Table 1. Means and effects of QTL genotypes in multi-environmental trials.

QTL genotype	Environment					Genotypic mean	Genotypic effect
	E1	…	Eh	…	Ee
QQ	${\widehat{\mu }}_{11}$	…	${\widehat{\mu }}_{1h}$	…	${\widehat{\mu }}_{1e}$	${\widehat{\mu }}_{1\cdot }$	${\widehat{\mu }}_{1\cdot }-{\widehat{\mu }}_{\cdot \cdot }$
qq	${\widehat{\mu }}_{21}$	…	${\widehat{\mu }}_{2h}$	…	${\widehat{\mu }}_{2e}$	${\widehat{\mu }}_{2\cdot }$	${\widehat{\mu }}_{2\cdot }-{\widehat{\mu }}_{\cdot \cdot }$
Environmental mean	${\widehat{\mu }}_{\cdot 1}$	…	${\widehat{\mu }}_{\cdot h}$	…	${\widehat{\mu }}_{\cdot e}$	${\widehat{\mu }}_{\cdot \cdot }$(grand or overall mean)
Environmental effect	${\widehat{\mu }}_{\cdot 1}-{\widehat{\mu }}_{\cdot \cdot }$	…	${\widehat{\mu }}_{\cdot h}-{\widehat{\mu }}_{\cdot \cdot }$	…	${\widehat{\mu }}_{\cdot e}-{\widehat{\mu }}_{\cdot \cdot }$

The grand mean and two weighted means were calculated as, ${\widehat{\mu }}_{\cdot \cdot }={\displaystyle \sum _{k=1}^2{\displaystyle \sum _{h=1}^e{f}_{kh}{\widehat{\mu }}_{kh}}}$, and ${\widehat{\mu }}_{k\cdot }={\displaystyle \sum _{h=1}^e{f}_{\cdot h}{\widehat{\mu }}_{kh}}=\frac{1}{e}{\displaystyle \sum _{h=1}^e{\widehat{\mu }}_{kh}}$ and ${\widehat{\mu }}_{\cdot h}={\displaystyle \sum _{k=1}^2{f}_{k\cdot }{\widehat{\mu }}_{kh}}$, respectively, where f _kh is the frequency of the k ^th QTL genotype (QQ or qq) in the h ^th environment; $f_{\cdot h}=\frac{1}{e}$ is the frequency of h ^th environment; f _k. is the frequency of the k ^th QTL genotype.