In this post, I will discuss about existence and detection of collinearity among regressors.

The problem of multicollinearity plagues the numerical stability of regression estimates. It also causes some serious problem in validation and interpretation of the regression model. Consider the usual multiple linear regression model,

$latex y = X \beta+u$,

where $latex y$ is an $latex n\times 1$ vector of observation on dependent variable, $latex X$ is known design matrix of order $latex \times p$, having full-column rank $latex p$, $latex \beta$ is $latex p \times 1$ vector of unknown parameters and $latex u$ is an $latex n\times 1$ vector of random errors with mean zero and variance $latex \sigma^2 I_n$, where $latex I_n$ is an identity matrix of order $latex n$.

Collinearity Among Regressors

The existence of linear dependence (relationship) between regressors can affect the regression model ability to estimate the model’s parameters (regression coefficients). Therefore, multicollinearity is lack of independence or the presence of interdependence signified by usually high inter-correlations within a set of regressors (predictors).

In case of severe multicollinearity (mathematically when matrix is ill-conditioned) the $latex X’X$ matrix cannot be inverted, implausible signs of coefficients, low t-ratios, high R-squared values, inflated standard errors, wider confidence intervals, very large condition number (CN) and non-significant and/or magnitude of regression coefficient estimates are some of the possible issues.

There are many diagnostic methods are available to check the existence of collinearity among regressors, such as variance inflation Factor (VIF), values of pair-wise correlation among regressors, eigenvalues, CN, Farrar and Glauber tests, Theil’s measure, and Klein’s rule, etc.

Our recently developed R package mctest computes several diagnostic measures to test the existence of collinearity among regressors. We classified these measures as individual collinearity diagnostic and overall collinearity diagnostics. Overall collinearity diagnostic includes determinant of $latex X’X$ matrix, red indicator, Farrar Chi-Square test, Theil indicator, CN, and the sum of lambda inverse values. Individual collinearity diagnostics include VIF/ TOL, Farrar and Glauber Wi test, the relationship between $R^2$ and F-test, corrected VIF (CVIF) and Klein’s rule.

How to use mctest package

You must have installed and load the mctest package to start with testing of collinearity among regressors. As an example, we used Hald data which is already bundled in mctest package.

mctest package have 4 functions namely, mctest(), omcdiag(), imcdiag() and mc.plot() functions. The mctest() function can be used to have overall and/or individual collinearity diagnostic. The mc.plot() can be used to draw graph of VIF and eigenvalues to have graphical judgement of among collinearity among regressors.

mctest illustrative Example

Arguments of mctest are with syntax are

mctest(x, y, type = c(“o”, “I”, “b”), na.rm = TRUE, Inter = TRUE, method = NULL, corr = FALSE, detr = 0.01, red = 0.5, theil = 0.5, cn = 30, vif = 10, tol = 0.1, conf = 0.95, cvif = 10, leamer = 0.1, all = all)

See  the detail of each argument see the mctest package documentation. Following are few commands that can be used get different collinearity diagnostics.

> x<-Hald[ ,-1]  # X variables from Hald data
> y<-Hald[ ,1]   # y variable from Hald data
> mctest(x, y)   # default collinearity diagnostics
> mctest(x, y, type = “i”)  # individual collinearity diagnostics
> mctest(x, y, type = “o”) # overall collinearity diagnostics

Overall collinearity diagnostics

For overall collinearity diagnostics, eigenvalues and condition numbers are also produced either intercept term is included or not. The syntax of omcdiag() function is

omcdiag(x, y, na.rm = TRUE, Inter = True, detr = 0.01, red = 0.5, conf = 0.95, theil = 0.5, cn = 30, …)

Determinant of correlation matrix, Farrar test of Chi-square, Red indicator, sum of lambda inverse values, Theils’ indicator and CN.

> omcdiag(x, y, Inter=FALSE)
> omcdiag(x, y)[1]
> omcidag(x,y, detr=0.001, conf=0.99)

For the last command (with threshold for determinant and confidence interval for Farrar and Glauber test). the output is,

Individual collinearity diagnostics

imcdiag(x, y, method = NULL, na.rm = TRUE, corr = FALSE, vif = 10, tol = 0.1, conf = 0.95, cvif = 10, leamer = 0.1, all = all)

Function imcdiag() detects the existence of multicollinearity due to certain X-variable. This includes VIF, TOL, Klein’s rule, CVIF, F&G test of Chi-square and F-test.

> imcdiag(x = x, y)
> imcdiag(x = x, y, corr = TRUE) # correlation matrix
> imcdiag(x = x, y, vif = 5, leamer = 0.05)   # with threshold of VIF and leamer method

> imcdiag(x = x, y, all = True)
> imcdiag(x = x, y, all = TRUE, vif = 5, leamer = 0.2, cvif = 5)

Graphical representation of VIF and Eigenvalues

> mc.plot(x, y, Inter = FALSE, vif = 10, ev = 0.01)
> mc.plot(x, y)
> mc.plot(x, y, vif = 5, ev =  0.2)

For further detail about collinearity diagnostic see

• Documentation file of R-Package
• mctest package on CRAN
• mctest package published Research paper

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