Can cluster analysis cluster variables that both positively and negatively correlate with each other? I received this question by email from a Neuroscience PhD student.

I would greatly appreciate if you
  could please let me know whether
  Factor Analysis could load positively
  and inversely correlated variables
  onto the same latent factor, whereas
  Cluster Analysis can only cluster into
  the same factor either positively or
  inversely correlated variables (e.g.,
  they would be segregated into 2
  factors).

 A: General interpration of question:
The question is a bit confusing, but I interpret it as follows.


*

*Factor Analysis: When a survey has multiple items and some are positively worded (e.g., "I am the life of the party") and others are negatively worded (e.g., "I avoid social interaction"), factor analysis often assigns such items to the same factor.
For example, extraversion items load positively on one factor and introversion items load negatively on the same factor.

*Cluster Analysis: It is possible to cluster by variables.

*

*In R you can use dist to generate a distance matrix and then send it to hclust to perform a hierarchical cluster analysis.

*In SPSS the hierarchical cluster analysis procedure allows you to cluster by variables. The procedure uses the proximities command to generate the distance matrix.



Variables and clusters:
When running a cluster analysis on variables, you need to think about what formula you will use to generate your distance matrix.


*

*Common ways of calculating distances between variables (i.e., the items in your survey) include the squared euclidean distance or one minus the correlation.
In both the cases, the distance between negative and positively worded items on the same dimension will appear distant. Thus, when you run a cluster analysis, such items will not cluster together.

*However, if you want such items to cluster together, you can choose a measure of distance between items that sees negatively correlated items as similar. A few ideas include:

*

*Reverse the negatively worded items before entering them into the cluster analysis.

*Use 1 minus absolute correlation as the measure of distance



In general, cluster analysis is an algorithmic and atheoretical way of examining groupings in your variables. You have a lot of freedom in how you define distances and how you aggregate based on those distances. The important consideration is that you align your conceptual definition of distance with your operational algorithm for measuring distance.
