Chapter 8
Factorial Experiments
Factorial experiments involve simultaneously more than one factor and each factor is at two or more
levels. Several factors affect simultaneously the characteristic under study in factorial experiments and
the experimenter is interested in the main effects and the interaction effects among different factors.
First, we consider an example to understand the utility of factorial experiments.
Example: Suppose the yield from different plots in an agricultural experiment depends upon
1. (i) variety of crop and
(ii) type of fertilizer.
Both the factors are in the control of the experimenter.
2. (iii) Soil fertility. This factor is not in the control of the experimenter.
In order to compare different crop varieties
- assign it to different plots keeping other factors like irrigation, fertilizer, etc. fixed and the same
for all the plots.
- The conclusions for this will be valid only for the crops grown under similar conditions with
respect to the factors like fertilizer, irrigation etc.
In order to compare different fertilizers (or different dosage of fertilizers)
- sow single crop on all the plots and vary the quantity of fertilizer from plot to plot.
- The conclusions will become invalid if different varieties of the crop are sown.
- It is quite possible that one variety may respond differently than another to a particular type of
fertilizer.
Suppose we wish to compare
- two crop varieties – a and b, keeping the fertilizer fixed and
- three varieties of fertilizers – A, B and C.
This can be accomplished with two randomized block designs ( RBD ) by assigning the treatments at
random to three plots in any block and two crop varieties at random.
Analysis of Variance | Chapter 8 | Factorial Experiments | Shalabh, IIT Kanpur
1
,The possible arrangement of the treatments may appear can be as follows.
bB bA bC and aA aB aC
bC bB bA aC aA aB
bA bC bB aB aC aA
With these two RBDs ,
- the difference among two fertilizers can be estimated
- but the difference among the crop varieties cannot be estimated. The difference among the crop
varieties is entangled with the difference in blocks.
On the other hand, if we use three sets of three blocks each and each block having two plots, then
- randomize the varieties inside each block and
- assign treatments at random to three sets.
The possible arrangement of treatment combinations in blocks can be as follows:
bB aB , aC bC and aA bA
aB bB bC aC bA aA
bB aB aC bC bA aA
Here the difference between crop varieties is estimable but the difference between fertilizer treatment is
not estimable.
Factorial experiments overcome this difficulty and combine each crop with each fertilizer treatment.
There are six treatment combinations as
aA, aB, aC, bA, bB, bC.
Keeping the total number of observations to be 18 (as earlier), we can use RBD with three blocks with
six plots each, e.g.
bA aC aB bB aA bC
aA aC bC aB bB bA
bB aB bA aC aA bC
Now we can estimate
- the difference between crop varieties and
- the difference between fertilizer treatments. Factorial experiments involve simultaneously more
than one factor each at two or more levels.
Analysis of Variance | Chapter 8 | Factorial Experiments | Shalabh, IIT Kanpur
2
, If the number of levels for each factor is the same, we call it is a symmetrical factorial experiment.
If the number of levels of each factor is not the same, then we call it as a symmetrical or mixed
factorial experiment.
We consider only symmetrical factorial experiments.
Through the factorial experiments, we can study
- the individual effect of each factor and
- interaction effect.
Now we consider a 22 factorial experiment with an example and try to develop and understand the theory
and notations through this example.
General notation for representing the factors is to use capital letters, e.g., A, B, C etc. and levels of a
factor are represented in small letters. For example, if there are two levels of A, they are denoted as a0
and a1 . Similarly, the two levels of B are represented as b0 and b1 . Another alternative representation
to indicate the two levels of A is 0 (for a0 ) and 1 (for a1 ). The factors of B are then 0 (for b0 ) and 1
(for b1 ).
Note: An important point to remember is that the factorial experiments are conducted in the design of an
experiment. For example, the factorial experiment is conducted as an RBD.
Factorial experiments with factors at two levels ( 22 factorial experiment):
Suppose in an experiment, the values of current and voltage in an experiment affect the rotation per
minutes ( rpm) of fan speed. Suppose there are two levels of current.
- 5 Ampere, call it as level 1 (C1 ) and denote it as a0
- 10 Ampere, call it as level 2 (C1 ) and denote it as a1 .
Similarly, the two levels of voltage are
- 200 volts, call it as level 1 (V0 ) and denote it as b0
- 220 volts, call it as level 2 (V1 ) and denote it as b1 .
The two factors are denoted as A, say for current and B, say for voltage.
