You are discussing the problem of multicollinearity. This arise from the inclusion of highly correlated independent variables (see Dohoo et al., 1997). Both correlation coefficient and VIF are used to verify multicollinearity. Dohoo et al. (1997) argued that multicollinearity is certain at the 0.9 level of a correlation coefficient or higher (see also discussion by Chen and Rothschild, 2010). On the other hand, the VIF values for included variables should be below 10 (see suggestion of Lin, 2008). If any (correlation coefficient or VIF) critical value is higher than described above, you should remove one of correlated independent variables from your set, or estimate two separated models. As an example you can follow my paper (Pawlicz and Napierala, 2017).

Literature:

• Chen, C.F. and Rothschild, R. (2010), “An application of hedonic pricing analysis to the case of hotel rooms in Taipei”, Tourism Economics, Vol. 16 No. 3, pp. 685-694.
• Dohoo, I., Ducrot, C., Fourichon, C., Donald, A. and Hurnik, D. (1997), “An overview of techniques for dealing with large numbers of independent variables in epidemiologic studies”, Preventive Veterinary Medicine, Vol. 29 No. 3, pp. 221-239.
• Lin, F.J. (2008), “Solving multicollinearity in the process of fitting regression model using the nested estimate procedure”, Quality and Quantity, Vol. 42 No. 3, pp. 417-426.
• Pawlicz A. and Napierala T., (2017),"The determinants of hotel room rates: an analysis of the hotel industry in Warsaw, Poland ", International Journal of Contemporary Hospitality Management, Vol. 29 Iss 1 pp. 571 - 588.

You are discussing the problem of multicollinearity. This arise from the inclusion of highly correlated independent variables (see Dohoo et al., 1997). Both correlation coefficient and VIF are used to verify multicollinearity. Dohoo et al. (1997) argued that multicollinearity is certain at the 0.9 level of a correlation coefficient or higher (see also discussion by Chen and Rothschild, 2010). On the other hand, the VIF values for included variables should be below 10 (see suggestion of Lin, 2008). If any (correlation coefficient or VIF) critical value is higher than described above, you should remove one of correlated independent variables from your set, or estimate two separated models. As an example you can follow my paper (Pawlicz and Napierala, 2017).

Literature:

• Chen, C.F. and Rothschild, R. (2010), “An application of hedonic pricing analysis to the case of hotel rooms in Taipei”, Tourism Economics, Vol. 16 No. 3, pp. 685-694.
• Dohoo, I., Ducrot, C., Fourichon, C., Donald, A. and Hurnik, D. (1997), “An overview of techniques for dealing with large numbers of independent variables in epidemiologic studies”, Preventive Veterinary Medicine, Vol. 29 No. 3, pp. 221-239.
• Lin, F.J. (2008), “Solving multicollinearity in the process of fitting regression model using the nested estimate procedure”, Quality and Quantity, Vol. 42 No. 3, pp. 417-426.
• Pawlicz A. and Napierala T., (2017),"The determinants of hotel room rates: an analysis of the hotel industry in Warsaw, Poland ", International Journal of Contemporary Hospitality Management, Vol. 29 Iss 1 pp. 571 - 588.

Dear Olabode

1. Correlation is used to test relationships between quantitative variables or categorical variables. In other words, it’s a measure of how things are related. The study of how variables are correlated is called correlation analysis.

A correlation coefficient gives a numerical summary of the degree of association between

two variables - e,g, to what degree do high values of one variable go with high values of

the other one? Correlation coefficients vary from -1 to +1, with positive values indicating an increasing relationship and negative values indicating a decreasing relationship. A “0” means there is no relationship between the variables at all, while -1 or 1 means that there is a perfect negative or positive correlation (negative or positive correlation here refers to the type of graph the relationship will produce).

2. Linear regression model/analysis is a technique used to predict the value of one quantitative variable by using its relationship with one or more additional quantitative variables. For example, if we know the relationship between height and weight in adult. males, we can use regression analysis to predict weight given a particular value for height.

3. VIF: variance inflation factors.

Collinearity test in SPSS provide a table containing three measures, the first is Tolerance, the second is VIF and the third is Minimum Tolerance. Each measure has a meaning.

For more details, see Landau and Everitt (2004). A hand Book in Statistical Analysis using SPSS, pages 103-107.

Regards,

Zuhair

Many thanks for the answers

I suggest you'd better try to drop one of the two correlated predictors and see how the model "behaves"...

Even if the VIF values is below the common thresholds, with such a great correlation between those two predictors I would also suggest to drop one of them from the model.

I would like to draw your attention to the fact that accurate or statistical multicollinearity is also possible when the correlations between pairs of variables are small. I attach the file with some examples.

VIF is a transformation of the R^2 resulting from predicting X by other other predictors in the model. As R is the multiple correlation coefficient then there is a form of relationship between the two concepts.

specifically VIF = 1/tolerance where t(tolerance) = 1/R^2

and R^2 is for predicting X1 from X2 to Xk in a model with K predictors.

If X1 and X2 are correlated .98 then VIF can not be as low as 1.034. So I suspect an error in your post, your model set-up or your reading of the model output.

just to add - if you want two variables in your model that are correlated at 0.98, I would ask: why? The meaning of your predictor is the association of Y and X1 controlling for the other Xs. If those other Xs include a variable that is almost identical to X1, and so measures the same underlying social construct, you are effectively controlling X1 for itself which is not something you want to do.

I guess my point is that collinearity is a substantive thing, not (just) a technical thing. Variables should be included in your model for sensible theoretical reasons, not just because they happens to be in your dataset.

Tomasz Napierala Thank you very much for helpful answer. May I continue this question? I got similar issue, the correlation value between control variable and independent variable is 0,78, but vif is low. Could I still control it. I read some materials showing that multicollinearity (high VIF ) would not be an issue if they are control variables . But I am not sure whether it is not issue as well when correlation value is high between control variable and independent variable. Could you give me some suggestion on it. Thank you very much.

Qian Chen, the problem of multicollinearity of control variables is not a problem you need to fix. It must be noted that multicollinearity disables interpretation of separately considered (ceteris paribus) experimental variables. But, the interpretation is not requested for control variables. Thus, you can just skip the issue. Very detailed explanation you can find on a website: http://statisticsbyjim.com/regression/multicollinearity-in-regression-analysis/.

Moreover, if solving the multicollinearity is requested, the easiest way (except of removing variables) is to run the Principle Component Analysis before regression.

Tomasz Napierala Thank you very much for your kind and helpful answer. Best/ Qian

There are instances that a high vif is not a problem. One of such is where the variable with high vif is a control variable and not the variable of interest. So long as the collinear variable is used as control variable and is not collinear with the variable of interest there is no need to worry. A close examination will reveal that it will not affect the coefficient of the variable of interest.

This depends on which correlation method you're using. Probably, you're not using pearson correlation since VIF is computed from R2 which is basically Pearson Correlation squared. You might be using spearman or another type of correlation.