~ ~~
Biom. J. S (1984) 1,8848
Heryeae Agrioultural University, Department of Genetics Hisser, India
Some Theoretical Aspects of Analysis in Hall Diallel
M. SINQH and R. K. SINGH
Abstract
The analysis of Pi and Fz half dialiel has been discussed as compamd to full dhllel without reci- procal effects given by MATEEB and Jrms (1971). In practice, mostly the half diallel (one-way) crosses are,preparedanderroreously analyeed using the leaat squam estimah of full diallel without reoiprocal effects. The Hi and Hz, however, appeared to be over estimated when full diallel without reciprocal effects estimetes were used in place of those of half diellel. Leest square estimates were also obtained assuming heterogeneity of error components between parents and FJFZ families.
Key WOTds: Half diallel, Full diallel without reciprocal effects, F2 half diallel, Numerical analysis.
Introduction
The genetic components of variance analysis of MATHER (1 949) was extended by JINKS (1954) a n d H ~ w (1954) to diallel cross mating system. In their origi- nal models and also in their later works (HAYMILN 1958; JINKS 1956; MATHER and JINKS 1971,. 1977) only the full diallel with or without reciprocal effects were considered. It appears that method of analysis of half diallel for components of variances is not easily ascertained by common user following principles laid down by the pioneer works of JINKS (1954) and HAYMAN (1954). As EL result half diallel crosses are often analysed using HAYMAN’S model for full diallel without reciprocal effects. Keeping this in view, estimates of components ‘of variances for hrtlf diallel crosses were derived and the results thus obtained are presented here.
Derivations and Results
Fi half diallel: The differences between the full, diallel without reciprocal effects and half diallel are illustrated in Table .1 and 2.
64 M. SINQH, R. K. SINQX
Table 1 Full diallel without recliprocel effects
Here, 4 2 =- x 1 2 ~ x a i and 80 on for 4 3 etc.
Tabla 2 Completed table for half diallel
1 2 3 ... n
It may be noted that in case of full diallel when there are no reciprocal differ- ences, the average values of P1s and their respective reciprocal Pis are put in the table of full diallel without reciprocal effects (Table 1). Accordingly, the en-
vironmental component of Pis will be halved -& and hence the weighted
mean environmental component of each entry in the diallel table will be 1 [ (.. + +z (n - 1) (A$))] which is the environmental contribution of the interaction
variance, Vr (MATHER and JINICS, 1971). Here, &, is the environmental component of parents andZF the environmental components of Fls families. Since completed table of half diallel also has the identical off-diagonal values like full diauel without reciprocal effects (Table 2); one follows paradoxically the same expectations of environmental components in Vr as that of full diallel without reciprocal effects. But in fact, the identical off-diagonal values in the completed table of half diallel have not been reproduoed by taking the average of the Pis and their respective reciprocal Pi values as in full diallel without reciprocal effect and hence environ- mental component will not be halved. It follows, therefore, that the weighted mean environmental component of each entry in the completed table of half
diallel is - (Bp + (n - 1) Idp) which is the environmental contribution of the inter-
c 4 n 1
I n
Analysis of half diallel 65
Table 3 Expectations in Fi half diclllel
Y D Hi a2 F E
Table 4 Estimates by Least Square Method
Pa DHiHflE=(X'X)-' X'Y Para- V,, Vr vt w r E meter
D = 1 0 0 0 -1
H,= 0 4 -4 0
F = 2 0 0 -4
4 (n-1) .n
2 (n-2) n
-- E = 0 0 0 0 1
4b D = 1 0 0 0 -1 ~
3n-2 -- -4 n Full Hi= 1 4 0
without H2=. 0 4 -4 0
effect F = 2 0
Diallel 2 (n2-1) -- n2
2 (n-2) reciprocal
-- 0 -4 n E = 0 0 0 0 1
action variance (V,) from which the environmental contribution of other statistics like V,, P,, etc. can be derived following M ~ I I E I L and JINKS (1971). The ex- pectations of various statistics thus derived for Pi half diallel me given in Table 3. Keeping these expectations in view, a set of least square estimates of various parameters obtainable from half-diallel were derived and are given in Table 4. For comparison, the least square estimates for full diallel without reciprocal effects were also given in this table.
Fz b l f Ctidlel: Keeping in view the fact that F2 is a segregating generation whileFi is not, the
coefficient of the dominance contribution to means and varianoes must be changed to allow for the halving of the frequency of heterozygotes ineach of the off- 6 Biom. J., Vol. Z6, No. 1
66 M. SINOH, R. K. SINQH
diagonal entries. Therefore, the coefficients of all terms in the expectation were halved. The contribution of the dominance component to family and generation means was also halved, so the coefficient of Hi and H2 (terms in h2)' was quartered and the coefficient E" (terms in h) was halved (MATHER and J m s , 1971). JINKS (1956) and HAYW (1957) derived the least square estimates for various para- meters in Fz and backcross generations. Them derivations were, however, valid for full diallel'croaaes. No such statistics are available for F2 half diallel. Therefore on the principle of Fl half diallel, the expectations of genetio parameters were derived for F2 half diallel and are given in Table 5 and'6, respectively.
