Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 39 / 2 / Solution

The calculation is principal component analysis on a correlation matrix, rather than on an unscaled covariance matrix. For each of the eleven rating variables, subtract its column mean and divide by its column standard deviation. If $Z$ is the resulting $43\times 11$ matrix, use the divisor convention of the fitted object to writeHere the scaling standard deviations use divisor $43$, consistently with this convention. Using sample standard deviations with divisor $42$ gives the same correlation matrix and directions, with the corresponding score scale. Every diagonal entry of $R$ is one, so ${\sum }_k{\lambda }_k=\mathrm{tr}R=11$.

A principal component is the linear combination of the standardized ratings along $a_k$. Its principal component score for judge $i$ is $t_{ik}=z_i^T{a}_k$, and its variance is ${\lambda }_k$. The displayed component standard deviations are $\sqrt{{\lambda }_k}$; the explained variance of a principal component is ${\lambda }_k/11$. In particular,The first component explains about $92.122\%$ of the standardized variation, and the first two jointly explain $95.982\%$. Components three, four and five add about $2.321\%$, $0.830\%$ and $0.339\%$. The cumulative line adds these proportions successively; it is not the proportion explained by that component alone. Retaining two components discards about $4.018\%$ of the total standardized squared variation.

The displayed principal component loadings are the unit eigenvector coefficients $a_{jk}$. The first direction has positive, broadly similar weights for all ratings. It measures an overall favorable-rating tendency: a judge high on most ratings has a high first score. The second direction compares integrity and demeanour, with large positive coefficients, against such attributes as decision promptness, case-flow management and physical ability, with negative coefficients. Thus among similarly rated judges, a positive second score indicates relatively stronger integrity/demeanour compared with those operational ratings. This is a description of a data direction, not a causal interpretation. Multiplying a whole eigenvector and its scores by minus one reflects the display but changes nothing substantive.

These principal component loadings are not themselves variable-component correlations. With standardized variables,For example, the integrity correlation with component one is approximately $3.1833\times 0.289\simeq 0.920$. Blank loadings in the printed second column indicate suppressed small entries, not exact zeros. The complete unrounded loading vectors in the fitted object must be used to calculate scores.

A two-dimensional representation plots the 43 pairswith each point labeled by its judge. In the fitted software object these are the first two columns of the score matrix. Nearby points have similar projections of their standardized ratings. The rank-two reconstruction ${\hat{z}}_i=t_{i1}{a}_1+t_{i2}{a}_2$ is the orthogonal projection onto the leading principal plane and minimizes the total squared reconstruction error over all two-dimensional linear subspaces. To see this, for any orthonormal pair of directions the captured variance is the sum of its two Rayleigh quotients, at most ${\lambda }_1+{\lambda }_2$ by diagonalizing $R$; the remaining variance is the total minus that captured sum. Distances in the plot omit the smaller components, so a close projected pair need not be equally close in every original rating. The actual 43 coordinates cannot be recovered from the summary alone; the supplied data matrix or fitted score matrix is needed.

The printed descriptive list omits INTG, although the loading table includes it. INTG denotes integrity and supplies the eleventh variable. The standard data set also contains a contacts variable; the stated eleven-rating analysis uses the ratings, not contacts.