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Math Help - How to isolate impact of factors in an equation

  1. #1
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    How to isolate impact of factors in an equation

    Hello,

    I am trying to isolate the effect of two different drivers/factors over time. For example, consider the following:

    Factor 1 x Factor 2 = Value

    Year 1: Factor 1 = 10, Factor 2 = 40, Value = 400
    Year 2: Factor 1 = 12, Factor 2 = 41, Value = 492

    The Year over year change of value is 23%

    I am trying to figure out what portion of that change is accounted for by each factor. The approach I have been trying to take is to hold one factor at a time constant and see what the resulting value would be. For example:

    Holding factor 1 constant gives value = 10 * 41 = 410
    - 410/400 - 1 = 2.5% Year over year change

    Holding factor 2 constant gives value = 12 * 40 = 480
    - 480/400 - 1 = 20% Year over year change

    I take this to mean that factor 1 accounted for 2.5% of the growth and factor 2 accounted for 20%. However, adding these two together gives a year over year change of 22.5% when the actual change was 23.0%.

    Could someone please let me know why this analysis does not seem to be very exact and if there is a better way to approach it?

    Thank You,

    Eric
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  2. #2
    Super Member malaygoel's Avatar
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    Quote Originally Posted by jesterea View Post
    Hello,

    I am trying to isolate the effect of two different drivers/factors over time. For example, consider the following:

    Factor 1 x Factor 2 = Value

    Year 1: Factor 1 = 10, Factor 2 = 40, Value = 400
    Year 2: Factor 1 = 12, Factor 2 = 41, Value = 492

    The Year over year change of value is 23%

    I am trying to figure out what portion of that change is accounted for by each factor. The approach I have been trying to take is to hold one factor at a time constant and see what the resulting value would be. For example:

    Holding factor 1 constant gives value = 10 * 41 = 410
    - 410/400 - 1 = 2.5% Year over year change

    Holding factor 2 constant gives value = 12 * 40 = 480
    - 480/400 - 1 = 20% Year over year change

    I take this to mean that factor 1 accounted for 2.5% of the growth and factor 2 accounted for 20%. However, adding these two together gives a year over year change of 22.5% when the actual change was 23.0%.

    Could someone please let me know why this analysis does not seem to be very exact and if there is a better way to approach it?

    Thank You,

    Eric
    total change(in %) =change due to 1st(in%)
    +change due to 2nd(in%)
    +[change due to 1st(in%)*change due to 2nd(in%) ]

    23% = 20% +2.5% + 20%*2.5%
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  3. #3
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    Thank you Malay, that is very helpful explains why the numbers weren't adding up exactly. Is there any way to completly capture the change due to a particular factor, or will you always have some "left over" change due to the interplay between the factors?
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  4. #4
    Super Member malaygoel's Avatar
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    Quote Originally Posted by jesterea View Post
    Thank you Malay, that is very helpful explains why the numbers weren't adding up exactly. Is there any way to completly capture the change due to a particular factor, or will you always have some "left over" change due to the interplay between the factors?
    yes,there will always some "left over" change due to the interplay between the factors. however, it can be seen that which factor plays how much part

    20/23=87% ...change due to fist is 87% of total change
    2.5/23=11%...change due to 2nd is 11% of total change
    .5/23=2%.....change due to interplay is 2% of total change

    hence, you can see that change due to interplay is considerably less than others.
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  5. #5
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    Isolating impact of multiple drivers in equation

    Thanks again Malay. It appears this method does not scale beyond more than two factors. For example, if we change the function to look like the following, I am no longer able to isolate the effect of the different factors.

    Factor1 x Factor2 x Factor3= Value

    Year 1: Factor1 = 10, Factor2 = 40, Factor3 = 20, Value = 8000
    Year 2: Factor1 = 12, Factor2 = 41, Factor3 = 17, Value = 8364

    Year over Year change = 4.6%

    Factor 1 isolation:
    Holding Factor2 & 3 constant yields 12 * 40* 20 = 9600
    - 9600/8000 - 1 = 20% change attributed to Factor1

    Factor 2 isolation:
    Holding Factor1 & 3 constant yields 10 * 41* 20 = 8200
    - 8200/8000 - 1 = 2.5% change attributed to Factor2

    Factor 3 isolation:
    Holding Factor1 & 2 constant yields 10 * 40 * 17 = 6800
    - 6800/8000 - 1 = -15% change attributed to Factor3

    If I sum the impact of these changes up I get 7.5%. If I then try to add the "interplay" of the three factors (20% * 2.5% * -15%), it comes to 7.4%, which is significantly different than the actual total year over year change of 4.6%.

