Effects of variables' change on total change in a product function

Question

Prelude

My topic is related to existing posts found here and here; although, my questions are different.

My post explains my objective and what has and has not worked for me.

I have met my objective and provided my solution.

My questions pertain to why the solution works in the hopes that I may apply my one-off solution here more readily to other instances.

Background

I am comparing two numbers that are products of multiple variables. My objective is to determine how much of the total change is attributable to each variable's change.

As a simple example, consider:

x1 = a1 * b1 = 10 * 10 = 100 
x2 = a2 * b2 = 11 * 10 = 110

How much of the total difference (110 - 100 = 10) is attributable to the change in a and b?

My first approach was to use a binary (on/off) approach to determine what happens if each variables' change were the only change to occur. $$\Delta = \Delta a * b_1 + \Delta b * a_1$$ $$\Delta = (11 - 10) * 10 + (10 - 10) * 10$$ $$10 = 10 + 0$$

As shown, the change in "a" is responsible for 10 (100%) of the total change.

However, as additional variables are added and/or change, this appears to stop working.

Consider a more complicated example.

Year	A	B	C	D	Total (A * B * C * D)
Y1	32	200	25	365	58,400,000
Y2	33	215	28	360	71,517,600
$\Delta$	1	15	3	-5	13,117,600

If I use the previously described approach ($\Delta = \Delta A * B_1 * C_1 * D_1 + ...$), the sum of the individual effects does not equal the total change.

A	B	C	D	Total Change (A+B+C+D)
1,825,000	4,380,000	7,008,000	-800,000	12,413,000

As shown, if A were the only variable to change, it would increase the value by 1.8 million, and so on. The sum total of isolated challenges in this method is approximately 12.4 million, which is less than the observed total of 13.1 million.

My next attempt tried the reverse. What would happen if each individual variable were the only variable not to change ($\Delta = \Delta A * B_2 * C_2 * D_2 + ...$)?

A	B	C	D	Total Change (A+B+C+D)
2,167,200	4,989,600	7,662,600	-993,300	13,826,100

The sum total for this approach (approximately 13.8 million) is too great.

The true answer exists somewhere between these two methods. And through trial and error, I found a solution.

Solution

Let T be the total product for a given year, and $\Delta$ T be the total difference between T_{year 1} and T_{year 2}.

Objective:

Total change is equal to the change caused by each variable.

$\Delta T = \Delta_A(T) + \Delta_B(T) + \Delta_C(T) + \Delta_D(T)$

Solution:

$\Delta_A(T) = (A_2 - A_1) * B_1 * C_2 * D_1$

$\Delta_B(T) = (B_2 - B_1) * A_2 * C_2 * D_1$

$\Delta_C(T) = (C_2 - C_1) * A_1 * B_1 * D_1$

$\Delta_D(T) = (D_2 - D_1) * A_2 * B_2 * C_2$

Check:

A	B	C	D	Total Change (A+B+C+D)
2,044,000	5,058,900	7,008,000	-993,300	13,117,600

As shown, the sum total of effects for the variables is equal to $\Delta$ T. Objective [presumably] met.

The Questions

Notice in the solution that function differs for each variable.

The effect of $\Delta$ A is $\Delta$ A * the year 1 value for B and D, but year 2 value for C.

The effect of $\Delta$ B is $\Delta$ B * the year 1 value for D, but year 2 value for A and C.

The effect of $\Delta$ C is $\Delta$ C * the year 1 value for A, B and D.

The effect of $\Delta$ D is $\Delta$ D * the year 2 value for A, B and C.

The functions are specific to the variable. Year 1 variables are applied to $\Delta$ C, and year 2 variables are applied to $\Delta$ D, these are not interchangeable functions. If year 2 variables are applied to $\Delta$ C, and year 1 variables are applied to $\Delta$ D, the sum total no longer equals the observed change ($\Delta$ T).

Q1: Why does this work? Why do we reference different years for different variables?

Q2: Is there a way (other than trial and error) to determine the appropriate function (i.e., what would the function be for an added variable "E"?)

Q3: Does this solution actually isolate variables if they do not use a consistent approach? Is this really an accurate depiction of variables' effect on total change or just a coincidental / convenient distribution?

