Society of Actuaries (SOA) PA Practice Exam

How can PCA bi-plot loadings be interpreted when two vectors are close together?

1. The corresponding variables are likely independent

2. The two variables are negatively correlated

3. The two variables are positively correlated

4. There is no correlation present

The correct answer is: The two variables are positively correlated

When interpreting PCA bi-plot loadings, the proximity of two vectors is an important aspect that reveals the relationship between the corresponding variables. When two vectors are close together in this context, it indicates that the associated variables are positively correlated. This can be understood through the geometry of the vector representation in the PCA bi-plot. Vectors that are positioned close to one another suggest that as one variable increases, the other variable also tends to increase, reflecting a strong positive correlation. The angle formed between the vectors is acute, further reinforcing the idea of positive correlation. The magnitude of the correlation can also be gleaned from how close the vectors are. If they are pointing in nearly the same direction, the correlation is likely strong and positive. This aids in identifying relationships between multiple variables in the dataset, simplifying complex multivariate data analysis. Given this understanding, the conclusion about the relationship between the two closely positioned vectors leads to the interpretation of positive correlation.