# Loss Tools

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### Example 16: Iman-Conover Method of Inducing Correlation

 Freq Dist Freq Mean Var Mult Sev Dist Layer Attach 1 Poisson Neg. Bin. Poisson-Inv. G. PO 1 PO 2 PO 3 Prod A Prod B Prod C CA Zone CA L&M CA H CA EH CA AO 1 Poisson Neg. Bin. Poisson-Inv. G. PO 1 PO 2 PO 3 Prod A Prod B Prod C CA Zone CA L&M CA H CA EH CA AO 3 Poisson Neg. Bin. Poisson-Inv. G. PO 1 PO 2 PO 3 Prod A Prod B Prod C CA Zone CA L&M CA H CA EH CA AO 4 Poisson Neg. Bin. Poisson-Inv. G. PO 1 PO 2 PO 3 Prod A Prod B Prod C CA Zone CA L&M CA H CA EH CA AO Sample Size Num_L2 Correlation Matrix 1.000 0.000 0.150 0.350 1.000 0.250 0.450 1.000 0.550 1.000 Computed Correl n/a Score Type Normal Uniform Exponential

### Notes

• The Iman Conover method induces correlation between given marginal distributions by appropriately shuffling each marginal. Positive correlation is induced by a tendency to shuffle larger values together, negative correlation by combining smaller losses from one marginal with larger losses from the other.
• The shuffle is determined by ranking the input marginals the same as a sample with the desired correlation. Thus, strictly, the method gives the desired rank correlation, not Pearson correlation. For reasonably symmetric distributions the approximation is very good. For more skewed distributions there is a greater difference.
• The "test" sample is computed in the usual way by applying the Choleski decomposition of the correlation matrix to a random sample of scores. Iman and Conover's method adjusts for the random correlation in the score distribution.
• The method can produce radically different contour plots depending on the distribution used for the scores. Selecting a normal score type results in elliptical contours. Selecting uniform or exponential scores will produce very different countour plots.

### References

• Iman, Ronald L. and W. J. Conover,  A Distribution-Free Approach to Inducing Rank Correlation Amoung Input Variables, Commun. Statist.-Simula. Computation 11(3), pp. 311-334 (1982)
• Vitale, R. A. On Stochastic Dependence and a Class of Degenerate Distributions, in Topics in Statistical Dependence, ed. Henry W. Block, Allan R. Sampson and Thomas H. Savits, Institiute of Mathematical Statistics, Hayward CA (1990)