Percentage Contribution

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imrek

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Jun 13, 2014
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I have a seemingly simple problem, that is giving me some trouble in solving. I have a 4 variable equation and want to determine the contribution of each variable in moving the dependent variable from time0 to time1.
The equation looks like: y = x1 * x2/x3 *x4
I need to assign a percentage of each X variable (totaling to 100%) responsible for the change in Y.
at time 0:
-0.4*205,000/640*.6 = -76
At time 1: 3.94*916,000/320*1.85 = 20,864
There is a total delta of 20,940, and I need to assign a percentage of "responsibility" for this delta among the four X variables changing. My approach was to "take away" each of the X variable changes individually, then add up the resulting deltas and simply take each delta from each individual X and divide by the sum of the deltas to determine percentage.
As I worked it out I get X1 = 36% X2 = 16% X3 = 22% X4 = 25%
It's a very simple sensitivity analysis, but I really don't have theoretical backup. Is there a better way that I am missing?
 
imrek said:
I have a seemingly simple problem, that is giving me some trouble in solving. I have a 4 variable equation and want to determine the contribution of each variable in moving the dependent variable from time0 to time1.
The equation looks like: y = x1 * x2/x3 *x4
I need to assign a percentage of each X variable (totaling to 100%) responsible for the change in Y.
at time 0:
-0.4*205,000/640*.6 = -76
At time 1: 3.94*916,000/320*1.85 = 20,864
There is a total delta of 20,940, and I need to assign a percentage of "responsibility" for this delta among the four X variables changing. My approach was to "take away" each of the X variable changes individually, then add up the resulting deltas and simply take each delta from each individual X and divide by the sum of the deltas to determine percentage.
As I worked it out I get X1 = 36% X2 = 16% X3 = 22% X4 = 25%
It's a very simple sensitivity analysis, but I really don't have theoretical backup. Is there a better way that I am missing?
Click to expand...
The function is non-linear so the percentage "contribution" of each variable depends on the value of the variables themselves- you cannot just give specific numbers that are always true. With \(\displaystyle y= \frac{x_1x2}{x_3x_4}\), \(\displaystyle dy= \frac{x_2}{x_3x_4}dx_1+ \frac{x_1}{x_3x_4}dx_2- \frac{x_1x_2}{x_3^2x_4}dx_3- \frac{x_1x_2}{x_3x_4^2}dx_4\)

The "percentage contribution" of each variable at given values of \(\displaystyle x_1\), \(\displaystyle x_2\), \(\displaystyle x_3\), and \(\displaystyle x_4\) is the value of each of \(\displaystyle \frac{x_2}{x_2x_4}dx_1\), \(\displaystyle \frac{x_1}{x_3x_4}dx_2\), \(\displaystyle -\frac{x_1x_2}{x_3^2x_4}dx_3\), and \(\displaystyle -\frac{x_1x_2}{x_3x_4^2}dx_4\), where \(\displaystyle dx_1\), \(\displaystyle dx_2\), \(\displaystyle dx_3\), and \(\displaystyle dx_4\) are the changes in each variable over dy, the change in y itself. Note that the last two are negative for positive \(\displaystyle x_1\), \(\displaystyle x_2\), \(\displaystyle x_3\), \(\displaystyle x_4\). That is because they in the denominator: increases in either \(\displaystyle x_3\) or \(\displaystyle x_4\) result in decreases in y.
 
Last edited:
HallsofIvy said:
The function is non-linear so the percentage "contribution" of each variable depends on the value of the variables themselves- you cannot just give specific numbers that are always true. With \(\displaystyle y= \frac{x_1x2}{x_3x_4}\), \(\displaystyle dy= \frac{x_2}{x_3x_4}dx_1+ \frac{x_1}{x_3x_4}dx_2- \frac{x_1x_2}{x_3^2x_4}dx_3- \frac{x_1x_2}{x_3x_4^2}dx_4\)

The "percentage contribution" of each variable at given values of \(\displaystyle x_1\), \(\displaystyle x_2\), \(\displaystyle x_3\), and \(\displaystyle x_4\) is the value of each of \(\displaystyle \frac{x_2}{x_2x_4}dx_1\), \(\displaystyle \frac{x_1}{x_3x_4}dx_2\), \(\displaystyle -\frac{x_1x_2}{x_3^2x_4}dx_3\), and \(\displaystyle -\frac{x_1x_2}{x_3x_4^2}dx_4\), where \(\displaystyle dx_1\), \(\displaystyle dx_2\), \(\displaystyle dx_3\), and \(\displaystyle dx_4\) are the changes in each variable over dy, the change in y itself. Note that the last two are negative for positive \(\displaystyle x_1\), \(\displaystyle x_2\), \(\displaystyle x_3\), \(\displaystyle x_4\). That is because they in the denominator: increases in either \(\displaystyle x_3\) or \(\displaystyle x_4\) result in decreases in y.
Click to expand...

Thanks, that is helpful. However, the derivative basically a rate of change, the different order of magnitudes prevent this from developing percentage.
 
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