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.

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.

Methodology

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.

Mathematical formulation

The least significant difference for two groups $i$ and $j$ is calculated as:

${\text{LSD}}_{i,j}=t_{\alpha /2,{\text{df}}_E}\sqrt{{\text{MS}}_E(\frac{1}{n_i}+\frac{1}{n_j})}$

where:

• $t_{\alpha /2,{\text{df}}_E}$ is the critical value from the t-distribution for a given significance level $\alpha$ and the error degrees of freedom ${\text{df}}_E$ from the ANOVA.
• ${\text{MS}}_E$ is the mean square error from the ANOVA.
• $n_i$ and $n_j$ are the sample sizes of the groups being compared.

Comparison with other methods

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.

Versus Bonferroni

Unlike the Bonferroni correction, which divides the significance level by the number of comparisons $m$, Fisher's LSD maintains the per-comparison error rate at $\alpha$. 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.

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.

Criticisms and limitations

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 $\alpha$, 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.

See also

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