In factorial experiments, the effect of two or more factors, each at two or more discrete possible levels are simultaneously investigated for all possible combinations using some suitable basic experimental design.

The experiment allows us to estimate the effect of each factor and the interaction effect of factors on the response.

A factorial design was initially used in the 19th century at Rothamsted Experimental Station (one of the oldest research stations in the UK). Ronal Fisher and Frank Yates are the pioneers of factorial design.

Experiments are often planned to investigate the effects of (say), different rates of fertilizers, different dates of planting, different categories of education, different intensities of a stimulus, etc.

The independent variables such as fertilizers, planting, education, and stimulus, etc., are called factors. In contrast, the values such as rates, dates, categories, or intensities at which a factor is held fixed, are known as levels. In the figure below, the “Amount of Seeding” is a factor variable, while 60kg, 50kg, and 80kg are levels of the factor “Amount of Seeding”. There are three levels of this factor variable.

## Types of Factorial Experiments

There are two types of factorial experiments

### Full Factorial Experiment

The experimental units of such an experiment take on all possible combinations of all levels across all the factors. Therefore, a full factorial design is also called a fully crossed design.

### Fractional Factorial Experiment

If the full factorial design includes too many combinations (runs) to be logically feasible, a fractional factorial may be used. The fractional factorial design may include half, one-third, etc. runs of a full factorial experiment.

In factorial experiments, we try to perform one rather than two, three, or more single-factor experiments. The single experiment involves a factorial set of treatments, that is, the treatments are all possible combinations of various levels of different factors.

The effect of a factor is defined to be the change in response produced by a change in the level of factors. This is called the main effect.

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## Design of Experiments Objective Questions 3

MCQs about the Design of Experiments Objective Questions for the preparation of PPSC, FPSC Lecturer Statistics job, and BS, M.Phil, Ph.D. Statistics Degree Programs. Let us start with the Design of Experiments Objective Questions quiz.

Multiple Choice Questions about Design of Experiments.

1. An experiment is performed in CRD with 3 replications to compare four treatments. Then, the total experimental units will be

2. In systematic designs, the treatments are applied to the experimental units by some

3. If the experimental material is not homogenous and there is one source of variation in the experiment then we use

4. Under one-way variability in environmental conditions, the appropriate design for experimenting will be – – – – – – – -.

5. What will be the F ratio if an experiment is performed in CRD with 3 replications to compare four treatments? The treatment mean sum of squares = 96, error mean sum of squares = 12.

6. There are – – – – – – – – basic principles of sound statistical design.

7. To test the hypothesis about one population variance, the test statistic will be – – – – – – -.

8. The random process of assigning treatments to the experimental units can be done by

9. What will be a degree of freedom (df) for column-wise blocking in $5 \times 5$ Latin Square Design?

10. For a $7 \times 7$ Latin Square Design, there will be ___________ observations.

11. Planning an experiment to obtain appropriate data and drawing inferences out of the data for any problem under investigation is known as _______.

12. The experiment is performed on

13. Manova stands for

14. An experiment is performed in CRD with 3 replications to compare four treatments, then what will be the degrees of freedom for error?

15. One of the purposes of experimental design is to

16. An experiment is performed in CRE with 3 replications to compare four treatments. The treatment sum of squares = 8. If the error sum of square = 12 then what will be the total sum of square

17. Which one provide the estimate of experimental error in the design of the experiment

18. Analysis of variance is a statistical method of comparing the – – – – – – – – of several populations.

19. An experiment is performed in CRD with 3 replications to compare four treatments. The treatment sum of squares = 9. If the error sum of squares = 12 then what will be the mean sum of squares for treatment?

20. The scientific method for the construction of a statistical layout plan for an experiment is – – – – – -?

Planning an experiment to obtain appropriate data and drawing inferences from the data concerning any problem under investigation is known as the design and analysis of the experiments.

An experiment deliberately imposes a treatment on a group of objects or subjects to observe the response. The experimental unit is the basic entity or unit on which the experiment is performed. It is an object to which the treatment is applied and the variable under investigation is measured and analyzed.

