- 4) Yiik. is t h e m e a n of the cross (ixj)k, y.... is the t o t a l of all the crosses, yi... is t h e t o t a l of all the crosses which involve line i as g r a n d - p a r e n t , y..i. is the sum of all the crosses which i n v o i v e line i as parent, yq.. is t h e sum of aU t h e D
=
(V ~ - -
crosses which i n v o l v e lines i and j as grand-parents,
yid. is t h e t o t a l of all t h e crosses which i n v o l v e line i as g r a n d - p a r e n t and line j as p a r e n t ; as usual, bar (--) a b o v e y's stands for t h e mean, average being t a k e n over all t h e subscripts which are replaced b y dots(.), and (A) o v e r t h e p a r a m e t e r s stands for t h e estimates. The v a r i a n c e s and covariances of these e s t i m a t e s and o t h e r related statistics for t e s t i n g the hypothesis i n v o l v i n g these p a r a m e t e r s h a v e also been w o r k e d out [c. f. Ponnusw a m y and Das 1973]. F o r t h e sake of r e a d y reference, t h e s t a n d a r d errors (SE) of the e s t i m a t e s are p r e s e n t e d in the Appendix. Comparisons of the p e r f o r m a n c e of t h e i n d i v i d u a l lines as well as t h e c o m b i n e d p e r f o r m a n c e of pairs or t r i p l e t s of lines are of considerable interest. No doubt, comparisons, i n v o l v i n g i n d i v i d u a l effects of general and specific effects of a set of lines w i t h those of s a m e o t h e r set, or t h e same set in different order, will p r o v i d e sufficient i n f o r m a t i o n in this direction; and t tests can be easily devised to m e e t the situation. H o w e v e r , in t h e e v a l u a t i o n of lines, n o t only t h e general line effects of the p a r t i c u l a r line b u t also all t h e two-line and three-line specific effects i n v o l v i n g t h a t line should be considered together. This
Theoret. Appl. Genetics, Vol. 45, No. 4
4v 4- l)
~, (14)
~,, '
(15)
( v - - a)z ]o" (t6) r~(v :- i) (~ - 3) where ~.z is t h e e s t i m a t e of error v a r i a n c e (cf. Ponnusw a m y and Das 1973). The r e l a t i v e m a g n i t u d e of these statistics will indicate t h e r e l a t i v e i m p o r t a n c e of the general and specific effects and will also show w h e t h e r t h e r e is m u c h v a r i a b i l i t y in the specific effects i n v o l v i n g a p a r t i c u l a r line, a pair or triplet of lines. F o r example, the presence of r e l a t i v e l y large v a r i a t i o n s in the specific effects i n v o l v i n g a p a r t i c u l a r line would indicate t h a t t h e r e are specific c o m b i n a t i o n s of t h e iine in question w i t h certain inbreds which yield c o n s i d e r a b l y more t h a n w o u l d be e x p e c t e d on t h e basis of t h e a v e r a g e p e r f o r m a n c e of t h e lines involved, and o t h e r c o m b i n a t i o n s w h i c h yield m u c h less t h a n expected. On the o t h e r hand, if t h e r e is no v a r i a b i l i t y in t h e specific effects associated w i t h a line, say i, t h e n it would a p p e a r t h a t line i u n i f o r m l y t r a n s m i t s its high-yielding (or lowyielding) ability to all of t h e crosses which involve line i. Suppose the v a r i a n c e s a,h, as~ and aq are s u b s t a n t i a l l y larger t h a n as.i, m..i, then, in order to get a high-yielding c o m b i n a t i o n , it m a y be b e t t e r to use line i as a g r a n d p a r e n t t h a n as a parent. Of course these decisions should t a k e into a c c o u n t the general and specific effects of t h e o t h e r lines too. In o r d e r to d e t e r m i n e e x a c t l y t h e lines which should be considered, and the order in which t h e y should be c o m b i n e d w i t h the given line, the general and specific effects m a y be tested t h r o u g h the use of approp r i a t e l test. T h u s the r e l a t i v e m a g n i t u d e of the r e l e v a n t statistics, for example, ~ , and 3~ as c o m p a r e d w i t h those of ~s, ~2 ~2 ~ ~g . . . ad~ and at~ (as, & o~..i), will p r o v i d e sufflcmnt k n o w l e d g e to decide w h e t h e r t h e general line effects alone are
~,~hi= (~i) ~-
~~ ~
( 2(v - 1) ~7~(~7-Z!
