Fisher's least significant difference

Mathematical formulation

The least significant difference for two groups ${\displaystyle i}$ and ${\displaystyle j}$ is calculated as:

${\displaystyle {\text{LSD}}_{i,j}=t_{\alpha /2,{\text{df}}_{E}}{\sqrt {{\text{MS}}_{E}\left({\frac {1}{n_{i}}}+{\frac {1}{n_{j}}}\right)}}}$

where:

• ${\displaystyle t_{\alpha /2,{\text{df}}_{E}}}$ is the critical value from the t-distribution for a given significance level ${\displaystyle \alpha }$ and the error degrees of freedom ${\displaystyle {\text{df}}_{E}}$ from the ANOVA.
• ${\displaystyle {\text{MS}}_{E}}$ is the mean square error from the ANOVA.
• ${\displaystyle n_{i}}$ and ${\displaystyle n_{j}}$ are the sample sizes of the groups being compared.^[3]

Versus Bonferroni

Unlike the Bonferroni correction, which divides the significance level by the number of comparisons ${\displaystyle m}$, Fisher's LSD maintains the per-comparison error rate at ${\displaystyle \alpha }$. While this increases the probability of finding a true effect (power), it also increases the risk of a false positive when the number of groups is large.^[4]

Versus Tukey's HSD

Tukey's Honest Significant Difference (HSD) controls the family-wise error rate for all possible pairwise comparisons. Fisher's LSD is generally more powerful than Tukey's HSD but is only considered valid for controlling the family-wise error rate when comparing exactly three groups.^[3]