# Count data with three response types

I administered a test to three groups of people. They had to choose the correct answer between four choices provided. Only one choice is correct and the other three are wrong answers. There are forty questions in total. There is a difference between the type of wrong answers: for example, the wrong answer type 1 is the most preferred and the wrong answer type 3 is the least preferred and the type 2 is in between.

Now I want to see how can I compare the wrong answers across the three groups. What kind of analysis should I conduct? how to check for the distribution? what kind of post hoc test is most suited for this? here is the sample data: So for example, the first participant in Group1, had 18 instances of wrong answer type 1, 3 instances of wrong answer type 2, and 3 instances of type 3. The sum is always below 40, because also the participant might have had some correct answers as well. I am using SPSS to conduct this.

Group    Wrong Ans Type 1   Wrong Ans Type 2    Wrong Ans Type 3
Group1   18                 3                   3
Group1   18                 4                   5
Group1   8                  5                   6
Group1   9                  4                   12
Group1   4                  1                   15
Group1   15                 3                   10
Group1   13                 3                   6
Group1   14                 1                   10
Group1   10                 1                   19
Group1   18                 6                   3
Group1   5                  2                   22
Group1   16                 4                   1
Group2   6                  3                   4
Group2   7                  6                   11
Group2   5                  3                   10
Group2   7                  2                   16
Group2   1                  3                   4
Group2   5                  6                   9
Group2   5                  4                   4
Group2   17                 9                   0
Group2   6                  5                   2
Group2   12                 5                   5
Group2   6                  6                   4
Group2   7                  2                   5
Group3   5                  7                   3
Group3   3                  2                   14
Group3   3                  5                   5
Group3   2                  9                   1
Group3   4                  3                   2
Group3   7                  4                   2
Group3   11                 5                   0
Group3   4                  5                   2
Group3   4                  5                   6
Group3   1                  2                   4
Group3   4                  5                   1
Group3   4                  10                  1


You would want to run a multinomial regression. In this case, your response ($x$) for each individual would be a vector the number of answers in each category (Correct, Wrong1, Wrong2, Wrong3), and your predictor ($x$) would be the group. The analysis should give you the probability of an individual answering a question in each of the categories, and how those probabilities are affected by group membership. I'm not sure how to implement one in SPSS, but I know that it is possible. There are also a number of answered questions on this site regarding the method.