Analysts W, X, Y, and Z are each performing Principal Components Analysis on the same data set with three variables. They use different programs with their default settings and discover that they have different factor loadings for the first principal component. Their loadings are:

	Variable 1	Variable 2	Variable 3
W	–0.549	–0.594	0.587
X	–0.549	0.594	0.587
Y	0.549	–0.594	–0.587
Z	0.140	–0.570	–0.809

Determine which of the following is/are plausible explanations for the different loadings.

• Loadings are unique up to a sign flip and hence X’s and Y’s programs could make different arbitrary sign choices.
• Z’s program defaults to not scaling the variables while Y’s program defaults to scaling them.
• Loadings are unique up to a sign flip and hence W’s and X’s programs could make different arbitrary sign choices.

• None
• I and II only
• I and III only
• II and III only
• The correct answer is not given by (A), (B), (C), or (D).

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: B

I is true. Uniqueness up to a sign flip means all three signs must be flipped. This is true for X and Y.

II is true. The presence of absence of scaling can change the loadings.

III is false. Uniqueness up to a sign flip means all three signs must be flipped. For W and X only the second loading is flipped.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.