Methods and formulas for the grouping information table in One-Way ANOVA

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Grouping information table for multiple pairwise comparisons

Minitab uses the confidence interval results for the difference between two level means to obtain the grouping information. The grouping information is in a matrix. Suppose a term has k levels, then the maximum dimension of the matrix is k x k. If all levels are in one group, then the dimension is k x 1, with letter "A" for all factor levels. If all levels are in different groups, the dimension is k x k with letters on the diagonal only.

Minitab uses this algorithm to determine the content of the matrix:
  1. Sort all the least square means at different levels of a term in descending order, denoted as 1, 2, ... , k.
  2. Define a k x k matrix with value 0 in every cell where k = the number of factor levels.
  3. For column j, where j = 1, ... , Minitab does the following:
    1. Checks the confidence intervals of mean j – mean r, where r = j + 1, .. , k. If the interval for r contains 0, set the rth row and the jth column cell, j) as 1.
    2. Sets (j, j) cell as 1 if at least one cell in column j has value 1.
    3. Calculates the row sums from column 1 to column j for row I = j + 1, ... , k. If min (all row sums) >= 1, terminate the loop; otherwise, increase j by 1 and go to step a.
  4. For every row i, Minitab checks the sum of all column values for the row ≥ 1. If the sum is zero, set the cell of row i and column j = 1, where column j is the first column in the matrix with 0 values. This procedure produces a matrix with values 1 and 0. The total number of groups is the number of nonzero columns.
  5. Minitab matches letters to columns (e.g. A to Column 1, B to Column 2, etc.) and assigns cells with value 1 the correct letter.

Grouping information table for multiple comparisons with a control

Minitab uses the confidence interval results for the difference between each level mean and the control level to obtain the grouping information. The grouping information is in a matrix with one column.

Minitab assigns the letter "A" to the control level.

If an interval contains 0, then the level mean is in the same group as the control level. Minitab assigns the letter "A" to the level mean.

If an interval does not contain zero, then no letter is assigned.

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