Specifying and Understanding Correlations in @RISK

Applies to: @RISK, all releases

Question:
How do I specify correlations between input variables, and how does @RISK apply them during a simulation?

Specifying Correlations

You can specify correlations in @RISK in several ways:

1. Ribbon: Click the Define Correlations icon.

2. Model Definition Window: Enter correlations for your variables.

3. Worksheet formulas: Add RiskCorrmat() functions directly to the distribution formulas for the variables.

Important: @RISK uses Spearman rank-order correlations, not Pearson linear correlations.

• Spearman correlations are based on rankings of values, not their actual magnitudes.

• Pearson correlations assume linearity, which is less appropriate for most non-linear distributions.

How @RISK Draws Correlated Random Numbers

1. Generate rank scores:

   • For each variable, generate a set of rank scores (one per iteration).

   • Example: For 100 iterations, generate 100 scores per variable.

   • Scores are based on van der Waerden scores (inverse normal function).

2. Pair rank scores to match the specified correlation:

   • The scores are rearranged to produce the desired rank-order correlation coefficient.

3. Generate random numbers for sampling:

   • Generate a set of random numbers (0–1) for each variable.

   • Rank them smallest to largest.

   • Assign the smallest random number to the iteration with the smallest rank score, the second smallest to the second rank, and so on.

This results in paired random numbers used for sampling values from the correlated distributions.

Distribution-free approach:

• Works with any distribution type.

• Preserves the original distribution shapes while inducing the specified rank correlation.

Use of Cholesky Decomposition

• @RISK uses Cholesky decomposition to validate the correlation matrix.

• If decomposition fails, the matrix is not self-consistent, and @RISK proceeds with an adjusted approach.

• If decomposition succeeds, @RISK follows the method in:

Iman, R. L., & Conover, W. J. (1982). A Distribution-Free Approach to Inducing Rank Correlation Among Input Variables. Commun. Statist.-Simula. Computa., 11, 311–334.

Why Excel Shows a Different Correlation

• Excel’s =CORREL() calculates Pearson coefficients, not Spearman rank-order correlations.

• Pearson and Spearman values may differ, especially for non-linear or discrete distributions.

• Differences are normal and expected.

Notes:

• Correlations for discrete distributions may behave differently and require special attention.

• The integrity of each original distribution is maintained during simulation.

 

Related Articles:

@RISK Computing of RankOrder Correlation

A Distribution-free Approach to Rank Correlation

Correlation of Discrete Distributions

Correlation in @RISK

How @RISK Adjusts an Invalid Correlation Matrix

 

 

Attachments:
RankOrderComparison.xls