Analysis of Variance | Chapter 8 | Factorial Experiments | Shalabh, IIT Kanpur
3
Factorial Experiments
Factorial experiments involve simultaneously more than one factor and each factor is at two or more
levels. Several factors affect simultaneously the characteristic under study in factorial experiments and
the experimenter is interested in the main effects and the interaction effects among different factors.
First, we consider an example to understand the utility of factorial experiments.
Example: Suppose the yield from different plots in an agricultural experiment depends upon
1. (i) variety of crop and
(ii) type of fertilizer.
Both the factors are in the control of the experimenter.
2. (iii) Soil fertility. This factor is not in the control of the experimenter.
In order to compare different crop varieties
- assign it to different plots keeping other factors like irrigation, fertilizer, etc. fixed and the same
for all the plots.
- The conclusions for this will be valid only for the crops grown under similar conditions with
respect to the factors like fertilizer, irrigation etc.
In order to compare different fertilizers (or different dosage of fertilizers)
- sow single crop on all the plots and vary the quantity of fertilizer from plot to plot.
- The conclusions will become invalid if different varieties of the crop are sown.
- It is quite possible that one variety may respond differently than another to a particular type of
fertilizer.
Suppose we wish to compare
- two crop varieties – a and b, keeping the fertilizer fixed and
- three varieties of fertilizers – A, B and C.
This can be accomplished with two randomized block designs ( RBD ) by assigning the treatments at
random to three plots in any block and two crop varieties at random.
Analysis of Variance | Chapter 8 | Factorial Experiments | Shalabh, IIT Kanpur
1
,The possible arrangement of the treatments may appear can be as follows.
bB bA bC and aA aB aC
bC bB bA aC aA aB
bA bC bB aB aC aA
With these two RBDs ,
- the difference among two fertilizers can be estimated
- but the difference among the crop varieties cannot be estimated. The difference among the crop
varieties is entangled with the difference in blocks.
On the other hand, if we use three sets of three blocks each and each block having two plots, then
- randomize the varieties inside each block and
- assign treatments at random to three sets.
The possible arrangement of treatment combinations in blocks can be as follows:
bB aB , aC bC and aA bA
aB bB bC aC bA aA
bB aB aC bC bA aA
Here the difference between crop varieties is estimable but the difference between fertilizer treatment is
not estimable.
Factorial experiments overcome this difficulty and combine each crop with each fertilizer treatment.
There are six treatment combinations as
aA, aB, aC, bA, bB, bC.
Keeping the total number of observations to be 18 (as earlier), we can use RBD with three blocks with
six plots each, e.g.
bA aC aB bB aA bC
aA aC bC aB bB bA
bB aB bA aC aA bC
Now we can estimate
- the difference between crop varieties and
- the difference between fertilizer treatments. Factorial experiments involve simultaneously more
than one factor each at two or more levels.
Analysis of Variance | Chapter 8 | Factorial Experiments | Shalabh, IIT Kanpur
2
, If the number of levels for each factor is the same, we call it is a symmetrical factorial experiment.
If the number of levels of each factor is not the same, then we call it as a symmetrical or mixed
factorial experiment.
We consider only symmetrical factorial experiments.
Through the factorial experiments, we can study
- the individual effect of each factor and
- interaction effect.
Now we consider a 22 factorial experiment with an example and try to develop and understand the theory
and notations through this example.
General notation for representing the factors is to use capital letters, e.g., A, B, C etc. and levels of a
factor are represented in small letters. For example, if there are two levels of A, they are denoted as a0
and a1 . Similarly, the two levels of B are represented as b0 and b1 . Another alternative representation
to indicate the two levels of A is 0 (for a0 ) and 1 (for a1 ). The factors of B are then 0 (for b0 ) and 1
(for b1 ).
Note: An important point to remember is that the factorial experiments are conducted in the design of an
experiment. For example, the factorial experiment is conducted as an RBD.
Factorial experiments with factors at two levels ( 22 factorial experiment):
Suppose in an experiment, the values of current and voltage in an experiment affect the rotation per
minutes ( rpm) of fan speed. Suppose there are two levels of current.
- 5 Ampere, call it as level 1 (C1 ) and denote it as a0
- 10 Ampere, call it as level 2 (C1 ) and denote it as a1 .
Similarly, the two levels of voltage are
- 200 volts, call it as level 1 (V0 ) and denote it as b0
- 220 volts, call it as level 2 (V1 ) and denote it as b1 .
The two factors are denoted as A, say for current and B, say for voltage.
Analysis of Variance | Chapter 8 | Factorial Experiments | Shalabh, IIT Kanpur
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