A new partial diallel model adapted to analyze reciprocal effects in grain yield of maize

Barata, Nazaré Mateus; Scapim, Carlos Alberto; Guedes, Terezinha Aparecida; Janeiro, Vanderly; Pinto, Ronald José Barth; Soto, Rodrigo Ivan Contreras; Kuki, Maurício Carlos

doi:10.1590/1984-70332019v19n1a04

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Crop Breeding and Applied Biotechnology

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ARTICLE • Crop Breed. Appl. Biotechnol. 19 (01) • Jan-Mar 2019 • https://doi.org/10.1590/1984-70332019v19n1a04 copy

A new partial diallel model adapted to analyze reciprocal effects in grain yield of maize

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

Comparison between two distinct groups using a partial diallel and considering reciprocal effects has not been described in the scientific literature, limiting diallel analysis results. This study compared two groups of parents using a partial diallel, and analyzed the general and specific combining ability and reciprocal effects, divided into maternal and cytoplasmic. A partial diallel was established with nine maize hybrids, including the F₁ and reciprocal hybrids for grain yield. For this application, the reciprocal effects were not significant. The proposed model would provide plant breeders with additional information to analyze partial diallel mating designs.

Keywords:
Combining ability; cytoplasmic effects maternal effects

Parents	Hybrids F₁’s and reciprocals
1 8330	1 x 6 (8330 x 30F80)	9 x 3 (CD3121-2 x AG8080)
2 CD3121-1	6 x 1 (30F80 x 8330)	4 x 6 (Flash x 30F80)
3 AG8080	1 x 7 (8330 x Strike)	6 x 4 (30F80 x Flash)
4 Flash	7 x 1 (Strike x 8330)	4 x 7 (Flash x Strike)
5 Dekalb 350	1 x 8 (8330 x Speed)	7 x 4 (Strike x Flash)
6 30F80	8 x 1 (Speed x 8330)	4 x 8 (Flash x Speed)
7 Strike	1 x 9 (8330 x CD3121-2)	8 x 4 (Speed x Flash)
8 Speed	9 x 1 (CD3121-2 x 8330)	4 x 9 (Flash x CD3121-2)
9 CD3121-2	2 x 6 (CD3121-1 x 30F80)	9 x 4 (CD3121-2 x Flash)
	6 x 2 (30F80 x CD3121-1)	5 x 6 (Dekalb 350 x 30F80)
	2 x 7 (CD3121-1 x Strike)	6 x 5 (30F80 x Dekalb 350)
	7 x 2 (Strike x CD3121-1)	5 x 7 (Dekalb 350 x Strike)
	2 x 8 (CD3121-1 x Speed)	7 x 5 (Strike x Dekalb 350)
	8 x 2 (Speed x CD3121-1)	5 x 8 (Dekalb 350 x Speed)
	2 x 9 (CD3121-1 x CD3121-2)	8 x 5 (Speed x Dekalb 350)
	9 x 2 (CD3121-2 x CD3121-1)	5 x 9 (Dekalb 350 x CD3121-2)
	3 x 6 (AG8080 x 30F80)	9 x 5 (CD3121-2 x Dekalb 350)
	6 x 3 (30F80 x AG8080)
	3 x 7 (AG8080 x Strike)
	7 x 3 (Strike x AG8080)
	3 x 8 (AG8080 x Speed)
	8 x 3 (Speed x AG8080)
	3 x 9 (AG8080 x CD3121-2)

