Example 1.2:
Analyzing a 26
Full Factorial
This example is based on the data file Textile.sta.
Open this data file:
Ribbon
bar. Select the Home tab.
In the File group on the Open menu, select Open
Examples to display the Open
a Statistica Data File dialog box. Doubleclick the Datasets
folder, and then open the data set.
Classic
menus. On the File menu, select Open Examples to display the
Open a Statistica Data File dialog
box. The data file is located in the Datasets
folder.
Box and Draper (1987, page 115) report a study of the manufacture of
certain dyestuff. The dependent variables of interest are the Strength,
Hue, and Brightness
of the resulting product. The particular process under investigation was
not very well understood in terms of the underlying chemical mechanism;
therefore, an empirical study (experiment) was conducted with 6 factors.
To view these 6 factors, display the Variable Specification Editor:
Ribbon bar. Select the Data
tab. In the Variables group,
click All Specs.
Classic menus. On the Data menu, select All Variable Specs.
The low and high settings (levels) for the factors were as follows:

Factor Setting 

Low 
High 
Polysulfide index 
6 
7 
Reflux rate 
150 
170 
Moles polysulfide 
1.8 
2.4 
Time 
24 
36 
Solvent 
30 
42 
Temperature 
120 
130 
Shown below is a partial listing of the data file Textile.sta:
Following are the original settings for the factors; note that text
values (Low, High)
were also entered to identify the respective settings. You can toggle
between text and numeric values in the data file:
Ribbon bar. Select the View
tab. Click Display Options, and
select Text Labels.
Classic menus. Click the Show/hide
text labels button on the toolbar.
Specifying the design. Start the Experimental
Design (DOE)
analysis:
Ribbon
bar. Select the Statistics tab, and in the Industrial Statistics group,
click DOE to display the Design
& Analysis of Experiments dialog boxl.
Classic
menus. On the Statistics  Industrial Statistics
& Six Sigma submenu, select Experimental
Design (DOE) to display the Design
& Analysis of Experiments dialog box.
Select
2**(kp) standard
designs (Box,
Hunter, & Hunter) and click the OK button. In the Design
& Analysis of Experiments with TwoLevel Factors
dialog box, select the Analyze design tab.
Click the Variables button
to display a standard variable
selection dialog box. Confirm that the Show
appropriate variables only is not selected.
Select Strength,
Hue, and Brightness as
the Dependent
variables, Variables 1
through 6 as the
Indep. (factors), and click the
OK button. The dialog
box will now look like this.
Click the OK button to display
the Analysis
of an Experiment with TwoLevel Factors dialog box.
Printing all results. Instead of reviewing results
variable by variable, let's send all key results to a
workbook and
a report.
Ribbon bar. Select the File
tab. Select Settings, and click
Output Manager.
Classic menus. On the File
menu, select Output Manager.
Select the Workbook option
button as well as Single Report (common
for all Analyses/Graphs).
Click the OK
button.
To send the results for
all the dependent variables, in the Analysis
of an Experiment with TwoLevel Factors dialog box, select
the All variables check box.
The dialog box will now look like this:
Before sending the results,
because this is a full factorial design, let's estimate all 2way and
3way interactions. Select the Model tab.
In the Include in model group
box, select the 3way
interactions option button.
Now, click the Print
results button. As you can see, a large amount of information is
sent to the workbook and report. However, if you review carefully
the ANOVA tables and tables of ANOVA parameters (and their statistical
significance), it appears that none of the 2way and 3way interaction
effects are statistically significant.
Normal probability plot of
effects. You can also
quickly "cut through the clutter" by displaying the normal
probability plot of effects.
In the Analysis of an Experiment
with TwoLevel Factors dialog box, select the ANOVA/Effects tab.
Select Strength
as the current Variable.
In the
Plots of Effects group box, select the Label
points in normal plot check box
and the Plot standardized effects
check box.
In the
Plots of Effects group box, click the Normal
probability plot button.
This plot is constructed
by first ranking the (standardized) effects; the ranks are then converted
into relative ranks or percentiles, which are converted into the respective
values for the standard normal distribution (plotted against the left
yaxis in this graph).
The majority of the main
effects and interactions effects are close together, plotted along a line.
The sizes of these effects are distributed in the way that one would expect
if they were (normal) random around zero. However, the main effects for
Polysulfide, Temperature,
and Time are clearly separate
in the upperright corner of the plot.
Note that you can use the
Interactive
Graphics Controls at the bottom of the graph window to adjust the
plot areas transparency and to scroll and pan in order to interactively
scale the graph.
Pareto chart of effects.
It appears that a simple
maineffects model is sufficient for dependent variable Strength. Therefore, In the Analysis
of an Experiment with TwoLevel Factors
dialog box,
select the Model tab. Select the No interactions option button. Then, select
the Quick tab,
and click the Pareto
chart of effects
button.
