Design and Analysis of Multi
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Transcript Design and Analysis of Multi
Design and Analysis of
Multi-Factored Experiments
Advanced Designs
-Hard to Change FactorsSplit-Plot Design and Analysis
L. M. Lye
DOE Course
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Hard-to-Change Factors
• Assume that a factor can be varied , with great difficulty, in
an experimental setup (such as a pilot plant), although it
cannot be freely varied during normal operating
conditions.
• Assume further that each of two factors has two levels and
the design is to have a factorial structure, and it is
imperative that the number of changes of the hard-tochange factor be minimized.
• We can minimize the number of level changes of one
factor simply by keeping the level constant in pairs of
consecutive runs. That is, either the high level is used on
consecutive runs and then the low level on the next two
runs, or the reverse.
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DOE Course
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• This means we that we have restricted randomization, as
there are 6 possible run orders of that one factor without
any restrictions, but with the restriction, there are only 2
possible run orders (+ +, - -) or (- -, + +)
• Restricted randomization increases the likelihood that
extraneous factors (i.e. factors not included in the design)
could affect the conclusions that are drawn from the
analysis.
• Furthermore, this will also cause bias in the statistics that
are used to assess significance. i.e. normal ANOVA based
on a completely randomized design may give the wrong
conclusions.
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DOE Course
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• Although hard-to-change factors have not been discussed
extensively in textbooks, it is safe to assume that such
factors occur very frequently in practice.
• Sometimes there may be no hard-to-change factors at all in
the experiments, but the experimenters or technician who
wants to save time may not have followed the randomized
design as prescribed by the experimental design.
• Hence it is very important for the analyst performing the
statistical analysis to know exactly how the experiments
were performed. Were the runs randomized as prescribed
or were the runs made “convenient” to save time.
• How the experiments were carried out can have serious
consequences on the results. Significant effects may turn
out to be insignificant or vice versa if is not properly
analyzed. The software will not know unless you tell it.
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DOE Course
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Split-Plot Design with Hard-to-Change Factors
• For example, all of you know a 23 full factorial design. Most
would choose to run the 8 treatment combinations in a
completely randomized order as given say by Design-Expert.
• Unfortunately, limitations involving time, cost, material, and
experimental equipment can make it inefficient and, at times,
impossible to run a completely randomized design.
• In particular, it may be difficult to change the level for one of
the factors. E.g oven temperature may take many hours to
stabilize.
• In this case, the hard-to-change factor is typically fixed at one
level and then run the combinations of the other factors – the
split-plot design (or Split-Unit design)
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DOE Course
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Recognizing a Split-Plot Design
• Split-plot experiments began in the agricultural industry
because one factor in the experiment usually a fertilizer or
irrigation method can only be applied to large sections of
land called “whole plots”.
• The factor associated with this is therefore called a whole
plot factor.
• Within the whole plot, another factor, such as seed variety,
is applied to smaller sections of the land, which is obtained
by splitting the larger section of land into subplots. This
factor is therefore referred to as the subplot factor.
• These same experimental situations are also common in
industrial settings.
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DOE Course
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3 Main Characteristics of Split-Plot Designs
• The levels of all the factors are not randomly determined
and reset for each run.
– Did you hold a factor at a particular level and the run all the
combinations of the other factors?
• The size of the experimental unit is not the same for all
experimental factors.
– Did you apply one factor to a larger unit or group of units
involving combinations of the other factors?
• There is a restriction on the random assignment of the
treatment combinations to the experimental units.
– Is there something that prohibits assigning the treatments to the
units completely randomly?
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DOE Course
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Effect of restricted randomization on
statistical analysis
• Consider a very simple example of 2 factors each at 2
levels. If the 4 combinations are run in random order, we
have a 22 design.
• Now assume that one of the factors is hard-to-change. One
of the levels of this factor is selected randomly and then
used in combination with each of the 2 levels of the other
factor, which are also randomly selected.
• Then this process is repeated for the second level of the first
factor.
• So assuming A is the hard-to-change factor the run
combinations could be as follows: A2B2, A2B1, A1B1, A1B2.
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• Notice that this sequence of runs could have of course risen
from the completely randomized experiment, but the data
would still have to be analyzed differently because of the
restricted randomization in the second case.
• That is, there are only 8 possible sequences of treatment
combinations with the restriction, whereas there are 24
possible sequences without the restriction.
• Another key point: With complete randomization, each run is
completely reset, whereas, with restricted randomization, the
hard-to-change factor was not reset.
• If the data were analyzed as a 22 completely randomized
design using Design-Expert, we will get the wrong answer!
• Why is this so?
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DOE Course
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Split-Plot Designs have 2 error terms
• Recall that in a 2k design, each effect is estimated with the
same precision. i.e. they have the same standard error.
• This does not happen with a split-plot design as subplot
factors are generally estimated with greater precision (smaller
errors) than are whole plot factors.
• This is because there is greater homogeneity among subplots
than are the whole plots, especially if the whole plots are
large.
• E.g. Smaller pieces cut from a sheet of plywood are more
homogeneous than between 2 different sheets of plywood. i.e.
pieces within a sheet has less variability than between 2 sheets
of plywood.
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• In fact for agricultural experiments, it can be
shown that the variance of any whole plot effect
estimate must exceed the variance of any subplot
effect estimate for any 2k design.
• The statistical model for a split-plot design can be
written as:
yij i WP j ij SP
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DOE Course
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• It has been shown mathematically that the whole plot factors
and their interactions have a variance of:
4 k 2
2 1 02
N
• Whereas subplot factors and their interactions have a variance
of:
4 2
0
N
• Here the subscript 1 and 0 denote the whole plot and subplot,
respectively.
