Designs and Factorial Structures
We have been discussing designs in general and I would
like to lay out the major designs that we will use in this
class. I will use a very simple example:
I am interested in comparing mean yield of 3 (V1, V2, V3)
varieties of wheat. To do the experiment I have 9 plots of
land available.
When the layout is as follows:
At this point we assume that our 9 plots of land are
homogenous.
How to randomize?
We can random numbers between 1 and 9.
Lets say, 7, 2,
4,
3,
8,
1,
6,
9,
5.
V1 V1 V1 V2 V2 V2 V3 V3 V3
So our design might look as follows:
1 V2
2 V1
3 V2
4 V1
5 V3
6 V3
7 V1
8 V2
9 V3
This is called a COMPLETELY RANDOMIZED
DESIGN (CRD). HERE the randomization is
COMPLETE here without any constraints.
This is the simplest design possible and easiest to analyze.
This is also the most desirable design. However, it
requires homogenous units which is often not possible in
practice.
Let us consider a slight variation of this:
Consider the situation if you knew that a River was
flowing at the bottom of the plots as the picture denotes.
Of course we do not have “homogenous units” any more.
So we cannot do an unconstrained randomization on all 9
units. But we can do it on the units of same variability or
groups of homogenous units (blocks). A possible lay out
might be:
Block 1
Block 2
Block 3
V1
V2
V1
V3
V3
V2
V2
V1
V3
River
Here we do not have “homogenous units” but have groups
or block of units that are homogenous. So we randomize
within a BLOCK. So within a BLOCK we randomize as
if it was a CRD.
This is called a RANDOMIZED COMPLETE BLOCK
DESIGN (RCBD)
This is also quite easy to analyze and quite a desirable
DESIGN. This is widely used in practice.
Now let us consider that there is a slope going down on
the plots, i.e the plots are on a hill side:
A possible way to take care of the block effects (River
and slope) is as follows:
slope
V1
V2
V3
V3
V1
V2
V2
V3
V1
River
Here each treatment appears once in each block and all
treatments appear in every block.
Again we need to do a more constrained randomization.
This type of design is called a LATIN SQUARE DESIGN
(LSD).
Not used much in practice, gaining popularity because of
SUDOKU.
Now, let us consider the scenario that we are back to the
set- up of having a plot of 9 homogenous plots but now
we are interested in looking at yield over the next 3 years.
Since each plot of land is looked at repeatedly (so no
randomization is done on the years) this is called
REPEATED Measures design.
Year1, Year 2, Year 3
V1
V3
V2
V1
V2
V1
V3
V3
V2
Again this is constrained randomization as the structure of
the design does not change over the years. A very
commonly used design as often feasibility does not allow
us to re-randomize each year. This is fairly easy to
analyze.
Now let us consider a bit more realistic situation. In this
case, suppose the treatment structure consisted of three
(3) varieties of wheat ( V , V and V ) as before, each planted
on four (3) randomly selected farms (large experimental
units). The response for this experiment is wheat yield in
bushels per acre. This design layout might appear as
follows:
1
2
3
1 V2
4 V1
7 V1
2 V1
5 V3
8 V2
3 V2
6 V3
9 V3
However, the researcher might also be interested in the
effects of two different fertilizers ( F and F ) on yield. The
CRD presented earlier can be modified by splitting each
farm (exp. unit) in half and then randomly assigning the
fertilizers: one fertilizer to each half experimental unit
(sub-unit). This modified design might appear as follows:
1
2
1 V2F1 1V2F2 4 V1F2 4 V1F1 7 V1F1 7 V1F2
2 V1F2 2V1F1 5 V3F1 5 V3F2 8 V2F2 8 V2F1
3 V2F2 3V2F1 6 V3F2 6 V3F2 9 V3F1 9 V3F2
In this experiment there are two different sizes of
experimental units: the large units are the farms; and the
small units (sub-units) are the half-farms.
This type of design with constrained randomization is
called a SPLIT PLOT DESIGN.
Randomized Incomplete Block Designs. These designs
are similar to the RCBD except that each block contains
fewer than t units and thus, only a portion of the t
treatments are applied to the units within each block.
There are several arrangement structures for application
of treatments within blocks:
blocks
1
1, V1
2
3
5, V2
6, V2
3, V1
10, V3
11, V3
Balanced Incomplete Block Designs (BIBD); and
Partially Balanced Incomplete Block Designs (PBIB).
Other than these we will also be looking at STRIP PLOT
designs, Cross-over designs and Unbalanced designs over
and above factorial designs.
I WOULD like to cover if time permits concepts of
optimality in a design context. Which design is the best
in any given scenario?
The point to understand is that in most of the situations
we would LIKE a CRD. However, we might have to use
a more complicated Design because of practical concerns.
For example, if the units are NOT homogenous and you
assume them to be, you have made your random errors
bigger than what you need for it to be. Since in most
ANOVA scenarios the error is the denominator of the F
statistic, we may not be able to detect significance, or
have a LOSS of power.
So the point is, if you have non-homogenous units, or
other constraints design your experiment in such a way
that you take care of these issues.
Once, you have designed your experiment the specific
design will dictate the type of model you have, so the type
of ANOVA you use.
So we want you to start thinking about what YOUR
model is for a particular design and so we can write out
the correct model. The model will dictate the type of
ANOVA you perform (One-way, Two-way, Factorial,
with fixed, random factors).
Recall the hierarchy here:
The constraints in your units for your experiment
determines the DESIGN, this dictates the model, which in
turn dictates the type of analysis used.
Experiment
ANALYSIS.
DESIGN
MODEL
So our next step will be modeling so we first need to talk
about the general ANOVA model using multiple
FACTORS.
But we need to set up some definitions and terms for now,
that will allow us to look at general models and pursue
ANOVA.
Fixed and Random Factors
Factor: any substance or item whose effect on the data is
to be studied. An experiment involving two or more
treatment is called factorial experiment.
Levels: values of the factor used in the experiment. The
levels of a factor are the specific types or amounts of the
factor that will actually be used in the experiment.
The levels can be fixed or random. This allows us to have
FIXED or RANDOM factors.
FIXED FACTOR: We are inherently interested in the
levels of this type of FACTOR and follow up our
significant analysis by multiple comparisons. Here we
compare the MEANS of the levels.
RANDOM FACTOR: We are not inherently interested in
the levels of the factor but the levels represent a random
selection of levels from a large number of potential levels.
We are more interested in seeing if there is an overall
difference. We are not interested in specific differences
among the levels. Here we are interested in the variability
among levels.
Definitions
o A factor is random if its levels consist of a random
sample from a population of possible samples.
o A factor is fixed if its levels are selected by a nonrandom process or if its levels consist of the entire
population of possible levels.
o A factor A is said to be crossed with respect to a
second factor, B, if each level of factor A is exactly
the same for each level of factor B. Otherwise, the
factor is said to be nested.
Fixed Effects Models
o Interested in only the t treatments used
o Desire to make inferences about parameters 1 , 2
,…, t
o Conclusions refer only to the t conditions
o If the experiment is repeated, the same levels for the
treatments would be selected
Random Effects Models
o Interested in a population of treatments
o Desire to make inference about  2

