The correlation coefficient (r) and variance inflation factor (VIF) are measuring different things. r tells you the strength and direction of the association between the two variables. The VIF is not a measure of the correlation between two variables, rather it indicates what percentage of the variance is inflated for each coefficient. In other words, are the two variables accounting for some of the same variance (since they are correlated), and thus, inflating the actual amount of variance explained by each predictor? VIF has a minimum value of 1 (no collinearity) and a higher value suggests greater collinearity.

There is debate regarding what is an acceptable VIF value. However, even by conservative estimates, your values (2.391, 2.297) are not problematic. Let's take the first value (2.39). You interpret this as follows: the variance of this particular coefficient is 1.39 (or 139%) larger than you would expect if there was no collinearity.

In sum, just because there is a correlation between two predictors does not necessarily mean that variance is (largely) inflated.

Thank you! But in regard to collinearity also the correlation coefficient is consulted and with a value of -.701 shouldn't I worry about a possible multicollinearity?

Merina Perocco I would go by VIF over the correlation coefficient.

If you have 2 different VIF values, I assume you have more than these 2 predictors, othwise they shoud be the same value. But I would like to slightly correct what Blaine Tomkins wrote:

1) In case of only two predictors, the VIF and the bivariate correlation are the same measure, but only a transformation into each other, since VIF = 1/(1-R^2), where R^2 is equal to the squared bivariate correlation (in the two predictor case). But since your VIF is not 1/(1-0.701^2) = 1.966, I conclude you have more predictors in your model, which increase your value.

2) The variance of your regression coefficient is 2.39 or 2.297 times larger than it would be with 0 intercorrelation (not 1.39 times), but this equals 139% or 129.7%, respectively.

3) You can think of it differently, the square root of the VIF (i.e. 1.55 and 1.51) indicates how many times larger the standard error of the regression coefficient is (since the s.e. of beta is the square root of the variance of beta).

4) To assess the impact, I would also look at the VIF (since it accounts for all other predictors, not only bivariate relationships). There is this rule of thumb that VIF < 10 is ok, but you have to assess individually for your application and research field if 2.39 or 2.29 is ok or not.

Rainer Duesing I don't follow your reasoning that VIF and bivariate correlation are the same thing. A correlation coefficient indicates nothing about variance explained in the outcome variable. The coefficient of determination, on the other hand, does get at this information. Although it is easy to to convert one into the other with a simple transformation, it seems incorrect to say that they are the same thing.

Blaine Tomkins , maybe we use the phrase "the same thing" differently. But since you do not need any other infomation (no factor, no constant, no additional parameter) to convert one into the other, they are deterministically, basically the same (at least for me). The same holds for correlation and coefficient of determination. If you know one you automatically know the other. Maybe, the one or the other is more comfortable to interpret for different people, but the coefficients do not give you different or any additional information per se.

In case of a regression with two predictors (and only then) you could say instead of VIF 2 predictors.

If your concern is the m​u​l​t​i​c​o​l​l​i​n​e​a​r​i​t​y​, then you should use r.

Noha A. Mostafa wow, after everything Blaine Tomkins and I wrote, you simply say " then you should use r "....

Rainer Duesing I agree that VIF is not very informative with only two predictors. As you say, it basically just reiterates the extent to which your predictors are correlated. I couldn't agree more. However, I still don't think the correlation coefficient and coefficient of determination (included in the VIF equation) should be considered the same measure - if that's what you were saying. If not, my apologies for misunderstanding. As I mentioned above, r tells you nothing more than the strength and direction of the association, whereas R2 taps into the shared variance of the two variables. Very different kinds of information regarding the relationship between x and y.

