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It’s been a while since I’ve done any statistics. The material I've read online and in books seem to be conflicting - so I would love a sense check to ensure that I am using the appropriate method for the results I have collected and the question I am trying to answer.

I have recently conducted a questionnaire with likert-type items (available for reference here). I asked participants to rate from 1 to 5 the usefulness of different datasets they may use. I presented them with 16 different datasets to consider. Of the 189 responses, 123 were fully completed. I also ask them what sector they work in.

The question I want to answer is: Are there distinctive groups of people who find similar datasets useful?

I’ve therefore decided to run a simple k-means clustering to try and identify these groups. However, I’ve found it difficult to define a cluster number using the elbow method as well as the silhouette method. Using the elbow method, there is not a distinct "break". Likewise, for the silhouette coefficient, as we increase the number of clusters, the mean silhouette coefficient just hovers around 0.24 to 0.27.

I’ve also tried to run a factor analysis using Principal Axis Factoring and orthogonal Varimax rotation following the Yong & Pearce 2013 tutorial. This gave me three “factors” which the datasets could load to. Despite being interesting, I don’t think it helps to answer my question of finding the groups of people who find similar datasets useful.

Have I misused statistics? If you were to approach the results of my questionnaire from fresh, how would you analyse it? I apologise for any statistics naivety and thank you in advance for any advice.

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I don't see how factor analysis answers your question.

Cluster analysis does seem to be the method you want. There are many, many variations on cluster analysis, you seem to have tried only one method. Nevertheless, the answer to your question may be "No. There are no such groups."

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