-~~ ~'ri = ( g i P -
2)(v 2 -
' (13)
[
t 72
K. N. P o n n u s w a m y et al. : Combining Ability T y p e of Analysis~ or Triallel Crosses in Maize
sufficient, or w h e t h e r two-line and three line specific effects should also be t a k e n into consideration in the e v a l u a t i o n of t h e b r e e d i n g w o r t h of lines. The inform.ation p r o v i d e d by these variances, s u p p l e m e n t e d b y knowledge gained f r o m t h e p a r t i c u l a r comparisons, should be a d e q u a t e to d e t e r m i n e w h e t h e r it is advisable to go for s y n t h e t i c s by using specific lines, or to go for the devel o p m e n t of hybrids, and if deciding on hybridization, which are the lines t h a t should be considered and the order in which t h e y should be c o m b i n e d in a t h r e e - w a y cross. M e t h o d s to s t u d y the Q u a n t i t a t i v e G e n e t i c A s p e c t o f T r i a l l e l
Considering the set of lines as a r a n d o m sample from an infinite population, and assuming the p a r t i c u l a r v a r i a n c e c o v a r i a n c e s t r u c t u r e for t h e p a r a m e t e r s in m o d e l (1) [viz., E ( h ~ ) = a~, E ( h i g , ) = agh E (gi)2 = a~, E(d~) = % E(d~js~i) = E(d~jsi~) = ,,d,, ~(s~j) = ,,'~, E ( s q s i i ) = ass, E(t~j~) = a~, E(tijktikj) = E(tijktiki ) = art,
E(e~.m) = a s and all the o t h e r covariances being zero@ P o n n u s w a m y and Das (1973) h a v e shown t h a t the variances and covariances c o m p o n e n t s of the general effects (viz. a~, a~ and agh) are functions of a d d i t i v e and a d d i t i v e • a d d i t i v e t y p e of epistasis only. The c o m p o n e n t s a~ and ads are functions of a d d i t i v e • a d d i t i v e t y p e of epistasis only. as2 and ass are the o n l y two c o m p o n e n t s which involve d o m i n a n c e component. The c o m p o n e n t s a~ and art are functions of epistatic c o m p o n e n t s other t h a n a d d i t i v e • additive. Thus, t h e non-significance of the three-line specific effects will p r o v i d e positive evidence of the absence of all epistasis o t h e r t h a n a d d i t i v e • additive. The pooled m e a n square of d effects and t effects when tested against error will show t h e absence of all t y p e s of epistatic gene action. Similarly, t h e pooled m e a n square of s and t effects when tested against error will provide an indirect t e s t for t h e absence of all gene action other t h a n additive, and a d d i t i v e • a d d i t i v e t y p e of epistasis. Likewise, the non-significance of the pooled m e a n square of s, d and t Table 1. A n a l y s i s
effects when tested against error will show t h e absence of all t y p e s of n o n - a d d i t i v e gene action. However, for the e s t i m a t i o n of all the nine design components, the n u m b e r of lines should be at least 6. M a t e r i a l f o r the S t u d y
H a n d o o ' s (1964) d a t a p e r t a i n i n g to 30 t h r e e - w a y h y b r i d s formed f r o m 5 inbred lines of maize c o n s t i t u t e d t h e m a t e r i a l for the present study. The d a t a were subjected to the analysis discussed above and the salient features of the 'results (presented in Tables I and 2) are now discussed.