No need for apologies in a scientific exchange ;-)

Something I did not consider in my original answer was the sign of the correlation, idicating the direction, fair enough. But for the rest: maybe I am missing something essential, but I still think both deliver the same basic information. Of course you are right, the correlation is no indicator for the shared variance, but on a broader level, what do we really want with this parameters (and other parameters in general)? We try to explain and make sense out of our data(-structure). For me there is no inherit better parameter, under some cirmumstances one may be better than the other to coonvey the relevant information. I can say that there is a correlation of r=0.4 or that the coefficient of determination was R^2 = .16 or that explained variance/shared variance was 16%. Would you say that the transformation in percent delivers different/more/less information that the other two?

Again, maybe I am missing something important here, but I do not agree with your statement that they are "Very different kinds of information regarding the relationship". What is so different? I am really looking forward to hear from you.

Rainer Duesing Let's take the simple example of the correlation between 'height' and 'weight'. There's bound to be a strong positive correlation (r = .88). From this we can conclude conceptually that there is a fairly strong, positive linear relationship between a person's height and weight. Were we to ask at this point: to what extent does the way that people vary in our sample with respect to their height account for (explain) the way that the same people vary with respect to their weight? The coefficient cannot tell us, since r does not take into account variance.

Now, we compute r2 = .882 = .774 - we are now equipped to answer the question. Approximately 77% of the variance in our sample's weight is explained by their height. Statistically, this is a subtle difference. Yet, conceptually a big difference - at least, in my opinion.

I still do not see the "conceptually a big difference" (emphasis on "big" here). We both agree about calculation and so on, so no need to discuss this point I think. ;-)

As I mentioned earlier, in different contexts shared variance or correlation may be more appropriate to present, respectively. But homestly, where in a practical application do you really need the information in shared variance instead of correlation (or vice versa), to assess the model fit? To take your example, is there a situation where you would say r = .88 is "a fairly strong" relationship, but you would not conclude this, when you see 77% shared variance, or where you would even say that 77% indicates a weak fit, whereas at the same time r = .88 is a strong fit? What do you (really) do with the information that 77% of the variance is shared or explained, where do you need it in an application to know this specific transformation (besides assessing the fit)?

Maybe, I can't think of a reasonable example, because I haven't seen one yet, but I am curious. Furthermore, from an application point of view, I even think r or r^2 alone are rather useless, I (personally) am more interested how much the DV changes with changes in the IV(s). So for me r is just the indicator for the fit, but I am really interested in the regresion coefficient (similar in MR). See the picture from Wilcox book, where the same regression coefficient may have different correlation coefficients. So I need both to understand the data.

But this again may depend on the research area and I am open to hear different views or examples from other field, with different applications.

I'm not speaking in terms of model fitting, rather with respect to what you as the researcher want to know. The question you are out to answer outside the context of the model.

Blaine Tomkins please give an example/question, where you specifically need the information of shared variance, which can't be answered with the knowledge of the correlation.

Rainer Duesing I fear we may be speaking past one another. I've never stated that information of shared variance can be obtained without first measuring correlations. Surely there are instances where a researcher's goal is to examine a correlation, but not R2, or vice versa.

Scenario 1: John, a professor, wants to know whether there is a correlation between student attendance and exam scores in his class. So, he analyzes the data and computes the correlation. He could easily derive the shared variance as well, but he's not interested in explaining the variance in student exam scores; he only wishes to see if there's a considerable correlation between attendance and grades.

Scenario 2: Sally has been hired by the W.H.O. to examine the relative influence of several predictors on outcomes for patients with disease x. She throws in all predictors she can think of (age, ethnicity, diet, sex, SES, etc,). Her correlation matrix is bound to show some intercorrelations among the quantitative predictors, so that won't be very helpful since her goal is not to simply predict a patient's outcome, but tease apart the relative contribution of each predictor. Correlations, while a necessary step, are not the end goal here.

Blaine Tomkins you have got a PM, no need to bother other people with our private discussion ;-)

Catalina Trujillo

One could compare a restricted model (1 predictor) to the full model (2 predictors) and see if R2 significantly increases.

Catalina Trujillo

If you are interested examining the relative importance of each predictor, you'll need to be mindful of collinearity and deal with it accordingly.

No problem :)