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```					Analysis of Variance for
Standard Designs

Chapter 15
Completely Randomized
Design with a Single Factor
Table 15.3: A Completely
Randomized Design

Treatment                        Mean
1           y11   y12    y1n1       y1.
2           y21   y22    y2 n2      y2.

t           yt1   yt 2   ytnt       yt .
The Model for a Completely
Randomized Design
yij     i   ij
i  1, ..., t ; j  1, ..., ni

where

yij :   Observation on j th experimental unit
receiving treatment i
     : Overall treatment mean, an unknown
constant
The Model for a Completely
Randomized Design, cont.

i     : An effect due to treatment i, an unknown
constant
 ij: A random error associated with the response
from the j th experimental unit receiving
treatment i. We require that the ij s have a
normal distribution with mean 0 and common
variance   . In addition, the errors must be
2

independent.
Sums of Squares
Total Sum of Squares (TSS)


TSS    yij  y.. 
2

ij

   Partition of TSS

 y        y..    ni  yi.  y..     yij  yi. 
2                    2                     2
ij
ij                    i                    ij
Sums of Squares, cont.
Between-treatment sum of squares (SST)


SST  n  yi.  y.. 
2

i

Sum of squares for error (SSE)


SSE=  yij  yi. 
2

ij
Table 15.4: Analysis of Variance Table
for a Completely Randomized Design

Source       SS    df         MS                 F

Treatments   SST   t 1   MST=SST/  t  1    MST/MSE
Error        SSE   N t   MSE=SSE/  N  t 
Total        TSS   N 1
Unbiased Estimates

When                        is true, both MST
and MSE are unbiased estimates of      , the
variance of the experimental error. That is,
under Ho, both have a mean value in
repeated sampling, called the expected
mean squares, equal to        .
Expected Mean Squares

Under   Ha   ,

   E  MSE       2

   E  MST      nT
2
Randomized Design

1. The design is extremely easy to construct.
2. The design is easy to analyze even though
the sample sizes might not be the same for
each treatment.
3. The design can be used for any number of
treatments.
Randomized Design

1. Although the completely randomized design can be
used for any number of treatments, it is best suited
for situations in which there are relatively few
treatments.
2. The experimental units to which treatments are
applied must be as homogeneous as possible. Any
extraneous sources of variability will tend to inflate
the error term, making it more difficult to detect
differences among the treatment means.
Randomized Complete Block
Design
Location
1         2        3           4
P1       P2       P3      P4
Confounded factors:
P1       P2       P3      P4
location & type of paint
P1       P2       P3      P4
P1       P2       P3      P4

Table 15.6: Random
assignment of 4 paints to 16
Randomized Complete Block
Design, cont.
Table 15.7: Randomized              Location
complete block assignment of
4 paints to 16 sections of     1     2     3    4
P1   P4    P3   P2
P3   P1    P4   P4
P4   P3    P2   P3
Definition

A randomized complete block design is an
experimental design for comparing t treatments
in b blocks. The blocks consist of t
homogeneous experimental units. Treatments
are randomly assigned to experimental units
within a block, with each treatment appearing
exactly once in every block.
Complete Block Design

1. The design is useful for comparing t
treatment means in the presence of a single
extraneous source of variability.
2. The statistical analysis is simple.
3. The design is easy to construct.
4. It can be used to accommodate any number
of treatments in any number of blocks.
Complete Block Design

1. Because the experimental units within a block must
be homogeneous, the design is best suited for a
relatively small number of treatments.
2. This design controls for only one extraneous source
of variability (due to blocks). Additional extraneous
sources of variability tend to increase the error term,
making it more difficult to detect treatment
differences.
3. The effect of each treatment on the response must
be approximately the same from block to block.
Figure 15.1: Treatment Means in
a Randomized Block Design
Plot of Treatment Mean by Treatment
100
3
90              4           3
80              1           4

70   3          2           1
4                      2
60
1
50
2
40

1           2         3
Treatment
The Hypotheses for Testing
Treatment Mean Differences
The null hypothesis is no difference among
treatment means versus the research
hypothesis treatment means differ.
Total Sum of Squares

Partition of TSS:
Table 15.11: Analysis of Variance
Table for a Randomized Complete
Block Design

