Edits for Case Study 1: Eddy Current


-----Section 0: Problem Description-----

Background & Project Goal:
The goal of this project is to develop a
nondestructive portable device for detecting
cracks and fractures in metals.  A primary
application for such would be for the detection of
defects in airplane wings.  The internal mechanism
for the detector would be to sense crack-induced
changes in the detector's electromagnetic field
which would in turn result in changes in the
impedance level of the detector.  This change of
impedance is termed "sensitivity" and it is a
sub-goal of this experiment to maximize such
sensitivity as the detector is moved from an
unflawed region to a flawed region on the metal.

Statistical Goals:
The specific statistical goals of the experiment
are the usual:

   1. Factors : determine important factors which
                affect sensitivity;
   2. Settings: determine settings which
                maximize sensitivity;
   3. Model   : determine a prediction equation which
                functionally relates sensitivity
                to various factors.

There were 3 detector wiring component factors
under consideration:

   X1 = Number of wire turns
   X2 = Wire winding distance
   X3 = Wire guage

Since the maximum number of runs that could be
afforded timewise and costwise in this experiment
was n = 10, a 2^3 full factoral experiment
(involving n = 8 runs) could be afforded and so
was chosen to be run.  With an eye to the usual
monotonicity assumption in 2-level factorial
designs, the selected settings for the 3 factors
were as follows:

   X1 = Number of wire turns :
   X2 = Wire winding distance:
   X3 = Wire guage           :

The experiment was run (with the 8 settings
executed in random order). The following data
resulted:

<data>
   <add additional column: run sequence>
   <2 8 3 6 7 1 4 5>

5.6.1.2. Main Effects


-----Section 1: DEX Scatter Plot-----

<label the plot as a DEX Scatter Plot>
<impedance => sensitivity     everywhere>

...

<dex scatter plot>

Conclusions:

1. Important Factors:
   Factor 1 is clearly important ; when X1 = -1,
   all 4 sensitivities are low; when X1 = +1, all
   4 impedances are high.  Factor 2 is less
   important; the 4 sensitivities for X2 = -1 are
   slightly higher than the 4 sensitivities for X2
   = +1.  Factor 3 does not appear to be important
   at all--the sensitivity is about the same (on
   the average) regardless of the settings for X3.

2. Best Settings:
   The best settings depend on the definition of
   "best" for the response.  In this experiment,
   we are using the device as a detector, and so
   high sensitivities are desirable.  Given this,
   our first pass at best settings yields (X1 =
   +1, X2 = -1, X3 = either).

-----Section 2: DEX Mean Plot-----

<label the plot as a DEX Mean Plot>

...

<main effects plot>

Conclusions:

The main effects plot reaffirms the ordering of
the DEX scatter plot, but additional information is
gleaned because the eyeball distance between the
mean values gives an approximation to the least
squares estimate for the factor effects:

1. Important Factors:
   X1 (effect = large   : about 3 ohms)
   X2 (effect = moderate: about -1 ohm)
   X3 (effect = small   : about 1/4 ohm)

2. Best Settings:
   As before, choose those factor settings which
   (on the average) maximize the sensitivity:

      (X1,X2,X3) = (+,-,+)

-----Section 3: Interaction Effects Plot-----

In addition to the main effects, it is also
important to check for interaction
effects--especially 2-term interaction effects.
The DEX Mean Interaction Effects Plot is an
effective tool for this.  The 3 diagonal plots are
a replotting of the main effects-- they are plots
of the average response versus each of the 3
factors.  The off-diagonal plots have the average
response (vertically) versus the interaction (=
cross-product here) horizontally.  Thus for the
middle plot on the top, the horizontal axis is
X1*X2, where X1 takes on values -1 and +1, X2
takes on -1 and +1, and thus X1*x2 takes on values
-1 and +1.

The legend within each plot is the least squares
estimate of the factor (or interaction) effect; in
this orthogonal design case, the estimated effect
is identical to the difference of the mean values.

