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A definition of orthogonality in the context of statistics is

An experimental design is orthogonal if the effects of any factor balance out (sum to zero) across the effects of the other factors.

Being a non-native speaker, I am not sure I understand that. With my favourite example of a latin square, how can I see this condition is satisfied?

  123
1 ABC
2 BCA
3 CAB

How does this show the variables are orthogonal?

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Well, treatments A,B and C are orthogonal to rows and columns. Lets say there is an linear effect on yield $Y$ by rows (columns treated equally). Code that linear effect of row by $-r,0,r$ (we can center it since the mean effect is modelled by the intercept in a linear model). Write a linear model (without interactions) as $$ Y_{ij}=\mu + r i + c j + \beta_t + \text{error} $$ where rows $i$ and columns $j$ are coded as $-1,0,1$. Then the mean observation of the three plots with treatment A (others are equal) is given by $$ \mu +\frac13 (-r +0 + r) + \frac13 (-c+0+c)+\beta_A = \mu + \beta_A $$ which is just the definition of orthogonality that you gave. We only used that the plots are organized in a latin square, whatever latin square will do.

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