Single-Factor Design: In a single-factor experiment only a single factor varies while all others are constant. The CRD, RCBD, and LSD are examples of single-factor designs.

Multi-Factor Design: Multi-factor designs are also known as factorial experiments. When several factors are investigated simultaneously in a single experiment, such experiments are known as factorial experiments.

Systematic Designs: In systematic designs treatments are applied to the experimental units by some systematic pattern, i.e., by the choice of the experimenter. For example, the experimenter wishes to test three treatments and he decides to have four repetitions of each treatment.

Randomized Designs: In randomized designs, as the treatments are applied randomly, therefore the conclusions drawn are supported by statistical tests. In this way, inferences are applicable in a wider range and the random process minimizes the systematic error. The analysis of variance techniques is also suitable for randomized designs only.

The purpose of the Design of Experiments is:

• Get maximum information for minimum expenditure in the minimum possible time.
• Helps to reduce the experimental error.
• To ignore spurious effects, if any.
• To evaluate and examine the outcomes critically and logically.

### Design of Experiments Objective Questions

• Which one provides the estimate of experimental error in the design of the experiment
• There are – – – – – – – – basic principles of sound statistical design.
• If the experimental material is not homogenous and there is one source of variation in the experiment then we use
• An experiment is performed in CRD with 3 replications to compare four treatments. Then, the total experimental units will be
• An experiment is performed in CRD with 3 replications to compare four treatments, then what will be the degrees of freedom for error?
• An experiment is performed in CRE with 3 replications to compare four treatments. The treatment sum of squares = 8. If the error sum of square = 12 then what will be the total sum of square?
• An experiment is performed in CRD with 3 replications to compare four treatments. The treatment sum of squares = 9. If the error sum of squares = 12 then what will be the mean sum of squares for treatment?
• What will be the F ratio if an experiment is performed in CRD with 3 replications to compare four treatments? The treatment mean sum of squares = 96, error mean sum of squares = 12.
• Analysis of variance is a statistical method of comparing the – – – – – – – – of several populations.
• To test the hypothesis about one population variance, the test statistic will be – – – – – – -.
• Under one-way variability in environmental conditions, the appropriate design for experimenting will be – – – – – – – -.
• The scientific method for the construction of a statistical layout plan for an experiment is – – – – – -?
• Manova stands for
• For a $7 \times 7$ Latin Square Design, there will be ___________ observations.
• The experiment is performed on
• The random process of assigning treatments to the experimental units can be done by
• In systematic designs, the treatments are applied to the experimental units by some
• Planning an experiment to obtain appropriate data and drawing inferences out of the data for any problem under investigation is known as ___________
• One of the purposes of experimental design is to
• What will be a degree of freedom (df) for column-wise blocking in $5 \times 5$ Latin Square Design?

## Classification of Randomized Designs (2023)

Randomized designs are a type of experimental design where randomization process is used to assign the units (like people or objects) to different treatment groups. The randomization process helps to control for bias and ensures that any observed differences between the groups are likely due to the treatment itself, rather than some other factors.

### Randomized Designs

In randomized designs, the treatments are applied randomly, therefore the conclusions drawn are supported by statistical tests. The classification of randomized designs for single-factor are:

Example: A market gardener wants to test three types of peas, $A$, $B$, and $C$, on his land. He divides a square plot into nine equal squares, three to be planted with each type of pea. The problem he then faces is which square to plant with which type.

One method is a Completely Randomized Design (CRD) which might,

This would be all right if all the plots were equally desirable. If however, there were prevailing north wind so that the northernmost plots were exposed, he might decide to use, a Randomize Complete Block Design (RCBD).

Randomized Complete Block Design, where each of the types $A$, $B$, and $C$ is planted once in each west-east block.

If the gardener also felt that the soil to the east was rather better than that to the west, he would use, a Latin Square Design (LSD).

A Latin Square design, where each type of pea is planted once in each row (west-east), and once in each column (north-south).

