# Multicollinearity between predictors used to simulate data

I have simulated a dataset containing individual level variables that results from two processes.

In the first process there is a selection of individuals according to one variable, say "indQual". There is another individual variable that is a categorical variable, say "indBin", with three levels, say, "A", "B", and "C". At the end of this (first) process, the individuals with high values of "indQual" will have a much higher chance of being indBin=B and indBin=C, viceversa, those with low values of "indQual" will be more likely indBin=A.

In the second process, I create a variable, say "indResp" according to this formula:

indResp ~ Intercept + beta1 x indQual + beta2 x indBin(B) + beta3 x indBin(C) + eps (eq.1)

Note that: (i) indQual is normally distributed and indResp follows a Poisson distribution; (ii) beta1 is positive while beta2 and beta 3 are negative and beta2 > beta3 (i.e. indQual has a positive effect on indResp which in turn is lower for indBin=B and even lower for indBin=C); (iii) in both the processes there is random noise (eps in the above formula).

I create lot of datasets under different scenarios (e.g. under stronger or weaker selection in the 1st process and/or different values of beta1 and beta2) and then, analyze the datasets comparing (by AIC) the following three models:

Mod1{indResp ~ indQual, family=poisson}

Mod2{indResp ~ indBin, family=poisson}

Mod3{indResp ~ indQual + indBin, family=poisson}

As expected the Mod3 performs much better in AIC and makes accurate predictions of indResp whereas I am especially interested (for the purpose of my study) to show that Mod2 may lead to wrong conclusions. In particular, if indQual is not accounted for, the effects of indBin(B,C) tend to be biased high.

However, indQual and indBin, the predictors of indRespo, are collinear (more or less depending on how I set the 1st process scenario). I am aware that "collinear" should be a term used for continuous predictors but I do not want to enter in the semantic debate, call it "associated" if you prefer. I am interested in the effect of that collinearity/association. My problem is that data are generated by using both the predictors (see eq.1) and if I drop one out of the model I obtain wrong results (e.g. in Mod2 biased estimated of beta2 and beta3).

Questions:

1) I use vif::car in R on the Poisson models I run and, in general, I get variance inflation factors < 3 (the threshold used in my field for collinearity) but if I plot indQual against indBin they are very much separated, do you know whether the variance inflation factor is reliable for collinearity/association between a categorical and a continuous variable?

2) I confess I am somehow lost about the importance of collinearity in the analyses of my simulated dataset, do you have any suggestion/opinion on it?

• Regarding the question 1) I have found that GVIF should be used in my case. This statistic is available in R from different packages like {car} and {DAAG}. Look at this: stats.stackexchange.com/questions/70679/… – user3844454 Feb 27 '18 at 11:44