Experimental Design and
Analysis of Variance: Basic
Design
M. Ali Akbar
Professor
Fakulti of Agro Industri dan Sumber Asli
Meaning of the term “Experiment”
Experiment: Planned research conducted for
certain period to obtain new facts or to confirm
or refute the results of previous study
Characteristics of Experiment
• Needs a hypothesis to be nullified
• Involves observation, collection and measurements
of data on the experimental object (s) & its
analyses and interpretation
• Frequently needs controlled environment for data
collection/observation
• Needs appropriate designing to get authentic
results
• Helps researchers to get an answer to some
questions or to make an inference about some
phenomenon
Steps in constructing an experiment:
• Selection of problem based on current issues and its
rationale
• Gathering information related to the problem
• Setting the hypothesis and the objectives
• Planning and designing the experiment
• Defining the materials and methods
• Commencement of work and collection of data and
information, analyses and interpretation of results
considering observations of other researchers
• Conclusions and/or recommendations on the findings
Experimental Units and Replications
• Experimental unit is a material to which the
treatment is applied; crop plots, cattle etc
Characteristics of exptl. unit is that they are
independent of one another
• Sample may be part of exptl. unit; no. of chicken in a
cage denotes chick as sample and cage as unit; again in
expt of weight gain of calves, calf is both experimental unit
as well as sample
• Replication is the application of treatment to more
than one experimental unit in an experiment
– They are several exptl. units all treated similarly
Experimental error
• Variability is the characteristics of biological
materials
• Total variability can be partitioned into or due to (a)
explained and (b) unexplained causes
• So observations in an exptl. unit differ because
they belong to explained & unexplained effects
• Experimental error is of two types: systematic and
random.
• Systematic error change the measurements in a
consistent way & comes from definite source e.g.
ununiformity in the units such as uncalibrated
instruments
• Random error occur due to random and
unpredictable phenomena
• They produce variability which can not be
explained
• In biological material there are always random
errors in measurements e.g. individual variation in
livestock sample
• Random errors can be minimised by increasing the
number of replicates
Experimental Design
The term experimental design refers to a plan for assigning
experimental units to treatment conditions in a systematic
way so that the results can give meaningful output.
It is not just describing the natural event through observation
and measurement, but it is more of treating the observation
and measurement in a planned way to illuminate the effect
of any change in conditions.
The importance of experimental design also stems from the
quest for inference about effect of causes or relationships
as opposed to simply description. Researchers are rarely
satisfied to simply describe the events they observe. They
want to make inferences about what produced, contributed
to, or caused events.
“Cause & Effect” Relationship
Important terminology
Treatment: A condition (or set conditions) that is imposed on
a group of elements (subjects) by the experimenter is called
treatment
Variable: Any concept, or thing, or event that varies or can be
made to vary, and is related to the research can be called a
variable.
Dependent variable: The variable that is changed by the
change of another variable is called dependent variable
Independent variable: The variable that is not changed due
to change in another variable rather when it changes also
causes change in another variable
Randomisation: The procedure in which elements (subjects)
are assigned to different groups at random (without any bias)
Replication: Repetition of experimental units in a treatment
which is intended to minimise the experimental error is called
replication.
Null hypothesis: The assumption that there is no effect of
independent variable on the dependent one.
Factor: Factors can be classified as either controllable or
uncontrollable variables. For example, in cake making the
controllable factors are the ingredients for the cake and the
oven that the cake is baked in.
Levels: or settings of each factor. Examples include the oven
temperature setting and the amounts of sugar, flour, and eggs.
Response: or output of the experiment. In the case of cake
baking, the taste, consistency, and appearance of the cake
are measurable outcomes potentially influenced by the factors
and their respective levels.
Why Experimental Design?
• Obtain maximum information from fewest experiments
— minimize time spent in gathering data
• Quantify effects from different factors using analysis
• Determine if a factor’s effect is significant
—differences might be random variations caused by
– measurement errors
– parameters not controlled
A good experimental design serves three purposes:
Causation. It allows the experimenter to make causal