Source   SS   df    MS                  F
Expected Mean Squares
When                          is true, both MST
and MSE are unbiased estimates of        ,
the variance of the experimental error.
Relative Efficiency
 RE(RCB, CR): the relative efficiency
of the randomized complete block
design compared to a completely
randomized design
 Did blocking increase our precision
for comparing treatment means in a
given experiment?
Latin Square Design
Secretary                        Secretary
Problem    1      2   3    4     Problem    1      2   3    4
I         A      A    C    A     I         A      B    C    D
II        B      D    A    D     II        B      C    D    A
III       D      B    D    B     III       C      D    A    B
IV        C      C    B    C     IV        D      A    B    C
Table 15.14: A Randomized       Table 15.15: A Latin Square
Complete Block Design for the   Design for the Spreadsheet
Square Design

1. The design is particularly appropriate for
comparing t treatment means in the
presence of two sources of extraneous
variation.
2. The analysis is quite simple.
Square Design

1. Although a Latin square can be constructed for any
value of t, it is best suited for comparing t
treatments when
2. Any additional extraneous sources of variability tend
to inflate the error term, making it more difficult to
detect differences among the treatment means.
3. The effect of each treatment on the response must
be approximately the same across rows and
columns.
Definition 15.2

A t x t Latin square design contains t rows
and t columns. The t treatments are randomly
assigned to experimental units within the rows
and columns so that each treatment appears in
every row and in every column.
Test for Treatment Effects

We can test specific hypotheses concerning
the parameters in our model. In particular, we
may wish to test for differences among the t
treatment means.
Table 15.17: Analysis of Variance
Table for a t x t Latin Square Design

Source   SS   df     MS               F
Relative Efficiency

RE(LS, CR): the relative efficiency of the
Latin square design compared to a completely
randomized design
Did accounting for row/column sources of
variability increase our precision in estimating
the treatment means?
Factorial Treatment Structure in a
Completely Randomized Design

A factorial experiment is an experiment in
which the response y is observed at all factor-
level combinations of the independent variables.
Figure 15.6a: Illustration of the
Absence of Interaction in a 2 x 2
Factorial Experiment
Mean response

Level 1, factor B
Level 2, factor B

Level 1   Level 2

Factors A and B do not interact
Figure 15.6b,c: Illustration of the
Presence of Interaction in a 2 x 2
Factorial Experiment
Mean response

Level 2, factor B

Level 1, factor B

Level 1   Level 2
Factors A and
B interact
Mean response

Level 1, factor B
Level 2, factor B

Level 1   Level 2
Table 15.25: Expected Values for
a 2 x 2 Factorial Experiment

Factor B
Factor A   Level 1          Level 2
Table 15.26: Expected Values for a
2 x 2 Factorial Experiment, with
Replications

Factor B
Factor A   Level 1          Level 2
Definition 15.4

Two factors A and B are said to interact if the
difference in mean responses for two levels of
one factor is not constant across levels of the
second factor.
Profile Plot

See Figure 15.6
 Used to amplify the notion of
interaction: when no interaction is
present, the difference in the mean
response between two levels of one
factor is the same for levels of the other
factor.
Table 15.27: AOV Table for a
Completely Randomized Two-Factor
Factorial Experiment

Source   SS   df   MS             F
Illustration of Significant,
Orderly Interaction
Figure 15.8:
Profile plot in              100                                      Level 3, factor B
which interactions                   90
are present, but
Mean response
80
interactions are
orderly                              70
Level 2, factor B
60
50
Level 1, factor B
40

Level 1 Level 2 Level 3 Level 4
Factor A
Illustration of Significant,
Disorderly Interaction
Figure 15.9:
Profile plot in              100
which interactions                   90
are present, and
Mean response
80                               Level 2, factor B
interactions are
disorderly                           70                               Level 1, factor B
60
50
40                               Level 3, factor B

Level 1 Level 2 Level 3 Level 4
Factor A
Factorial Treatment Structure in a
Randomized Complete Block Design
Estimation of Treatment Differences
and Comparisons of Treatment
Means
100(1-)% Confidence Interval for the
Difference in Treatment Means

where s is the square root of MSE in the AOV table
and t/2 can be obtained from Table 2 in the Appendix
for a = /2 and the degrees of freedom for MSE.
Multiple Comparison
Procedures

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 views: 37 posted: 6/29/2012 language: English pages: 43