Interpret principal components analysis (PCA) software outputs.

R hands you prcomp() output with five pieces: sdev, rotation, x, center, scale. The exam tests whether you can read those numbers and pick the right number of components without re-running the code.

The geometry first, the printout second. PCA rotates the $p$-variable data cloud to a new orthogonal basis where PC1 points along the direction of maximum variance, PC2 along the next, and so on. Software returns four things you must read: the eigenvalues, the loadings, the scores, and the centering/scaling metadata.

prcomp reports sdev, the standard deviations of the principal components. Square them to get eigenvalues ${\lambda }_j$ of the covariance (or correlation) matrix.

$${\text{PVE}}_j=\frac{{\lambda }_j}{{\sum }_{k=1}^p{\lambda }_k}$$

If you ran PCA on the correlation matrix (scale. = TRUE), then ${\sum }_k{\lambda }_k=p$ exactly. On a raw covariance matrix, the sum equals total sample variance ${\sum }_k\text{Var}(X_k)$.

Cumulative PVE. Sort PCs descending and accumulate. The retained dimension $k$ is the smallest index where cumulative PVE crosses your threshold.

Common mistakes

• Reading sdev as variance. sdev is the standard deviation of each PC; square it for the eigenvalue. Treating sdev = 1.732 as if PC1 explains 1.732 of the variance instead of 3.000 halves your PVE.
• Reversing sign interpretation. When PC1 loadings come out all negative, observations with low raw values get high PC1 scores. Readers often flip the conclusion when the only thing that changed was an arbitrary software-chosen sign.
• Using Kaiser on covariance PCA. ${\lambda }_j>1$ is calibrated to a correlation matrix whose diagonal is one. On dollar-variance inputs, eigenvalues can be in the millions, so the cutoff is meaningless. Use cumulative PVE or scree instead.

Bottom line

• Eigenvalues equal sdev^2 and give the variance captured by each principal component. When inputs are standardized, they sum to $p$, the number of variables.
• Proportion of variance explained (PVE) for PC $j$ is ${\lambda }_j/{\sum }_k{\lambda }_k$. Cumulative PVE drives the 80% to 90% retention rule.
• Loadings (columns of rotation) are unit-length eigenvectors. Squared loadings on a PC sum to one. Signs are arbitrary; only relative magnitudes and sign patterns matter.
• Scores (matrix x) are the rotated, centered (and possibly scaled) observations. Columns are orthogonal, each with sample mean zero and variance ${\lambda }_j$.

Exam shortcut

If the question gives sdev values, square them first and only then divide for PVE; reversing the order is the most common rounding trap. If the question asks "how many components" without naming a rule, default to the 80% cumulative PVE rule unless the data is standardized and Kaiser is mentioned.

The full lesson (about 1,763 words, 12 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• C6

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