Multiple Comparisons by Fisher’s LSD Test

  Fisher’s LSD (Least Significant Difference) test is a post-hoc procedure for multiple comparisons of paired treatment means based on Student’s t distribution. A pairwise absolute difference between treatments is considered significant whenever it exceeds the critical LSD value (called DMS in Portuguese).


1 Equation:

\[ LSD = t_{(\alpha,, df)} \sqrt{\frac{2 \cdot qmr}{r}} \]

Where

  • LSD = least significant difference (minimum significant difference);
  • k = number of groups (with degrees of freedom for treatments (df_{tr} = k - 1));
  • r = number of observations per group;
  • qmr = residual mean square;
  • t = critical value from Student’s t distribution.

2 Files:

  1. Fish1matrix and 1ANAVA1matrix programs
  2. Program source code

3 Usage and example

# Input:
1. Data matrix containing the dataset;
2. Run the program "1ANAVA1matrix" from the main page ("hp50");
3. Run the program "Fish1matrix".

# Output:
1. "lsd" – least significant difference value;
2. Matrix containing comparison values for each pair of groups.


  Pairwise comparisons are automated in the program. Pairs are considered significantly different when:

\[ \Delta AB > LSD \]

(a) Insertion of data matrix.
(b) ANOVA results.
(c) Results of Fisher’s LSD test.
(d) Pairwise comparison matrix. The group identifiers A and B appear on the left (not visible in the image).
Figure 1: Fish1matrix program running on the Android version of the HP50G calculator (Go49gp), showing data input, ANOVA output, Fisher’s LSD results, and the pairwise comparison worksheet. The matrix contains columns A, B, values of \(\Delta AB\), and the corresponding LSD thresholds.


  The results are identical to those reported in the reference source (Vieira, 2000, p. 69).

4 References:

  1. Vieira, Sônia. Analysis of Variance: ANOVA. Atlas Publishing, 2000.
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