Source variation	df	Sum of squares	E(LS)
Source variation	df	Sum of squares	Fixed	Random
Group	1	SS( $\hat{g} r_{l}$ )	$\frac{upq (p + q + 2)}{2N} ϕ_{g r} + σ^{2}$	$a_{1} u σ_{g r}^{2} + a_{2} u σ_{g 1}^{2} + a_{3} u σ_{g 2}^{2} + a_{4} u σ_{s}^{2} + σ^{2}$
GCAgr₁	p - 1	SS(CGCgr₁)	$(4 + 2 q) u ϕ_{g 1} + σ^{2}$	$(4 + 2 q) u σ_{g 1}^{2} + \frac{2 (p + 1)}{2 + q} u σ_{s}^{2} + σ^{2}$
GCAgr₂	q - 1	SS(CGCgr₂)	$(4 + 2 p) u ϕ_{g 2} + σ^{2}$	$(4 + 2 p) u σ_{g 2}^{2} + \frac{2 (p + 1)}{2 + q} u σ_{s}^{2} + σ^{2}$
SCA	pq	SS(SCA)	$2 u ϕ_{s} + σ^{2}$	$a_{5} u σ_{s}^{2} + σ^{2}$
MATgr₁	p -1	SS(MATgr₁)	$2 u q {φ_{d}}_{1} + \frac{2 u (q - 1)}{p q} ϕ_{r} + σ^{2}$	$2 q u σ_{d 1}^{2} + 2 u σ_{r}^{2} + σ^{2}$
MATgr₂	q - 1	SS(MATgr₂)	$2 u p {ϕ_{d}}_{2} + \frac{2 u (p - 1)}{p q} ϕ_{r} + σ^{2}$	$2 p u σ_{d 2}^{2} + 2 u σ_{r}^{2} + σ^{2}$
CIT	pq - p - q+1	SS(CIT)	$\frac{2 u (2 p q + 1)}{p q} ϕ_{r} + σ^{2}$	$2 u σ_{r}^{2} + σ^{2}$
Residual	N(u-1) -u+1	SSR	(²	(²

Source of variation	Hypotheses	F-Statistics	df
Fixed effects model			df
Group	H₀: gr₁=gr₂	$\frac{M S (G R)}{Q M R}$	[1, N(u -1)-u+1]
GCAgr₁	H₀: g₁=g₂=...=g_p=0	$\frac{M S (G C A g r_{1})}{Q M R}$	[(p-1), N(u-1)-u+1]
GCAgr₂	H₀: g₁=g₂=...=g_q=0	$\frac{M S (G C A g r_{2})}{Q M R}$	[(q-1), N(u-1)-u+1]
SCA	H₀: s_i0=s_0j=s_ij=0	$\frac{M S (S C A)}{Q M R}$	[pq, N(u-1)-u+1]
MATgr₁	H₀: d₁=d₂=...=d_p	$\frac{[M S (M A T g r_{1}) + b_{1} M S (C I T)]}{b_{2} Q M R}$	[(₁*, (p-1)(q-1)]
MATgr₂	H₀: d₁=d₂=...=d_q	$\frac{[M S (M A T g r_{2}) + b_{3} M S (C I T)]}{b_{4} Q M R}$	[(₂*, (p-1)(q-1)]
CIT	H₀: r_ij=0	$\frac{M S (C I T)}{Q M R}$	[(p-1)(q-1),N(u-1)-u+1]
Random effects model			[(p-1)(q-1),N(u-1)-u+1]
Group	H₀: (² _gr = 0	$\frac{M S (G R)}{[c_{1} M S (G C A g r_{1}}) + c_{2} M S (G C A g r_{2}) + c_{3} M S (S C A) + c_{4} Q M R]$	[1, (₃*]
GCAgr₁	H₀: (² _g1 =0	$\frac{M S (G C A g r_{1})}{[d_{1} M S (S C A}) + (1 - d_{1}) Q M R]$	[(p-1), (₄*]
GCAgr₂	H₀: (² _g2 = 0	$\frac{M S (G C A g r_{2})}{[d_{2} M S (S C A}) + (1 - d_{2}) Q M R]$	[(q-1), (₅*]
SCA	H₀: (² _s = 0	$\frac{M S (S C A)}{Q M R}$	[pq, N(u -1)-u+1]
MATgr₁	H₀: (² _d1 =0	$\frac{M S (M A T g r_{1})}{M S (C I T)}$	[(p-1), (p-1)(q-1)]
MATgr₂	H₀: (² _d2 = 0	$\frac{M S (M A T g r_{2})}{M S (C I T)}$	[(q-1), (p-1)(q-1)]
CIT	H₀: (² _r = 0	$\frac{M S (C I T)}{Q M R}$	[(p-1)(q-1),N(u -1) - u +1]

