22-22-11
Multiple Comparisons in Factorial Experiments
If Main Effects are significant AND Interactions are NOT significant:Use multiple comparisons on factor main effects (factor means).
If Interactions ARE significant:1) Multiple comparisons on main effect level means
should NOT be done as they are meaningless.2) Should instead perform multiple comparisons among
all factorial means of interest.
In addition, interactions must be decomposed to determine what they mean
A significant interaction between two variables means that one factor value changes as a function of the other, but gives no specific information
The most simple and common method of interpreting interactions is to look at a graph
Multiple Comparisons in Factorial Experiments
Problems in factorial experiment
1. In some two-factor experiments the level of one factor, say B, is not really similar with the other factor.
2. There are multifactor experiments that address common economic and practical constraints encountered in experimentation with real systems.
3. There is no link from any sites on one area to any sites on another area.
Nested and Split-plot design
Cross and nested The levels of factor A are said to be crossed
with the level of factor B if every level of A occurs in combinations with every level of B
Factorials design The levels of factor B are said to be nested
within the level of factor A if the levels of B can be divided into subsets (nests) such that every level in any given subset occurs with exactly one level of A
Nested design
Fertilizers can be applied to individual fields;Insecticides must be applied to an entire farm
from an airplane
Agricultural Field Trial Investigate the yield of a new variety of crop Factors
• Insecticides• Fertilizers
Experimental Units• Farms• Fields within farms
Experimental Design ?
Agricultural Field Trial
Insecticides applied to farms One-factor ANOVA
Main effect: Insecticides MSE: Farm-to-farm
variability
Farms
Agricultural Field Trial Fertilizers
applied to fields
One-factor ANOVAMain Effect:
FertilizersMSE: Field-to-
field variability
Fields
Agricultural Field Trial
Insecticides applied to farms, fertilizers to fields
Two sources of variability Insecticides subject to
farm-to-farm variability Fertilizers and
insecticides x fertilizers subject to field-to-field variability
Farms
Fields
Nested DesignFactorial design when the levels of one factor (B) are similar, but not identical to each other
at different levels of another factor (A).
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Nested Design
Nested Design A factor B is considered nested in
another factor, A if the levels of factor B differ for different levels of factor A.
The levels of B are different for The levels of B are different for different levels of A.different levels of A.
Synonyms indicating nesting:Hierarchical, depends on, different for,
within, in, each
Examples - Nested
Examples - Nested
Examples - Crossed5-1 The yield of a chemical process is being studied. The two most important variables are thought to be the pressure and the temperature. Three levels of each factor are selected, and a factorial experiment with two replicates is performed. The yield data follow:
Pressure Temperature 200 215 230
150 90.4 90.7 90.2 90.2 90.6 90.4
160 90.1 90.5 89.9 90.3 90.6 90.1
170 90.5 90.8 90.4 90.7 90.9 90.1
Examples - Crossed5-2 An engineer suspects that the surface finish of a metal part is influenced by the feed rate and the depth of cut. She selects three feed rates and four depths of cut. She then conducts a factorial experiment and obtains the following data:
Depth of Cut (in) Feed Rate (in/min) 0.15 0.18 0.20 0.25
74 79 82 99 0.20 64 68 88 104
60 73 92 96 92 98 99 104
0.25 86 104 108 110 88 88 95 99 99 104 108 114
0.30 98 99 110 111 102 95 99 107
Examples - Nested
Two-Stage Nested DesignStatistical Model and ANOVA
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• Fixed.Fixed.
Expected Mean SquaresExpected Mean SquaresAssume that fixtures and layouts are fixed,
operators are random – gives a mixed model (use restricted form).
Alternative Analysis If the need detailed analysis is not available, start with multi-factor
ANOVA and then combine sum of squares and degrees of freedom. Applicable to experiments with only nested factors as well as
experiments with crossed and nested factors.Sum of squares from interactions are combined with the sum of squares for a
nested factor – no interaction can be determined from the nested factor.
Alternative Analysis
Split-Plot Design
Factorial experiment can have either of these features:Two hierarchically nested factors, with additional crossed factors occurring within levels of the nested factorTwo sizes of experimental units, one nested within the other, with crossed factors applied to the smaller units
In a single factor experiment has different features, such as:1.Multi-locations2.Repeated measurements
Further phenomena in Experimental Design
Split-plot DesignThere are numerous types of split-plot designs, including the Latin square split plot design, in which the assignment of the main treatments to the main plots is based on a Latin square design.A split-plot design can be conceptualized as consisting of two designs: a main plot design and a subplot design.The main plot design is the protocol used to assign the main treatment to the main units. In a completely randomized split-plot design, the main plot design is a completely randomized design, in a randomized complete block design, by contrast, the main plot design is a RCBD.The subplot design in a split-plot experiment is a collection of a RCBD, where a is the number of main treatment. Each of these RCBDs has b treatments arranged in r blocks (main plots), where b is the number of sub treatment.
