Variance Inflation Factor (VIF) and Multicollinearity

The variance inflation factor (VIF) measures how much the variance of a regression coefficient is inflated because that predictor is correlated with the other predictors. It is the standard diagnostic for multicollinearity. A VIF of 1 means no inflation; values above 5 or 10 are common warning thresholds, but they are conventions, not rules.

This guide explains the formula, walks through a worked example, and shows what to do and how to report it.

What multicollinearity is

Multicollinearity occurs when two or more predictors in a regression model are strongly linearly related. Typical examples:

• Age and years of work experience in the same model.
• Several subscales of the same questionnaire that measure overlapping constructs.
• A variable and its square, or a variable and an interaction term built from it, without centering.
• Dummy variables for every category plus an intercept (perfect multicollinearity, the "dummy variable trap").

When predictors overlap, the model has trouble separating their individual contributions. The overall fit (R², predictions) is not harmed, but the individual coefficients become unstable:

• Standard errors grow, so coefficients that matter can look non-significant.
• Estimates can change sharply when you add or drop a variable or a few observations.
• Coefficients can take implausible signs.

Multicollinearity is a problem only for interpreting individual coefficients. If you only care about prediction, it matters much less.

The VIF formula

For predictor j, regress it on all the other predictors in the model and take the R² of that auxiliary regression, written R²_j. Then:

VIF_j = 1 / (1 − R²_j)

R²_j is the share of predictor j's variance that the other predictors already explain. If they explain none of it (R²_j = 0), the VIF is 1. As R²_j approaches 1, the VIF heads toward infinity.

Two related quantities:

• Tolerance is the reciprocal: Tolerance_j = 1 − R²_j = 1 / VIF_j. SPSS reports both. A tolerance below 0.10 corresponds to a VIF above 10; a tolerance below 0.20 corresponds to a VIF above 5.
• Square root of VIF tells you how much larger the standard error is than it would be if predictor j were uncorrelated with the others. A VIF of 4 doubles the standard error (√4 = 2).

With only two predictors, R²_j is simply the squared correlation between them, so VIF = 1 / (1 − r²) for both. A correlation of .60 gives a VIF of 1 / (1 − .36) = 1.56. A correlation of .90 gives 1 / (1 − .81) = 5.26.

Note that VIF depends only on the predictors. The outcome variable plays no part.

Interpreting VIF values: thresholds and caveats

The cutoffs of 5 and 10 are widely used rules of thumb, and you will see both in published papers and textbooks. Treat them with care:

• No threshold is universal. A VIF of 10 means the standard error is about 3.2 times larger than it would be without the overlap. Whether that matters depends on your sample size and effect sizes. In a large sample with strong effects, a VIF of 8 may cause no practical trouble; in a small sample, a VIF of 3 can already make a coefficient unstable.
• High VIF is not always a problem. If the high-VIF variables are control variables you do not interpret, or if they are a variable and its square or interaction term, the inflation may be harmless for the coefficients you care about.
• Low VIF does not guarantee a good model. VIF says nothing about omitted variables, nonlinearity, or heteroscedasticity.

A good practice is to report the VIFs, name the threshold you used, and explain why it is appropriate for your design rather than presenting it as a fixed rule.

A worked example

A researcher predicts job satisfaction from three predictors: monthly salary, years of work experience, and age. The correlations among the predictors are:

You can check pairwise correlations like these with the correlation calculator. Pairwise correlations are a useful first look, but VIF goes further because it accounts for all the other predictors at once.

Step 1: Auxiliary regressions. Regress each predictor on the other two:

For experience: 1 / (1 − .826) = 5.74 (computed with the unrounded R²_j of .8258). Its standard error is about 2.4 times what it would be if experience were unrelated to salary and age.

Step 2: Interpret. Salary is fine. Experience and age both exceed 5 but not 10. The problem is clearly the pair experience and age (r = .90): given someone's age and salary, their experience is largely predictable.

Step 3: Decide. If the research question is about experience, age adds little separate information and inflates the uncertainty. Dropping age leaves salary and experience, with r = .60, so both VIFs fall to 1 / (1 − .36) = 1.56. If the theory requires both, keep them and interpret their individual coefficients with caution, or combine them (see below).

