Collinearity and stepwise VIF selection

Collinearity, or excessive correlation among explanatory variables, can complicate or prevent the identification of an optimal set of explanatory variables for a statistical model. For example, forward or backward selection of variables could produce inconsistent results, variance partitioning analyses may be unable to identify unique sources of variation, or parameter estimates may include substantial amounts of uncertainty. The temptation to build an ecological model using all available information (i.e., all variables) is hard to resist. Lots of time and money are exhausted gathering data and supporting information. We also hope to identify every significant variable to more accurately characterize relationships with biological relevance. Analytical limitations related to collinearity require us to think carefully about the variables we choose to model, rather than adopting a naive approach where we blindly use all information to understand complexity. The purpose of this blog is to illustrate use of some techniques to reduce collinearity among explanatory variables using a simulated dataset with a known correlation structure.

A simple approach to identify collinearity among explanatory variables is the use of variance inflation factors (VIF). VIF calculations are straightforward and easily comprehensible; the higher the value, the higher the collinearity. A VIF for a single explanatory variable is obtained using the r-squared value of the regression of that variable against all other explanatory variables:

\displaystyle VIF_j=\frac{1}{1-R_j^2}

where the VIF for variable j is the reciprocal of the inverse of R^2 from the regression. A VIF is calculated for each explanatory variable and those with high values are removed. The definition of ‘high’ is somewhat arbitrary but values in the range of 5-10 are commonly used.

Several packages in R provide functions to calculate VIF: vif in package HH, vif in package car, VIF in package fmsb, vif in package faraway, and vif in package VIF. The number of packages that provide VIF functions is surprising given that they all seem to accomplish the same thing. One exception is the function in the VIF package, which can be used to create linear models using VIF-regression. The nuts and bolts of this function are a little unclear since the documentation for the package is sparse. However, what this function does accomplish is something that the others do not: stepwise selection of variables using VIF. Removing individual variables with high VIF values is insufficient in the initial comparison using the full set of explanatory variables. The VIF values will change after each variable is removed. Accordingly, a more thorough implementation of the VIF function is to use a stepwise approach until all VIF values are below a desired threshold. For example, using the full set of explanatory variables, calculate a VIF for each variable, remove the variable with the single highest value, recalculate all VIF values with the new set of variables, remove the variable with the next highest value, and so on, until all values are below the threshold.

In this blog we’ll use a custom function for stepwise variable selection. I’ve created this function because I think it provides a useful example for exploring stepwise VIF analysis. The function is a wrapper for the vif function in fmsb. We’ll start by simulating a dataset with a known correlation structure.



We’ve created fifteen ‘explanatory’ variables with 200 observations each. The mvrnorm function (MASS package) was used to create the data using a covariance matrix from the genPositiveDefMat function (clusterGeneration package). These functions provide a really simple approach to creating data matrices with arbitrary correlation structures. The covariance matrix was chosen from a uniform distribution such that some variables are correlated while some are not. A more thorough explanation about creating correlated data matrices can be found here. The correlation matrix for the random variables should look very similar to the correlation matrix from the actual values (as sample size increases, the correlation matrix approaches cov.mat).


Now we create our response variable as a linear combination of the explanatory variables. First, we create a vector for the parameters describing the relationship of the response variable with the explanatory variables. Then, we use some matrix algebra and a randomly distributed error term to create the response variable. This is the standard form for a linear regression model.

y<-rand.vars %*% matrix(parms) + rnorm(num.obs,sd=20)

We would expect a regression model to indicate each of the fifteen explanatory variables are significantly related to the response variable, since we know the true relationship of y with each of the variables. However, our explanatory variables are correlated. What happens when we create the model?

lm.dat<-data.frame(y,rand.vars)<-paste('y ~',paste(names(lm.dat)[-1],collapse='+'))


The model shows that only four of the fifteen explanatory variables are significantly related to the response variable (at \alpha = 0.05), yet we know that every one of the variables is related to y. As we’ll see later, the standard errors are also quite large.