R e s u l t s and D i s c u s s i o n s W e s h a l l c o n s i d e r t h e r e s u l t s of t h e c h a r a c t e r ' m e a n g r a i n y i e l d p e r p l a n t ' for d e t a i l e d d i s c u s s i o n a n d b r i e f l y i n d i c a t e t h e s a l i e n t f e a t u r e s of t h e r e s u l t s for the other characters. Mean
Grain
Yield per Plant (in gins/at)
!5%
moisture)
T h e a n a l y s i s of v a r i a n c e is p r e s e n t e d in t a b l e 1, a n d t h e e s t i m a t e s of t h e g e n e r a l a n d s p e c i f i c e f f e c t s a n d o t h e r r e l a t e d s t a t i s t i c s for m e a n y i e l d p e r p l a n t are p r e s e n t e d in t a b l e 2. T h e a n a l y s i s s h o w s t h a t t h e g e n e r a l line e f f e c t s ( b o t h first a n d s e c o n d kinds) as w e l l as t h e t w o - l i n e s p e c i f i c e f f e c t s of t h e first k i n d are h i g h l y s i g n i f i c a n t , t h e t w o - l i n e s p e c i f i c e f f e c t s of t h e s e c o n d k i n d (s~is) are s i g n i f i c a n t , w h e r e a s t h e t h r e e - l i n e s p e c i f i c e f f e c t s are n o t s i g n i f i c a n t . T h i s s h o w s t h e a b s e n c e of all t y p e s of e p i s t a s i s o t h e r t h a n a d d i t i v e • a d d i t i v e . I t is also e v i d e n t t h a t t h e n a t u r e of g e n e a c t i o n is predominantly additive. However, because the numb e r of lines w a s less t h a n six, t h e m a g n i t u d e s of
o f v a r i a n c e f o r c h a r a c t e r s 1 to 7 ( m e a n s q u a r e s )
Characters Source 1) General line effect of t h e first k i n d ( h ' s e l i m . g's) General line effect of t h e 2nd k i n d ( g ' s e l i m . h's) T w o line specific effects of t h e first kind ( d ' s e l i m . s's) T w o line specific effects of t h e second k i n d (s's elim. d's) Three-line specific effects Crosses Error
d.f.
Mean yield
Mean number of days
Mean ear length (cm.) (7)
girth (cm.) (8)
Mean plant height (cm.) (9)
gms./plant
kg./per ha.
to tassel
to silk
(2)
(3)
(4)
(5)
(6)
v1 = 4
1471.41"*
2632220**
3.631"*
3.417"*
3.798**
2.569**
109.18"*
v1 = 4
1865.71"*
2859570**
56.322**
55.5430**
3.366**
4.181"*
3084.37**
v2 = 5
552.05**
457134*
2.135"
3.290**
2.t44"*
0.122NS
219.70"
va = II
270.73*
474425*
0.705NS
1.227NS
0.972*
0.279**
130.55NS
155821NS
0.496NS
0.670NS
0.603NS 0.125NS
139.63NS
797579** 161 057
8.836** 0.907
9.133"*
1.490"* 0.521
554.61"* 89.62
v4 = 5 v5 = 29 v6 = 177
58.46NS 506.49** 89.34
0.947
0.863** 0.096
where v1 ~ (v -- 1), v 2 ~ v(v -- 3)/2, v 3 = (v 2 - - 3v + 1), v 4 = (v ~ -- 6v + 7)/2, v 5 ~ [v(v -- ~) (v -- 2)/2 -- 11 and v 6 ~ vs(r--~ ) r being the number of replications and v the number of lines. ** denotes significance at t % level, * denotes significance at 5% level and NS stands for not significant. T h e o r e t . A p p l . Genetics, Vol. 45, N o . 4
K. N. P o n n u s w a m y et al. ; C o m b i n i n g A b i l i t y T y p e of Analysis for Triallel Crosses in Maize
~73
