Common Misinterpretations of Principal Components Analysis

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Any particular Rotated Component Matrix is one of an infinite number of possible rotated component matrixes that explain the data to the same extent. For example, the following two component matrices are both identical in terms of how well they describe the correlations in the example presented on the main principal components analysis page. Yet, most people would interpret them as being qualitatively different, with the solution on the left implying that watching of sports programs is not correlated with watching of current affairs programs while the solution on the right implies that they are uncorrelated (note: neither of these interpretations is correct and it is the ease of misinterpretation that makes the use of principal components analysis dangerous).

SPSSRotatedCompentMatrix.png ComponentMatrix.png

The "cluster" interpretation

A particularly common interpretation of a component matrix is that there will be groups of people that correspond to the components. In the example below, this error in interpretation would be that there are a group of people that prefer sports programs and another that prefer current affairs programs. Not only is this a misinterpretation of the principal components analysis, this interpretation is unlikely to be true, as if there really were two such groups of people we would only identify a single component (i.e., as it would lead to positive correlations between viewing of the sports programs and negative correlations between the sports and current affairs programs).