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DOE Course
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Example
• Assume that factor A is a hard-to-change factor and factor
B is not hard to change, with the experiment such that
material (e.g. a board) is divided into two pieces and the
two levels of factor A applied to the two pieces, one level
to each piece.
• Then the pieces are further subdivided and each of the two
levels of factor B and applied to the subdivided pieces.
Three pieces of the original length (e.g. three full boards)
are used.
• The data will be analyzed assuming a fully randomized
design like a regular 22 design, and then correctly using a
split-plot design.
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DOE Course
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Data and Analyses
A
-1
-1
+1
+1
B
-1
+1
-1
+1
2.5
2.7
2.3
2.7
Observations
2.4
2.6
2.3
2.7
2.6
2.5
2.4
2.8
If the data are improperly analyzed as 22 design with three replications,
the results are as follows:
Two-way ANOVA: Y versus A, B
Source
A
B
AB
Error
Total
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DF
1
1
1
8
11
SS
0.000833
0.187500
0.067500
0.053333
0.309167
MS
F
0.000833 0.12
0.187500 28.12
0.067500 10.12
0.006667
DOE Course
P
0.733
0.001
0.013
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• The proper analysis of the data as having come from a splitplot design is not easily achieved.
• Most DOE software cannot handle split-plot design directly.
Design-Expert can handle some types of split-plot design but
it must be done manually.
• However, statistical packages can be tricked into performing
the correct analysis by assuming a nested model and forcing a
nested model analysis.
• Minitab’s General Linear Model routine can do the analysis if
the data are set up properly.
• Best papers to read are: Kowalski and Potcner (2003), Kevin
Potcner and Kowalski (2004), Bisgaard, Fuller, and Barrios
(1995). You can download these from the course website.
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DOE Course
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Proper statistical analysis: split-plot analysis
General Linear Model: Y versus A, B, WP
Factor
A
WP(A)
B
Type Levels
fixed
2
random 6
fixed
2
Values
-1, 1
1, 2, 3, 4, 5, 6
-1, 1
need to set up this column
Analysis of Variance for Y, using Sequential SS for Tests
Source DF
A
1
WP(A) 4
B
1
A*B
1
Error
4
Total
11
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Seq SS
0.000833
0.023333
0.187500
0.067500
0.030000
0.309167
Seq MS
F
P
0.000833 0.14 0.725
0.005833
*
*
WP error term
0.187500 25.00 0.007
0.067500
9.00 0.040 3 times higher than CRD
0.007500 subplot error term
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• A somewhat different picture emerges when the data are
analyzed correctly.
• The p-value for AB is more than 3 times than of for CRD.
• The difference in the conclusions drawn with the wrong
analysis and the conclusions made with the proper analysis
can be much greater than the difference in this example.
• As illustrated by Potcner and Kowalski (2004), a
significant main effect in the complete randomization
analysis can become a non-significant whole-plot main
effect when the split-plot analysis is performed.
• And, a non-significant main effect in the complete
randomization analysis can become a significant subplot
main effect when the split plot analysis is performed.
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DOE Course
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Several hard-to-change factors
• Sometimes there may be instances where there are several
hard-to-change factors and one or more easy to change
factors.
• For example, 4 of the factors (A, B, C, and D) may be
hard-to-change whereas E may be easy to change. Or we
may have 3 hard-to-change factors and say 6 easy to
change factors, etc.
• In addition, there is only one replication. This means that
there are no pure error terms.
• For these situations, the split plot design can be analyzed
using two separate half-normal probability plots. One for
the whole plot effects and one for the subplot effects. This
can be done on Design-Expert manually.
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DOE Course
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Dividing into Whole Plots and Subplots
• Let’s consider 2 examples:
• Example 1: 5 factors (A, B, C, D, E). A, B, C, D are hard-tochange factors, and E is easy to change.
• Whole plot group: A, B, C, D and interactions involving only
these factors.
• Subplot group: E, and all interactions involving E only. E.g.
AE, BE, CDE, etc.
• Example 2: 9 factors (A, B, C, D, E, F, G, H, J). A, B, C are
hard-to-change, and D, E, F, G, H, J are easy to change.
• Whole plot group: A, B, C, and all interactions involving only
these 3 factors
• Subplot group: D, E, F, G, H, J and all interactions involving
these factors. E.g. AD, DE, etc, but not AB, BC, or ABC.
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DOE Course
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Effects and Half-normal plots
• The effects of each factor and its interaction are determined
in exactly the same way as in regular factorial design.
• Once the whole plot group and subplot group have been
decided, a half-normal plot of effects are used to determined
the significant effects for each group.
• Hence, two half-normal plots are constructed.
• The significant effects from both groups are then combined
to give the final model and prediction equation.
• If the half normal plot is done for all the effects without
separating them into the two groups, it is likely that the
subplot effects will be buried in the whole plot error terms.
• Hence significant subplot effects maybe missed if the split
plot nature of the experiment is not taken into account by the
analysis. Example – Plasma.dx7
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DOE Course
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Summary
• When it is not convenient or not economical to do a
completely randomize experiment due to one or more
hard-to-change factors, we have a restricted randomization
case.
• A common and often used approach is a split plot
experiment which has a whole plot group of effects and a
subplot group of effects leading to two error terms in the
ANOVA or two half-normal plots for the unreplicated case.
• If not analyzed properly, significant subplot effects may be
masked by the larger whole plot errors thus giving the
wrong conclusions and wrong model.
• It is thus crucial that you know exactly how the experiment
was carried out in practice.
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DOE Course
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