Example 4: Designing
and Analyzing a 2332 Experiment
Connor and Young (in McLean and Anderson, 1984) report an experiment
(taken from Youden and Zimmerman, 1936) on various methods of producing
tomato plant seedlings prior to transplanting in the field. The experiment
is a 2332 factorial design
(3 factors at 2 levels, 2 factors at 3 levels). We will first design the
experiment, and then analyze the yield (in pounds) as a function of the
different factor settings.
The factors in the design are:
Factor 
Levels 
Value 
Soil condition 
Field soil 
0 

+fertilizer 
1 
Size of pot 
Threeinch 
0 

Fourinch 
1 
Variety of tomato 
Bony Best 
0 

Marglobe 
1 
Method of production 
Flat 
0 

Fibre 
1 

Fibre+NO3 
2 
Location on field 

0,1,2 
Note that in principle, the design and analysis of mixed level designs
proceeds much like the analysis of 3(kp)
designs (see Example 3).
Thus, we will only briefly review the steps to design and analyze this
experiment. If you have not reviewed any of the previous examples, we
suggest that, before proceeding, you review Example
1.1 and Example 1.2,
where some basic conventions and plots are explained.
Designing the experiment.
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 Startup Panel.
Classic
menus. On the Statistics  Industrial Statistics
& Six Sigma submenu, select Experimental
Design (DOE) to display the Design
& Analysis of Experiments Startup Panel.
Select Mixed 2 and 3 level designs
and click OK to display the Design
& Analysis of Experiments with Two and ThreeLevel Factors
dialog box.
The design reported in Connor and Young consists of 3 twolevel factors,
and 2 threelevel factors. Thus, select the 3/2
design on the Design
Experiment tab.
Click the OK button
to display the Design
of Experiments with Two and ThreeLevel Factors dialog box.
Click the Change factor names, values,
etc. button. Enter the values into the Summary
for Variables (Factors) dialog box as shown below.
Click the OK button.
Reviewing unconfounded effects.
Before saving the design, let's look at the unconfounded factor effects
for this design.
On the Generator
& aliases tab, select the Include
interactions by quadratic components check box. Then, click the
Correlation matrix (main effects and
interactions) button; two spreadsheets
will be produced. The first one contains the correlation matrix for the
coded main effects and interactions.
The second one lists the main effects and twoway interactions that
are unconfounded with other main effects and interactions.
Note that all main effects are unconfounded with all other main effects.
Moreover, the two 3level factors are also unconfounded with the twoway
interactions. Also, several twoway interactions are unconfounded with
the main effects and other twoway interactions.
Displaying and saving the design.
To display and save the design, select the Display
design tab.
Select the By Names option
button in the Denote factors group
box, and select the Text
labels option button in the Show
(in spreadsheet) group box.
Click the Summary: Display design
button. Shown below is a portion of the design.
Even though this spreadsheet
shows the text values that we entered to denote the factor levels, "underneath"
these are stored the numeric factor highs and lows. Thus, when you save
this design, both the numeric values and text values will be saved to
the data file.
Analyzing the design.
The design along with the values for the dependent variable (Pounds)
are stored in the data file Tomatoes.sta.
Open that data file.
Start the Experimental
Design module, and select Mixed
2 and 3 level designs in the Startup Panel.
In the Design
& Analysis of Experiments with Two and ThreeLevel Factors
dialog box, select the Analyze
design tab. Click the Variables
button, and select the variable Pounds
as the Dependent variable and
variables Soil Condition through
Location as the
Indep. (Factors). Click the OK
button.
In the Design & Analysis
of Experiments with Two and ThreeLevel Factors dialog box, click
OK to display the Analysis
of an Experiment with Two and ThreeLevel Factors dialog box.
Analysis of variance. Click
on the Model
tab, and select the 2way
interactions (linear, quadr.) option button in the Include
in model group box.
Select the Quick
tab, and click the ANOVA
table button.
The first spreadsheet
shows the ANOVA for each singledegree of freedom test, that is, it shows
separate tests for the linear and quadratic components of the 3level
factors. Since the factors in this experiment are qualitative in nature,
we will ignore the values in that spreadsheet and focus on the results
reported in the second spreadsheet, which reports the results for the
combined effects (i.e., multiple degreeoffreedom tests).
None of the interaction effects are statistically significant. The soil
condition by production method (1*4)
interaction is marginally significant.
In the Observed marginal means
group box, click the Means plot
button. In the Compute
marginal means for dialog box, select Soil
Condition and Production
Method as the Factors,
and click OK.
In the Arrangement
of Factors dialog box, select Production
Method as the xaxis, upper
and Soil Condition as the Line pattern, and click OK
to produce the graph of the interaction, as shown below.
It appears that for the fieldplusfertilizer (Plus)
condition, the fiber pots with added NO3
did not further improve the yield. However, this effect was not statistically
significant.
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.
Plot of means. You can also
use the Means plot button to
plot the of means for the significant main effects (Soil
Condition, Variety, Production Method, and Location).
If you produce these four graphs, the patterns of means will clearly show
how the four factors are related to the resulting tomato yield. [Shown
here is a compound graph produced from the four means plots, one for each
main effect. This graph can be created using MultiGraph
Layouts in the Tools group
on the Graphs tab (ribbon bar)
or Multiple Graphs Layouts on
the Graphs menu (classic menus).]
The patterns of means clearly shows how the four factors are related
to the resulting tomato yield.
Probability plot of residuals.
Finally, we will examine the normal probability plot of residuals for
any indication that the residual values are not normally distributed.
In the Analysis of an Experiment with
Two and ThreeLevel Factors dialog box, select the Residual
plots tab.
In the Probability plots of residuals
group box, click the Normal plot
button to produce the graph shown below.
It appears that the residuals closely follow the line, indicating the
normal distribution. Thus, we can conclude that the normality assumption
is probably not violated in the data.
See also, Experimental
Design Index.