Table 2. The estimates of general and specific effects of lines and other related statistics
(mean yield per plant)
General line effects Two line specific effect [dq, sq] Lines
of first kind
of 2nd kind
PI
8.719
t3.78o
P~
--5.733
-9.646
P~
6.499
6.020
P~
--4.619
--6.173
P~
--4.866
--3.979
Lines 1
2
--
3 5.625
(6.650)
5.625 (--1.859) -- 9599 (1.444) 3.o58 (6.694) --8.083 (--6.280)
-- 5.975 (3.420) --4.742 (--4.642) 5.092 (--5.429)
4
-- .599
3.058
(-2.395) --5.975 (-- .129) -2.633 9 (--1.088) 3.942 (3.612)
5
(2.964) --4.742 (--5.432) 2.633 (-- 5.628) --- .950 (8.096)
--8.083
(-7.220) 5.092 (7.420) 3.942 (0.762) --.950 (--.963) --
Standard Errors:
SE(h,) = 1.544, SE(ffi) = t.894, SE('hi-- "hi~~ 2.4t7, SE(hi~-- g i ) ~ 1.890, SE(gi -- g"/) = 2.989 SE(dq) = 2,363, SE(~'q) = 2.745, SE(dii -- dik) = 3.858, SE(dij -- det) = 2,728 SE(dij -- ~ii) = 2.745, SE(dii--('si j + "sii)/2) = 2.046, SE('~ii -- 2ii) = 3.66t, SE(2ij -- ~i/c) = 4.484 SE(~q -- ~'ei) = 3.661, SE(~'ii -- "2kl) = 3.504. Figures in the bracket stand for (sq) effects. Three-line specific effects. tx~3 = 0.629, tlz4 = 0.866, ta~5= -- t.495, tla~ = 2.291, t x s ~ = - 2.145, tx.~5= -- 0.t45, ~14,= -2.083 t'x,. = .44t, t~,,5= 1.64t, t'152= -- 0.208, t;,, = -- 1.070, txs, = 1.279, t'2,1= -- 2.920, t;41 = 1.216, f2sl = t .704
SE(~ie) = 1.930 Note 1: Estimates for only 15 out of 30 parameters are given here, since when the number of lines is 5, the parameter tqe = time for all i, j, k, m, i #/" ~ k 9a t ~ m. Also for the estimation of a~ and ate, v should be greater than 5 and hence genetic components could not be estimated in this case. Note 2: Also when the number of lines v = 5, SE('gi) and SE(h| -- "gi) are equal, so also SE(~q -- "did and SE('~q -- "d~i). Lines
ahi
agi
~di
~si.
~Ts.i
~ti..
fft..i
Px P~ P3 P4 -P5
73.638 30.485 39.855 18.953 21.295
t86.3Ol 89,472 32.667 34.532 12.259
25.767 30.800 tl.725 5.439 28.116
27.866 t9.294 5.295 t2.772 39.117
19.879 25.596 --3.389 35.120 26.180
--3.562 --3.125 --4.090 --4.346 --5.666
1.16t --1.025 --6.288 --2.787 --4.146
different genetic components could not be worked out. F r o m t h e r e s u l t s p r e s e n t e d i n t a b l e 2, i t is e v i d e n t t h a t lines P1 a n d P8 a r e s i g n i f i c a n t l y s u p e r i o r to t h e r e s t of t h e lines i n t h e i r a v e r a g e p e r f o r m a n c e , Also, there are no marked differences in the average perf o r m a n c e of lines P1 a n d P3 as g r a n d - p a r e n t s or a m o n g t h e r e s t of t h e t h r e e l i n e s b o t h as p a r e n t s a n d as g r a n d - p a t e n t s . L i n e Pa is b e t t e r t h a n l i n e P3 i n p a r e n t a l p e r f o r m a n c e . L i n e Pa p e r f o r m s b e t t e r as p a r e n t t h a n as g r a n d - p a r e n t , w h i l e t h e r e v e r s e is t r u e for l i n e P z ; i n t h e case of t h e o t h e r s t h e r e are n o m a r k e d differences. T h e c o m p a r a t i v e l y large m a g n i t u d e of { ~ } a n d {~h,) c o m p a r e d w i t h { ~ i , ~ i , a n d ~.i} i n all t h e lines
