Principal Components Analysis
Principal components analysis identifies variables that are highly correlated with each other.
This example analyzes seven variables measuring the extent of agreement with the following statements:
- Technology is fascinating
- I am often surprised by the size of bills
- I find it difficult to determine best deal
- I spent a lot of time shopping for best deal
- I closely monitor the time I spend on the phone
- Cost is a factor when deciding where to SMS or phone
- I try to keep calls kept short and to the point
Please refer to Cluster Analysis for more detail on this data and how it has been processed prior to being used in analysis. The key output of principal components analysis is the rotated component matrix, such as the one shown below (computed using SPSS). The seven variables that have been analyzed can be reduced to three variables. The seven original variables are shown in the rows and the three new variables are represented by the columns and are referred to as components. The numbers in the table are correlations but when conducting principal components analysis they are typically referred to as loadings.
This output is interpreted as follows:
- The first component is highly correlated with Closely monitors time on phone and reasonably strongly correlated with Calls kept short and to the point and Cost is a factor when deciding to SMS or phone. Thus, this first component seems to relate to reflect a cost sensitivity dimension (that is, it suggests that a key distinction between people is that they vary in terms of how concerned they are about price).
- The second dimension is a bit more difficult to interpret . There are relatively high loadings for fascination with technology and being surprised by bill size. Why these two variables would be correlated is difficult to explain and is perhaps more of a weird fluke than a profound insight (not that because these two variables are both correlated with the second component this implies that they are also correlated with each other).
- The third component essentially only reflects agreement with the idea that it is difficult to determine the best deal.
- The variable Spent a lot of time shopping or the best deal is not highly correlated with any of the components and thus is largely ignored.
The real conclusion to draw from this analysis is that the principal components analysis has failed to identify much that is interesting. Perhaps a more interesting solution could be found by investigating more components (e.g., four or five), but it is unlikely that this would be so useful as having, say, a five component solution instead of the original seven variables is not much in the way of a simplification of the data.
Almost all applications of principal components analysis in survey analysis employ a varimax rotation.
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