Understanding confidence interval for VIF/tolerance when looking at multicollinearity

Question

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1. What the student is asking (in plain language)

The student fitted a linear mixed‑effects model that contains a predictor called Eyes.
To check whether any of the predictors are highly collinear, the software printed a table that includes:

Term	VIF	VIF 95 % CI (adjusted)	Tolerance	Tolerance 95 % CI
Eyes	1.00	[1.00, ∞]	1.00	[0.00, 1.00]

The student is surprised to see that the confidence interval (CI) for the VIF stretches to infinity (and the CI for tolerance stretches down to 0).
Question: Does an infinite (or unbounded) confidence interval mean that there is “uncertainty” about multicollinearity, or does it tell us something else?

2. Step‑by‑step explanation

2.1. What are VIF and tolerance?

Quantity	Definition	Interpretation
VIF (Variance Inflation Factor)	( \displaystyle \text{VIF}_j = \frac{1}{1-R_j^{2}} ) where (R_j^{2}) is the coefficient of determination when regressing predictor (X_j) on all the other predictors.	A VIF of 1 means no linear relationship with the other predictors. Larger values indicate that the variance of the estimated coefficient for (X_j) is inflated because of multicollinearity. Rough rules of thumb: VIF > 5 (or 10) may be worrisome.
Tolerance	( \displaystyle \text{Tol}_j = 1 - R_j^{2} = \frac{1}{\text{VIF}_j} )	Tolerance near 0 (e.g., < 0.1) signals a problem; tolerance close to 1 signals little or no collinearity.

Thus VIF and tolerance are deterministic functions of the sample‑based (R_j^{2}). If we could observe the exact population correlation structure, the VIF would be a fixed number (not a random variable). The CI reflects sampling variability of the estimated (R_j^{2}).

2.2. How are confidence intervals for VIF/tolerance obtained?

Most statistical packages compute a CI for VIF (or tolerance) by:

Fit the auxiliary regression (X_j = \beta_0 + \sum_{k\neq j}\beta_k X_k + \varepsilon).
Obtain the residual sum of squares and the corresponding (R_j^{2}) (or equivalently the error variance estimate (s^{2})).
Treat the residual variance estimate as a scaled chi‑square random variable: [ \frac{(n-p) \, s^{2}}{\sigma^{2}} \sim \chi^{2}_{n-p}, ] where (n) = total observations, (p) = number of predictors (including the intercept) in the auxiliary regression.
Derive a CI for the error variance (\sigma^{2}) by inverting the chi‑square quantiles: [ \frac{(n-p) s^{2}}{\chi^{2}{\alpha/2,\,n-p}} \le \sigma^{2} \le \frac{(n-p) s^{2}}{\chi^{2}{1-\alpha/2,\,n-p}} . ]
Transform the CI for (\sigma^{2}) into a CI for (R_j^{2}) (or directly for tolerance). The relationship between the error variance and (R_j^{2}) is [ R_j^{2}=1-\frac{\sigma^{2}}{\text{Var}(X_j)} . ]
Finally invert the VIF–tolerance relationship: [ \text{VIF}= \frac{1}{\text{Tolerance}}. ]

When the point estimate of (R_j^{2}) is exactly zero (i.e., the auxiliary regression finds no linear relationship between (X_j) and the other predictors), we have:

( \hat{R}_j^{2}=0 )
( \widehat{\text{Tolerance}} = 1-0 = 1)
( \widehat{\text{VIF}} = 1/1 = 1)

2.3. Why does the CI become unbounded (∞ for VIF, 0 for tolerance)?

The CI for a proportion (here, (R_j^{2}) or tolerance) is based on the variability of the estimate. If the sample estimate lies at the boundary of the possible parameter space (0 or 1), the usual (Wald‑type) CI may “run off the edge”.

Consider tolerance as an estimate of a probability that lies in ([0,1]). If the point estimate is exactly 1, any plausible lower limit must be ≥ 0, but there is no upper bound beyond 1. When we convert that interval to VIF, the upper bound becomes (1 / (\text{lower bound of tolerance})). If the lower bound of tolerance is exactly 0, the corresponding VIF upper bound is (1/0 = \infty).