inferences about the relationship between independent
variables and a dependent variable.
Control. It allows the experimenter to rule out alternative
explanations due to the confounding effects of extraneous
variables (i.e., variables other than the independent variables).
Variability. It reduces variability within treatment conditions,
which makes it easier to detect differences in treatment
outcomes.
Principles of Design of Experiment
According to Prof. R. A. Fisher, the basic principles of design of
experiments are:
(a) Randomisation
(b) Replication and
(c) Error control
Randomisation and Replication are described earlier
Error control:
Although every experiment would provide an estimate of error
variance, it is not desirable to have large number of error. There are
measures to reduce error variance. This measure for reducing the
error variance in an experiment is called error control.
There are two methods of reducing error control: one is to make
experimental unit homogenous, the other is to form experimental
units into several homogenous groups called blocks
Important steps in designing an experiment
• The statement of problem should be clearly defined. The
experimenter must understand what to do and how to tackle
the problems
• Formulation of hypothesis should be done properly and thus
the method of collection of data can be determined
• Experiment should be conducted accordingly and proper
statistical techniques are to be applied on data
• Drawing of valid conclusion is a crucial part of the design of
experiment so careful consideration to be given to the validity
of the conclusion for the population of events to which they
are to apply
Basic Experimental Designs
1.
2.
3.
4.
5.
6.
7.
8.
9.
One-Shot
One-Group, Pre-Post
Static Group
Random Group
Pre-Post Randomised Group
Completely Randomised
Randomized Block
Factorial
Latin Square
One-shot
• The One-Shot is a design in which a group of subjects are
administered a treatment and then measured (or observed).
• Usually, with this design, an intact group of subjects is given the
treatment and then measured or observed.
• No attempt is made to randomly assign subjects to the groups,
nor does the design provide for any additional groups as
comparisons.
• Thus, one group will be given one treatment and one
"observation." This design is diagramed as follows: GP--T--O
• The One-Shot Design is highly useful as an inexpensive measure
of a new treatment of the group in question. If there is some
question as to whether any expected effects will result from the
treatment, then a one-shot may be an economical route.
• This design answers only one question and that is in reference to
post-treatment behaviour.
One-Group, Pre-Post
In this design, one group is given a pre-treatment measurement or
observation, the experimental treatment, and a post-treatment
measurement or observation. The post-treatment measures are
compared with their pre-treatment measures. This design is
diagramed as follows: GP--O--T--O
The usefulness of this design is similar to that of one-shot, except
that an additional class of information is provided, i.e., pretreatment condition or behaviuor. This design is frequently used in
clinical and education research to determine if changes occurred. It
is typically analyzed with a matched pairs t-test.
Static Group
In this design, two intact groups are used, but only one of them is
given the experimental treatment. At the end of the treatment, both
groups are observed or measured to see if there is a difference
between them as a result of the treatment. The design is
diagrammed as follows: GP--T—O, GP------O
• This design may provide information on some rival hypotheses.
Whether it does or not depends on the initial comparability of the
two groups and whether their experience during the experiment
differs in relevant ways only by the treatment itself.
• Whether the groups were comparable or not is crucial in
determining the extent of information yielded by this design. The
design is could be used to compare the value of a drug. If the
designer cannot, on the basis of information outside the experiment
itself, assume the comparability of the groups.
Random Group
This design is similar to the Static Group design except than an
attempt is made to ensure similarity of the groups before treatment
begins.
• Since it is difficult to have exactly similar subjects in each of two
groups (unless you separate identical twins), the design works
toward a guarantee of comparability between groups by assigning
subjects to groups at random.
• If the researcher does this there is likely to be reasonable
comparability between the two groups. This design can be
diagrammed as: R--GP--T—O, R---GP------O
• It is economical. It provides fairly clear-cut information as to the
relationship between treatment and post-treatment measurement or
behaviours. Since this is often the sole reason for the research, the
randomized group design is frequently the appropriate selection.
Pre-Post Randomized Group
This design adds a pre-test to the previous design as a check on the
degree of comparability of the control and experimental groups
