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

**CollinearityTest**(

**X**,

**Mask**,

**Method**,

**Column Index**)

**X** is the independent variables data matrix, such that each column represents one variable.

**Mask** 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

- The sample data may include missing values.
- Each column in the input matrix corresponds to a separate variable.
- Each row in the input matrix corresponds to an observation.
- Observations (i.e. row) with missing values are removed.
- In the variance inflation factor (VIF) method, a series of regressions 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.

- A tolerance of less than 0.20 or 0.10 and/or a VIF of 5 or 10 and above indicates a multicollinearity problem.
- 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.

- As a rule of thumb, a condition number ($\kappa$) greater or equal to 30 indicates a severe multi-collinearity problem.
- The CollinearityTest function is available starting with version 1.60 APACHE.

## Files Examples

## References

- Farrar Donald E. and Glauber, Robert R (1967). "Multicollinearity in Regression Analysis: The Problem Revisited". The Review of Economics and Statistics 49(1):92-107.

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