In practice, the software often uses the F‑distribution to build a CI for VIF directly (e.g., vifci in car package). The formula for the lower and upper limits of VIF is

[ \text{VIF}{\text{lower}} = \frac{\text{VIF}}{F{1-\alpha/2,\,df_{1},\,df_{2}}}, \qquad \text{VIF}{\text{upper}} = \frac{\text{VIF}}{F{\alpha/2,\,df_{1},\,df_{2}}} ]

where the denominator involves an F‑quantile. If the degrees of freedom for the auxiliary regression are small (or the residual variance is zero), the lower F‑quantile can be zero, leading to an infinite upper bound for VIF.

Bottom line: The infinite upper bound does not indicate that the predictor is “somewhat” collinear. It is a mathematical consequence of the fact that the estimated VIF equals its theoretical minimum (1). Because the estimate sits at the boundary, the usual method cannot produce a finite upper confidence limit.

2.4. What does this tell us about multicollinearity for Eyes?

Point estimate: VIF = 1, tolerance = 1 → perfectly non‑collinear with the other predictors.
Confidence interval:
- VIF 95 % CI = ([1,\,\infty))
- Tolerance 95 % CI = ([0,\,1])

Interpretation:

Parameter	Meaning of CI
VIF	We are certain that the true VIF is at least 1 (the minimum possible). The data do not give us enough information to place a finite upper limit; mathematically the interval must extend to infinity.
Tolerance	Likewise, the true tolerance is at most 1 and could be as low as 0. Because the estimate is right at the upper boundary, the lower bound is forced to 0.

In practical terms, the result tells us there is no evidence of multicollinearity involving Eyes. The infinite upper bound is a technical artifact, not a warning sign.

2.5. When does an infinite CI arise in other situations?

Exact zero residual variance in the auxiliary regression (perfect prediction) → VIF = ∞, tolerance = 0 (the opposite extreme).
Very small sample size relative to the number of predictors → the F‑distribution quantile in the denominator can be zero (or numerically very close), producing an unbounded interval.
Perfect separation in logistic regression (analogous situation) leads to infinite standard errors and unbounded CIs.

3. Final answer

The confidence interval ([1,\infty)) for the VIF (and ([0,1]) for tolerance) does not indicate uncertainty about the presence of multicollinearity. It simply reflects the fact that the point estimate of VIF is at its theoretical minimum (1). Because the estimate lies on the boundary of the permissible range, the usual method for constructing a confidence interval cannot produce a finite upper bound, so it reports “infinity”. In substantive terms, there is no evidence of multicollinearity for the predictor *Eyes; the infinite upper limit is a mathematical artifact, not a diagnostic concern.*

4. Common mistakes when interpreting VIF/tolerance confidence intervals

Mistake	Why it’s wrong	Correct approach
Thinking “∞” means “high multicollinearity”.	An infinite upper bound occurs when the VIF estimate is at its minimum (1), not when it is large.	Remember that VIF ≥ 1; only values much larger than 1 raise concern.
Assuming the CI gives a range of likely VIF values.	The interval is derived from an asymptotic (Wald) approximation that breaks down at the boundary; the lower bound is reliable, the upper bound may be infinite.	Use the point estimate (VIF = 1) as the practical indicator; treat the infinite bound as “cannot be bounded given the data”.
Ignoring the sample‑size effect.	With few observations relative to predictors, the CI may be very wide or infinite even when the point estimate is modest.	Check the degrees of freedom of the auxiliary regression; if they are small, consider collecting more data or reducing the number of predictors.
Confusing VIF with correlation coefficient.	VIF is a function of the multiple R² from regressing one predictor on all others, not a simple pairwise correlation.	Examine the auxiliary regression or a correlation matrix to understand which predictors contribute to any collinearity.
Using the CI to decide whether to drop a variable.	Because the CI can be unbounded, the decision rule “drop if CI includes > 5” is meaningless.	Base the decision on the point estimate and substantive knowledge; a VIF of 1 is as low as it gets.

Original question: Understanding confidence interval for VIF/tolerance when looking at multicollinearity on Cross Validated (Stats Stack Exchange), licensed CC BY-SA.