Discussion
When the conditional effect (also known as sample effect) for one of the factors in a \(2^{2}\) design differ only in sign, then the corresponding main effect estimate will be :
Answer: A
The main effect estimate is zero.
In a \(2^{2}\) design, the main effect of A is the average of its two simple (conditional) effects, one at each level of B:
\[ A = \frac{1}{2}\left[(a - (1)) + (ab - b)\right] \]
If the two conditional effects are equal in size and opposite in sign, say \(+d\) and \(-d\), their average is 0. The interaction, which is half their difference, is then \(\pm d\): the whole effect of A shows up as interaction, and the main effect hides it.
'One' has no basis, and \((a + b)/2\) and \((a - b)/2\) treat the treatment-combination symbols as if they were the two simple effects.
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