Split-Plot Design
Whole-Plot Experiment : Whole-Plot Factor = A
Level a1 Level a2 Level a2 Level a1
Split Plot DesignsAnalysis of Variance Table
Source dfWhole-Plot Analysis
Factor A a-1Whole-Plot Error a(r-1)
Split-Plot DesignSplit-Plot Design
Split-Plot Experiment : Split-Plot Factor = B
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Split Plot DesignsSplit Plot DesignsAnalysis of Variance TableAnalysis of Variance Table
Source dfWhole-Plot Analysis
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Split-Plot AnalysisFactor B b-1A x B (a-1)(b-1)Split-Plot Error a(b-1)(r-1)Total abr-1
Agricultural Field Trial
Agricultural Field TrialAgricultural Field TrialInsecticide 2 Insecticide 2
Insecticide 2
Insecticide 1 Insecticide 1
Insecticide 1
Agricultural Field TrialAgricultural Field Trial
Fert B Fert AFert A
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Insecticide 2 Insecticide 2
Insecticide 2
Insecticide 1 Insecticide 1
Insecticide 1
Agricultural Field TrialAgricultural Field Trial
Whole Plots = Farms
Split Plots = Fields
Large Experimental Units
Small Experimental Units
Agricultural Field TrialAgricultural Field Trial
Whole Plots = Farms
Split Plots = Fields
Large Experimental Units
Small Experimental Units
Whole-Plot Factor = InsecticideWhole-Plot Error = Whole-Plot Replicates
Split-Plot Factor = FertilizerSplit-Plot Error = Split-Plot Replicates
The Split-Plot Designa multifactor experiment where it is
not practical to completely randomize the order of the runs.
Example – paper manufacturing• Three pulp preparation methods.• Four different temperatures.• The experimenters want to use three
replicates.• How many batches of pulp are
required?
The Split-Plot Design Pulp preparation method is a hard-to-
change factor. Consider an alternate experimental design:
• In replicate 1, select a pulp preparation method, prepare a batch.
• Divide the batch into four sections or samples, and assign one of the temperature levels to each.
• Repeat for each pulp preparation method.• Conduct replicates 2 and 3 similarly.
The Split-Plot Design Each replicate has been divided into three
parts, called the whole plots.• Pulp preparation methods is the whole plot
treatment. Each whole plot has been divided into four
subplots or split-plots.• Temperature is the subplot treatment.
Generally, the hard-to-change factor is assigned to the whole plots.
This design requires 9 batches of pulp (assuming three replicates).
The Split-Plot Design
Tensile StrengthRep (Day) 1 Rep (Day) 2 Rep (Day) 3
Pulp Prep Method 1 2 3 1 2 3 1 2 3Temperature
200 30 34 29 28 31 31 31 35 32225 35 41 26 32 36 30 37 40 34250 37 38 33 40 42 32 41 39 39275 36 42 36 41 40 40 40 44 45
The Split-Plot DesignThere are two levels of randomization
restriction.• Two levels of experimentation
Tensile StrengthRep (Day) 1 Rep (Day) 2 Rep (Day) 3
Pulp Prep Method 1 2 3 1 2 3 1 2 3Temperature
200 30 34 29 28 31 31 31 35 32225 35 41 26 32 36 30 37 40 34250 37 38 33 40 42 32 41 39 39275 36 42 36 41 40 40 40 44 45
Experimental Units in Split Plot Designs
Possibilities for executing the example split plot design.• Run separate replicates. Each pulp prep method
(randomly selected) is tested at four temperatures (randomly selected).
Large experimental unit is four pulp samples. Smaller experimental unit is a an individual sample.
• If temperature is hard to vary select a temperature at random and then run (in random order) tests with the three pulp preparation methods.
Large experimental unit is three pulp samples. Smaller experimental unit is a an individual sample.Tensile Strength
Rep (Day) 1 Rep (Day) 2 Rep (Day) 3Pulp Prep Method 1 2 3 1 2 3 1 2 3
Temperature200 30 34 29 28 31 31 31 35 32225 35 41 26 32 36 30 37 40 34250 37 38 33 40 42 32 41 39 39275 36 42 36 41 40 40 40 44 45
The Split-Plot Design Another way to view a split-plot Another way to view a split-plot
design is a RCBD with replication.design is a RCBD with replication.• Inferences on the blocking factor can be Inferences on the blocking factor can be
made with data from replications.made with data from replications.
The Split-Plot Design Model The Split-Plot Design Model and Statistical Analysisand Statistical Analysis
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There are two error structures; the whole-plot error and the subplot error
Split-Plot Design
Whole-Plot Experiment : Whole-Plot Factor = A
Level a1 Level a2 Level a2 Level a1
Split-Plot DesignSplit-Plot Design
Split-Plot Experiment : Split-Plot Factor = B
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Split-Plot DesignSplit-Plot Design
Split-Plot Experiment : Split-Plot Factor = B
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