Trinity College Dublin
Diploma in Quality Improvement
Computer Laboratory 3
Analysis of Fractional and Blocked Factorial Experiments
Invitations to consider the results of Minitab analysis and their statistical and substantive
interpretations are printed in italics. Take some time for this; consult your neighbour or
tutor. Enter your responses in a Word document, as draft contributions to a report on
the experiment and its analysis.
Factorial Experiments
Experiments involving factors whose variation is confined to two values (levels or categories)
are important because
(i)
(ii)
(iii)
(iv)
(v)
they are relatively simple to set up
they are relatively simple to analyse
they permit several factors to be investigated in relatively few experimental runs,
they permit even more factors to be investigated by using carefully chosen
fractions of a full experiment,
they provide clues to seeking better conditions.
In this laboratory, you will analyse an unreplicated fractional factorial experiment and a blocked
factorial experiment.
Learning Objectives
On completion of this laboratory, students should be able to
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set up and analyse 2-level factorial experiments in Minitab;
interpret effects plots, both formally and informally;
apply the Pareto Principle informally,
use simulation to set up reference plots for Normal effects plots,
produce and interpret numerical summaries of effects;
fit reduced models to provide error estimates as a basis for formal inference,
produce and interpret main effect and interaction plots,
produce stratified two-factor interaction plots to assist with interpretation of 3-factor
interactions
interpret diagnostic plots and relate the results to effects patterns;
make and interpret a categorised table of summary statistics;
prepare management reports based on results of detailed analyses.
Exercise 1: Analysis of a 25–1 fractional factorial experiment
A food processing company was having a problem with its soup mix packet filling machine; it
was inconsistent in the weights it delivered, with the result that excessive overfill was needed to
ensure achieving the nominal weight. The problem was isolated to the uneven flow of a
component mix of vegetable oil, salt and other ingredients that was added to the soup mix
during the mixing process. Five factors were identified that were thought to influence this part
of the process and experimental settings were agreed, as follows:
A:
B:
C:
D:
E:
the number of ports for adding oil to the mix,
mixer vessel temperature,
mixing time,
batch weight,
delay between mixing and packaging,
1 or 3,
ambient or water cooled,
60 seconds or 80 seconds,
1500 lbs or 2000 lbs,
1 day or 7 days.
A 25–1 experiment was run, with the 16 chosen factor level combinations being run in random
order. The response variable was a measure of the spread of the actual weights of 5 sample
packets, one taken every 15 minutes, during the production run for each factor level
combination. The results, together with the run order, are shown below.
RunOrder
A
B
C
D
E
Y
5
3
2
3
6
4
6
4
7
0
1
5
9
8
11
2
–
+
–
+
–
+
–
+
–
+
–
+
–
+
–
+
–
–
+
+
–
–
+
+
–
–
+
+
–
–
+
+
–
–
–
–
+
+
+
+
–
–
–
–
+
+
+
+
–
–
–
–
–
–
–
–
+
+
+
+
+
+
+
+
+
–
–
+
–
+
+
–
–
+
+
–
+
–
–
+
0.97
1.70
1.13
1.25
1.18
0.98
1.47
1.28
1.85
0.62
0.78
1.36
0.76
2.10
1.09
1.10
The data are stored in an Excel file named Soup.xls in the course webpage; copy and paste
into Minitab.
Confirm that the sign pattern for allocating levels of Factor E to design points is the
product of the corresponding sign patterns for the other four factors.
Use the Stat / DOE / Factorial / Analyze Factorial Design to set up the design and analyse the
data. In setting up the analysis,
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from Terms, select all available Terms (the default),
from Graphs, select the Normal and Pareto effects plots, ignore residuals,
from Results, select Coefficients and ANOVA table (the latter being useless!).
Identify the statistically significant factors and the values of their effects. Noting the
Alias Structure in the Session window output, identify what is actually estimated by
the corresponding estimated effects.
Discuss the application of the Pareto Principle in this case; does it influence your
choice of active effects?
Use simulation to check the interpretation of the Normal effects plot:
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use the Start menu to open a new Minitab worksheet (and keep the existing one open),
use the Calculator to generate 15 standard Normals into C1,
Use the Graph menu to produce a Normal plot of the 15 values,
page 2
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when choosing the distribution, click on the Data Display tab and uncheck "Show confidence
interval,
right-click in the finished graph and select Update Graph Automatically,
use the calculator again to generate a second sample of random numbers,
use Ctrl+E to simulate repeated samples.
In the light of these simulated reference samples and their Normal plots, how do you
interpret the Normal effects plot?