o Conclusions refer to the population of treatments
o If the experiment is repeated, a different random
sample of treatment levels would be selected.
Example 1 - A Fixed Effect:
Consider a genetics study beef cattle where the primary
interest is the assessment of sire contributed traits to meat
quality. Most quality sires of a specific breed of cattle
contribute disproportionately to the next generation
through artificial insemination of the dams. For the study
in question, the experimental protocol involves only 3
sires (factor Sire: levels Sire1, Sire2 and Sire3), each of
which is mated to three randomly selected dams (factor
Dams: levels Dam1, Dam2 and Dam3). Total of 9 dams
were used. These dams were randomly selected from a
very large population of dams.
In this example, the sires were of specific interest, and
therefore constitute the entire population. Sires are a fixed
effect.
The dams were chosen at random from a larger
population of dams. Dam is a random effect.
The factor Dam is nested within Sires because Dam1 for
Sire1 is not the same dam as Dam1 for Sire2, and so on.
The experimental design is simply a completely
randomized design with a one-way treatment structure:
Sire is the treatment factor and Dams are the replications.
Therefore, the model is:
𝑌𝑖𝑗 = 𝜇 + 𝛼𝑖 + 𝜀(𝑖)𝑗
where m is the overall mean
ai is the fixed factor due to Sire.
e(i)j is the random error effects nested within the ith
level of the fixed factor.
However, in general we don’t make a distinction of the
nested effects for replication and so the model is written
as:
𝑌𝑖𝑗 = 𝜇 + 𝛼𝑖 + 𝜀𝑖𝑗
Which as we know is the model for I way ANOVA
Now, before we write out the ANOVA table we need to
discuss Expected Mean Square Errors.
Recall:
2
2
2
 (Yij  Y.. )   (Yi.  Y.. )   (Yij  Yi. )
TotalSS = SSFactor + SSError
MSE= SSE/(t-1)
MSF=SSF/(N-t)
Now, if you work out the expectation (this is an exercise
in Math Stat) you will find:
E(MSE) = 2
2
2


