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I need to prove the following identities for a factorial design experiment of the form $2^k$.

  1. $\overline{Y}(AB+)-\overline{Y}(AB-)=0.5[A(B+)-A(B-)]$
  2. $A(B+)=A+AB$
  3. $A(B-)=A-AB$
  4. $A=[A(B+)+A(B-)]/2$

($\overline{Y}(AB+)$ is the mean of all observations where A*B has a positive sign, A(B+) is the effect of A when B is positive +, and so on..)

I know that for a factorial design experiment of the form $2^3$ $AB=[A(B+)-A(B-)]/2$, but I'm not sure it's applicable for a $2^k$ factorial design experiment. We've mostly talked about the $2^2$ factorial design experiment and briefly mentioned the $2^k$ factorial design

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