Fisher's Least Significant Difference Test

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A Fisher's Least Significant Difference Test is a two-steps post-hoc statistical test for detecting pairs of means differences.

References

2016

In each t-test, a pooled standard deviation is computed from only the two groups being compared, while the Fisher's LSD test computes the pooled standard deviation from all groups - thus increasing power.
Fisher's LSD does not correct for multiple comparisons [1].

2016

  • (PSU online courses, 2016) ⇒ 3.3 - Multiple Comparisons. (n.d.). Retrieved August 28, 2016, from https://onlinecourses.science.psu.edu/stat503/node/15 Copyright: 2016, The Pennsylvania State University.
    • Fisher’s LSD -- which is the F-test, followed by ordinary t-tests among all pairs of means, but only if the F-test rejects the null hypothesis. The F-test provides the overall protection against rejecting [math]\displaystyle{ H_0 }[/math] when it is true. The t-tests are each performed at [math]\displaystyle{ \alpha }[/math] level and thus likely will reject more than they should, when the F-test rejects. A simple example may explain this statement: assume there are eight treatment groups, and one treatment has a mean higher than the other seven, which all have the same value, and the F-test will rejects [math]\displaystyle{ H_0 }[/math]. However, when following up with the pairwise t-tests, the 7 × 6 / 2 = 21 pairwise t-tests among the seven means which are all equal, will by chance alone reject at least one pairwise hypothesis, [math]\displaystyle{ H_0:\;\mu_i = \mu_i' }[/math] at [math]\displaystyle{ \alpha = 0.05 }[/math]. Despite this drawback Fisher's LSD remains a favorite method since it has overall [math]\displaystyle{ \alpha }[/math] level protection, and offers simplicity to understand and interpret.

2010

  • (Williams et al., 2010) ⇒ Williams, Lynne J., and Hervé Abdi. “Fisher’s least significant difference (LSD) test." Encyclopedia of research design (2010): 1-5. http://ftp.utdallas.edu/~herve/abdi-LSD2010-pretty.pdf
    • When an analysis of variance (ANOVA) gives a significant result, this indicates that at least one group differs from the other groups. Yet, the omnibus test does not indicate which group differs. In order to analyze the pattern of difference between means, the ANOVA is often followed by specific comparisons, and the most commonly used involves comparing two means (the so called “pairwise comparisons”).
The first pairwise comparison technique was developed by Fisher in 1935 and is called the least significant difference (LSD) test. This technique can be used only if the ANOVA F omnibus is significant. The main idea of the LSD is to compute the smallest significant difference (i.e., the LSD) between two means as if these means had been the only means to be compared (i.e., with a t test) and to declare significant any difference larger than the LSD.

1939

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