    Is this a flawed approach when there are more than two variables (e.g., is it applying a linear approach to a non-linear problem, or something similar to that). If so, is there a better approach to isolate the impact of changes in factors, when three factors are involved?

    Thank you again for the help,

    Eric
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  6. #6
    Super Member malaygoel's Avatar
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    Quote Originally Posted by jesterea View Post
    Thanks again Malay. It appears this method does not scale beyond more than two factors. For example, if we change the function to look like the following, I am no longer able to isolate the effect of the different factors.

    Factor1 x Factor2 x Factor3= Value

    Year 1: Factor1 = 10, Factor2 = 40, Factor3 = 20, Value = 8000
    Year 2: Factor1 = 12, Factor2 = 41, Factor3 = 17, Value = 8364

    Year over Year change = 4.6%

    Factor 1 isolation:
    Holding Factor2 & 3 constant yields 12 * 40* 20 = 9600
    - 9600/8000 - 1 = 20% change attributed to Factor1

    Factor 2 isolation:
    Holding Factor1 & 3 constant yields 10 * 41* 20 = 8200
    - 8200/8000 - 1 = 2.5% change attributed to Factor2

    Factor 3 isolation:
    Holding Factor1 & 2 constant yields 10 * 40 * 17 = 6800
    - 6800/8000 - 1 = -15% change attributed to Factor3

    If I sum the impact of these changes up I get 7.5%. If I then try to add the "interplay" of the three factors (20% * 2.5% * -15%), it comes to 7.4%, which is significantly different than the actual total year over year change of 4.6%.

    Is this a flawed approach when there are more than two variables (e.g., is it applying a linear approach to a non-linear problem, or something similar to that). If so, is there a better approach to isolate the impact of changes in factors, when three factors are involved?

    Thank you again for the help,

    Eric
    Isolated changes:20%,2.5%,-15%

    Interplay: (1)20%*2.5%=.5%
    (2)20%*(-15%)=-3%
    (3)(-15%)*2.5%=-.375%
    (4)20% * 2.5% *( -15%)=-.075%

    Adding all of the above: 4.55%

    How do I get this?
    Let value=V,
    factor1=x,
    factor2=y,
    factor3=z

    We have,
    V=xyz
    V+v_1=(x+x_1)(y+y_1)(z+z_1)

    V+v_1=xyz+x_1yz+y_1xz+z_1xy+x_1y_1z+x_1z_1y+z_1y_1  x+x_1y_1z_1

    v_1=x_1yz+y_1xz+z_1xy+x_1y_1z+x_1z_1y+z_1y_1x+x_1y  _1z_1

    Now, total error is \frac{v_1}{V}

    hence,
    \frac{v_1}{V}=\frac{x_1yz+y_1xz+z_1xy+x_1y_1z+x_1z  _1y+z_1y_1x+x_1y_1z_1}{V}

    \frac{v_1}{V} =\frac{x_1yz+y_1xz+z_1xy+x_1y_1z+x_1z_1y+z_1y_1x+x  _1y_1z_1}{xyz}

    \frac{v_1}{V} =\frac{x_1}{x}+\frac{y_1}{y} +\frac{z_1}{z} +\frac{x_1}{x}\cdot\frac{z_1}{z}+\frac{x_1}{x}\cdo  t\frac{y_1}{y}+\frac{y_1}{y}\cdot\frac{z_1}{z}+\fr  ac{x_1}{x}\cdot\frac{y_1}{y}\cdot\frac{z_1}{z}
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  7. #7
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    Thanks again Malay. It seems the logic of trying to use this approach for analayzing the relative impact of the factors may breakdown somewhat for when there are three factors since the interplay effect is getting so large, but it is very helpful to at least be able to explain the interplay effect and bridge it back to the total change as you have illustrated.

    Thanks,

    Eric
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