Why Two Collinearity Diagnostics Are Read Together
A condition index signals that a regression design has a direction with relatively little variation. It does not, by itself, say which predictors participate in that direction. Variance-decomposition proportions supply that information: they show which coefficient variances are associated with each eigenvalue dimension. The useful diagnosis comes from reading the two measures together.
The methods below apply to the predictor columns after a consistent scaling and centering convention has been chosen. To focus on slope coefficients, a common approach is to center and scale the predictors and assess the resulting predictor matrix separately from the intercept. The interpretation of the coefficients in the fitted model is unchanged by this diagnostic calculation; the scaling is used to make the predictors’ contributions comparable.
The eigenvalue dimensions are ordered from the greatest eigenvalue to the smallest. Thus, the first condition index is 1, and later indices increase as their eigenvalues decrease. A large index identifies a relatively weak direction in the design. A large proportion \(\pi_{jk}\) means that a substantial share of the decomposition for coefficient \(j\) is associated with that dimension.
A common diagnostic convention treats condition indices in the range of roughly 10 to 30 as potentially concerning and values above 30 as more serious. These are guidelines, not universal cutoffs. A useful signal of a shared near-dependency is a high condition index on a dimension where at least two predictors have large variance-decomposition proportions. A single large proportion does not identify a group of coefficients sharing that dimension.
Why the Proportions Use Inverse Eigenvalues
The definition follows from the covariance structure of the least-squares coefficients. Under the linear regression model with independent errors of mean 0 and common variance \(\sigma^2\), the covariance matrix of the least-squares coefficient estimates is \(\sigma^2(X^\mathsf{T}X)^{-1}\), when the design matrix \(X\) has full column rank. For a scaled predictor matrix with \(R=Z^\mathsf{T}Z\), write its eigenvalue decomposition as \(R=Q\Lambda Q^\mathsf{T}\). Then
The diagonal element \((R^{-1})_{jj}\) gives the scaled variance contribution for coefficient \(j\), apart from the common error-variance factor. Each eigenvalue dimension contributes \(q_{jk}^{2}/\lambda_k\). Dividing that contribution by the sum across all dimensions gives a proportion, so the common variance factor cancels. The inverse eigenvalue explains why a small eigenvalue can dominate a coefficient’s decomposition even when the corresponding eigenvector entry is not especially large.
This result assumes the predictor matrix has full rank, so all its eigenvalues are positive. With exact multicollinearity, at least one eigenvalue is zero and the usual inverse and coefficient-variance decomposition do not exist. The condition diagnostics here describe near-dependencies in a full-rank design.
Worked Example: Finding the Pair Behind a High Condition Index
A hypothetical pricing analysis has four centered, standardized predictors, \(x_1,x_2,x_3,x_4\). Its scaled cross-product matrix has two independent correlation blocks: \(x_1,x_2\) have correlation 0.99, and \(x_3,x_4\) have correlation 0.80. Find the condition indices and variance-decomposition proportions, then identify which pair shares the strongest near-dependency.
Method. Each two-predictor correlation block with correlation \(r\) has eigenvalues \(1+r\) and \(1-r\), with squared eigenvector entries \(1/2\) for each predictor in that block. Thus, the four eigenvalues, in descending order, are \(1.99, 1.80, 0.20, 0.01\). The eigenvalue sum is \(4\), as expected for a four-predictor correlation matrix with diagonal entries 1.
For the condition indices, divide the largest eigenvalue, 1.99, by each eigenvalue and take the square root:
The values are rounded to two decimal places. For either \(x_1\) or \(x_2\), the contributions on dimensions 1 and 4 are \(0.5/1.99=0.251256\) and \(0.5/0.01=50\). The proportions are therefore
For either \(x_3\) or \(x_4\), the contributions on dimensions 2 and 3 are \(0.5/1.80=0.277778\) and \(0.5/0.20=2.5\). Hence,
The resulting table is consistent with the two correlation blocks and their eigenvectors:
| Dimension | Eigenvalue | Condition index | \(x_1\) | \(x_2\) | \(x_3\) | \(x_4\) |
|---|---|---|---|---|---|---|
| 1 | 1.99 | 1.00 | 0.0050 | 0.0050 | 0 | 0 |
| 2 | 1.80 | 1.05 | 0 | 0 | 0.1000 | 0.1000 |
| 3 | 0.20 | 3.15 | 0 | 0 | 0.9000 | 0.9000 |
| 4 | 0.01 | 14.11 | 0.9950 | 0.9950 | 0 | 0 |
Dimension 4 has the largest condition index, 14.11, and both \(x_1\) and \(x_2\) have large proportions on it. This identifies the \(x_1,x_2\) pair as sharing the strongest near-dependency. The proportions for each predictor sum to 1, subject to displayed rounding. By contrast, dimensions 2 and 3 concern \(x_3,x_4\), but their condition indices are much smaller.
Common mistake avoided: The raw contribution \(0.251256\) for \(x_1\) on dimension 1 is not its variance-decomposition proportion. Calling it a proportion would ignore the contribution 50 on dimension 4. The correct proportion divides by the total, \(0.251256+50\), giving approximately 0.0050.
Reading a Diagnostic Table in Order
For a supplied table, start with the eigenvalue dimensions that have the largest condition indices. Then scan across each such row and identify predictors with large proportions on that same dimension. Do not infer a shared near-dependency by looking at large proportions in different rows: two predictors can each have a large proportion, but on separate dimensions. Those results point to different directions in the design.
Confirm which predictors were centered or scaled and whether the intercept is included. Interpret the table using its stated convention.
Since a condition index uses the reciprocal scale of an eigenvalue, small eigenvalues produce the largest indices.
