Fisher's Least Significant Difference (LSD) test

Following one-way analysis of variance (ANOVA), you may want to explore further and compare the mean of one group with the mean of another. One way to do this is by using  Fisher's Least Significant Difference (LSD) test.

How the Fisher's LSD test works

The Fisher's LSD test begins like the Bonferroni multiple comparison test. It takes the square root of the Residual Mean Square from the ANOVA and considers that to be the pooled SD. Taking into account the sample sizes of the two groups being compared, it computes a standard error of the difference between those two means. Then it computes a t ratio by dividing the difference between means by the standard error of that difference. 

To compute a P value and confidence interval, the Fisher's LSD test does not account for multiple comparisons (but see the section on the protected LSD test below). In this respect, it is quite different than the Bonferroni, Tukey and Dunnett methods. The Fishers LSD test is basically a set of individual t tests. The only difference is that rather than compute the pooled SD from only the two groups being compared, it computes the pooled SD from all the groups. If all groups are sampled from populations with the same SD, using all the data to compute the pooled SD gives a more accurate value for the SD (usually) and this shows up as more degrees of freedom.

The protected Fisher's LSD test

Protection means that you only perform the calculations described above when the overall ANOVA resulted in a P value less than 0.05. If the P value for the ANOVA is greater than 0.05 (or whatever significance level you set), you conclude that the data are consistent with the null hypothesis that all population means are identical, and you don't look further. 

This first step sort of controls the false positive rate for the entire family of comparisons. While the protected Fisher's LSD test is of historical interest as the first post test ever developed, it is no longer recommended. It pretends to correct for multiple comparisons, but doesn't do so very well. 

Note that other multiple comparison tests (Bonferroni, Tukey, etc.) do not require this first step -- do not need to be protected. The results of these multiple comparisons tests are valid even if the overall ANOVA has a P value greater than 0.05. 

The unprotected Fisher's LSD test

Unprotected simply means that you do the calculations regardless of the results of the one-way ANOVA. The unprotected Fisher's LSD test is essentially a set of t tests, without any correction for multiple comparisons. The results are not  quite the same as truly doing individual t tests, because the Fisher's LSD test uses the pooled SD from all the groups and not just the two being compared. Some scientists prefer not to correct for multiple comparisons while doing calculations, but prefer that the corrections be done while interpreting data. The unprotected Fishers LSD test does this. 

The unprotected Fisher LSD test is built in as one of many choices for followup (multiple comparison) tests after one- and two-way ANOVA in Prism 6.