5. Process Improvement
5.6. Case Studies
5.6.1. Eddy Current Probe Sensitivity Case Study

5.6.1.5.

Estimate Main and Interaction Effects

Effects Estimation

Although the effect estimates were given on the dex interaction plot on a previous page, they can also be estimated quantitatively.

The full model for the 2³ factorial design is

Data from factorial designs with two levels can be analyzed using the Yates technique, which is described in Box, Hunter, and Hunter. The Yates technique utilizes the special structure of these designs to simplify the computation and presentation of the fit.

Dataplot Output

Dataplot generated the following output for the Yates analysis.

  
(NOTE--DATA MUST BE IN STANDARD ORDER)
NUMBER OF OBSERVATIONS           =        8
NUMBER OF FACTORS                =        3
NO REPLICATION CASE
  
PSEUDO-REPLICATION STAND. DEV.   =    0.20152531564E+00
PSEUDO-DEGREES OF FREEDOM        =        1
(THE PSEUDO-REP. STAND. DEV. ASSUMES ALL
3, 4, 5, ...-TERM INTERACTIONS ARE NOT REAL,
BUT MANIFESTATIONS OF RANDOM ERROR)
  
STANDARD DEVIATION OF A COEF.    =    0.14249992371E+00
(BASED ON PSEUDO-REP. ST. DEV.)
  
GRAND MEAN                       =    0.26587500572E+01
GRAND STANDARD DEVIATION         =    0.17410624027E+01
  
99% CONFIDENCE LIMITS (+-)       =    0.90710897446E+01
95% CONFIDENCE LIMITS (+-)       =    0.18106349707E+01
99.5% POINT OF T DISTRIBUTION    =    0.63656803131E+02
97.5% POINT OF T DISTRIBUTION    =    0.12706216812E+02
  
IDENTIFIER    EFFECT        T VALUE      RESSD:     RESSD:
                                         MEAN +     MEAN +
                                         TERM    CUM TERMS
----------------------------------------------------------
   MEAN       2.65875                   1.74106    1.74106
      1       3.10250         21.8*     0.57272    0.57272
      2      -0.86750         -6.1      1.81264    0.30429
     23       0.29750          2.1      1.87270    0.26737
     13       0.24750          1.7      1.87513    0.23341
      3       0.21250          1.5      1.87656    0.19121
    123       0.14250          1.0      1.87876    0.18031
     12       0.12750          0.9      1.87912    0.00000

Description of Yates Output

In fitting 2-level factorial designs, Dataplot takes advantage of the special structure of these designs in computing the fit and printing the results. Specifically, the main effects and interaction effects are printed in sorted order from most significant to least significant. It also prints the t-value for the term and the residual standard deviation obtained by fitting the model with that term and the mean (the column labeled RESSD MEAN + TERM), and for the model with that term, the mean, and all other terms that are more statistically significant (the column labeled RESSD MEAN + CUM TERMS).

Of the five columns of output, the most important are the first (which is the identifier), the second (the least squares estimated effect = the difference of means), and the last (the residuals standard deviation for the cumulative model, which will be discussed in more detail in the next section).

Conclusions

In summary, the Yates analysis provides us with the following ranked list of important factors.

X1 (Number of Turns):	effect estimate = 3.1025 ohms
X2 (Winding Distance):	effect estimate = -0.8675 ohms
X2*X3 (Winding Distance with Wire Guage):	effect estimate = 0.2975 ohms
X1*X3 (Number of Turns with Wire Guage):	effect estimate = 0.2475 ohms
X3 (Wire Guage):	effect estimate = 0.2125 ohms
X1*X2*X3 (Number of Turns with Winding Distance with Wire Guage):	effect estimate = 0.1425 ohms
X1*X2 (Number of Turns with Winding Distance):	effect estimate = 0.1275 ohms