Preface to First Edition - lib

290 PRINCIPAL COMPONENT ANALYSISlongjump 0.91 0.78 0.74 0.82 1.00 0.07 0.70javelin 0.01 0.00 0.27 0.33 0.07 1.00 -0.02run800m 0.78 0.59 0.42 0.62 0.70 -0.02 1.00Examination of these numerical values confirms that most pairs of events arepositively correlated, some moderately (for example, high jump and shot) andothers relatively highly (for example, high jump and hurdles). And we see thatthe correlations involving the javelin event are all close to zero. One possibleexplanation for the latter finding is perhaps that training for the other sixevents does not help much in the javelin because it is essentially a ‘technical’event. An alternative explanation is found if we examine the scatterplot matrixin Figure 16.1 a little more closely. It is very clear in this diagram that forall events except the javelin there is an outlier, the competitor from PapuaNew Guinea (PNG), who is much poorer than the other athletes at these sixevents and who finished last in the competition in terms of points scored. Butsurprisingly in the scatterplots involving the javelin it is this competitor whoagain stands out but because she has the third highest value for the event.It might be sensible to look again at both the correlation matrix and thescatterplot matrix after removing the competitor from PNG; the relevant Rcode isR> heptathlon round(cor(heptathlon[,-score]), 2)hurdles highjump shot run200m longjump javelin run800mhurdles 1.00 0.58 0.77 0.83 0.89 0.33 0.56highjump 0.58 1.00 0.46 0.39 0.66 0.35 0.15shot 0.77 0.46 1.00 0.67 0.78 0.34 0.41run200m 0.83 0.39 0.67 1.00 0.81 0.47 0.57longjump 0.89 0.66 0.78 0.81 1.00 0.29 0.52javelin 0.33 0.35 0.34 0.47 0.29 1.00 0.26run800m 0.56 0.15 0.41 0.57 0.52 0.26 1.00The correlations change quite substantially and the new scatterplot matrix inFigure 16.2 does not point us to any further extreme observations. In the remainderof this chapter we analyse the heptathlon data with the observationsof the competitor from Papua New Guinea removed.Because the results for the seven heptathlon events are on different scales weshall extract the principal components from the correlation matrix. A principalcomponent analysis of the data can be applied using the prcomp functionwith the scale argument set to TRUE to ensure the analysis is carried out onthe correlation matrix. The result is a list containing the coefficients definingeach component (sometimes referred to as loadings), the principal componentscores, etc. The required code is (omitting the score variable)R> heptathlon_pca print(heptathlon_pca)© 2010 by Taylor and Francis Group, LLC