We can try an alternative approach to building the model that accounts for collinearity among the explanatory variables. We can implement the custom VIF function as follows.


 var vif             
 X1  27.7352782054202
 X2  36.8947196546879
 X3  12.5694198086941
 X4  50.7385544899845
 X5  8.35069942629285
 X6  114.685122241139
 X7  67.3415420139211
 X8  153.597012767649
 X9  48.226662808638 
 X10 50.7371404106266
 X11 33.9720046917178
 X12 43.2541022358368
 X13 12.0823286959991
 X14 74.6186892947576
 X15 29.8722459010406

removed:  X8 153.597 

 var vif             
 X1  6.67306561938667
 X2  7.98347501302268
 X3  4.56187657632574
 X4  8.03048468634153
 X5  7.70736760805487
 X6  19.6743072270573
 X7  52.9521670096974
 X9  17.8683960730445
 X10 46.2484642889962
 X11 18.2479446141727
 X12 42.133697798185 
 X13 10.8973377491163
 X14 37.9296952803818
 X15 21.5847028917955

removed:  X7 52.95217 

 var vif             
 X1  6.54376168051204
 X2  7.68236114754164
 X3  4.04873004990332
 X4  5.08958904348524
 X5  2.65685239947949
 X6  9.12685384862522
 X9  2.89940351012031
 X10 4.24712217472346
 X11 4.45202381077724
 X12 12.8835110845825
 X13 1.92759852488083
 X14 6.02382346000219
 X15 8.33332235677386

removed:  X12 12.88351 

 var vif             
 X1  4.21690743815539
 X2  6.88058249786417
 X3  3.88265091747854
 X4  4.48146995293666
 X5  2.20144300270463
 X6  7.76775127218149
 X9  2.71324446993905
 X10 2.90517847805482
 X11 4.43541888871566
 X13 1.78221291029774
 X14 5.02289299193397
 X15 3.02196822776011

removed:  X6 7.767751 

 [1] "X1"  "X2"  "X3"  "X4"  "X5"  "X9"  "X10" "X11" "X13" "X14" "X15"

The function uses three arguments. The first is a matrix or data frame of the explanatory variables, the second is the threshold value to use for retaining variables, and the third is a logical argument indicating if text output is returned as the stepwise selection progresses. The output indicates the VIF values for each variable after each stepwise comparison. The function calculates the VIF values for all explanatory variables, removes the variable with the highest value, and repeats until all VIF values are below the threshold. The final output is a list of variable names with VIF values that fall below the threshold. Now we can create a linear model using explanatory variables with less collinearity.

keep.dat<-vif_func(in_frame=rand.vars,thresh=5,trace=F)<-paste('y ~',paste(keep.dat,collapse='+'))


The updated regression model is much improved over the original. We see an increase in the number of variables that are significantly related to the response variable. This increase is directly related to the standard error estimates for the parameters, which look at least 50% smaller than those in the first model. The take home message is that true relationships among variables will be masked if explanatory variables are collinear. This creates problems in model creation which lead to complications in model inference. Taking the extra time to evaluate collinearity is a critical first step to creating more robust ecological models.

Function and example code:

38 thoughts on “Collinearity and stepwise VIF selection

  1. Visualizing neural networks from the nnet package – R is my friend

  2. Visualizing neural networks from the nnet package | spider's space

  3. This is a good idea, but why compute VIF = 1/(1-R^2) ? This is a strictly increasing function of R^2, so why not just use R^2 ?

  4. This is an excellent method for automating feature selection in multivariate linear models. I was able to significantly improve predictability by applying this to the input frame before LM model fitting. Thanks.

  5. Thank you for sharing your code, I find it very helpful! I used it to detect collinearity among variables calculated for about 50 objects. The code worked perfectly when the initial number of variable was less than 30. When I tried with more variables (up to 100) for the same 50 objects, I got the following error:

    Error in if (vif_max < thresh) { : missing value where TRUE/FALSE needed

    I am not expert at using R and I'd like to ask you whether you know the possible cause of this problem. Could it really depend on the number of variables included? Thank you again for the code!!

    • Hi there, sorry you’re having problems. I can’t reproduce the error using my code:

      #remove collinear variables and recreate lm   

      Could you provide an example with your code?

      • Hi, I did not create the ‘explanatory’ variables (rand.vars), but I used an experimental dataset (X) where each variable represents a certain characteristic of the objects. So I used your code as such on my X matrix, but first I removed all variables with standard deviation equal to zero (same value across all objects). Successively I ran the code on all 100 remaining variables; as I got that error, I only included the first 30 variables and then it worked.
        Anyway, as it is working with your 100 ‘random’ variables, the problem must lie in the values of my variables. I will control the dataset better and try again!
        Thank you very much for your help and your fast reply!