Tip: the VIF of each predictor equals the corresponding diagonal element of the inverse of the predictors' correlation matrix, which is a quick way to check software output.

What to do about multicollinearity

Pick a remedy that fits your research question, not the one that makes the numbers look best.

1. Do nothing, if it does not affect your question. If the high-VIF variables are controls, or your key predictor has a low VIF and a clear result, you can report the VIFs and move on.
2. Remove a redundant predictor. If two variables measure nearly the same thing, keep the one that matches your theory. Explain the choice in the methods section.
3. Combine predictors. Average or sum items that measure one construct (check reliability first, for example with Cronbach's alpha), or create a composite score or factor.
4. Center variables before creating interaction or polynomial terms. Centering reduces the correlation between X and X² or between X and X × W. It does not change the interaction coefficient or the model fit; it mainly makes the lower-order terms interpretable and the VIFs meaningful.
5. Collect more data. Larger samples reduce standard errors, which offsets some of the inflation. Multicollinearity is partly a small-sample problem.
6. Use a different method. Ridge regression or principal component regression handle correlated predictors, at the cost of biased or harder-to-interpret coefficients. These are more common in prediction-focused work.

Do not drop variables one by one until every VIF is below a threshold without any theoretical justification. That can create omitted-variable bias, which is a bigger problem than inflated standard errors.

Special cases

• Categorical predictors with several dummy variables. The VIFs of individual dummies depend on which category is the reference group. For a multi-category variable, the generalized VIF (GVIF; Fox and Monette, 1992), available in R's car::vif(), is more appropriate.
• Interaction terms. High VIFs for X, W, and X × W are expected when variables are not centered. This is sometimes called "nonessential" multicollinearity because centering removes it.
• Logistic and other generalized linear models. The same logic applies, since VIF depends only on the predictors. Many researchers compute VIFs from a linear regression with the same predictors.

How VIF fits with other regression checks

VIF is one of several standard assumption checks. Check also for heteroscedasticity, where residual spread changes with the fitted values (see the heteroscedasticity guide), normality of residuals, and influential cases. The regression and mediation analysis guide summarizes the full workflow.

How to report VIF in APA 7 style

Methods section:

Results section, when VIFs are acceptable:

Results section, using the worked example:

In a regression table, add a VIF or tolerance column next to the coefficients, or report the range in a table note:

Common mistakes

• Treating 10 (or 5) as a law. Report the value and justify your threshold.
• Using only pairwise correlations. Multicollinearity can involve three or more variables even when no single pairwise correlation is very high. VIF catches this; a correlation matrix may not.
• Panicking about high VIFs on interaction terms. Center the variables and check again.
• Dropping theoretically important variables. Removing a true confounder to lower VIFs trades a variance problem for a bias problem.
• Confusing multicollinearity with correlation between a predictor and the outcome. Strong predictor-outcome correlations are what you want.

FAQ

What is a good VIF value?

Values close to 1 indicate little overlap. Values under 5 are usually considered acceptable, and many papers use 10 as the upper limit. More conservative authors use 2.5 or 3, particularly in smaller samples.

What is the difference between VIF and tolerance?

They contain the same information. Tolerance is 1 − R²_j, and VIF is its reciprocal. A tolerance of 0.20 equals a VIF of 5, and 0.10 equals a VIF of 10.

Does multicollinearity affect R² or predictions?

No. The overall model fit and the predicted values are not affected. Multicollinearity affects the precision and stability of individual coefficients.

Can VIF be less than 1?

No. Since R²_j is between 0 and 1, the VIF is always 1 or greater.

Should I check VIF before or after running the regression?

Either works, since VIF depends only on the predictors. Most software reports it alongside the regression output (in SPSS, tick "Collinearity diagnostics" under Statistics in the Linear Regression dialog).

Turning your analysis into a paper

If you have a dataset and need a full draft with regression results, VIF and other assumption checks, tables, and interpretation text, Data2Paper Generate Paper produces one from your CSV or XLSX file. You can preview the first half for free and pay a one-time $19.90 to unlock the full paper. It is an AI-generated draft, so review the variables, analysis choices, and numbers before using it.