Mean comparisons: insecticides example
First, inspect the F test in the ANOVA table.
P<0.001 which is < 0.05, and so reject the null hypothesis of no difference between treatments.
Second, make all pairwise comparisons:
LSD=t_crit √((2s^2)/n)=2.080√((2∗0.095)/4)=0.453, noting that the t_crit=2.080 is obtained from a table corresponding to the t value with α/2=0.025 and df=21, the degrees of freedom associated with the residual. We divide α by 2, because under the alternative hypothesis either of the two treatments can superior (2-sided test).
Using the table of treatment means (shown in the upper right corner), we can compare every pair of means using the LSD as a yardstick. For example:
Control vs Dimecron-Knap: |1.316-1.681|=0.365<0.453 (LSD) so are not significantly different between each other.
Control vs Dimecron-Boom: |1.316-1.796|=0.480>0.453 (LSD) so are significantly different between each other.
Because we make the pair-wise comparison only after rejecting H_0 by an F test these are called Fisher’s protected LSD.