Use the DOE command to make relevant interaction plots,
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navigate to the Factorial Plots dialog window, check Interaction Plot, click on Setup, add two
relevant Available factors to the Selected factors box,
click OK, OK,
repeat to get a second two factor interaction plot,
ensure that both plots have the same scale of the vertical axis:
 double click each scale in turn,
 if necessary, adjust Position of ticks in one or both scales,
 establish the maxima and minima, then set the most inclusive maximum and minimum
pair in both scales.
Provide an interpretation of these plots for management.
Noting that reducing Y is desirable, identify the best operating conditions
Assuming that a seven day delay in filling is not feasible, identify the next best.
Record your analysis in a Word document, including plots and verbal descriptions.
Use the Analyze Factorial Design feature to produce relevant numerical summaries:
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navigate to the Analyze Factorial Design dialog window,
click on Results,
under Display of Least Squares Means, select relevant terms for display,
click OK, OK.
Enhance your management report with a numerical description of the effects and the
results of using the recommended sets of operating conditions.
Given the parsimonious model implicit in the above analysis, re-analyse the raw data using the
DOE command with a reduced set of Selected terms:
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navigate to the Analyze Factorial Design dialog window,
click on Terms,
clear the Selected Terms box and add back relevant terms, including all main effects
corresponding to two-factor interactions added,
click on Graphs and include residual plots,
click on Results and include relevant terms for Display of Least Squares Means,
click OK.
Review the diagnostic plots.
Iterate the analysis in light of the diagnostics.
page 3
Interpret the final analysis.
Adjust the numerical summaries in your Management Report accordingly.
Add relevant confidence intervals, including a confidence interval (and significance
test) for the difference between means under the competing operating conditions.
Exercise 2:
A widely quoted study carried out at the UK Agricultural Experimental Station at Rothamstead,
North of London, was concerned with the effect of combinations of different fertilisers on the
yield of beans, measured in pounds. Four fertilisers were to be studied, Dung (D), Nitrochalk
(N), SuperPhosphate (P) and Muriate of Potash (Potassium, K). The levels of each factor were
as follows:
Low
High
Dung (D):
none
10 tons per acre
Nitrochalk (N):
none
0.4 cwt per acre
SuperPhosphate (P):
none
0.6 cwt per acre
Muriate of Potash (K):
none
1.0 cwt per acre
The experiment was organised in two blocks of 8 plots each, with a half fraction of the 25 being
run in each block. The run order was randomised within each block. Details of the
experimental design and results follow.
Block
D
N
P
K
RunOrder
Yield
1
1
1
1
1
1
1
1
–
+
–
+
–
+
–
+
–
–
+
+
–
–
+
+
–
–
–
–
+
+
+
+
+
–
–
+
–
+
+
–
2
3
8
5
1
7
4
6
55
53
42
41
45
55
36
48
2
2
2
2
2
2
2
2
–
+
–
+
–
+
–
+
–
–
+
+
–
–
+
+
–
–
–
–
+
+
+
+
–
+
+
–
+
–
–
+
6
3
2
7
4
1
8
5
58
43
44
41
51
50
50
44
The data are stored in an Excel file named BeanYields.xls in the course webpage. Copy the
data into Minitab.
Check the product of signs of D, N and P; note the correspondence with K.
Check the product of signs of the four factors; note the correspondence with blocks.
Use the DOE command to analyse, first setting up the design:
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select D, N, P, K as the factors,
enter "-" and "+" as the factor levels (Low/High),
page 4
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enter the Design details,
Standard Order as order of the data (or C1),
Run Order as in C7,
Blocks as in C6,
click OK,
select Y as the response,
click on Terms, confirm Default selected terms and "Block included in the model",
click on Graphs and check Normal and Pareto Effects Plots,
click on Results and check Coefficients and ANOVA table,
Click OK.
Interpret the Normal effects plot and the Pareto effects plot. What effects do you
think you should recognise as active?
Check the values of the active effects.
Is there a block effect?
Assuming that inactive effects reflect chance variation, repeat the analysis:
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in the Terms window, remove all terms from the selected terms list, then add active terms,
including all lower order terms corresponding to higher order interactions,
exclude the Block effect,
in the Graphs window, check appropriate diagnostics.
Review the diagnostics.
Review your earlier interpretations.
To display the active effects, use the Factorial Plots subcommand of the DOE command. To
display a 3-factor interaction, use a Cube plot. Alternatively, split the data in two subsets
according to one of the three factors and display the interaction plots for other two factors plot
for the two subsets. When the two plots are produced, ensure that the vertical axes have the
same scales, including maximum, minimum and tick marks. To do this, double click each scale
in turn, establish the maxima and minima, then set the most inclusive maximum and minimum
pair in both scales.
Provide a detailed interpretation of the interactions.
Make recommendations for fertiliser levels to apply for increased yield.
This ends the formal laboratory.
Review the Learning Objectives. Have they been achieved?
page 5