Design of Experiments

Variance Inflation Factors (VIF)

Measures multicollinearity between model terms in an experimental design.

Statusvalidated

Version1.0.0

Minitab equivalentStat > DOE > Factorial > Analyze Factorial Design > VIF

Description

The Variance Inflation Factor (VIF) quantifies how much the variance of an estimated regression coefficient is inflated due to correlation with other model terms. For orthogonal designs (e.g. full factorial 2^k designs), all VIFs are exactly 1 — no inflation. VIF > 1 indicates multicollinearity: VIF ≤ 5 is generally acceptable, VIF > 10 is problematic. High VIFs mean that effects cannot be cleanly separated and parameter estimates become unstable.

Formulas

VIF_j

VIF_{j} = [(X^{⊤} X)^{- 1}]_{jj} \cdot n

j-th diagonal element of the inverse information matrix, scaled by the number of runs n. Intercept (j=0) is skipped.

X'X

X^{⊤} X = n I (orthogonal)

For orthogonal designs, X'X = nI, so (X'X)⁻¹ = (1/n)I and VIF_j = 1 for all terms.

Assumptions

Coded design matrix with values in {−1, 0, +1}
Linear model with main effects and 2-factor interactions
X'X must be invertible (non-singular design)

Limitations

VIF only indicates the strength of multicollinearity, not its cause
For singular X'X (e.g. more terms than runs), all VIFs are set to ∞

References

Montgomery, D.C., Design and Analysis of Experiments, 10th Ed., Wiley — Chapter 10
Kutner, M.H. et al., Applied Linear Statistical Models, 5th Ed., McGraw-Hill — Chapter 10