before the treatment is given. This experimental design could be
diagrammed as: R--GP--O--T—O, R--GP--O------O
This yields information on post-treatment behavior and a
comparison of post-treatment behavior between groups. It answers
most of the class I questions, that is, threats to internal validity. The
groups are comparable because they are randomized.
Completely Randomised
The design deals with only two principles viz. replication and
randomisation are used in the experiment. In this design the
experimental materials are randomly assigned to the
treatments to make them homogenous in nature and assume
that there is no significant variation among them. It is
diagrammed as follows: R----GR----T----O
It is divided into number of experimental plots depending on
the number of treatments and number of replications for each
treatment. The randomisation must be as complete as
possible.
Randomized Block
This design is of particular value when the experimenter wishes to
determine the effect of a treatment on different types of subjects
within a group. This design can be diagrammed as:
----------BLK--T—O, R—GP, ----------BLK--T—O, ---------BLK------O,
R—GP, ----------BLK------O
Typically, this design refers to blocking or grouping of subjects with
similar characteristics into treatment subgroups. The group to be
used in an experiment is usually given some pre-treatment
measure, or previous records are examined, and the entire group is
blocked or sorted into categories. Then equal numbers from each
category are assigned to the various treatment and/or control
groups.
Factorial
As you saw above in the blocking design, the subjects were
assigned to different groups on the basis of some of their own
characteristics such as age, weight, gender etc.
• Sometimes we wish to assign different variations of the treatment
as well, and the procedure is similar.
• For example, we may wish to try two kinds of treatments varied in
two ways (called a 2x2 factorial design). Some factorial designs
include both assignment of subjects (blocking) and several types of
experimental treatment in the same experiment. When this is done
it is considered to be a factorial design. A diagram of a 2x2 factorial
design would look like:
R--GP--T-------O, A1 B1, R--GP--T-------O, A1 B2
R--GP--T-------O, A2 B1, R--GP--T-------O, A2 B2
• The factorial design as we are describing is really a complete
factorial design of which there are several variations. The factorial is
used when we wish information concerning the effects of different
kinds or intensities of treatments.
• The factorial design provides relatively economical information not
only about the effects of each treatment, level or kind, but also
about interaction effects of the treatment.
• In a single 2x2 factorial design information can be gained about
the effects of each of the two treatments and the effect of the two
levels within each treatment, and the interaction of the treatments.
• If all these are questions of interest, the factorial design is much
more economical than running separate experiments.
Latin Square
A researcher may wish to use several different treatments in the
same experiment, for example the relative effects of an assortment
of perhaps three or more drugs in combination in which the
sequence of administration may produce different results. A diagram
of a three treatment Latin Square design is:
R--GP--TA--O--TB--O--TC--O
R--GP--TB--O--TC--O--TA--O
R--GP--TC--O--TA--O--TB--O
If all possible sequencing permutations had to be addressed,
there would be 3x2=6 possible arrangements--a doubling of
cost. The original 3-treatment design would, however, answer
the question whether sequencing made a difference without
testing all possible sequences.
Types of experimental Design
(commonly used)
1. Completely Randomised Design (CRD)
2. Randomised Complete Block Design (RCBD) Or
Randomised Block Design (RBD)
3. Latin Square Design
4. Factorial Design
5. Split Plot Design
An Example of Experimental Designs
Consider the following hypothetical experiment:
Acme Medicine is conducting an experiment to test a new
vaccine, developed to immunize people against the common
cold. To test the vaccine, Acme has 1000 volunteers: 500 men
and 500 women. The participants’ age ranges from 21 to 70.
In this lesson, we can describe two experimental designs: - a
completely randomized design, a randomized block design,
and a matched pairs design. And we show how each design
might be applied by Acme Medicine to understand the effect of
the vaccine, while ruling out confounding effects of other
factors.
Experimental Design and Analysis of
Variance: Completely Randomised Design
What is CRD?
It is the simplest type of the basic designs, may be defined as a
design in which the treatments are assigned to experimental units
completely at random, that is the randomization is done without any
restrictions.
The design is completely flexible, i.e., any number of treatments and
any number of units per treatment may be used.
A completely randomized design is considered to be most useful in
situations where (i) the experimental units are homogeneous, (ii) the