Theoret. Appl. Genetics, Vol. 45, No. 4
e x c e p t P4 a n d P~ s h o w s t h a t t h e g e n e r a l l i n e effects are m o r e i m p o r t a n t t h a n t h e specific effects i n lines P1, P2 a n d Pz, w h e r e a s for l i n e s P~ a n d P5 t h e specific effects are m o r e i m p o r t a n t . A l t h o u g h l i n e s P1 a n d P3 are s i m i l a r i n t h a t b o t h e x h i b i t h i g h g e n e r a l line effects, t h e y a t t a i n t h e i r h i g h average performance by entirely different means. T h e r e l a t i v e l y low v a r i a t i o n s i n t h e specific effects [ d3., a~3. a n d a~.3~ a s s o c i a t e d w i t h l i n e P8 i n d i c a t e t h a t line P3 u n i f o r m l y t r a n s m i t s i t s h i g h - y i e l d i n g a b i l i t y to all i t s t h r e e - w a y crosses. O n t h e o t h e r h a n d , t h e h i g h v a r i a t i o n s a s s o c i a t e d w i t h l i n e P1 s h o w t h a t t h e r e are specific c o m b i n a t i o n s of l i n e P1 w i t h c e r t a i n inbreds which yield considerably more than would be expected and certain other combinations which
t74
K. N. Ponnuswamy et al. : Combining Ability Type of Analysis for Triallel Crosses in Maize
yield much less than expected on the basis of the average performance. For this reason line Pz is probably superior to line P1 for inclusion in the production of a synthetic variety, line P1 is probably superior to line P3 if specific high-yielding combinations are desired. Also, since there are no specific effects involving lines P1 and P3, probably a cross involving both the lines might be used for the development of a synthetic variety. If one is interested in developing superior threeway hybrids from the given material, Px and P3 cannot be ignored as they are the only two lines which have high general effects, which means that a third line has to be selected from the remaining three lines which will have high positive interactions with lines P1 and Pa- The significantly high negative values associated with the specific effects of line P5 with line Pa, as well as the negative general effects of line Ps, indicate that line P5 is not a desirable one. Lines P2 and P4 have both negative and positive interactions with lines P1 and Pz. By considering the two-line specific effects, it can be shown that the triplets of lines (P,, P2 and Pa) or (P,, Pa and P4) m a y be chosen and combined as (13)2 and (34)1, respectively, to produce superior three-way hybrids. Comparison of the yields of the different inbred lines shows line P 3 i s the highest yielder and line P1 is the lowest yielder. This observation, and information on the average performance of lines P1 and Pa in the three-way hybrid Combinations, indicate that inbred line P1 probably contains a large number of unfavourable alleles whereas line P3 contains a larger number of favourable alleles. This aspect needs to be examined further.