Table 6 Expectations in F2 half diallel
1 0 0 0 1 1 1 8
0 1 1
4 16 -- -
VP Vr -
1 1 1 4 16 16 8 n
- 1 -- -- 1 - - V;
E2 0 0 0 0 1
Table 6 Eetimates of component in F2 diallel by Least Square Method
D =
Fa half Hi = diallel
612 =
F = E2 =
6b D =
a, = F2 half $allel without reoiprowl effects F =
E2 =
As evident of dominance
1
4
0
4
0
1
4
0
4
0
0 0 0
16 0 - 16
16 -16 0
0 0 - 8
0 0 0
0 0 0
16 0 -16
16 -16 0
0 0 -8
0 0 0
- 1 4 (an-4)
n - 16 (n-1)
n 4 (n-2)
n
--
-- 1
-1 12 n-8
n 8 (n2-1)
n2
n
-- --
4 (n-2)
1
--
from the expectations given in Table 4a and 4b the estimates components (Hl, H2) were different in two casa. Similar was the
Analysis of half diallel 67
case for F2 as indicated by Hi and H2 expression in 6a and 6b. Obviously, by using expectations of full diallel without reciprocal effects in half diallel, an over estimate of dominance components will be obtained. The magnitude of over-
(2n-2) E (2n2-4n+2) E estimation for HI and H 2 in Pi was by and , respec-
n n2 tively. In P3 these magnitudes of overestimations were amplified four times (Table
~
(89% - 8) E2 6 s and 6b). Precisely, the overestimation for HI and Ha was by n
(8nz-l6n+8) E2 n2
and respectively. Evidently, the actual amount of over-
estimation will depend on the number of parents (n) used and error component (E) in the diallel experiment. From above, one concludes that the use of least square estimates as given in Table 4b and 6 b for Pi and P2, respectively, will not be valid for analysing one-way Fi/F2 crosses.
Another aspect which needs mention is the error component. Expectations of second degree statistics are available with separate as well as combined estima- Table 7 The Least Square Estimates when E p and E F are not homoge- neow in Pi half didlel
Para- V p V , V; W, E P EP meter
D = 1 0 0 0 0 -1 4 (n-1)
n HI = 1 4 0 -4 -- -1
4 (n2-2n+1) 4 (n-1) H 2 = 0 4' -4 0 - n2 n2
2 (n-2) F = 2 0 0 -4 0 -- n
-~
E p = . O 0 O O 1 E p = O O 0 O O
0 1
Table 8 The Least Square Estimates when E p and Ep, are not homogeneous in F2 half diallel
Para- V p V, V; W, Ep, Z P meters
D = 1 0 0 0 0 -1 16 (n-1)
n Hi = 4 16 0 -16 -
16 (n2-2n+l) 16 (n-1) - n2 HZ = 0 16 -16 0 -
lt2
F = 4 0 0 - 8 0
E p , = O 0 0 0 1 E p = 0 0 0 0 0 6.
4 (n-2) n
0 1
--
68 M. SINGE. R; K. SINGE
tea of error variance for parents and their crosses in full diallel (HAYDUN, 1964 b). As such statistics were not available for half dialle1,'the least square estimates were derived assuming heterogeneity of error variance between parents and crosses for Pi (Table 7) and Fz (Table 8) separately. As evident from these tables, it would be useful first to test the heterogeneity of error varianoe between parents and their crosses and then proceed for diallel analysis accordingly. If heterogeneity for error component between Fl/Fz families and the parent population is found sig- nificant one should use the least square estimates for genetic components as given in Table 7 and 8. If, however, the heterogeneity is not significant, use of estima- tes as given in Table 4a and 6 a for Pi and P,, respectively, will be valid.
Acknowledgementa
The authors express their gratitude to Professor J. L. JINKS, Univerity of Birmingham, U.K. for his valuable guidanae during the course of present study.
References
HA-, B. I., 1964i The theory and analysis of diallel crosses. Genetics 89, 789-809. HAYBIAW, B. I., 1967 : Interaction, heterosis and diallel c r o m . Genetioa 42, 336-366. HAYINAN, B. I., 1968: The theory and analysis of diallel crosses 11. Genetics 48, 63-85. JINXS, J. L., 1964: The enalyais of continuous veriation in a diallel cross of Nicotiaana rzcetica
JINKB, J. L., 1968: The F2 and beok cross generations from a set of diallel orosses. Heredity 10,
MATHEB, K., 1949: Biometrical Genetics. London, Dover Publ. MATED, K., JINKS, J. L., 1971 : Biometrical Genetics (2nd edn.) Chapman and Hall Ltd., London. MAT-, K., JINKS, J. L., 1977: Introduction t o Biometricel Genetios. Chapman and Hall, London.
Manuserfpt received: 7.1.1982
varieties. Genetios 89, 767-788.
1-30.
Author's address: Prof. Dr. R. K. SINGH Dr. M. SINGE, Astt. Geneticist Department of Genetics Haryanr Agricultural University Hissar - 126004, India