experiments are relatively small and (iii) the number of treatments is
relatively small.
• A completely randomized design (CRD) relies on
randomization to control for the effects of extraneous
variables.
• The experimenter assumes that, on average,
extraneous factors will affect treatment conditions
equally
• So, any significant differences between conditions
can fairly be attributed to the independent variable.
Usage of CRD
• Simplest design to use.
• Design can be used when experimental units are essentially
homogeneous.
• Because of the homogeneity requirement, it may be difficult
to use this design for field experiments.
• The CRD is best suited for experiments with a small
number of treatments.
Randomization Procedure
-Treatments are assigned to experimental units completely at
random.
-Every experimental unit has the same probability of receiving
any treatment.
-Randomization is performed using a random number table,
computer, program, etc.
Advantages of a CRD
1. Very flexible design (i.e. number of treatments and
replicates is only limited by the available number of
experimental units).
2. Statistical analysis is simple compared to other designs.
3. Loss of information due to missing data is small compared
to other designs due to the larger number of degrees of
freedom for the error source of variation.
Disadvantages
1. If experimental units are not homogeneous and you fail to
minimize this variation using blocking, there may be a loss of
precision.
2. Usually the least efficient design unless experimental units
are homogeneous.
3. Not suited for a large number of treatments.
An Example of Completely Randomised
Design
Consider the following hypothetical experiment:
ACME Medicine is conducting an experiment to test a new
vaccine, developed to immunize people against the common
cold. To test the vaccine, ACME has 1000 volunteers: 500
men and 500 women. The participants’ age ranges from 21 to
70.
In this example we show how the design might be applied by
ACME Medicine to understand the effect of the vaccine, while
ruling out confounding effects of other factors.
Treatment
Population
The completely randomized
design is probably the simplest
Vaccine
500
experimental design, in terms of
data analysis and convenience. With this design, participants are
randomly assigned to treatments.
Placebo
500
A CRD layout for the Acme Experiment is shown in the above table.
In this design, the experimenter randomly assigned participants to
one of two treatment conditions.
• They received a Placebo or they received the Vaccine
• The same number of participants (500) were assigned to each
treatment condition (although this is not required).
• The dependent variable is the number of colds reported in each
treatment condition.
If the vaccine is effective, participants in
the "vaccine" condition should report significantly fewer colds than
participants in the "placebo" condition.
Let us take another example:
Suppose you have 4 different diets A, B, C & D and you want
to examine the effect of diets on weight gain of 16 rabbits. The
rabbits should be kept in cages of four shelves one above
another and each shelf has four chambers.
• How should you use randomization to assign the rabbits to
the 4 treatments and for which to put them into 16 cages?
• You have to make complete randomisation, not part of it.
One way of doing is:
• Label the cages 1-16. In a bowl put 16 strips of paper each with
one of the integers 1-16 written on it.
• In a second bowl put 16 strips of paper, four each labelled A,
B, C, and D.
• Catch a rabbit. Select a number and a letter from each bowl. Place
the rabbit in the cage indicated by the number and feed it the diet
assigned by the letter.
• Repeat without replacement of the used paper strips until all
rabbits have been assigned a diet and cage. That will give you:
Shelf 1
C
A
B
D
Shelf 2
D
B
D
C
Shelf 3
C
B
A
D
Shelf 4
A
A
C
B
Notice that the completely randomized
design does not account for the
difference in heights of the cages nor
position of cages in the room. It is just
as the name suggests, a completely
random assignment.
DATA ANALYSIS
One-Way ANOVA
To analyze the results of the experiment, we use a one-way
analysis of variance.
The measured live weight of rabbit for each diet are given below:
Diet A Diet B
Diet C
Diet D
Shelf 1 62
63
68
56
Shelf 2 60
67
66
62
Shelf 3 63
71
71
60
Shelf 4 59
64
67
61
Mean
66.25
68
59.75
61
The null hypothesis is
H0 : µA = µB =µC =µD (all treatment means the same)
and the alternative is
Ha : at least one mean different.
The CRD model is: Yij    T j  eij
Where  is the overall mean
Yij is the i th observation under the j th treatment of
the factor feed
T j is the effect of the j th treatment of the factor feed
eij is the random error associated with the i th
observation under the j th treatment of the factor
feed
k = number of treatments. In our example, there are 4
treatment classes, Diet A, Diet B, Diet C, and Diet D.
nj = number of observations for j th treatment.
Each of the treatments in this experiment have four
observations,
n1 =n2 = n3 = n4 = 4 .
N = total number of observations,
In this case, N = 16.
n
SS total =
t
Y
i 1 j 1
t
SS treatments =

