In several kinds of software the prior probabilities are "equal" or "size". With "equal" the goal it to assign the same number of cases to each group. With "size" the goal is to assign cases to groups according to the original size of the group.
If one has 3 groups with 10, 20, and 30 cases, the software tries to assign 20,20,20 when "equal" is specified. So when crosstabbing the original group by the assigned group there have to be cases on the off-diagonal of the table. Off-diagonal cells are "incorrect" assignments.
However, when the size of the assigned groups is the same as the size of the original groups, it is at least possible that all of off-diagonal cells be empty.
A thought experiment.
take 10 red chips, 20 white chips, and 30 blue chips. Put them in a bag. Shake the bag. Put the first 20 draws in pile 1, the next 20, in pile 2, and the last 20 in pile 3. Call pile 1 red, pile 2 white, pile3 blue.
Draw a 3 by 3 table. Label the row direction actual color, and the column direction assigned color. call the row value labels red, white, blue. The same with the column labels. Place the chips in the cells.
count how many are correct. Of necessity you have at least 20 chips in the off diagonal.
Do it again. Except, put 10,20,30 in the 3 piles. Call the first pile "red", the next pile "white", the third "blue". Place the pile in the 9 cells.
However, I do not know of a DFA program that takes into account that the levels of skill are ordered.
How did you measure skill in the first place?
If you do not have more fine-grained skill scores consider using Categorical Regression or ther software that takes into account that the skill levels are ordered.
Did you try drawing a profile graph of the raw variables? What does the scatterplot of the cases look like in the space defined by the functions?
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Arthur Kendall
Social Research Consultants
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