Thanks,

Andrew

AWaddington · Accepted Answer · 2021-05-18T16:06:47.910

[It's amazing how one can work on something for days and not find an answer, but the process of asking someone else brings an answer to light]

Q1: Why does the solution above work?

First: why the binary (on/off) approach does not work. When taking a product of the variables, the variables are not individually responsible for the outcome. The outcome only exists in aggregate. A variable's change influences the aggregate change, but the magnitude of its influence is affected by changes in other variables.

Therefore, one variable's influence is only true in the context of a fixed or assumed state for the other variables. To apply the same function to each variable's $\Delta$ ignores the changes in other variables and results in over or underestimation.

Q2: Is there a way to determine the appropriate function?

Yes.

My "aha" moment was a consideration of [Type I?] ANOVA analysis where variables are loaded in sequential order (greatest to least effect) and the impact is incremental change, not individual.

Consider the example again:

Year	A	B	C	D	Total (A * B * C * D)
$Y_1$	32	200	25	365	58,400,000
$Y_2$	33	215	28	360	71,517,600
$\Delta (Y_2 - Y_1)$	1	15	3	-5	13,117,600

To reflect the incremental nature of the change, each variable's function is basically answering the question, "Moving from left to right across the table, what is the incremental change related to this variable, given that the variables to the left have already changed (and the variables to the right have not yet changed)?"

$\Delta A(T) = \Delta A * B_{Y1} * C_{Y1} * D_{Y1}$

$\Delta B(T) = A_{Y2} * \Delta B * C_{Y1} * D_{Y1}$

$\Delta C(T) = A_{Y2} * B_{Y2} * \Delta C * D_{Y1}$

$\Delta D(T) = A_{Y2} * B_{Y2} * C_{Y2} * \Delta D$

Year	A	B	C	D	Total (A * B * C * D)
$Y_1$	32	200	25	365	58,400,000
$Y_2$	33	215	28	360	71,517,600
$\Delta (Y_2 - Y_1)$	1	15	3	-5	13,117,600
Incremental Effect	2,044,000	5,058,900	7,008,000	-993,300	13,117,600

In my "trial and error" approach, the variable order did not match the incremental order, making the functions appear somewhat random. Aligning the variable order and incremental order, the method is clear.

Because this is incremental, the order of the variables matters. Any order should yield the same total change as the observed; however, the magnitude of the individual variables' contribution to the change will vary based on their position in the sequence. So how does one determine the order of the sequence?

One approach is to order the sequence based on variables' individual weighted potential effect (i.e., the effect of $\Delta$ variable units assuming no other variables change).

Q3: Does this solution actually isolate variables if they do not use a consistent approach?

As a model, I think so, yes. Independently, maybe.

However, when calculating the percentage share of effect, the share is the share of total "movement," not the share of total change. The total value is the sum of variable effects that may be negative or positive. An individual effect may be greater than 100% of the total change.

So, to measure the percentage share of effect for each variable, use the absolute effect value for the variable, divided by the sum of absolute effects for all variables.

The Final Example (putting it all together):

Reordered Variables Based on their Weighted Potential Effect Rank

Value	A	B	C	D
Weighted Potential	1825000	4380000	7008000	-800000
Absolute Weighted Potential	1825000	4380000	7008000	800000
Weighted Potential Rank	3	2	1	4

Calculate Incremental Share of Total Movement from $Y_1$ to $Y_2$

Year	C	B	A	D	Total
$Y_1$	25	200	32	365	58,400,000
$Y_2$	28	215	33	360	71,517,600
$\Delta (Y_2 - Y_1)$	3	15	1	-5	13,117,600
Incremental Effect	7,008,000	4,905,600	2,197,300	-993,300	13,117,600
Absolute Incremental Effect	7,008,000	4,905,600	2,197,300	993,300	15,104,200
Share of Total Movement	46%	32%	15%	7%	100%

Interpreting this, the change observed in Variable C has the greatest effect on the Total product and accounts for nearly half (46%) of the incremental movement (positive or negative) between the $Y_1$ total and $Y_2$ total.