Which statement about orthogonality in statistics is true?

Which statement about orthogonality in statistics is true?

• A. It means variables are identical.
• B. It implies perfect collinearity.
• C. It means two vectors are perpendicular and uncorrelated.
• D. It implies nonlinearity.

Answer: (C)

Orthogonality in statistics describes two vectors that are perpendicular, which makes their dot product zero and their correlation zero. When two vectors are orthogonal, one provides no linear information about the other, and in a regression context their design columns are uncorrelated, simplifying interpretation of coefficients. The statement that two vectors are perpendicular and uncorrelated best captures both the geometric and statistical meaning of orthogonality: they point in completely independent directions in the data space, with no linear relationship between them. Identities or identical variables would show perfect correlation, not orthogonality. Perfect collinearity means one variable is a exact linear combination of another, which breaks orthogonality. Nonlinearity refers to relationships that aren’t captured by a straight line, which is separate from the idea of being orthogonal (zero linear association).

Orthogonality in statistics describes two vectors that are perpendicular, which makes their dot product zero and their correlation zero. When two vectors are orthogonal, one provides no linear information about the other, and in a regression context their design columns are uncorrelated, simplifying interpretation of coefficients.

The statement that two vectors are perpendicular and uncorrelated best captures both the geometric and statistical meaning of orthogonality: they point in completely independent directions in the data space, with no linear relationship between them.

Identities or identical variables would show perfect correlation, not orthogonality. Perfect collinearity means one variable is a exact linear combination of another, which breaks orthogonality. Nonlinearity refers to relationships that aren’t captured by a straight line, which is separate from the idea of being orthogonal (zero linear association).