# Thread: Contribution of Each Factor to the Total Change

1. ## Contribution of Each Factor to the Total Change

I apologize for the title not being very clear, but that is the best I could think of to describe my question.

So, assume that I have three factors:

$\displaystyle a_0=1.1$
$\displaystyle b_0=0.7$
$\displaystyle c_0=1.31$

$\displaystyle T_0=a_0*b_0*c_0 = 1.0087$

Then the 3 factors are changed:

$\displaystyle a_1=1.22$
$\displaystyle b_1=0.91$
$\displaystyle c_1=1.04$

$\displaystyle T_1=a_1*b_1*c_1 = 1.154608$

How can I express how much the change of $\displaystyle a_0=1.1$ to $\displaystyle a_1=1.22$ effected the overall change from $\displaystyle T_0=1.0087$ to $\displaystyle T_1=1.154608$?

I'd like to be able say something like 45% of the change from $\displaystyle T_0$ to $\displaystyle T_1$ came from the change from $\displaystyle a_0$ to $\displaystyle a_1$,
25% of the change from $\displaystyle T_0$ to $\displaystyle T_1$ came from the change from $\displaystyle b_0$ to $\displaystyle b_1$,
30% of the change from $\displaystyle T_0$ to $\displaystyle T_1$ came from the change from $\displaystyle c_0$ to $\displaystyle c_1$

Thanks,

2. ## Re: Contribution of Each Factor to the Total Change

$\Delta T_0 = \dfrac{dT_0}{da_0} \Delta a_0 + \dfrac{dT_0}{db_0} \Delta b_0+\dfrac{dT_0}{d c_0} \Delta c_0$

$\Delta T_0 =b_0 c_0 \Delta a_0 + a_0 c_0 \Delta b_0 + a_0 c_0 \Delta c_0 = T_0\left(\dfrac{\Delta a_0}{a_0} + \dfrac{\Delta b_0}{b_0}+ \dfrac{\Delta c_0}{c_0}\right)$

3. ## Re: Contribution of Each Factor to the Total Change

Originally Posted by romsek
$\Delta T_0 = \dfrac{dT_0}{da_0} \Delta a_0 + \dfrac{dT_0}{db_0} \Delta b_0+\dfrac{dT_0}{d c_0} \Delta c_0$

$\Delta T_0 =b_0 c_0 \Delta a_0 + a_0 c_0 \Delta b_0 + a_0 c_0 \Delta c_0 = T_0\left(\dfrac{\Delta a_0}{a_0} + \dfrac{\Delta b_0}{b_0}+ \dfrac{\Delta c_0}{c_0}\right)$
Thanks for the response, but I'm a bit hung up on the method. What is $\displaystyle \Delta{a_0}$? is it $\displaystyle \frac{a_1-a_0}{a_0}$?

4. ## Re: Contribution of Each Factor to the Total Change

Originally Posted by downthesun01
Thanks for the response, but I'm a bit hung up on the method. What is $\displaystyle \Delta{a_0}$? is it $\displaystyle \frac{a_1-a_0}{a_0}$?
just $a_1-a_0$

5. ## Re: Contribution of Each Factor to the Total Change

Originally Posted by romsek
just $a_1-a_0$
I must be missing something.

You're saying:

$\displaystyle 1.154608-1.0087= 1.0087*(\frac{1.22-1.1}{1.1}+\frac{0.91-0.7}{0.7}+\frac{1.31-1.04}{1.31})$

But:

$\displaystyle 0.145908 \neq 0.20475$

6. ## Re: Contribution of Each Factor to the Total Change

try this

$\displaystyle x y z=a b c+(x-a)b c+(y-b)a c+(z-c)a b+ c(x-a)(y-b)+b(x-a)(z-c)+a(y-b)(z-c)+(x-a)(y-b)(z-c)$

7. ## Re: Contribution of Each Factor to the Total Change

Originally Posted by Idea
$\displaystyle x y z=a b c+(x-a)b c+(y-b)a c+(z-c)a b+ c(x-a)(y-b)+b(x-a)(z-c)+a(y-b)(z-c)+(x-a)(y-b)(z-c)$
Thanks for the simpler variables!

Wouldn't it be simpler to forget about z and c;
calculate x,a and y,b effect, then z,c defaults to total effect - x,a - y,b ?