n
(

 i i ) /( r  1)
E(MSF)=
Under the null hypothesis the τi are all 0 so the second
𝑀𝑆𝐹
𝑀𝑆𝐸
𝑀𝑆𝐹
term drops out. So
÷ (
=
which follows
)
𝐸(𝑀𝑆𝐹)
𝐸 𝑀𝑆𝐸
𝑀𝑆𝐸
F.
Hence the corresponding ANOVA table will be:
Theoretical ANOVA Table with Expected Mean
Squares
Source
df
Expected Mean Square
Sire(F)
Dam (Error)
Total
3- 1=2
3(3- 1)=6
33 – 1=8
  2   ni ( i 2 ) /( r  1)
 e2
Hence we do our usual one-way ANOVA in this set-up.
Testing the hypothesis of no variation.
Ho: k=(0)
Ha: at least one inequality
𝐹=
𝑀𝑆𝐹
𝑀𝑆𝐸
Reject Ho if
F
0 >F
α,t-1,tr-1
or P-value < α
Example 2 - A Random Effect:
A manufacturer of cloth has a factory consisting of a large
number of looms which produce the same cloth fabric. A
quality control program is in place to maintain the quality
of the cloth produced by the looms. If a loom produces an
inferior quality fabric, it must be shut down and the loom
re-tuned. It is known that re-tuning is necessary for all
looms if the variation in the fabric from all of the looms
reaches a threshold. Furthermore, it is very expensive to
test all looms in the factory, so an experimental design is
developed to assess the variation among looms.
For the study in question, the experimental protocol
involves a random sample of four looms. From each
loom, four randomly selected samples of fabric are
selected during production and the strength of the fabric
assessed.
In this example, the loom variation was of specific
interest, and therefore the four looms constitute a sample
from the population of looms. Loom is a random effect.
The fabric samples were chosen at random from a larger
population. Fabric is a random effect with fabric nested
within loom because fabric1 for loom1 is not the same
fabric1 for loom2, and so on. The data resulting from this
experiment can be found in the following table:
Strength Data for the Loom Example
Loom
Fabric
Sample
1
2
3
4
1
98
91
96
95
2
97
90
95
96
3
99
93
97
99
4
96
92
95
98
A Statistical Model for Variance Components
The objective of the analysis of a random effects model is
to decompose the total variance into identifiable
components. Hence, another name for these models is
variance components models.
𝑌𝑖𝑗 = 𝜇 + 𝑎𝑖 + 𝜀(𝑖)𝑗
where m is the overall mean
ai is the random factor.
e(i)j is the random error effects nested within the ith
level of the random factor.
However, in general we don’t make a distinction of the
nested effects for replication and so the model is written
as:
𝑌𝑖𝑗 = 𝜇 + 𝑎𝑖 + 𝜀𝑖𝑗
The only difference of this model and the one we saw
before is that we use ROMAN letter a to denote the loom
effect as opposed to GREEK letter a, which is used to
denote a FIXED effect. We assume a distribution on “a”.
Assumptions:
ai are independent and identically distributed normal
random variables with mean zero and variance 𝜎𝑎2 .
𝜀𝑖𝑗 are independent and identically distributed normal
random variables with mean zero and variance 𝜎𝜀2 .
ANOVA Table of Expected Mean Squares
Source
df
Expected Mean Square
Random Effect. t - 1
e2 + r a2
Error
t(r - 1)
 e2
Total
tr - 1
Testing the hypothesis of no variation.
2
2
H
:


0
v
s
.
H
:

0
0 a
a a
𝐹=
𝑀𝑆𝐹
𝑀𝑆𝐸
Reject Ho if
F
0 >F
α,t-1,tr-1
or P-value < α
Example1: (cont)
Loom Example ANOVA Table
Source
Loom.
Error
Total
Sum of
Squares
89.19
22.75
111.94
df
Mean
F
Square
______________
4-1=3
29.73
15.68
4(4-1)=12 1.90
44-1=15
Fo = 15.68
Reject Ho if Fo > F(0.05, 3, 12) = 3.49
Conclusion: Reject Ho and assume that there is sufficient
evidence to conclude that the variation among looms is
not equal to zero. There is probable cause to retune the
population of looms.
So the technique of doing the ANOVA is not different.
Its just the assumptions that are different in fixed versus
random effect.
Also for random effects we DO NOT follow up with
Multiple Comparison methods.