Chapter 11
Fractional Replications
Consider the set up of complete factorial experiment, say 2k . If there are four factors, then the total
number of plots needed to conduct the experiment is 24 16. When the number of factors increases to
six, then the required number of plots to conduct the experiment becomes 26 64 and so on.
Moreover, the number of treatment combinations also become large when the number of factors
increases. Sometimes, it is so large that it becomes practically difficult to organize such a huge
experiment. Also, the quantity of experimental material needed, time, manpower etc. also increase and
sometimes even it may not be possible to have so many resources to conduct a complete factorial
experiment. The non-experimental type of errors also enters in the planning and conduct of the
experiment. For example, there can be a slip in numbering the treatments or plots or they may be
wrongly reported if they are too large in numbers.
About the degree of freedoms, in the 26 factorial experiment there are 26 1 63 degrees of freedom
which are divided as 6 for main effects, 15 for two-factor interactions and rest 42 for three or higher-
order interactions. In case, the higher-order interactions are not of much use or much importance, then
they can possibly be ignored. The information on main and lower-order interaction effects can then be
obtained by conducting a fraction of complete factorial experiments. Such experiments are called as
fractional factorial experiments. The utility of such experiments becomes more when the
experimental process is more influenced and governed by the main and lower-order interaction effects
rather than the higher-order interaction effects. The fractional factorial experiments need less number
of plots and lesser experimental material than required in the complete factorial experiments. Hence it
involves less cost, less manpower, less time etc.
It is possible to combine the runs of two or more fractional factorials to assemble sequentially a larger
experiment to estimate the factor and interaction effects of interest.
To explain the fractional factorial experiment and its related concepts, we consider here examples in
the set up of 2k factorial experiments.
One-half fraction of 23 factorial experiment
First, we consider the set up of 23 factorial experiment and consider its one-half fraction. This is a
very simple set up to understand the basics, definitions, terminologies and concepts related to the
fractional factorials.
Analysis of Variance | Chapter 11 | Fractional Replications | Shalabh, IIT Kanpur
1
, Consider the setup of 23 factorial experiment consisting of three factors, each at two levels. There is a
total of 8 treatment combinations involved. So 8 plots are needed to run the complete factorial
experiment.
Suppose the material needed to conduct the complete factorial experiment in 8 plots is not available or
the cost of total experimental material is too high. The experimenter has material or money which is
sufficient only for four plots. So the experimenter decides to have only four runs, i.e., ½ fraction of 23
factorial experiment. Such an experiment contains a one-half fraction of a 23 experiment and is called
231 factorial experiment. Similarly, 1/ 22 fraction of 23 factorial experiment requires only 2 runs
and contains 1/ 22 fraction of 23 factorial experiment and is called as 23 2 factorial experiment. In
general, 1/ 2 p fraction of a 2k factorial experiment requires only 2 k p runs and is denoted as 2 k p
factorial experiment.
We consider the case of ½ fraction of 23 factorial experiment to describe the various issues involved
and to develop the concepts. The first question is how to choose four out of eight treatment
combinations for conductive the experiment. In order to decide this, first we have to choose an
interaction factor which the experimenter feels can be ignored. Generally, this can be a higher-order
interaction which is usually difficult to interpret. We choose ABC in this case. Now we create the
table of treatment combinations as in the following table.
Arrangement of treatment combinations for one-half fraction of 23 factorial experiment
Factors I A B C AB AC BC ABC
Treatment
combinations
a + + - - - - + +
b + - + - - + - +
c + - - + + - - +
abc + + + + + + + +
*********** ******* ******* ******* ******* ******* ******* ******* ******
ab + + + - + - - -
ac + + - + - + - -
bc + + + + - - + -
(1) + + - - + + + -
Analysis of Variance | Chapter 11 | Fractional Replications | Shalabh, IIT Kanpur
2
Fractional Replications
Consider the set up of complete factorial experiment, say 2k . If there are four factors, then the total
number of plots needed to conduct the experiment is 24 16. When the number of factors increases to
six, then the required number of plots to conduct the experiment becomes 26 64 and so on.
Moreover, the number of treatment combinations also become large when the number of factors
increases. Sometimes, it is so large that it becomes practically difficult to organize such a huge
experiment. Also, the quantity of experimental material needed, time, manpower etc. also increase and
sometimes even it may not be possible to have so many resources to conduct a complete factorial
experiment. The non-experimental type of errors also enters in the planning and conduct of the
experiment. For example, there can be a slip in numbering the treatments or plots or they may be
wrongly reported if they are too large in numbers.
About the degree of freedoms, in the 26 factorial experiment there are 26 1 63 degrees of freedom
which are divided as 6 for main effects, 15 for two-factor interactions and rest 42 for three or higher-
order interactions. In case, the higher-order interactions are not of much use or much importance, then
they can possibly be ignored. The information on main and lower-order interaction effects can then be
obtained by conducting a fraction of complete factorial experiments. Such experiments are called as
fractional factorial experiments. The utility of such experiments becomes more when the
experimental process is more influenced and governed by the main and lower-order interaction effects
rather than the higher-order interaction effects. The fractional factorial experiments need less number
of plots and lesser experimental material than required in the complete factorial experiments. Hence it
involves less cost, less manpower, less time etc.
It is possible to combine the runs of two or more fractional factorials to assemble sequentially a larger
experiment to estimate the factor and interaction effects of interest.
To explain the fractional factorial experiment and its related concepts, we consider here examples in
the set up of 2k factorial experiments.
One-half fraction of 23 factorial experiment
First, we consider the set up of 23 factorial experiment and consider its one-half fraction. This is a
very simple set up to understand the basics, definitions, terminologies and concepts related to the
fractional factorials.
Analysis of Variance | Chapter 11 | Fractional Replications | Shalabh, IIT Kanpur
1
, Consider the setup of 23 factorial experiment consisting of three factors, each at two levels. There is a
total of 8 treatment combinations involved. So 8 plots are needed to run the complete factorial
experiment.
Suppose the material needed to conduct the complete factorial experiment in 8 plots is not available or
the cost of total experimental material is too high. The experimenter has material or money which is
sufficient only for four plots. So the experimenter decides to have only four runs, i.e., ½ fraction of 23
factorial experiment. Such an experiment contains a one-half fraction of a 23 experiment and is called
231 factorial experiment. Similarly, 1/ 22 fraction of 23 factorial experiment requires only 2 runs
and contains 1/ 22 fraction of 23 factorial experiment and is called as 23 2 factorial experiment. In
general, 1/ 2 p fraction of a 2k factorial experiment requires only 2 k p runs and is denoted as 2 k p
factorial experiment.
We consider the case of ½ fraction of 23 factorial experiment to describe the various issues involved
and to develop the concepts. The first question is how to choose four out of eight treatment
combinations for conductive the experiment. In order to decide this, first we have to choose an
interaction factor which the experimenter feels can be ignored. Generally, this can be a higher-order
interaction which is usually difficult to interpret. We choose ABC in this case. Now we create the
table of treatment combinations as in the following table.
Arrangement of treatment combinations for one-half fraction of 23 factorial experiment
Factors I A B C AB AC BC ABC
Treatment
combinations
a + + - - - - + +
b + - + - - + - +
c + - - + + - - +
abc + + + + + + + +
*********** ******* ******* ******* ******* ******* ******* ******* ******
ab + + + - + - - -
ac + + - + - + - -
bc + + + + - - + -
(1) + + - - + + + -
Analysis of Variance | Chapter 11 | Fractional Replications | Shalabh, IIT Kanpur
2
