I am not sure that I understand. The way your question reads could be interpreted that you are clustering based presumably on some set of variables, and then want test the null hypothesis that this classification is completely unrelated to these variable. Obviously no one would do this. Can you rephrase your question? I assume I am not the only one confused (though, I might be!).

Daniel Wright Hi, thank you for answering! Maybe I'm not clear because I'm confused myself and SPSS is not my forte. I have clustered using 3 variables. Because this is my thesis, the prof told me to try a 2 cluster model and 3 cluster model and see what model (2 or 3) is most optimal. I used a syntax for clustering (first hierarchical and then k-means). Afterwards, she gave me the instructions to do a discriminant analysis to compare the percentages (of the originally cases correctly classified) for 2 clusters and 3 clusters that I got in the output. Then she told me to see if these percentages are significantly different or not to determine which cluster model (2 or 3) to use. She suggested using a chi-squared or percentage test, if I understood correctly. (If not significant, she gave me other guidelines to use to determine the optimal amount of clusters). I'm a complete noob when it comes to using SPSS so I don't know how to do this last instruction. The guidelines where also quite vague for somebody like me. I hope this clarifies my previous question.

Hello Dewi,

There have been tests and descriptive statistics developed, with the express aim of identifying the "optimal" number of clusters for a given data set. For example, SAS includes a CCC statistic (cubic clustering criterion). None is universally recognized as ideal, and proposed significance tests/indicators often draw specific criticism about their performance or other limitations, subsequent to initial publication.

Given your query, however, none of the suggested methods you list (or describe) will identify the optimal number of clusters. CA is structure seeking; DA works with identified structure (groupings). If you're using, as IVs for the discriminant analysis, the same three variables used for the cluster analysis, then all you're doing is determining how well DA reproduces the group memberships that the CA yielded (which plodded along with very different aims, quantification, and algorithms). If forced to follow this path (it happens to students!), what you could do would be to record, for each case, the result of the two-cluster solution and subsequent DA as a single variable (matched = 1, did not match = 0), and likewise for the three-cluster solution as a second dichotomous variable. Then, a 2 x 2 contingency table could be constructed, and McNemar's test would tell you whether the instances of disagreement were comparable across the 2- vs. 3-cluster solutions.

For me, the more compelling evidence of whether alternate CA solutions differ would be how they performed in distinguishing among the cases when the 2-cluster set and 3-cluster set were compared on new variables, not used in the clustering process, but related to or important to decision-makers in the context of the sample and intended use of the results.

Good luck with your work (and getting final approval on your project.