# CollinearityTest - Test for the Presence of Multi-colinearity

Returns the p-value of the multi-collinearity test (i.e., whether one variable can be linearly predicted from the others with a non-trivial degree of accuracy).

## Syntax

X
is the independent variables data matrix, such that each column represents one variable.
is the boolean array to select a subset of the input variables in X. If missing, all variables in X are included.
Method
is the statistics to compute (1 = Condition Number (default), 2 = VIF, 3 = Determinant, 4 = Eigenvalues).
Method Description
1 Condition Number (Kappa).
2 Variance Inflation Factor (VIF).
Column Index
is a switch to designate the explanatory variable to examine (not required for condition number).

## Remarks

1. The sample data may include missing values.
2. Each column in the input matrix corresponds to a separate variable.
3. Each row in the input matrix corresponds to an observation.
4. Observations (i.e., row) with missing values are removed.
5. In the variance inflation factor (VIF) method, a series of regression models are constructed, where one variable is the dependent variable against the remaining predictors.$$\textrm{Tolerance}_i = 1-R_i^2$$ $$\textrm{VIF}_i =\frac{1}{\textrm{Tolearance}_i} = \frac{1}{1-R_i^2}$$ Where:
• $R_i^2$ is the coefficient of determination of a regression of explanator $i$ on all the other explanators.
6. A tolerance of less than 0.20 or 0.10 and/or a VIF of 5 or 10 and above indicates a multicollinearity problem.
7. The condition number ($\kappa$) test is a standard measure of ill-conditioning in a matrix; It will indicate that the inversion of the matrix is numerically unstable with finite-precision numbers (standard computer floats and doubles).$$X = \begin{bmatrix} 1 & X_{11} & \cdots & X_{k1} \\ \vdots & \vdots & & \vdots \\ 1 & X_{1N} & \cdots & X_{kN} \end{bmatrix}$$ $$\kappa = \sqrt{\frac{\lambda_{max}}{\lambda_{min}}}$$ Where:
• $\lambda_{max}$ is the maximum eigenvalue.
• $\lambda_{min}$ is the minimum eigenvalue.
8. As a rule of thumb, a condition number ($\kappa$) greater or equal to 30 indicates a severe multi-collinearity problem.
9. The CollinearityTest function is available starting with version 1.60 APACHE.