Question

I hope that this one is not going to be “ask-and-answer” question… here goes:
(multi)collinearity refers to extremely high correlations between predictors in the regression model. How to cure them… well, sometimes you don’t need to “cure” collinearity, since it doesn’t affect regression model itself, but interpretation of an effect of individual predictors.

One way to spot collinearity is to put each predictor as a dependent variable, and other predictors as independent variables, determine R², and if it’s larger than .9 (or .95), we can consider predictor redundant. This is one “method”… what about other approaches? Some of them are time consuming, like excluding predictors from model and watching for b-coefficient changes – they should be noticeably different.

Of course, we must always bear in mind the specific context/goal of the analysis… Sometimes, only remedy is to repeat a research, but right now, I’m interested in various ways of screening redundant predictors when (multi)collinearity occurs in a regression model.

Answer 1

The kappa() function can help. Here is a simulated example:

> set.seed(42)
> x1 <- rnorm(100)
> x2 <- rnorm(100)
> x3 <- x1 + 2*x2 + rnorm(100)*0.0001    # so x3 approx a linear comb. of x1+x2
> mm12 <- model.matrix(~ x1 + x2)        # normal model, two indep. regressors
> mm123 <- model.matrix(~ x1 + x2 + x3)  # bad model with near collinearity
> kappa(mm12)                            # a 'low' kappa is good
[1] 1.166029
> kappa(mm123)                           # a 'high' kappa indicates trouble
[1] 121530.7

and we go further by making the third regressor more and more collinear:

> x4 <- x1 + 2*x2 + rnorm(100)*0.000001  # even more collinear
> mm124 <- model.matrix(~ x1 + x2 + x4)
> kappa(mm124)
[1] 13955982
> x5 <- x1 + 2*x2                        # now x5 is linear comb of x1,x2
> mm125 <- model.matrix(~ x1 + x2 + x5)
> kappa(mm125)
[1] 1.067568e+16
> 

This used approximations, see help(kappa) for details.