For Randomized Designs, Note that

• Completely Randomized Design (CRD) is a statistical experimental design where the treatments are assigned completely at random so that each treatment unit has the same chance (equal chance) of receiving any one treatment.
• In CRD any difference among experimental units receiving the same treatment is considered as an experimental error.
• CRD is applicable only when the experimental material is homogeneous (eg., homogeneous soil conditions in the field).
• Since soil is heterogeneous in the field, the CRD is not a preferable method in field experiments. Therefore, CRD generally applies to the lab experimental conditions, as in labs, the environmental conditions can be easily controlled.
• The concept of “local control” is not used in CRD.
• CRD is best suited for experiments with a small number of treatments.

The best design for a study will depend on the specific research question and the factors that one needs to control for. By incorporating randomization, you can control for extraneous variables that might influence the outcome and improve the validity of the findings.

However, the choice of the randomized design depends on the specific research question(s) being asked. It is important to consider the strengths and weaknesses of each design before making a decision.

Learn about Poisson Regression in R Language

## Designs of Experiment Terminology (2023)

Planning an experiment to obtain appropriate data and drawing inferences from the data concerning any problem under investigation is known as the design and analysis of the experiment or simply the designs of experiment (DOE).

### Important Designs of Experiment Terminology are:

EXPERIMENT: An experiment deliberately imposes a treatment on a group of objects or subjects in the interest of observing the response.

EXPERIMENTAL UNIT: The experimental unit is the basic entity or unit on which the experiment is performed. It is an object to which the treatment is applied and in which the variable under investigation is measured and analyzed. For example, the experimental unit may be a single person, animal, plant, manufactured item, or country that belongs to a larger collection of such entities being studied.

### Identify the Experimental Units

• A teacher practices the different teaching methods on different groups in her class to see which yields the best results.
• A doctor treats a patient with a skin condition with different creams to see which is most effective.

The experimental unit is the physical entity or subject exposed to the treatment independently of other units. In other words, it is the basic unit on which the experiment is performed (smallest division of experimental material).

TREATMENTS: In experiments, a treatment is something that researchers administer to experimental units. For example, a corn field is divided into four, each part is ‘treated’ with a different fertilizer to see which produces the most corn.

Treatment is an experimental condition whose effect is to be measured and compared. For example, animal diets, temperature, soil types, and brands of tires.

FACTOR: A factor of an experiment is a controlled independent variable; a variable whose levels are set by the experimenter. A factor is a general type or category of treatments. Different treatments constitute different levels of a factor.

### EXPERIMENTAL ERROR

It describes the variation among identically and independently treated experimental units. In the designs of experiments, various origins of experimental error include:

• The natural variation among the experimental units.
• Inability to reproduce the treatment conditions exactly from one unit to another.
• Interaction of treatments and experimental units.
• Any other extraneous factors that influence the measured characteristics.

There are two types of errors:

1. Systematic Errors
Systematic Errors are caused by a consistent bias in one direction, consistently pushing your results away from the true value. Systematic errors can be caused by a variety of factors, such as a faulty instrument, an incorrect calibration, or an error in the experimental design. Systematic errors will cause data points to shift all in the same direction, away from the true value.
2. Random Error
The random error is caused by small and unpredictable variations that occur in every experiment. Random errors can come from a variety of sources, such as slight differences in how a measurement is made, or fluctuations in environmental conditions. Random errors tend to cause data points to scatter randomly around the true value.

The experimental error can be controlled by

• Blocking
• Proper plot technique
• Data Analysis

### EXPERIMENTAL DESIGN

An experimental design is a plan to collect the data relevant to the problem under investigation. In such a way as to provide a basis for valid and objective inferences about the stated problem.

The plan usually consists of the selection of the treatments, specifications of the experimental layouts, allocation of the treatments, and collection of observations for analysis.

Hence designs of Experiments are simply a sequence of steps taken ahead of time to ensure that the appropriate data will be obtained in a way that permits an objective analysis leading to a valid analysis concerning the problem.