Fixed effects
Source of Variation	df	Sum of Squares	Mean Square	F	P-value
Treatments	48	238306781	4964725	3.563	0.000000
Block	2	21608144	10804072	7.753	0.000756
Residual	96	41388973	431135
Total	146	393696024
Group	1	3995786	3995786	2.867	0.0936
GCAgr₁	4	46605185	11651296	8.361	0.0000
GCAgr₂	3	10761960	3587320	2.574	0.0584
SCA	20	162530585	8126529	5.832	0.0000
MATgr₁	4	1379999	345000	0.277^**	0.9375
MATgr₂	3	4307305	1435768	0.943^**	0.4530
CIT	12	8057475	671456	0.482	0.9209
Estimates of least square
${\hat{φ}}_{g r}$ =357875.39; ${\hat{φ}}_{g 1}$ =279602.55; ${\hat{φ}}_{g 2}$ =47659.89; ${\hat{φ}}_{s}$ =1090154.12; ${\hat{φ}}_{d 1}$ =0^; ${\hat{φ}}_{d 2}$ =0 ${\hat{φ}}_{r}$ =0^*
Random effects
Source of Variation	D.F.	Sum of Squares	Mean Square	F	P-value
Group	1	3995785.71	3995785.71	0.7714^***	0.3886
GCAgr₁	4	46605184.68	11651296.17	1.4706^***	0.2480
GCAgr₂	3	10761959.97	3587319.99	0.4424^***	0.7253
SCA	20	162530584.70	8126529.24	5.8315	0.0000
MATgr₁	4	1379999.37	344999.85	0.5138	0.7271
MATgr₂	3	4307304.57	1435768.20	2.1383	0.1486
CIT	12	8057475.27	671456.28	0.4818	0.9209
Residual	96	133781099.00	1393553.10
Estimates of components of variance
${\hat{σ}}_{g r}^{2}$ =0^; ${\hat{σ}}_{g 1}^{2}$ = 103563.41; ${\hat{σ}}_{g 2}^{2}$ = 0^; ${\hat{σ}}_{s}^{2}$ =2238672.16; ${\hat{σ}}_{d 1}^{2}$ =0^; ${\hat{σ}}_{d 2}^{2}$ =25477.06; ${\hat{σ}}_{r}^{2}$ =0^

Group1		Group2
Parameters	Estimates	Parameters	Estimates
gr₁	-377.27	gr₂	392·99
$S E [\hat{g} r_{1} - \hat{g} r_{2}]$	454.88	-	-
g1	-616.69	g6	285·89
g2	-598.53	g7	-73·47
g3	611.25	g8	164·48
g4	300.84	g9	-376·90
g5	303.15	-	-
$S E [{\hat{g}}_{i} - {\hat{g}}_{k}]$	278.24	$S E [{\hat{g}}_{j} - {\hat{g}}_{k'}]$	257·60
S11	1745.39	S66	169·34
S22	2010.99	S77	1864·21
S33	151.65	S88	1742·24
S44	1708.79	S99	1024·58
S55	-816.45	-	-
$S E [{\hat{s}}_{i 0} - {\hat{s}}_{k 0}]$	786.99	-	-
d1	31.55	d6	-47.79
d2	30.81	d7	223.65
d3	153.81	d8	-214.64
d4	-75.48	d9	-259.78
d5	232.49
$S E [{\hat{d}}_{i} - {\hat{d}}_{k}]$	340.78	$S E [{\hat{d}}_{j} - {\hat{d}}_{k'}]$	357.41
S16	-521.35	r16	-266.46
S17	-693.09	r17	142.21
S18	-637.44	r18	-157.89
S19	106.49	r19	-16.40
S26	1284.96	r26	-537.26
S27	-65.24	r27	-131.99
S28	119.95	r28	159.77
S29	-3350.67	r29	210.94
S36	-404.68	r36	87.24
S37	-260.35	r37	-255.71
S38	-86.03	r38	227.43
S39	599.41	r39	-357.51
S46	-592.35	r46	-49.57
S47	-651.56	r47	276.75
S48	-949.96	r48	-207.24
S49	485.08	r49	-318.49
S56	64.08	r56	392.86
S57	-193.97	r57	-404.44
S58	-188.77	r58	-395.25
S59	1135.12	r59	108.27
$S E [{\hat{s}}_{i j} - {\hat{s}}_{k j}]$	622.17	$S E [{\hat{r}}_{i j} - {\hat{r}}_{k j}]$	762.00
$S E [{\hat{s}}_{i j} - {\hat{s}}_{i k'}]$	630.99	$S E [{\hat{r}}_{i j} - {\hat{r}}_{i k'}]$	746.61
$S E [{\hat{s}}_{i j} - {\hat{s}}_{k k'}]$	585.54	$S E [{\hat{r}}_{i j} - {\hat{r}}_{k k'}]$	1180.49

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[1] *E-mail: mcarloskuki@gmail.com