j 1
2
ij
Y2

N
2
.j
Y
2
..
Y

n
N
SS error = SS total – SS treatments
4
4
Y..   Yij  1020
i 1 j 1
Y.1 = 244, Y.2 =265, Y.3 =272, Y.4 =239
2
Y
2
..
Y


ij
16
i 1 j 1
4
SS total =
4
= (62)2 + (63)2 + ….+ (61)2 + (67)2 – (1020)2/16 = 275
4
SS treatments =

j 1
2
.j
Y
2
..
Y

4 16
= {(244)2 + (265)2 + (272)2 + (239)2}/4 – (1020)2/16 = 191.5
SS error = SS total – SS treatments = 275 – 191.5 = 83.5
Treatment MS is SS(tret) divided by tret DF = 191.5/3
= 63.833
Divide SS(error) by the error degrees of freedom, N – k to get
Error MS, 83.5/12 = 6.95833
Now, divide Treatment MS by Error MS to get F value, here
63.833/6.95833 = 9.1737.
Now u have to compare it with Tabular F value to find the
significant variation among treatements
AOVA Table is essentially as follows:
Source
DF
SS
MS
F
Prob>F
Treatment
Error
Total
K–1
N–K
N–1
SS (Tret)
SS (Error)
SS (Total)
MS (Tret)
MS (Error)
-------
MS (Tret) /
MS (Error)
*
Source
DF
SS
MS
F
Treatment
Error
Total
3
12
15
191.5
83.5
275
63.8333
6.9583
----------
9.1737
Prob>F
Problem:
You have 3 different diets A, B and C. You have given 9
milking cows to see the effect of feeding diets on milk
production. Suppose you have the following data on milk yield
as affected by the diets:
Diets
A
B
C
1
15
18
17
2
16
19
17
3
18
19
18
Animals
Analyse data to see whether there are significance
differences among the treatment effects.
Experimental Design and Analysis of
Variance: Randomised Block Design OR
Randomised Complete Block Design
What is RBD or RCBD?
A randomized block design is a restricted randomization
design in which the experimental units are first sorted into
homogeneous groups, called blocks, and the treatments are
then assigned at random within the blocks.
The randomized block design is an improvement over the
completely randomized design. Both designs use
randomization to guard against confounding effect or error.
But only the randomized block design more accurately
controls for gender.
If you consider the example of ACME vaccine.
The experimenter divides participants into subgroups called
blocks, such that the variability within blocks is less than the
variability between blocks. Then, participants within each
block are randomly assigned to treatment conditions.
Because this design reduces variability and potential
confounding, it produces a better estimate of treatment effects
than CRD.
Gender
Gender
Treatment
Treatment
Placebo
Placebo Vaccine
Vaccine
The table to the left shows the randomised complete block design (RCBD).
Male
250
250
Male
250
250
Participants are assigned to blocks
Female
250
250
Female 250
250
based on gender. Then within each block
participants are randomly assigned to treatments. For this design,
250 men get the placebo, 250 men get the vaccine, 250 women get
the placebo, and 250 women get the vaccine.
