Design of Experiments (DOE)

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

Classification of Randomized Designs (2023)

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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.

Classification of Randomized Designs

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

123
CAC
BAA
BBC
Allocation of Different Types of Peas Randomly to plots

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.

123
ABC
ACB
CBA
Allocation of Different Types of Pease 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).

Block 1Block 2Block 3
ABC
BCA
CAB
Allocation of Different Types of Pease 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.
Design of Experients

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.

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Designs of Experiment Terminology (2023)

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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.

Designs of Experiment

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.

Designs of Experiment

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.

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Important MCQs Design of Experiments

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This test contains MCQs Design of Experiments (DOE). Click the links from the MCQS Design of Experiments list to start with the quiz. All the MCQs Designs of Experiments are from topics of Basic principles of Design of Experiments, concept of Randomization, Replication, types of Designs, Experimental Unit and Error, CRD, CRBD, LSD, Greco LSD, Factorial design and experiments, Response surface design, and balanced incomplete block designs. etc.

Online MCQs Design of Experiments

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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.

Online MCQs Design of Experiments

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.
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Important Design of Experiments Quiz 2

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The post contains MCQs on the Design of Experiments Quiz (DOE). Most of the MCQs on the Design of Experiments Quiz are from factorial Experiments. Let us start with Online MCQs on the Design of Experiments Quiz.

Multiple Choice Questions about the Design of Experiments for preparation of examinations related to PPSC, FPSC, NTS, and Statistics job- and education-related examinations

1. Let ADE and BCE be two effects confounded in blocks. Then generalized interaction is

 
 
 
 

2. The parameter $\lambda$ for a balanced incomplete block design with $a=4, b=4, k=3, r=3$ as usual notation is

 
 
 
 

3. A design with $v$ treatment labels, each occurring $r$ times, and with $bk$ experimental units grouped into $b=v$ blocks of size $k<v$ in such a way that the units within a block are alike and units in different blocks are substantially different is

 
 
 
 

4. If ABC is confounded in replicate I,
AB is confounded in replicated II,
BC is confounded in replicate III,
then the design technique is called

 
 
 
 

5. In a randomized complete block design, the block should be constructed so that

 
 
 
 

6. When a number of confounded arrangements for factorial designs are made in Latin Squares, the designs are called

 
 
 
 

7. In $3^k$ factorial design with $n$ replicates in the experiment, the $df$ of error are

 
 
 
 

8. An experiment was designed to investigate the effect of the amount of water and seed variety on the subsequent growth of plants. Each plant was potted in a clay plot, and a measured amount of water was given weekly. The height of the plant at the end of the experiment was measured. Which of the following is not correct?

 
 
 
 

9. The models in which the levels of treatment factors are specifically chosen are known as

 
 
 
 

10. The number of aliases of two-factor interactions in a $2^6$-factorial experiment (1/4 replicate) would be

 
 
 
 

11. For a two-factor factorial design, if there are ‘$a$’ levels of Factor-A and ‘$b$’ levels of Factor-B, then $df$ of interaction are:

 
 
 
 

12. In a $2^3$ factorial experiment with partial confounding in three replications of 6 blocks, the error degrees of freedom would be

 
 
 
 

13. An experiment was conducted where you analyzed the results of the plant growth experiment after you manipulated the amount of water and seed variety. Which of the following is correct?

 
 
 
 

14. When a factorial experiment is performed in fractional replication, the two factorial effects that are represented by the same comparisons are called

 
 
 
 

15. In a factorial experiment, if $r$ is the number of replicates then each factorial effect has the same variance, that is

 
 
 
 

16. When all pairs of treatments are compared with approximately the same precision, even though the differences among blocks may be large, called

 
 
 
 

17. The designs in which the number of treatments must be an exact square, the size of a block is the square root of this form separate replications are called

 
 
 
 

18. An important relationship between the coefficient of determination $R^2$ and the F-ratio used in ANOVA is

 
 
 
 

19. For a Latin Square design

 
 
 
 

20. Which of the following is NOT CORRECT about a randomized complete block experiment?

 
 
 
 

Design of experiments (DOE) is a systematic method used to plan, conduct, analyze, and interpret controlled tests to study the relationship between factors and outcomes. Design of Experiment is a powerful tool used in various fields, including science, engineering, and business, to gain insights and optimize processes.

Design of Experiments Quiz

Design of Experiments Quiz

  • The parameter $\lambda$ for a balanced incomplete block design with $a=4, b=4, k=3, r=3$ as usual notation is
  • For a two-factor factorial design, if there are ‘$a$’ levels of Factor-A and ‘$b$’ levels of Factor-B, then $df$ of interaction are:
  • In $3^k$ factorial design with $n$ replicates in the experiment, the $df$ of error are
  • Let ADE and BCE be two effects confounded in blocks. Then generalized interaction is
  • If ABC is confounded in replicate I, AB is confounded in replicated II, BC is confounded in replicate III, then the design technique is called
  • An important relationship between the coefficient of determination $R^2$ and the F-ratio used in ANOVA is
  • In a randomized complete block design, the block should be constructed so that
  • For a Latin Square design
  • In a factorial experiment, if $r$ is the number of replicates then each factorial effect has the same variance, that is
  • When all pairs of treatments are compared with approximately the same precision, even though the differences among blocks may be large, called
  • In a $2^3$ factorial experiment with partial confounding in three replications of 6 blocks, the error degrees of freedom would be
  • When a factorial experiment is performed in fractional replication, the two factorial effects that are represented by the same comparisons are called
  • When a number of confounded arrangements for factorial designs are made in Latin Squares, the designs are called
  • The number of aliases of two-factor interactions in a $2^6$-factorial experiment (1/4 replicate) would be
  • The designs in which the number of treatments must be an exact square, the size of a block is the square root of this form, and separate replications are called
  • A design with $v$ treatment labels, each occurring $r$ times, and with $bk$ experimental units grouped into $b=v$ blocks of size $k<v$ in such a way that the units within a block are alike and units in different blocks are substantially different is
  • An experiment was designed to investigate the effect of the amount of water and seed variety on the subsequent growth of plants. Each plant was potted in a clay plot, and a measured amount of water was given weekly. The height of the plant at the end of the experiment was measured. Which of the following is not correct?
  • The models in which the levels of treatment factors are specifically chosen are known as
  • Which of the following is NOT CORRECT about a randomized complete block experiment?
  • An experiment was conducted where you analyzed the results of the plant growth experiment after you manipulated the amount of water and seed variety. Which of the following is correct?
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