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ANOVA in DOE

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  • #33223

    ROSS
    Member

            Hello, everyone!
            Can someone tell me the theory and formular of the ANOVA analysis in DOE?  For example, there is no p-value in anova if we do not set any replicate or center point. pla see belowing:
    Full Factorial DesignFactors:        3   Base Design:           3, 8    Runs:           8   Replicates:               1    Blocks:      none   Center pts (total):       0All terms are free from aliasingFractional Factorial Fit: y versus A, B, CEstimated Effects and Coefficients for y (coded units)Term         Effect      CoefConstant              10.6173 A           -0.6864   -0.3432B            0.1356    0.0678C            0.6737    0.3368A*B         -1.5324   -0.7662A*C         -0.5254   -0.2627B*C         -0.0687   -0.0344A*B*C       -0.6437   -0.3218Analysis of Variance for y (coded units)Source                DF      Seq SS     Adj SS     Adj MS      F      PMain Effects           3      1.8866     1.8866     0.6289      *      *2-Way Interactions     3      5.2582     5.2582     1.7527      *      *3-Way Interactions     1      0.8286     0.8286     0.8286      *      *Residual Error         0      0.0000     0.0000     0.0000Total                  7      7.9734
         But, if we run a replicate, there is p-value in anova:
    Factorial DesignFull Factorial DesignFactors:        3   Base Design:           3, 8    Runs:          16   Replicates:               2    Blocks:      none   Center pts (total):       0All terms are free from aliasingFractional Factorial Fit: y versus A, B, CEstimated Effects and Coefficients for y (coded units)Term         Effect      Coef     SE Coef       T      PConstant              10.0526      0.1544   65.11  0.000A           -0.2719   -0.1360      0.1544   -0.88  0.404B           -0.1666   -0.0833      0.1544   -0.54  0.604C            0.0432    0.0216      0.1544    0.14  0.892A*B         -0.8646   -0.4323      0.1544   -2.80  0.023A*C          0.5737    0.2869      0.1544    1.86  0.100B*C         -0.8855   -0.4427      0.1544   -2.87  0.021A*B*C        0.9524    0.4762      0.1544    3.08  0.015Analysis of Variance for y (coded units)Source                DF      Seq SS     Adj SS     Adj MS      F      PMain Effects           3      0.4143    0.41427     0.1381   0.36  0.7822-Way Interactions     3      7.4434    7.44335     2.4811   6.50  0.0153-Way Interactions     1      3.6283    3.62834     3.6283   9.51  0.015Residual Error         8      3.0515    3.05152     0.3814  Pure Error           8      3.0515    3.05152     0.3814Total                 15     14.5375
        Or, we run some center points , we will get p-value:
    Fractional Factorial Fit: y versus A, B, CEstimated Effects and Coefficients for y (coded units)Term         Effect      Coef     SE Coef       T      PConstant               10.242      0.1459   70.19  0.009A             0.428     0.214      0.1459    1.47  0.381B            -0.448    -0.224      0.1459   -1.53  0.368C             0.713     0.357      0.1459    2.44  0.247A*B          -0.717    -0.358      0.1459   -2.46  0.246A*C           0.316     0.158      0.1459    1.08  0.475B*C          -0.292    -0.146      0.1459   -1.00  0.499A*B*C        -0.039    -0.019      0.1459   -0.13  0.916Ct Pt                  -1.246      0.3263   -3.82  0.163Analysis of Variance for y (coded units)Source                DF      Seq SS     Adj SS     Adj MS      F      PMain Effects           3     1.78552    1.78552    0.59517   3.49  0.3702-Way Interactions     3     1.39844    1.39844    0.46615   2.74  0.4123-Way Interactions     1     0.00301    0.00301    0.00301   0.02  0.916Curvature              1     2.48465    2.48465    2.48465  14.59  0.163Residual Error         1     0.17035    0.17035    0.17035  Pure Error           1     0.17035    0.17035    0.17035Total                  9     5.84197
     

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    #89569

    ROSS
    Member

          Can someone help me? Thank you very much!

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    #89570

    Breizh
    Participant

    In the first analysis (without a replicate), it looks like you do not have enough degrees of freedom to calculate the p value. If you do not run a replicate, try re-running the analysis after removing one or two of the main effects or interactions with the lowest effect/coefficients and see what happens.

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    #89573

    ROSS
    Member

            Breizh, thank you!
           I know it is lack of degreee, and if we remove some insignificant factors it will be some replicates.
           But, can you tell me the formular fo anova in doe? I want to know the hteory.

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    #89575

    Dr. Scott
    Participant

    Tony,
    In a nutshell, ANOVA divides the Sum of Squares Between by the Sum of Squares Within to determine if any factor accounts for a siginificant amount of the variation (using the F statistic). When you have saturated the ANOVA model the way you did in the first example (i.e., used all your degrees of freedom to see effects, but leaving none to estimate noise), you are unable to estimate the SSwithin (noise). That is, there is no way to estimate within variation or “noise” because all variation is considered to be “effect”. Therefore, there is no way to determine whether the variation you accounted for with your factors (effects) is big or little (significant or insignificant).
    Your training manual or a text on Six Sigma or ANOVA will show you the formulas used at a glance.

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    #89579

    Jose Luis Rivera
    Participant

    If you look all your analysis there something very interesting.  First without replicates (as already you know) you will not have an estimate for the p-value but you can obtain a pareto graph (if you are using Minitab) and this will give you an idea of the more influence factors on your model.  Second I look thru all your posted analysis and it tells you that your interactions have a lower p-value that your main factor which mean that there are more significant.  Then when you add central points no factors become significant but your central point present the highest absolute value for the coefficients of your equation on coded units.  I think what all this tell you about it is that you have to consider a surface response (Central composite or Box-Behnken experiment) experiment because your regresion analisis present quadratic elements.  The factorial experiment that you already development will not tell you too much of your experiment if quadratic element are driven your process.  I hope this help you.
    Jose Luis Rivera
     
     
     

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    #89588

    faceman
    Participant

    Tony,
    Here is a web page that will give you some background on ANOVA.  Keep taking the ‘next’ link until you have had enough.  It will take through the process.
    http://www.itl.nist.gov/div898/handbook/prc/section4/prc43.htm

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    #89596

    Hemanth
    Participant

    Hi Tony,
    The reason why you didnt get the p valu is bcos you did not have replicate in your first design. Replicate is essential to estimate the true error or noise as pointed before in the discussion. This error is required to calculate the f value and then the p value. Since u didnt had the error value the software did not give you the p value.
    More can be found on the link provided. To overcome this issue I suggest you take replicates (always recommended) and in case of extreme scenario where you cannot take the replicates, include your 3 level interaction into the residual error and then get the f value and p value (recommended only in case you have large number of factors, and you are doing fractional factorial, not recommended)
    hope this was helpful
    Hemanth

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