• It is known that men and women are physiologically different and
react differently to medication.
• This design ensures that each treatment condition has an equal
proportion of men and women.
• As a result, differences between treatment conditions cannot be
attributed to gender. This randomized block design removes gender
as a potential source of variability and as a potential confounding
variable.
Randomisation
Let us take the example of the rabbit experiment of CRD
In the previous section, we analyzed results from a
completely randomized design (CRD) with a one-way
analysis of variance. This design ignored the physical
layout of the cages and the potential effect of the height of
cage in which rabbits were housed.
• If we wanted to acknowledge the potential effect of
height of the cage on weight gain, we should organize the
experiment using a randomized complete block design.
• One diet of each type will be used on each of the 4
shelves. The randomization procedure would assign a
number 1-16 to each of the rabbits.
• Put four paper slips marked as 1, 2, 3 and 4 (as
shelf number) in a bowl.
• Put 16 paper slips numbering from 1 – 16 (rabbit
number) in a 2nd bowl. Take a number at random
from 2nd bowl to select a rabbit for Diet A
• Now take a number out of the 1st bowl to select a
position on the top row (e.g. Shelf 1).
• Repeat three times without replacement for Diets
B, C, and D to complete the assignment to the top
row. Follow the same procedure to assign the other
three rows.
An example is shown below:
• This is a randomized block design and should not be
analyzed with the one-way ANOVA.
• For the sake of illustration, the data collected with this
method is the same as with the completely randomized design.
Ordering the data so it is easier to read.
DATA ANALYSIS
One – way ANOVA
We have the following observations:
Shelf 1
Shelf 2
Shelf 3
Shelf 4
Mean for
diet
Diet A
Diet B
Diet C
Diet D
Mean for Shelf
62
63
68
56
62.25
60
67
66
62
63.75
63
71
71
60
66.75
59
64
67
61
62.75
61
66.25
68
59.75
Grand mean
63.75
Hypothesis with respect to treatment (Diets)
H0 : µA = µB =µC =µD (all treatment means the same)
and the alternative is
Ha : at least one mean different.
Hypothesis with respect to block (Shelf height)
H0 : µ1= µ2 =µ3 =µ4 (all treatment means the same)
and the alternative is
Ha : at least one mean different.
The RCBD model is:
Yij    Bi  T j  eij
where
• μ is the overall mean
• Yij is the observation under the j th treatment of the factor
Feed and the i th block (Shelf)
• Bi is the effect of the i th block (Shelf)
• Tj is the effect of the j th treatment of the factor Feed
• eij is the random error associated with the i th block (Shelf)
under the j th treatment of the factor Feed.
b
SS total =
a
Y
ij
i 1 j 1
2
i.
b
2
Y2

