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 time $t_0$ to time $t_1$.
The equation looks like: $y = a\cdot\frac{b}{c}\cdot d$
I need to assign a percentage of each variable (totaling to 100%) responsible for the change in $Y$. $$ \begin{align} t_0 &: -0.4\cdot\frac{205,000}{640}\cdot 0.6 = -76\\ t_1 &: 3.94\cdot\frac{916,000}{320}\cdot 1.85 = 20,864 \end{align} $$
There is a total delta of $20,940$, and I need to assign a percentage of "responsibility" for this delta among the four variables changing. My approach was to "take away" each of the 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 $a = 36\%, b = 16\%, c = 22\%, d = 25\%$
It's a very simple sensitivity analysis, but I really don't have theoretical backup. Is there a better way that I am missing?
Alright, just for the sake of documentation. I found justification for what I did, it's called swing weighting, and it is part of the multi attribute utility theory (MAUT).
Thanks!