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Fisher's Least Significant Difference Test
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 Ronald 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.
Notations
The data to be analyzed comprise A groups, and a given group is denoted a. The number of observations of the ath group is denoted Sa. If all groups have the same size, the notation S is used. The total number of observations is denoted N. The mean of Group a is denoted Ma+. From the ANOVA, the mean square of error (i.e., within group) is denoted MSS(A) and the mean square of effect (i.e., between group) is denoted MSA.
Least Significant Difference
The rationale behind the LSD technique value comes from the observation that when the null hypothesis is true, the value of the t statistics evaluating the difference between Groups a and a is equal to
and follows a Student's t distribution with N – A degrees of freedom. The ratio t therefore would be declared significant at a given α level if the value of t is larger than the critical value for the α level obtained from the t distribution and denoted tνα (where v = N – A is the number of degrees of freedom of the error; this value can be obtained from a standard t table). Rewriting this ratio shows that a difference between the means of Groups a and a′ will be significant if
When there is an equal number of observations per group, Equation 2 can be simplified as
In order to evaluate the difference between the means of Groups a and a′ (where a and a′ are the indices of the two groups under consideration), we take the absolute value of the difference between the means and compare it to the value of LSD. If
then the comparison is declared significant at the chosen α level (usually .05 or .01). Then, this procedure is repeated for all
Note that LSD has more power compared to other post hoc comparison methods (e.g., the honestly significant difference test, or Tukey test) because the? level for each comparison is not corrected for multiple comparisons. And, because LSD does not correct for multiple comparisons, it severely inflates Type I error (i.e., finding a difference when it does not actually exist). As a consequence, a revised version of the LSD test has been proposed by Anthony J. Hayter (and is known as the Fisher—Hayter procedure) where the modified LSD (MLSD) is used instead of the LSD. The MLSD is computed using the Studentized range distribution q
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