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Multiple Comparisons in Factorial Experiments

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Title: Multiple Comparisons in Factorial Experiments


1
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) Sh
ould instead perform multiple comparisons among
all factorial means of interest.
2
Multiple Comparisons in Factorial Experiments
  • 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

3
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
4
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

5
Agricultural Field Trial
  • Investigate the yield of a new variety of crop
  • Factors
  • Insecticides
  • Fertilizers
  • Experimental Units
  • Farms
  • Fields within farms

Experimental Design ?
Fertilizers can be applied to individual
fields Insecticides must be applied to an entire
farm from an airplane
6
Agricultural Field Trial
Farms
  • Insecticides applied to farms
  • One-factor ANOVA
  • Main effect Insecticides
  • MSE Farm-to-farm variability

7
Agricultural Field Trial
  • Fertilizers applied to fields
  • One-factor ANOVA
  • Main Effect Fertilizers
  • MSE Field-to-field variability

Fields
8
Agricultural Field Trial
Farms
Fields
  • 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

9
Nested Design
  • Factorial design when the levels of one factor
    (B) are similar, but not identical to each other
    at different levels of another factor (A).

b1
b3
a1
a2
b2
b4
10
Nested Design
11
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 different
    levels of A.
  • Synonyms indicating nesting
  • Hierarchical, depends on, different for, within,
    in, each

12
Examples - Nested
13
Examples - Nested
14
Examples - Crossed
15
Examples - Crossed
16
Examples - Nested
17
Two-Stage Nested DesignStatistical Model and
ANOVA
18
Two-Stage Nested DesignStatistical Model and
ANOVA
19
Residual Analysis
  • Calculation of residuals.

20
m-Stage Nested Design
21
m-Stage Nested Design
  • Test statistics depend on the type of factors and
    the expected mean squares.
  • Random.
  • Fixed.

22
Expected Mean Squares
Assume that fixtures and layouts are fixed,
operators are random gives a mixed model (use
restricted form).
23
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.

24
Alternative Analysis
25
Split-Plot Design
Further phenomena in Experimental Design
  • In a single factor experiment has different
    features, such as
  • Multi-locations
  • Repeated measurements
  • Factorial experiment can have either of these
    features
  • Two hierarchically nested factors, with
    additional crossed factors occurring within
    levels of the nested factor
  • Two sizes of experimental units, one nested
    within the other, with crossed factors applied to
    the smaller units

26
Split-plot Design
There 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.
27
Split-Plot Design
Whole-Plot Experiment Whole-Plot Factor A
Level a1
Level a2
Level a2
Level a1
28
Split Plot DesignsAnalysis of Variance Table
29
Split-Plot Design
Split-Plot Experiment Split-Plot Factor B
b2
b1
b1
b2
b1
b1
b2
b1
b2
b2
b2
b1
b1
b2
b1
b2
Level a1
Level a2
Level a2
Level a1
30
Split Plot DesignsAnalysis of Variance Table
31
Agricultural Field Trial
32
Agricultural Field Trial
Insecticide 2
Insecticide 1
Insecticide 2
Insecticide 2
Insecticide 1
Insecticide 1
33
Agricultural Field Trial
Insecticide 2
Insecticide 1
Insecticide 2
Fert A
Fert B
Fert A
Fert B
Fert B
Fert A
Fert B
Fert A
Fert B
Fert A
Fert B
Fert A
Fert A
Fert B
Insecticide 2
Fert A
Fert A
Fert B
Fert A
Fert B
Fert B
Fert B
Fert A
Fert B
Fert A
Fert B
Fert A
Fert B
Fert B
Fert A
Fert A
Fert B
Fert A
Fert A
Fert A
Fert A
Fert B
Fert B
Fert B
Fert A
Fert B
Insecticide 1
Insecticide 1
34
Agricultural Field Trial
Whole Plots Farms
Large Experimental Units
Split Plots Fields
Small Experimental Units
35
Agricultural Field Trial
Whole Plots Farms
Large Experimental Units
Whole-Plot Factor Insecticide Whole-Plot Error
Whole-Plot Replicates
Split Plots Fields
Small Experimental Units
Split-Plot Factor Fertilizer Split-Plot Error
Split-Plot Replicates
36
The Split-Plot Design
  • a 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?

37
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.

38
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).

39
The Split-Plot Design
40
The Split-Plot Design
  • There are two levels of randomization
    restriction.
  • Two levels of experimentation

41
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.

42
The Split-Plot Design
  • Another way to view a split-plot design is a RCBD
    with replication.
  • Inferences on the blocking factor can be made
    with data from replications.

43
The Split-Plot Design Model and Statistical
Analysis
Sum of squares are computed as for a three factor
factorial design without replication.
44
RCBD Model
45
The Split-Plot Design Model and Statistical
Analysis
There are two error structures the whole-plot
error and the subplot error
46
Split-Plot Design
Whole-Plot Experiment Whole-Plot Factor A
Level a1
Level a2
Level a2
Level a1
47
Split-Plot Design
Split-Plot Experiment Split-Plot Factor B
b2
b1
b1
b2
b1
b1
b2
b1
b2
b2
b2
b1
b1
b2
b1
b2
Level a1
Level a2
Level a2
Level a1
48
Split-Plot Design
Split-Plot Experiment Split-Plot Factor B

b1
b1
b2
b1
b1
b2
b1
b2
b2
b2
b1
b1
b2
b1
b2
Level a1
Level a2
Level a2
Level a1
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