This
chart also clearly identifies the main effects for Time,
Temperature, and Polysulfide
as the most important determinants of resultant Strength.
Plots of
marginal and predicted means.
Cube plot of predicted means.
To assess the effect of those three factors, on the Quick
tab in the Predicted (estimated)
means group box, click the Cube
plot of predicted means button to display the Factors for Cube Plot dialog box. Select
the three factors Time, Temperature, and Polysulfide
as the three variables for the plot. Then, click the OK
button to produce the cube plot for the predicted means.
By
default, this plot shows the predicted means and their confidence intervals
for the three factors, when all other factors are set at their respective
means. The highest predicted mean (14.834) occurs at the point where all
three factors are set at their respective High settings.
Plot of marginal means. Another plot that is particularly useful
for exploring the nature of interactions is the plot of marginal means.
Even though there is no indication of any interaction effects in this
study, let's look at this graph anyway. On the Quick
tab, click the Means
plot button to display the Compute
marginal means for dialog
box. Select the three variables Polysulfide,
Time, and Temperature.
Click the OK
button to display the Specify the arrangement of the factors in
the plot dialog box, where you can select the assignment of factors
to the axes and line patterns in the interaction plot. Select Polysulfide
as the Line pattern; Time
as the xaxis, upper; and Temperature as the xaxis,
lower.
Then click the OK
button to produce the interaction plot.
Effect estimates,
coefficients, and regression coefficients.
If you look back at the data file, you can see that the factor levels
were entered in their original metric, and not in their coded (±1) form.
Therefore you can for this analysis review 1) the ANOVA effect estimates,
2) the coefficients for the coded (±1) factors, and 3) the coefficients
for uncoded (raw) factors values.
ANOVA effect parameters. Click the Summary:
Effect estimates button (either
on the Quick
tab in the ANOVA group box or on the
ANOVA/Effects
tab) to produce the ANOVA effect estimates and
the coefficients for the coded factor levels. A portion of these results
are shown below.
The
first 6 columns of the spreadsheet
show the ANOVA effect estimates, their standard errors, confidence intervals,
etc. The interpretation of the effect estimates is discussed in the Introductory
Overview. Specifically, for the main effects, these values can be
interpreted as the differences between the low and high settings for the
respective factors. (If there were twoway interactions in this model,
the respective effect estimates could be interpreted as half the difference
between the main effects of one factor at the two levels of a second factor;
threeway interaction effect estimates can be interpreted as half the
difference between the twofactor interaction effect at the two levels
of a third factor, and so on; see, for example, Mason, Gunst, and Hess,
1989, page 127.)
The
last four columns of the spreadsheet
contain the estimates of the coefficients for the coded factors. These
can be interpreted as the regression coefficients for the recoded (to
±1) factor levels.
Regression coefficients. On the ANOVA/Effects tab, click
the Regression coefficients button
to produce the table of regression
coefficients for the original (untransformed) factors (i.e., for the factor
values in their original metric).
These
are the coefficients that you could use to make predictions from factor
values without having to recode those values. However, note that these
coefficients are no longer comparable to each other, because their scaling
depends on the scaling of the factors.
The
conclusions you would reach regarding which factors are important are,
of course, independent of the scaling of the factors. Indeed, if you review
the spreadsheet
shown above, you will see that the statistical significance tests (i.e.,
the t values) for the main effect
estimates and regression parameters are the same. Note, however, that
this will not necessarily be the case anymore when the model includes
quadratic effects and interactions, as will be discussed in the context
of 3(kp)
designs.
Diagnostic
checking of the fitted model.
The analysis so far for dependent variable Strength has revealed significant main effects for
factors Time, Temperature, and Polysulfide. Now let's see how well this model fits
the data, and whether the prediction residuals for this model are approximately
normally distributed (which is an assumption of the leastsquares estimation
method).
First, pool all nonsignificant
effects into the error term. On the Model
tab, select the Ignore some effects check box to display the Customized
(Pooled) Error Term dialog box.
Select all nonsignificant
main effects (Reflux, Moles,
and Solvent) to be pooled into
the error.
Click the OK
button. Select the Prediction
& profiling tab, and click the Predicted
vs. observed values button.
The
predicted values cluster fairly closely and homogeneously around the diagonal
line in this plot, indicating a good fit of the model.
Now, select the Residual
plots tab. In the Probability
plots of residuals group box, click the Normal
plot button to produce the normal probability
plot of the residuals
It
appears that the residuals follow the normal distribution very closely.
Thus, we can conclude that the threemaineffects model provides a good
fit for the dependent variable Strength.
See
also, Experimental
Design Index.