# Design of Experiments (DOE)

Lecture notes and quizzes related to the Design of Experiments (DOE).

## Factorial Experiment

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.

Online MCQs Test Website

## 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. The treatment sum of squares = 9. If the error sum of squares = 12 then what will be the mean sum of squares for treatment?

2. 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.

3. 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

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

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

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

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

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

9. One of the purposes of experimental design is to

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

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

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

13. Manova stands for

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

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

16. The experiment is performed on

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

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

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

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

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.

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 kept 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

The post is about the classification of Randomized Designs.

### 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 which 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 Desing (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.

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