Fisher's least significant difference

In statistics, Fisher's least significant difference (LSD) is a procedure used to identify statistically significant differences between the means of multiple groups. Developed by Ronald Fisher in 1935, it was the first post-hoc test designed to be performed following a significant analysis of variance (ANOVA) result.^[1]

The method is intended to control the Type I error rate while maintaining higher statistical power than more conservative adjustments, such as the Bonferroni correction. It remains widely used in fields like agronomy and the social sciences.^[2]

The LSD procedure is typically applied in two stages, a process often referred to as Fisher's protected LSD:

1. Omnibus test: an F-test is performed via ANOVA to determine if there are any statistically significant differences among the group means. If the F-test is not significant, the procedure stops to prevent inflating the family-wise error rate.
2. Pairwise comparisons: if the omnibus F-test from step 1 is significant, pairwise t-tests are conducted for all pairs of groups. These tests use a pooled variance estimated derived from the ANOVA in step 1.

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]

Fisher's LSD is categorized as an "anti-conservative" test because it does not directly adjust the Type I error rate for the total number of comparisons.

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]

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]

The primary criticism of Fisher's LSD is that the "protection" offered by the omnibus F-test diminishes as the number of groups increases. For four or more groups, the probability of at least one Type I error occurring among the pairwise comparisons can exceed the nominal ${\displaystyle \alpha }$, even if the F-test is significant. For this reason, for experiments involving many groups, many statisticians recommend more modern procedures like the Holm–Bonferroni method or Tukey's range test.^[5]

• Multiple comparisons problem
• Newman–Keuls method
• Post-hoc analysis
• Tukey's range test