Perfoming ANOVA test using orthogonal contrasts

[bonfire09]

I have to test treatments B,C,D and E where $H_0:\mu_B=\mu_C=\mu_D=\mu_E$ vs $H_1: not H_0$. Use $\alpha=0.05$ given this problem.

Now $\mu_A$ is not included in the hypothesis so I am trying to figure out how to go about this problem. I was thinking of using orthogonal contrasts and this is what I came up with.

$C_1-C_4$ are my 4 orthogonal contrasts (all pairwise) for the $5$ treatments. The part that confuses me is what to do with $\mu_A$ since it was never included in the null hypothesis? Also I'm guessing that I test C1-C4 to see if I can violate the null hypothesis but what if each of my contrasts support the null hypothesis? There should be a better way of doing this problem I think. I am not sure if I was to ask this question here, but since this deals with statistics I asked it here. Thanks.

Edit- I just noticed that my contrasts are not linear independent. I will have to come up with new ones.

 

[Jhero]

You are correct that you need to use orthogonal contrasts in order to test your hypothesis. However, you don't need to include $\mu_A$ in your contrasts since it was not part of the null hypothesis.Instead, you should focus on contrasting the 4 treatments included in the null hypothesis: B, C, D, and E. You can create 4 orthogonal (linear independent) contrasts of the form:$C_1=\frac{B-C}{\sqrt{2}}$$C_2=\frac{B+C-D-E}{2}$$C_3=\frac{D-E}{\sqrt{2}}$$C_4=\frac{B+C+D+E}{4}$To test your hypothesis, you can then conduct an ANOVA with the 4 contrasts as your independent variables and a significance level of $\alpha=0.05$. If any of the contrasts show a statistically significant difference, then you can reject the null hypothesis.