N
2
..
Y
Y

SS block = 
N
i 1 a
a
SS treatments =

j 1
Y. 2j
Y..2

b
N
SSerror = SStotal − SSblocks − SStreatments
So,
4
4
Y..   Yij 
1020
i 1 j 1
Y.1 = 244, Y.2 =265, Y.3 =272, Y.4 =239
Y1. = 249, Y2. =255, Y3. =265, Y4. =251
4
SS total =
2
..
4
Y
Y 

16
i 1 j 1
2
ij
= (62)2 + (63)2 = ….+ (61)2 + (67)2 – 1020/16 = 275
4
SS treatments =

j 1
2
.j
Y
2
..
Y

4 16
= {(244)2 + (265)2 + (272)2 + (239)2}/4 – (1020)/16 = 191.5
4
SS Block =
2
i.
2
..
Y
Y


16
i 1 4
= {(249)2 + (255)2 + (265)2 + (251)2}/4 – (1020)2/16 = 38
= 62001+65025+70225+63001/4 – (1020)2 /16
= 65063- 65025=38
SS error = SS total − SS blocks − SS treatments
SS Error = 275 – 38 – 191.5 = 45.5
Treatment MS is SS Treat divided by treat DF = 191.5/3
= 63.833
Block MS = SS Block divided by block DF, 38/3 = 12.6667
Divide SS Error by the error degrees of freedom, N – k –B to
get Error MS, 45.5/9 = 5.0556
Divide Treat MS by Error MS to get F value for treatment
effect, here 63.8333/ 5.0556 = 12.6263.
Divide Block MS by Error MS to get F value for Block effect,
here 12.6667/5.0556 = 2.5055
Now you have to compare it with Tabular F value to find the
significant variation among treatments
AOVA Table is essentially as follows:
Source DF
Treat
Block
Error
Total
K–1
B–1
N–K
N–1
SS
MS
F
Prob>F
SS (Tret) MS (Tret) MS (Tret)/MS (Error)
SS (Blok) MS (Blok) MS (Blok)/MS (Error)
SS (Error) MS (Error)
SS (Total)
-------
Source
DF
SS
MS
F
Treatment
Block
Error
Total
3
3
9
15
191.5
38
45.5
275
63.8333
12.6667
5.0556
----------
12.6263
2.5055
*
*
Prob>F
Problem:
You have 3 different diets A, B and C. You have given 15
milking cows of 3 age groups 1, 2 & 3. You are asked to see
the effect of feeding diets and also age of cows on milk
production. Suppose you have the following data on milk yield
as affected by the diets:
Age 1
Age 2
Age 3
Diets
Anim group
A
B
C
Mean of
Age group
1
11
12
14
12.3334
2
15
14
17
15.3334
3
19
19
20
19.3334
Mean of
animal group
15
15
17
Grand
Mean
Analyse data to see whether there are significance
differences among the treatment effects.
From the computer output, we see that there is again a statistically
significant difference in weight gain, with the p-value slightly smaller
( p = 0.0014) . The mean square error has been reduced to 5.0556
and the degrees of freedom are reduced to 9. In this case,
LSD = 2.262
2 * 5.0556
= 3.596
4
Any difference in means greater than 3.6 is considered significant.
The means for Diet B and Diet C are not significantly different, as
are those for Diet A and Diet D. However, Diets B and C have larger
mean weight gain than Diets A and D.
There is no evidence that there is a difference in mean weight gain
among shelves. We did reduce the variation by blocking, but there
was no significant shelf effect. If we were to perform this experiment
another time, a completely randomized design would be appropriate.
Factorial Experiment
In statistics, a full factorial experiment is an experiment
whose design consists of two or more factors, each with
possible levels, and whose experimental units take on all
possible combinations of these levels across all such factors.
• Such an experiment allows studying the main effect of each
factor on the response variable, as well as the interactions
between factors on the response variable.
• For vast majority of factorial experiments, each factor has
only two levels. For example, with two factors each taking two
levels, a factorial experiment have four treatment combinations
in total, and is usually called a 2×2 factorial design.
Example
The simplest factorial experiment contains two levels for each
of two factors. Suppose an engineer wishes to study the total
power used by each of two different motors, A and B, running
at each of two different speeds, 2000 or 3000 RPM.
• The factorial experiment would consist of four experimental
units: motor A at 2000 RPM, motor B at 2000 RPM, motor A at
3000 RPM, and motor B at 3000 RPM. Each combination of a
single level selected from every factor is present once.
• This experiment is an example of a 22 (or 2x2) factorial
experiment, so named because it considers two levels (the
base) for each of two factors (the power or superscript), or
#levels#factors, producing 22=4 factorial points.
Designs can involve many independent variables. As a further
example, the effects of three input variables can be evaluated
in eight experimental conditions shown as the corners of a
cube.
This can be conducted with or without replication, depending
on its intended purpose and available resources. It will
provide the effects of the three independent variables on the
dependent variable and possible interactions.
To save space, the points in a two-level factorial experiment
are often abbreviated with strings of plus and minus signs. The
strings have as many symbols as factors, and their values
dictate the level of each factor: conventionally, − for the first (or
low) level, and + for the second (or high) level. The points in
this experiment can thus be represented as − − , + − , − + , and
+ +.
A
B
2X2 Factorial experiment
(1)
-
-
a
+
-
b
-
+
ab
+
+
The factorial points can also be abbreviated by (1), a, b, and
ab, where the presence of a letter indicates that the specified
factor is at its high (or second) level and the absence of a
letter indicates that the specified factor is at its low (or first)
level (for example, "a" indicates that factor A is on its high
setting, while all other factors are at their low (or first) setting).
(1) is used to indicate that all factors are at their lowest (or
first) values.