      • I am receiving the same error message IaIe got: Error in if (vif_max < thresh) { : missing value where TRUE/FALSE needed. I am importing a .csv with 138 explanatory variables, which I don't believe is an issue given your example. There are some variables where every observation is the same. However, I cannot afford to kick out any variables with a standard devation equal to zero like IaIe did. I was wondering if you all found a solution to this by chance? Thanks!

      • Hi MTV,

        Again, I cannot reproduce the error using my code. I used the VIF function with a simulated dataset that contained constant variables (sd = 0) in addition to continuous random variables and did not have a problem:

        #continuous random variables, corrrelated
        #constant variables, sd=0
        #append all variables, randomize column order
        #run function

        Is there some way you could post a sample dataset that causes the error? I suspect the problem is not major but I can’t fix it unless I can reproduce the error.

    • The error:

      Error in if (vif_max < thresh) { : missing value where TRUE/FALSE needed

      Will occur if the variables include non-numeric (i.e. nominal variables).

      Compliments to the author :)

      • I also had this problem when I tried to use LOO validation and VIF, and the one row that I excluded made one of my regressors to be a fixed column. I did not used factors, only numeric.
        Think about a case where a binary column with 100 rows, has a 99 ‘1’ and 1 ‘0’. It will not work.
        After I switch some of the ‘1’ to become ‘0’ (not many, just to even it up a bit), it worked.

    • Hi, this function is very cool.
      However, I have a question with your VIF function. According to the fmsb VIF manual, when calculating the VIF for the val in the for loop, the lm function should not be regressed on the whole variable set (your function is
      form_in<-formula(paste(val,' ~ .'))
      But rather should be on all the other variables, excluding the val (
      var_names <-names(in_frame)
      regressors <- var_names[- which(var_names == val)]
      form <- paste(regressors, collapse = '+')
      form_in <- formula(paste(val, '~', form)
      Please let me know if I'm wrong or anything.

  6. Hi there, great blog and Happy New Years!
    The function is working fine with my dataset, but when trying to create your sample data, I get an error which is due to the variable/matrix/dataset “y” in your code is only a one column matrix with 200 rows – created in line 74 (y<-rand.vars %*% matrix(parms) + rnorm(num.obs,sd=20))
    Is this how it should be? And what is supposed to be %*% doing?
    Hope you have time to answer this.
    Thanks, M

  7. Sorry for bothering, I just found my mistake: The copy I received had in line 77 only lm.dat<-data.frame(y) and not lm.dat<-data.frame(y,rand.vars)…
    So please delete the following entry as well – all is fine

  8. Hi,
    I am running your code to try to reduce the number of bioclimatic variables that may be collinear for certain species. Sometimes it seems to work fine, but other times I have infinite values until about the last vif run, leaving usually 1 or 2 variables out of 20, when using a threshold of 10.
    Is this right to have these infinite values for so many of the vif runs?
    Perhaps my data frame requires transformation first?
    Any insight you have would be appreciated

    • Hi there,

      An infinite VIF will be returned for two variables that are exactly collinear… variables that are exactly the same or linear transformations of each other. Perhaps check your variables for exact collinearity, using the pairs function or something similar (e.g., ggpairs).


  9. Hi,
    Great code, it works well for me, however not well when i have ‘NA’ values. Any idea how i could deal with this?


    • Hi J,

      I tried running the function with missing data and it works fine for me. The part of the function that handles NA values would be the call to lm, specifically the na.action argument where the default is na.omit. I added an additional argument to the vif_func so you can change this option, though you shouldn’t need to. You could also remove the missing values in your data before using vif_func using na.omit, though you will get a slightly different result since this would remove all rows with incomplete data. Here’s the example I did, note the difference in the final VIF values for the two approaches.

      # create data
      y<-rand.vars %*% matrix(parms) + rnorm(num.obs,sd=20)
      # add missing values to the data
      nr <- nrow(rand.vars)
      nc <- ncol(rand.vars)
      rand.vars2 <- c(rand.vars)
      rand.vars2[sample(1:length(rand.vars2), 100)] <- NA
      rand.vars2 <- matrix(rand.vars2, ncol = nc, nrow = nr)
      # use function by removing missing values first
      # run on whole dataset, default for na.action is na.omit, so you don't have to specify
      vif_func(in_frame=rand.vars2,thresh=5,trace=T, na.action = na.omit)
  10. Hi,
    I am running your code and with your data it works fine.
    Now I have a data.frame with num and int Variables.
    And I am getting the message:
    “Error in if (vif_max < thresh) { : missing value where TRUE/FALSE needed"
    But there are only numbers, I dont have factors defined or characters.
    Do you have any idea, what might be the reason?
    The integer values are (0/1)-Results from questions, but I only use the numbers not the translation (no/yes).
    Any idea would be very apprechiated

    • Hi Mark,

      Could you provide some example data that causes the problem? This is hard to diagnose if I can’t reproduce the issue.