On a concerning dimension, look for two or more predictors with substantial proportions on that same dimension.
Name the dimension and predictors, and describe the evidence as a potential near-dependency rather than proof of a particular causal or predictive relationship.
Worked Example: Large Proportions Without a Large Condition Index
A two-predictor diagnostic has standardized predictor correlation 0.80. Determine whether the proportions alone indicate a serious collinearity signal.
Method. The correlation matrix is
Its eigenvalues are \(1.80\) and \(0.20\), and each eigenvector has squared entries \(1/2\). The condition indices are \(\sqrt{1.80/1.80}=1\) and
For either predictor, the contributions are \(0.5/1.80=0.277778\) and \(0.5/0.20=2.5\). The proportions are \(0.277778/(0.277778+2.5)=0.1000\) and \(2.5/(0.277778+2.5)=0.9000\). Thus both predictors have large proportions on the same dimension, but the condition index there is only 3. The proportions locate a shared direction; on their own they do not establish a severe condition-index signal.
Common mistake avoided: Treating a proportion of 0.9000 as a stand-alone severity measure. The proportion describes how the coefficient’s variance decomposition is distributed; the condition index describes how weak the design direction is.
Worked Example: Distinguishing Two Near-Dependencies
A second hypothetical analysis has four centered, standardized predictors in two independent pairs. The predictor correlations are 0.96 for \(x_1,x_2\) and 0.90 for \(x_3,x_4\). Calculate the indices and proportions, then decide whether the table suggests one shared problem or two separate directions.
Method. The eigenvalues for the first pair are \(1.96\) and \(0.04\); for the second pair they are \(1.90\) and \(0.10\). In descending order, the eigenvalues are \(1.96,1.90,0.10,0.04\), which sum to 4. The condition indices are
For \(x_1\) and \(x_2\), the contributions on dimensions 1 and 4 are \(0.5/1.96=0.255102\) and \(0.5/0.04=12.5\). Their proportions are \(0.255102/(0.255102+12.5)=0.0200\) and \(12.5/(0.255102+12.5)=0.9800\). For \(x_3\) and \(x_4\), the contributions on dimensions 2 and 3 are \(0.5/1.90=0.263158\) and \(0.5/0.10=5\), giving proportions \(0.0500\) and \(0.9500\), respectively.
| Dimension | Eigenvalue | Condition index | \(x_1\) | \(x_2\) | \(x_3\) | \(x_4\) |
|---|---|---|---|---|---|---|
| 1 | 1.96 | 1.00 | 0.0200 | 0.0200 | 0 | 0 |
| 2 | 1.90 | 1.02 | 0 | 0 | 0.0500 | 0.0500 |
| 3 | 0.10 | 4.43 | 0 | 0 | 0.9500 | 0.9500 |
| 4 | 0.04 | 7.00 | 0.9800 | 0.9800 | 0 | 0 |
The proportions identify two separate pairwise directions: \(x_1,x_2\) are concentrated on dimension 4, while \(x_3,x_4\) are concentrated on dimension 3. Neither condition index exceeds 10, so under the rough guideline stated earlier, this table does not show a strong condition-index warning. It still indicates which pairs are most closely associated with the weaker dimensions. The conclusion should preserve both facts rather than labeling all large proportions as equally serious.
Common mistake avoided: Combining the \(x_1,x_2\) proportions from dimension 4 with the \(x_3,x_4\) proportions from dimension 3 and reporting one four-predictor dependency. The evidence is on two distinct dimensions, so the pairs should be described separately.
Interpretation Limits and Exam Traps
Condition indices and variance-decomposition proportions are diagnostic summaries, not automatic decision rules. Their values depend on the scaling convention, so comparisons require a consistent setup. A table that includes an intercept or uses uncentered, differently scaled columns may look different from one based on centered, standardized predictors. Read the table’s construction before interpreting it.
A large proportion means that a coefficient’s variance decomposition is concentrated on a dimension; it does not mean that the coefficient itself is large, that its estimate has a particular sign, or that the model’s predictions must be poor. Likewise, a large condition index indicates a weak direction in the design, but it does not name the predictors responsible. The row-wise combination of a concerning index and multiple large proportions is the key evidence for a shared near-dependency.
- Do not use the condition index alone to name variables. Use the proportions on the associated dimension.
- Do not use proportions alone to assess severity. Check the condition index for that same dimension.
- Do not compare proportions across different dimensions as if they described one dependency. Interpret each dimension separately.
- Do not treat a conventional cutoff as a theorem. Thresholds are context-dependent guidelines.
Check Your Understanding
For each problem, use the stated scaling convention and interpret the condition indices and proportions together.
- A diagnostic has largest eigenvalue 3.20 and smallest eigenvalue 0.08. Calculate the condition index for the smallest eigenvalue to two decimal places. (Answer: 6.32)
- For one coefficient on a dimension, \(q_{jk}^{2}=0.40\) and \(\lambda_k=0.05\). The sum of its contributions from all other dimensions is 12. Find its variance-decomposition proportion on dimension \(k\), to four decimal places. (Answer: 0.4000)
- One dimension has condition index 18. Its proportions for \(x_1,x_2,x_3\) are 0.04, 0.82, and 0.79. Which predictors share the near-dependency indicated by this row? (Answer: \(x_2\) and \(x_3\))
- A two-predictor standardized design has correlation 0.95. Its eigenvalues are 1.95 and 0.05, with squared eigenvector entries 0.5 for each predictor on each dimension. Calculate the condition index for the smaller eigenvalue and each predictor’s proportion on that dimension. (Answer: condition index 6.24; proportion 0.9750 for each predictor)