HomeTechniques and Tips@RISK Simulation: Numerical ResultsCalculating Contribution to Variance

# 6.26. Calculating Contribution to Variance

Applies to: @RISK 7.5 and newer

The help file describes Contribution to Variance this way:

These values are calculated during the regression analysis. The sequential contribution to variance technique calculates how much more of the variance in an output is explained by adding each of a sequence of inputs to the regression model. The selection of the variables and the order in which they are added is determined by the stepwise regression procedure. As with any regression technique, when input variables are correlated, the regression can pick any of the correlated variables and ascribe much of the variance to it and not inputs correlated with it. Thus, caution in interpreting the contribution to variance results is critical when inputs are correlated.

Can you expand on that?

@RISK runs a stepwise regression on an output, to find several measures of sensitivity to the input distributions in the model. Stepwise regression is an iterative process where input variables enter into the regression sequentially. From the inputs that have not yet entered the regression, the next one to enter is the one with greatest significance to the output. However, rerunning the regression with that additional input variable can change the results for inputs that entered earlier. If an input no longer contributes significantly, it will leave the regression.

After performing the stepwise regression, @RISK performs a second regression, this time a forward regression. Variables enter this regression in the same order as before, but only the ones that did not leave the original stepwise regression; and no variables leave.

@RISK records the change in R² when each input enters the second, forward regression. (R² is between 0 and 1, and is a measure of how effectively the regression predicts output values. R² is the proportion of the output's total variance that is associated with input variables; 1–R² is the proportion associated with measurement errors, sampling variation, and random variations in general.)

The change in R² when an input enters is that input's percentage contribution to the total variance of the output. It's shown in the Contribution to Variance tornado graph. You can also place those numbers in your worksheet. The total of the percentages given by the worksheet functions will equal R². Because the number of bars on a tornado graph is limited, the total in the graph will be less than R² if not all contributing inputs fit on the graph.

A word on correlated variables: Some correlated variables may leave the first, stepwise regression, because some of their contribution to the output's variance overlaps with the contribution of the other correlated variables, and thus they don't add significant predictive power to the regression. In that case, they won't be part of the second, forward regression, and their contribution to variance is zero. The stronger the correlation, the stronger the tendency to omit some correlated variables. It's not easy to predict which variable is excluded in such cases; it could depend on slight changes in samples from one simulation to the next. But in that scenario it doesn't make much difference which of the correlated variables are used.