  11. I am getting memory errors that are similar to ones discussed in which I have copied here. Do you have any workaround on this ?

    Loading required package: fmsb

    *** caught segfault ***
    address 0x7f8e32c2d554, cause ‘memory not mapped’

    1: terms.formula(formula, data = data)
    2: terms(formula, data = data)
    3: model.frame.default(formula = form_in, data = in_frame, drop.unused.levels = TRUE)
    4: stats::model.frame(formula = form_in, data = in_frame, drop.unused.levels = TRUE)
    5: eval(expr, envir, enclos)
    6: eval(mf, parent.frame())
    7: lm(form_in, data = in_frame, …)
    8: summary(X)
    9: VIF(lm(form_in, data = in_frame, …))
    10: rbind(vif_init, c(val, VIF(lm(form_in, data = in_frame, …))))

    • Hm, I’ve never seen that error before. I don’t know what the issue is but it may help if the vectors in the function are pre-allocated. I was lazy when I wrote the initial function so the output for each loop simply makes a new copy with each iteration – an inefficient method, expecially in R. Check out this blog for an explanation. I rewrote the vif function with pre-allocated vectors:

      #stepwise VIF function with preallocated vectors
        if(class(in_frame) != 'data.frame') in_frame<-data.frame(in_frame)
        #get initial vif value for all comparisons of variables
        vif_init<-vector('list', length = ncol(in_frame))
        names(vif_init) <- names(in_frame)
        var_names <- names(in_frame)
        for(val in var_names){
            regressors <- var_names[-which(var_names == val)]
            form <- paste(regressors, collapse = '+')
            form_in<-formula(paste(val,' ~ .'))
        if(vif_max < thresh){
          if(trace==T){ #print output of each iteration
              cat(paste('All variables have VIF < ', thresh,', max VIF ',round(vif_max,2), sep=''),'\n\n')
          #backwards selection of explanatory variables, stops when all VIF values are below 'thresh'
          while(vif_max >= thresh){
            vif_vals<-vector('list', length = ncol(in_dat))
            names(vif_vals) <- names(in_dat)
            var_names <- names(in_dat)
            for(val in var_names){
              regressors <- var_names[-which(var_names == val)]
              form <- paste(regressors, collapse = '+')
              form_in<-formula(paste(val,' ~ .'))
            if(vif_max<thresh) break
            if(trace==T){ #print output of each iteration
              vif_vals <-'rbind', vif_vals)
              cat('removed: ', names(vif_max),unlist(vif_max),'\n\n')
            in_dat<-in_dat[,!names(in_dat) %in% names(vif_max)]

      Hope that helps.


  12. Can someone help in analysis of sequential path analysis using grain yield as dependent variable and its component traits as independent variables in R

  13. Hi,

    I just wondered if this VIF could help in the feature selection for classification type-models (neural networks) or is just limited to regression type-models?



    • Hi Alejandro,

      The VIF selection could help with fitting neural network models. It will basically tell you which variables are collinear and if they should be removed before creating a model. Collinearity is a problem in most models, not limited to regression. You could try running VIF selection to identify/remove collinear variables prior to creating a neural network. Alternatively, the RSNNS package provides a method for ‘pruning’ a neural network to remove variables that have little influence in the model. I suppose this is similar to variable selection with VIF because it might remove collinear variables, in addition to variables that aren’t related to the response. You could try it both ways to see if you get similar results. There is an example in the help file for the plotnet function that shows how to create and plot a pruned network. Again, you’ll have to download the development version to see this.


  14. Hi, this function is very cool.
    However, I have a question with your VIF function. According to the fmsb VIF manual, when calculating the VIF for the val in the for loop, the lm function should not be regressed on the whole variable set (your function is
    form_in<-formula(paste(val,' ~ .'))
    But rather should be on all the other variables, excluding the val (
    var_names <-names(in_frame)
    regressors <- var_names[- which(var_names == val)]
    form <- paste(regressors, collapse = '+')
    form_in <- formula(paste(val, '~', form)
    Please let me know if I'm wrong or anything.

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