Module 10 · Project 6 · Due the date on Canvas · PCA & Dimension Reduction · adaptive competency DV18
Read the loadings without inventing a hidden truth
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Begin with the five-minute linear-algebra refresh.
Module 10 · Step 4 of 9 · Challenge8 min
Loadings describe how variables contribute to a component. Scores locate observations in the new coordinate system. Complete all three interpretation cases to separate evidence from overclaim.
- Signs. A component score adds up the standardized variables, each weighted by its loading. A positive loading raises the score as that variable rises; a negative loading moves the score the opposite way, so a high score means less of that variable. Example: if PC1 has reading 0.70, writing 0.68 and lateness −0.22, a high PC1 score means strong reading and writing with less lateness.
- Contrasts and small loadings. Two large loadings with opposite signs make the component a contrast. If PC2 has speed 0.70 and accuracy −0.71, a high score means relatively faster than accurate, and someone high on both lands near zero. A loading near zero barely moves the score: that variable contributes little to this component.
- What a component is not. A component is a weighted summary of the measured variables, not a cause and not a validated construct. Naming PC1 “motivation” or “intelligence” claims something the loadings never measured; that needs separate evidence.
- Explained variance and the scree plot. The explained-variance ratio of component \( k \) is \( \lambda_k / \sum_j \lambda_j \). With standardized variables the eigenvalues sum to the number of variables. For four variables with eigenvalues 2.4, 0.9, 0.5 and 0.2, PC1 explains \( 2.4/4 = 60\% \), and PC1 and PC2 together explain \( 3.3/4 = 82.5\% \). A scree plot draws these ratios against the component number, so a reader can see how quickly the retained share levels off.
- What a reduced view leaves out. A two-component plot keeps only the retained share. In the example, 17.5% of the variance is omitted, so distances and patterns in the plane are approximations. A caption must state the retained share and that the omitted components may hide patterns; dropping later components silently hides that loss.
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