# Variance Inflation Factor for Meta-Analyses

I've conducted a Meta-Regression Analysis for my Masters Thesis. However, my supervisor told me that I should check if the proposed moderating variables highly correlate.

So I thought, that I can assess multicollinearity with the use of the Variance Inflation Factor. There are numerous examples on how to calculate VIF for "normal" Regression Analyses.

This requires the specification of a linear model (see this video as an example https://www.youtube.com/watch?v=pZsOn6wnGSo). However, in Meta-Regression I do not have a dependent variable. It's more that the effect size between the dependent and independent variable is the dependent variable for the meta-regression.

So my question is, how do I correctly calculate the VIF when dealing with Meta-Regression? I did not find any examples out there.

I would especially appreciate if someone could tell me what R-Packages I could use to calculate the VIF (and what I have to consider in order to do it correctly) or how I can calculate it with SPSS.

I hope that my question is clear. Thanks for considering my request.

Greetings!

• Can you clarify what you mean by dependent variable here? Most people do meta-regression by regressing the effect size on the moderators. Is that what you did? If so surely the effect size is what you are calling the dependent variable. Feb 16 '17 at 17:12
• Yes this is what I did. However, I dont have an idea on how to calculate the VIFs appropriately. Or do you suggest that I can adopt the procedure from the video, just taking the effect size as dependent varible? Feb 16 '17 at 17:17

The simplest thing would be to just compute the correlations between the predictor variables and present that. I suppose that technically speaking you should weight each observation by its meta-regression weight which you already have.

If your supervisor insist on VIF then this Q&A shows you how to calculate it from the value of $R^2$ from a multiple regression. Again I can see arguments for doing that regression with weights extracted from the meta-regression.

I do not use SPSS so I have no idea how to do this except that you must not use what are called frequency weights but what are called importance weights. Weighted regression is available in R in lm.

The variance inflation factors (VIFs) are functions of the predictor variables and do not involve the dependent variable. This page provides formulas. Although there are VIF formulas that involve $R^2$, this $R^2$ is for regressing one independent variable against the other independent variables, as shown on the linked page. It doesn't involve the dependent/outcome variable at all.

So you will have a problem in meta-analysis if you don't have access to the raw data on the predictors in each study that would be ideally suited to determining VIFs. If you only have the mean values for the predictors in each study, they probably will have a much narrower range of variation than the raw values of the predictors and thus be less suitable to this analysis. Also, although trying to calculate VIFs based on summary statistics of multiple studies might tell you something about multicollinearity among studies, it will not directly address multicollinearity among predictors in the underlying population of interest.

• The first sentence is only true for ordinary least squares models. See this page. With other types of models that are fit via maximum likelihood, the variance-covariance matrix used to estimate VIF will depend on the dependent-variable values, not just on the predictor values.
– EdM
Jul 31 '20 at 19:01