Analysis of heterosis components and prediction of hybrid means based on intergroup topcrosses in maize

Parameter or contrast	Variance
$\hat{μ} = \frac{1}{2} [{\bar{V}}_{. (1)} + {\bar{V}}_{. (2)}]$	$(\frac{n + m}{4 n m}) σ^{2}$
$\hat{d} = \frac{1}{2} [{\bar{V}}_{. (1)} - {\bar{V}}_{. (2)}]$	$(\frac{n + m}{4 n m}) σ^{2}$
${\hat{v}}_{i} = V_{i} - {\bar{V}}_{. (1)}$	$(\frac{n - 1}{n}) σ^{2}$
${\hat{v}}_{j} = V_{j} - {\bar{V}}_{. (2)}$	$(\frac{m - 1}{m}) σ^{2}$
$\hat{\bar{h}} = \frac{1}{2} \{[{\bar{T}}_{. (1)} + {\bar{T}}_{. (2)}] - [{\bar{V}}_{. (1)} + {\bar{V}}_{. (2)}]\}$	$(\frac{n + m}{2 n m}) σ^{2}$
${\hat{h}}_{i} = T_{i} - {\bar{T}}_{. (1)} - \frac{1}{2} [V_{i} - {\bar{V}}_{(1)}]$	$[\frac{5 (n - 1)}{4 n}] σ^{2}$
${\hat{h}}_{j} = T_{j} - {\bar{T}}_{. (2)} - \frac{1}{2} [V_{j} - {\bar{V}}_{(2)}]$	$[\frac{5 (m - 1)}{4 m}] σ^{2}$
$\hat{c} = \frac{1}{2} [{\bar{T}}_{. (1)} - {\bar{T}}_{. (2)}]$	$(\frac{n + m}{4 n m}) σ^{2}$
$v_{i} - v_{i'}$	$2 σ^{2}$
$v_{j} - v_{j'}$	$2 σ^{2}$
$h_{i} - h_{i'}$	$\frac{5}{2} σ^{2}$
$h_{j} - h_{j'}$	$\frac{5}{2} σ^{2}$

Source of variation	df	Sum of squares
Populations	$2 (m + n) - 1$	$\sum_{i = 1}^{m} T_{i}^{2} + \sum_{i = 1}^{m} V_{i}^{2} + \sum_{j = 1}^{n} T_{j}^{2} + \sum_{j = 1}^{n} V_{j}^{2} - \frac{Y_{. .}^{2}}{2 (m + n)}$
Varieties (1)	$m - 1$	$\frac{4}{5} \sum_{i = 1}^{m} {\{[V_{i} - {\bar{V}}_{. (1)}] + \frac{1}{2} [T_{i} - {\bar{T}}_{. (1)}]\}}^{2}$
Varieties (2)	$n - 1$	$\frac{4}{5} \sum_{j = 1}^{n} {\{[V_{j} - {\bar{V}}_{. (2)}] + \frac{1}{2} [T_{j} - {\bar{T}}_{. (2)}]\}}^{2}$
V (1) vs V (2)	1	$\frac{m n}{m + n} {[{\bar{V}}_{. (1)} - {\bar{V}}_{. (2)}]}^{2}$
Average heterosis	1	$\frac{1}{2 (m + n)} {\{[T_{. (1)} + T_{. (2)}] - [V_{. (1)} + V_{. (2)}]\}}^{2}$
Varieties heterosis (1)	$m - 1$	$\frac{4}{5} \sum_{i = 1}^{m} {\{[T_{i} - {\bar{T}}_{. (1)}] - \frac{1}{2} [V_{i} - {\bar{V}}_{. (1)}]\}}^{2}$
Varieties heterosis (2)	$n - 1$	$\frac{4}{5} \sum_{j = 1}^{n} {\{[T_{j} - {\bar{T}}_{. (2)}] - \frac{1}{2} [V_{j} - {\bar{V}}_{. (2)}]\}}^{2}$
TC (1) vs TC (2)	1	$\frac{m n}{m + n} ({\bar{T}}_{. (1)} - {\bar{T}}_{. (2)})^{2}$
Error	df _e	SS _e

Source of variation	df	FF	PH	GY
Partial diallel
Populations	118	30.296^**	3742.151^*	13.1581^**
Parents group 1 (G1)	9	56.608^**	8607.559^**	8.4989^**
Parents group 2 (G2)	9	145.521^**	4684.066^**	6.7861^**
G1 vs G2	1	151.250^**	3125.000^**	0.2289
Heterosis	99	16.207^**	3220.447^**	14.2915^**
Average heterosis	1	285.315^**	57859.437^**	314.5897^**
Parent heterosis (G1)	9	5.159^**	5208.223^**	1.8128^**
Parent heterosis (G2)	9	15.013^**	851.048^**	2.1618^**
Specific heterosis	80	14.045^**	2548.536^**	13.1419^**
Error	372	0.144	12.133	0.1382
Intergroup reciprocal topcrosses
Populations	39	11.33^**	1578.83^**	5.10955^**
Parents group 1 (G1)	9	4.20^**	184.50^**	0.46634^**
Parents group 2 (G2)	9	16.72^**	553.38^**	0.39885^**
G1 vs G2	1	37.81^**	781.25^**	0.05722
Average heterosis	1	160.00^**	51480.62^**	178.94723^**
Parent heterosis (G1)	9	1.87^**	143.83^**	0.49121^**
Parent heterosis (G2)	9	2.90^**	125.77^**	0.89124^**
TC1 vs TC2	1	12.80^**	245.00^**	0.03929
Error	126	0.148	17.379	0.12812

\hat{\bar{h}} = \frac{1}{2} \{[{\bar{T}}_{. (1)} + {\bar{T}}_{. (2)}] - [{\bar{V}}_{. (1)} + {\bar{V}}_{. (2)}]\}

\sum_{i = 1}^{m} T_{i}^{2} + \sum_{i = 1}^{m} V_{i}^{2} + \sum_{j = 1}^{n} T_{j}^{2} + \sum_{j = 1}^{n} V_{j}^{2} - \frac{Y_{. .}^{2}}{2 (m + n)}

\frac{4}{5} \sum_{i = 1}^{m} {\{[V_{i} - {\bar{V}}_{. (1)}] + \frac{1}{2} [T_{i} - {\bar{T}}_{. (1)}]\}}^{2}

\frac{4}{5} \sum_{j = 1}^{n} {\{[V_{j} - {\bar{V}}_{. (2)}] + \frac{1}{2} [T_{j} - {\bar{T}}_{. (2)}]\}}^{2}

\frac{1}{2 (m + n)} {\{[T_{. (1)} + T_{. (2)}] - [V_{. (1)} + V_{. (2)}]\}}^{2}

\frac{4}{5} \sum_{i = 1}^{m} {\{[T_{i} - {\bar{T}}_{. (1)}] - \frac{1}{2} [V_{i} - {\bar{V}}_{. (1)}]\}}^{2}

\frac{4}{5} \sum_{j = 1}^{n} {\{[T_{j} - {\bar{T}}_{. (2)}] - \frac{1}{2} [V_{j} - {\bar{V}}_{. (2)}]\}}^{2}