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Suppose that we perform two way ANOVA with interactions, with factors $A$ and $B$ with three level each $A_{1},A_{2},A_{3}$ and $B_{1},B_{2},B_{3}$.

We know that Fisher's LSD test is used for comparing means.

The most common hypothesis for Fishers test are

$H_{0}:\mu_{A_{1}}=\mu_{A_{2}} \ \ \ \ or \ \ \ \ H_{0}:\mu_{A_{1}}=\mu_{A_{3}} \ \ \ \ or \ \ \ \ H_{0}:\mu_{A_{2}}=\mu_{A_{3}} $

and

$H_{0}:\mu_{B_{1}}=\mu_{B_{2}} \ \ \ \ or \ \ \ \ H_{0}:\mu_{B_{1}}=\mu_{B_{3}} \ \ \ \ or \ \ \ \ H_{0}:\mu_{B_{2}}=\mu_{B_{3}} $

we also know that the comparison inequality of Fisher test is

$\left | \bar{Y_{i}}-\bar{Y_{j}} \right |>t_{a/2}\sqrt{MSE(\frac{1}{n_{i}}+\frac{1}{n_{j}})}$

My question is , if we can restrict Fisher's test for a specific level of B i.e.

$H_{0}:\mu_{A_{1}}|B_{2}=\mu_{A_{2}}|B_{2}$

My guess is that we can ,and the only change on Fisher inequality will be on $\bar{Y_{i}},\bar{Y_{j}},n_{i},n_{j}$ and $MSE$ will remain the same.

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