# FIsher's LSD test is appropriate for restricted mean comparisons

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.