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PPT – Multiple Comparisons in Factorial Experiments PowerPoint presentation | free to download - id: 56c8f8-NDViN

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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.

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

Problems in factorial experiment

- In some two-factor experiments the level of one

factor, say B, is not really similar with the

other factor. - There are multifactor experiments that address

common economic and practical constraints

encountered in experimentation with real systems. - 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

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

Agricultural Field Trial

Farms

- Insecticides applied to farms
- One-factor ANOVA
- Main effect Insecticides
- MSE Farm-to-farm variability

Agricultural Field Trial

- Fertilizers applied to fields
- One-factor ANOVA
- Main Effect Fertilizers
- MSE Field-to-field variability

Fields

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

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

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 different

levels of A. - Synonyms indicating nesting
- Hierarchical, depends on, different for, within,

in, each

Examples - Nested

Examples - Nested

Examples - Crossed

Examples - Crossed

Examples - Nested

Two-Stage Nested DesignStatistical Model and

ANOVA

Two-Stage Nested DesignStatistical Model and

ANOVA

Residual Analysis

- Calculation of residuals.

m-Stage Nested Design

m-Stage Nested Design

- Test statistics depend on the type of factors and

the expected mean squares. - Random.
- Fixed.

Expected Mean Squares

Assume 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

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

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.

Split-Plot Design

Whole-Plot Experiment Whole-Plot Factor A

Level a1

Level a2

Level a2

Level a1

Split Plot DesignsAnalysis of Variance Table

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

Split Plot DesignsAnalysis of Variance Table

Agricultural Field Trial

Agricultural Field Trial

Insecticide 2

Insecticide 1

Insecticide 2

Insecticide 2

Insecticide 1

Insecticide 1

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

Agricultural Field Trial

Whole Plots Farms

Large Experimental Units

Split Plots Fields

Small Experimental Units

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

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?

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

The Split-Plot Design

- There are two levels of randomization

restriction. - Two levels of experimentation

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.

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.

The Split-Plot Design Model and Statistical

Analysis

Sum of squares are computed as for a three factor

factorial design without replication.

RCBD Model

The Split-Plot Design Model and Statistical

Analysis

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 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

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