Exercise 5.1.1
Box, Hunter and Hunter (2005) describe an experiment of the split plots type designed to study
the corrosion resistance of steel bars treated with four different coatings C1, C2, C3 and C4 at
three furnace temperatures 360, 370 and 380°C. The results follow.
Furnace
Run
1
2
3
4
5
6
Temperature
( °C )
360
370
380
380
370
360
1
67
65
155
108
140
33
Coating
2
3
73
83
91
87
127
147
100
90
142
121
8
46
4
89
86
212
153
150
54
The positions of the coated steel bars in the furnace were randomised within each temperature
setting. However, because the furnace temperature was difficult to change, the temperature
settings were run in the systematic order shown. This facilitated raising the temperature from
360°C through 370°C to 380°C and back down again
Display the plot structure and treatment structure in this experiment.
Given that the furnace temperature was difficult to change, outline the practical advantage of a
split plots design as opposed to a completely randomised design in this case.
With reference to the systematic order of the temperature settings, Box, Hunter and Hunter
remark that "it would have been preferable to randomise the order had this been possible".
Explain why randomising the run order is preferable.
Box, Hunter and Hunter noted that what was of primary interest were the comparisons of
coatings and how they interacted with temperature; the main effects of changing temperature
were of secondary interest. Explain how the split plots design facilitates this.
Write down the Minitab model for a full split plot analysis, separating terms appropriate to the
whole plots and the split plots and distinguishing between fixed and random effects.
Minitab was used to calculate an analysis of variance with results as follows:
Source
DF
Temperature
2
Run(Temperature)
3
Coating
3
Temperature*Coating 6
Error
9
Total
23
SS
26519.2
14439.6
4289.1
3269.8
1120.9
49638.6
MS
13259.6
4813.2
1429.7
545.0
124.5
F
2.75
38.65
11.48
4.38
Report on the statistical significance of these results.
The following expected mean square formulas were also produced.
1
2
3
4
5
Source
Temp
Run(Temp)
Coating
Temp*Coating
Error
Expected Mean Square
(5) + 4.0000 (2) + Q[1]
(5) + 4.0000 (2)
(5) + Q[3]
(5) + Q[4]
(5)
P
0.209
0.000
0.002
0.024
The F-ratios for Temperature and Coating are the ratios of relevant mean squares. Indicate the
relevant mean squares, justify the formulas for the F-ratios by reference to the table of Expected
Mean Squares and confirm the values of F.
Outline how relevant entries in the MS column of the Analysis of Variance table support your
explanation of how the split plots design facilitates regarding the comparison of coatings and
how they interact with temperature as of primary interest with the main effects of changing
temperature being of secondary interest.
Make a TemperatureCoating interaction plot. (Hint: For ease of interpretation, put Temperature
on the horizontal axis). Provide a brief interpretation, including a comment on optimum
conditions (assuming high corrosion resistance is desirable).
Suppose that the first three furnace runs took place in one day and the second three runs took
place on a different day, with Days being regarded as blocks. Suppose further that the four
coatings C1, C2, C3 and C4 were all four combinations of two Base coats, B1, B2, and two Finish
coats, F1, F2.
Display the plot structure and treatment structure in this more elaborate experiment
Write down the Minitab model for a full split plots analysis, separating terms appropriate to the
whole plots and the split plots and identifying the error term for the whole plots analysis.
Minitab was used to calculate an analysis of variance using this model with results as follows:
Source
Day
Temperature
Day*Temperature
Base
Finish
Base*Finish
Temperature*Base
Temperature*Finish
Temperature*Base*Finish
Error
Total
DF
1
2
2
1
1
1
2
2
2
9
23
SS
782.0
26519.2
13657.6
852.0
1820.0
1617.0
601.1
787.6
1881.1
1120.9
49638.6
MS
782.0
13259.6
6828.8
852.0
1820.0
1617.0
300.5
393.8
940.5
124.5
F
0.11
1.94
54.83
6.84
14.61
12.98
2.41
3.16
7.55
P
0.767
0.340
0.000
0.028
0.004
0.006
0.145
0.091
0.012
Report on the statistical significance of these results. Comment on the need for regarding days
as blocks.
In view of the highly significant three factor interaction, the following table of average corrosion
resistance was produced.
Base
1
Base
2
Base
1
Base
2
Base
1
Base
2
Finish 1
50.0
40.5
102.5
116.5
131.5
113.5
Finish 2
64.5
71.5
104.0
118.0
118.5
182.5
Temperature
360°C
370°C
380°C
Construct plots to show the three factor interaction. Provide an interpretation. Identify the
optimum conditions for maximising corrosion resistance.
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