factor Interactions in a DOE

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    Nitin Saini

    I thought I knew all about DOE until this occured to me :
    a) Suppose  I am monitoring one output variable Y and I have two input variables X1 and X2 (say temp and pressure)
    b) I setup an expt with both X1 and X2 at two levels each from my historical knowledgs (X1 200 and 300 C and X2 : 4 and 6 bar).
    c) I assign codes to the levels -1 (200 C and 4 bar) and +1(300C and 6 bar).
    d) When I want to study interaction between factors I simply multiply the coded units for X1 and X2 and say that I have interaction also  at two levels (-1 and +1)
    e) What is bothering me here that infact I have four levels of interactions viz (200, 4); (200, 6); (300, 4) and (300, 6) and not two as explained in step d
    In fact by simply multiplying coded units (-1 and +1) of X1 and X2, I am treating factor interaction to be the same in following two entirely diffeent situations:
    1. when both factors are at minimium ( 200 & 4 or -1 & -1) for -1 x -1 = 1
    2. when both factors are at maximum level (300 & 6 or +1 & +1) for +1 x +1 = 1
    Hope I am making my query clear.



    do a simple ANOVA and your confusion will go away.
    If still in doubt, write indetail to [email protected]



    Hi nitin,
    I think you understood the definition of interaction wrongly .Interaction comes between factors not between levels. By considereing all 2*2 combinations we can find out is there any interaction between two factors not between any levels.
    if needs more clarification [email protected]


    Robert Butler

    It appears that you are using the term interaction in two different ways.  What makes it interesting is that both, by themselves, are correct.  Let’s try the following:
    Two factors  X1 and X2
    X1 low = 200C, X1 high = 300C
    X2 low = 4, X2 high = 6
    Experimental combinations for two levels, no reps or center points would be
    experiment    X1    X2   X1X2
           (1)           -1     -1          1
            a              1    -1         -1
            b             -1     1        -1
          ab              1     1          1
    The COLUMN corresponding to the interaction of X1 and X2 is derived by multiplying together the columns for X1 and X2.  If you look at the result for
    X1X2 for each experiment you see, exactly as you described, an “interaction” for each combination.  When it comes to running a regression and getting a model of the form :
    Y = a0 +a1*X1 +a2*X2 +a3*X1*X2
    you will have the second situation you described, namely that when you plug in the low, low and the high high combinations you will get, for the interaction term, the same value of 1.  It is also true that when you plug in low, high and high, low you will also get the same value which, for these combinations, will be -1.
      Thus, as you observed, the interaction term in the regression equation will treat the above listed combinations in the same manner.  The differences in these combinations, from the standpoint of the regression and the response, will make themselves apparent in the linear terms for X1 and X2.  If you should have a regression situation where the only thing that is significant is the interaction, you graph of the response vs X1 and X2 will be a large X. 
      If this last situation arises, your analysis is telling you that your process can have the same output for two different combinations of your X’s.  This could be a good or a bad thing.  For example, if you have been running X1 high and X2 low and it would be much cheaper to run X1 low and X2 high your analyis would tell you that, at least for that one response, you could save money just by reversing the levels of X1 and X2.



    The number of levels of the interaction is not the number of possible combinations of the factors involved.
    Factor                                       Levels                                                
    A                  1 (high)                                    -1 (low)
    B                 1 (high)                                    -1 (low)
    AB   1 (both the same, high or low)    -1 (both different, one high and one low)


    Nitin Saini

    Gabriel – that exacltly is causing confusion. As a regression is carried out between response variable and (X1, X2 and X1X2) to derive the equation, why I should not be considering 4 combinations of X1X2 or more imprtantly why cases involving both X at minimum and botht X at maximum are considered identical.



    Again: X1X2 is not a “combination”. It is an interaction. The interaction effect measure the difference between what happens if the factors change individually vs if the factors change together. Imagine the folowing:
    A           B            Y
    -1         -1            100
    1           -1           130
    -1          1            120
    1            1            150
    In this case the effect of the interaction effect is zero. Yet you are still taking into account all the 4 combinations in the mathematic model equation, even when the interaction does not even appear: Try:
    Another example:
    A           B            Y
    -1         -1            105
    1           -1           125
    -1          1            115
    1            1            155
    In this case the interaction is not 0 and the equation is almost the same, but with an added term for the interaction:
    Y=125+15*A+10*B+5*AB. (AB=1 for the first and last row and -1 for both middle rows).
    Do you feel there is acombination not represented by the equation?
    The four combinations are not supposed to be represented by the interaction AB alone, but by A, B and AB together. For example, for A+ B+ and A- B- the interaction has the same value, but A and B do change.
    I hope what I’m saying is correct (and understandable). I am not anexpert in DOE (not even close).
    I would like your feedbak.

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