<matrix of plots>

Conclusions:

1. Important Factors:
   Noting the plots which have steepest lines (that is,
   largest effects), we note that:
      X1 is most important     : estimated effect = -3.1025;
      X2 is next most important: estimated effect = -.8675;
      X3 is relatively unimportant;
      All 3 2-term interactions are relatively unimporant.

2. Best Settings:
   As with the main effects plot, the best settings
   (to maximize the sensitivity) are

      (X1,X2,X3) = (+1,-1,+1)

    but with the X3 setting of +1 mattering little.

-----Section 4: Block Plots-----

Block plots are a useful adjunct to confirm the importance of
factors, to establish the robustness of main effect conclusions,
and to determine the existence of interactions.

The first plot below answers the question: Is factor 1 important
(robust over all 4 settings of X2 and X3)?
The second plot answers the question: Is factor 2 important
(robust over all 4 settings of X1 and X3)?
The third plot answers the question: Is factor 3 important
(robust over all 4 settings of X1 and X2)?

For block plots, it is the height of the bars which is important--
not the relative positioning of each bar; hence we focus on the
size and internals of the blocks--not "where" the blocks are one
relative to another.

<block plots>

Conclusions:

1. Relative Importance of Factors:
   All of the bar heights in plot 1 are bigger
   than the bar heights in plots 2 and 3--hence
   factor 1 is more important than factors 2
   and 3.

2. Statistical Significance
   In plot 1, looking at the levels within each
   bar, we note that the response for level 2
   is higher than level 1 in each and all of
   the 4 bars.  By chance this happens with
   probability 1/(2^4) = 1/16 = 6.25%.  Hence
   factor 1 is near-statistically significant
   at the 5% level.  Similarly, for plot 2,
   level 1 > level 2 for all 4 bars, hence
   factor 2 is near-statistically significant.
   For factor 3, there is not consistent
   ordering within all 4 bars and hence factor
   3 is not statistically significant.

3. Interactions:
   For factor 1, the 4 bars do not change
   height in any systematic way and hence there
   is no evidence of X1 interacting with either
   X2 or X3.  Similarly for factor 2.

-----Section 5: Estimating Effects-----

<text as you have it>

<yates output>

<omit "Interpretation" and have your text as is>
<I think "Interpetation" and "Conclusions" are close enough
to be confusing>

<in addition to the good stuff you  have explaining the Dataplot
output, I would suggest adding the following line...>

Of the 5 columns of output, the most important are the first
(which is the identifier), the second (which is the least squares
estimated effect = difference of means), and the last (which
is the residuals standard deviation for the cumualtive model--more on
this in the next section).

Conclusions:

1. Important Factors:
   X1 (Number of Turns)           : effect est. =  3.1025 ohms
   X2 (Winding Distance)          : effect est. = -0.8675 ohms
   X2*X3 (Number x Guage)         : effect est. =   .2975 ohms
   X1*X3 (Number x Guage)         : effect est. =   .2475 ohms
   X3 (Wire Guage)                : effect est. =   .2125 ohms
   X1*X2*X3 (Num. x Dist. x Guage): effect est. =   .1425 ohms
   X1*X2 (Number x Distance)      : effect est. =   .1275 ohms

-----Modeling & Prediction Equations-----

The last column of the above table gives the 
residual standard deviation for 8 possible 
models--each one progressively more complicated.  
If none of the factors are important, then the 
prediction equation defaults to 

      Yhat = the average = 2.65875

From the last column of the table, it is seen that
this simplest of all models has a residual 
standard deviation (a measure of goodness of fit) 
of 1.74106 ohms.  Finding a good-fitting model was
NOT one of the stated goals of this experiment, 
but its determination is "free" along with the 
rest of the analysis and so it is included.  

Conclusions:
From the last column of the above table, we 
conclude the following: The prediction equation: