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**Interpreting Regression Coefficients in Tornado Graphs**

**Applies to:**

@RISK for Excel 4.x–7.x

@RISK for Project 4.x

How can I interpret the regression coefficients on the tornado diagram or sensitivity report produced by @RISK?

The regression coefficients are calculated by a process called stepwise multiple regression. (A technical paper on the computations is attached.)

The main idea is that the longer the bar or the larger the coefficient, the greater the impact that particular input has on the output that you are analyzing. A positive coefficient, with bar extending to the right, indicates that this input has a positive impact: increasing this input will increase the output. A negative coefficient, with bar extending to the left, indicates that this input has a negative impact: increasing this input will decrease the output.

**Regression Coefficients**

A graph with "regression coefficients" does not express them in terms of actual dollars or other units. Rather, they are scaled or "normalized" by the standard deviation of the output and the standard deviation of that input. Please have a look at the attached example TornadoCoefficients45.xls. (If you open the workbook in Excel without @RISK for Excel, you may see #NAME errors in a few cells but you can still follow the logic.)

For the output, Input A has a regression coefficient (standard b) of 0.772. That means that for every k fraction of a standard deviation increase in Input A, the output will increase by 0.772k standard deviations. You may find the coefficient easier to work with if you "descale" it by converting from standard deviations to units. To descale a standard b coefficient, multiply it by the standard deviation of the output and divide by the standard deviation of the input. 0.772 × 12961 ÷ 1000 = 10.006. This tells you that for an increase of 1 unit in Input A, you get an increase of about 10 units in the output.

**Regression – Mapped Values**

With @RISK 5.0 and newer, you can get mapped coefficients and reduce the amount of arithmetic needed to find impact on outputs. Please see the attached example TornadoCoefficients50.xls.

The mapped coefficients are in units of output per standard deviation of input. For example, Input A has a mapped coefficient of 10013.4, meaning that an increase of k fraction of a standard deviation in Input A produces an increase of 10013.4k units (not standard deviations) of the output. The standard deviation of Input A is 1001, and therefore every unit increase in Input A is 1/1001 of a standard deviation and impacts the output positively by 10013.4 × (1/1001) = 10.003 units of output.

**Actual (Unscaled) Regression Coefficients**

There is no option to show these on the graph, but in @RISK 5.0 and newer you can get them from worksheet functions. Please see Regression Coefficients in Your Worksheet.

**Additional keywords:** Sensitivity analysis

Last edited: 2015-07-06

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