After the F-test: pairwise comparisons
The rejection of the null hypothesis implies that at least one of the treatment means is different. However, that as such is not a very informative discovery, as still we do not know whether all treatment means are different from each other, or just a few of them are.
To answer this more specific question, we need to perform a pair-wise comparison of the different treatment means to find out which is(are) different.
In order to decide when a difference between two means is deemed to be large enough to claim that they are indeed significantly different, it is necessary to define a minimum difference, i.e. the least significant difference (LSD).
Effectively we perform a t-test for each pair of means: t=((y ?_1-y ?_2))/(√((2s^2)/n) ) and reject H_0 of the two means being equal whenever |t|>t_crit
the t_crit is a critical value from the Student t-distribution, t_(α/2,df), α the type I error rate (usually, but not necessarily so, set to 0.05), and df the degrees of freedom associated with the error mean squares.
Equivalently, we can define a minimum significance difference: LSD=t_crit √((2s^2)/n), and use that value to compare means, and reject H_0 of the two means being equal whenever |y ?_1-y ?_2 |>LSD.
The expressions above assume treatments with equal number of observations, the expressions can be generalized for unequal number of observations:
LSD=t_crit √(s^2 (1/n_1 +1/n_2 ) )