Other Characters The results for the character mean yield per hectare broadly agree with the conclusions drawn from character mean yield per plant. The results indicate that eloss (t3)2 is probably superior to other crosses. The results pertaining to m a t u r i t y characters, such as number of days to tassel, number of days to silk etc., show that lines t92 and Pa are always associated with early m a t u r i t y and P1 and Pa with late maturity. They also indicate that it is possible to combine earliness of m a t u r i t y and high-yielding potential in the three-way hybrid b y forming crosses which include lines P2 or P4 with P1 and Pa. It is evident from the results of the ear characteristics, such as mean ear length and mean ear girth, that the highyielding potential associated with lines -Pl and Pa can probably be ascribed to the large and lengthy ears associated with lines ,ol and Pa. The poor potential of line -P2 m a y be attributed largely to shorter ear length rather than smaller ear girth, the reverse being true for line P~. An interesting point
here is the absence of all epistasis other than additive • additive for all the characters considered. Conclusions
As pointed out earlier, the analysis carried out indicates that it m a y be possible to develop synthetic varieties using the materials at hand. It also shows that it may be possible to exploit the highyielding potential, early maturity and other desired characteristics which are present in different lines, to obtain superior three-way hybrids by selecting the suitable lines and combining them in appropriate order. AeknGwlcdgements
The authors are grateful to the referee for his valuable suggestions and comments. Literature
Griffing, B.: Concept of general and specific combining ability in relation to diallel systems. Aust. J. Biol. Sc. 9, 463--493 0956). Handoo, M. Iqbal: Studies on the combining ability, gene-action and performance of inbred lines in single, three-way and double cross combinations in Maize (Zea mays L.). Unpublished M. Sc. thesis I. A. R. I., N. Delhi (1964). Hinkelmann, K. : Partial triallel crosses. Sankhya, Series A. 27 , 173--195 (1965). Ponnuswamy, K. N. : Some contributions to design and analysis for diallel and triallel crosses. Ph. D. thesis, Indian Agricultural Research Institute, New Delhi (1971). Ponnuswamy, K. N., Das, M. N. : Design and analysis for triallel cross. Communicated to Biometrics (1973). Rawlings, J. O., Cockerham, C. C. : Triallel analysis. Crop Sci. 2, 228--23t 0962a). Rawlings, J. O., Cockerham, C. C.: Analysis of double cross hybrid populations. Biometrics 18, 229--244 (1962 b). Sprague, G. F., Tatum, L. A. : General vs. Specific combining ability in single cross of corn. Jour. Amer. Soc. Agron. 34, 923--932 0942). Appendix
The standard errors of the estimates are given below : --
=[
.....
-]
Ll/rv2(v-2) (v - 3)J a,
SU(g~)
SE(dq)
= l/2(v -- 1) V ~Tff~( --
= []/r(v
sE(&)-
~) •
o,
(_v ~ 3 )
(~-2)
] (Y,
1/!~-4~+')
1/
(v~ - 6v + 7) SE'(i,k) = V~(~-- 'Y(~ = ~ i x ~, S E ( h i - hi)
1/
= ~rv(V
SE(~i -- gi) --
2(v -- 1/ "
2) (V ~ 3)
X a,
2a
Vrv(v - 3)
TheoreL Appl. Genetics, Vol. 45, No. 4
K. N. P o n n u s w a m y et aI. : Combining A b i i i t y T y p e of Analysis for Triallel Crosses in Maize = l/3(~ SE(s
SE(~j
SE(~ i
- - '2~)
-
Vr~(~
^
1) -
V
2) ~ '
2
2(v - 3) ~
- - dik) = _ _ r ( v - - 1) (v -- 2) (v -- 4)
-
=
2(v - - 3) /~(~ - 1)(~-
• (7
SE(h~k -
~#.) = V
~
1/2(v
= g2r(v
~kj) = V r ( v
-
1) •
- - 1) 2 •
a,
~'
1/2(~ ~ 6~ +_ 71 x (7,
2) •
S E ( 2 q ~ - - hik) = g r ( v
~'
SE(hjk-
sE(~,.j -
175
BZm)
- - 1) (v - - 3)
1/2(v~ -- I iv ~ + 3Sv -- 39)
x (7, -
2) (v = -
4v + 1) 4i • (7,
= V r(v -
S E ( ~ I j - - s'ie) = Vr~(v 2_~) ( v - ~ j ) ( v ~
l ) (v -
3) (v -
4)
(7,
] / 2 ( v ~ - - t l v ~ - 39v ~ 4J)_ SE(hik SE(sq
- - $ki) = V r v ( v _
1) (v - 4) • a ,
1/ 2(~_ _- 3~ + 1)
S E ( ~ i i - - "s,~l) = g r v ( v
- - 1) (v -- 3) X
