Introduction

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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Data Collection Methods

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There are many methods to collect data. These Data Collection Methods can be classified into four main methods (sources) of collecting data: used in statistical inference.

Data Collection Methods

The Data Collection Methods are (i) Survey Method (ii) Simulation (iii) Controlled Experiments (iv) Observational Study. Let us discuss Data Collection Methods one by one in detail.

(i) Survey Method

A very popular and widely used method is the survey, where people with special training go out and record observations of, the number of vehicles, traveling along a road, the acres of fields that farmers are using to grow a particular food crop; the number of households that own more than one motor vehicle, the number of passengers using Metro transport and so on. Here the person making the study has no direct control over generating the data that can be recorded, although the recording methods need care and control.

(ii) Simulation

Simulation is also one of the most important data collection methods. In Simulation, a computer model for the operation of an (industrial)  system is set up in which an important measurement is the percentage purity of a (chemical) product. A very large number of realizations of the model can be run to look for any pattern in the results. Here the success of the approach depends on how well the model can explain that measurement and this has to be tested by carrying out at least a small amount of work on the actual system in operation.

(iii) Controlled Experiments

An experiment is possible when the background conditions can be controlled, at least to some extent. For example, we may be interested in choosing the best type of grass seed to use in the sports field.

The first stage of work is to grow all the competing varieties of seed at the same place and make suitable records of their growth and development. The competing varieties should be grown in quite small units close together in the field as in the figure below

Data Collection Methods: Controlled Experiments

This is a controlled experiment as it has certain constraints such as;

i) River on the right side
ii) Shadow of trees on the left side
iii) There are 3 different varieties (say, $v_1, v_2, v_3$) and are distributed in 12 units.

In the diagram below, much more control of local environmental conditions than there would have been if one variety had been replaced in the strip in the shelter of the trees, another close by the river while the third one is more exposed in the center of the field;

Data Collection Methods: Controlled Experiments 2

There are 3 experimental units. One is close to the stream and the other is to trees while the third one is between them which is more beneficial than the others. It is now our choice where to place any one of them on any of the sides.

(iv) Observational Study

Like experiments, observational studies try to understand cause-and-effect relationships. However, unlike experiments, the researcher is not able to control (1) how subjects are assigned to groups and/or (2) which treatments each group receives.

Note that small units of land or plots are called experimental units or simply units.

There is no “right” side for a unit, it depends on the type of crop, the work that is to be done on it, and the measurements that are to be taken. Similarly, the measurements upon which inferences are eventually going to be based are to be taken as accurately as possible. The unit must, therefore, need not be so large as to make recording very tedious because that leads to errors and inaccuracy. On the other hand, if a unit is very small there is the danger that relatively minor physical errors in recording can lead to large percentage errors.

Experimenters and statisticians who collaborate with them, need to gain a good knowledge of their experimental material or units as a research program proceeds.

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Basic Principles of DOE (Design of Experiments)

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The basic principles of doe (design of experiments or experimental design) are (i) Randomization, (ii) Replication, and (iii) Local Control. Let us discuss these important principles of experimental design in detail below.

Principles of DOE (Design of Experiments)

  1. Randomization

    Randomization is the cornerstone underlying the use of statistical methods in experimental designs.  Randomization is the random process of assigning treatments to the experimental units. The random process implies that every possible allotment of treatments has the same probability. For example, if the number of treatments = 3 (say, $A, B$, and C) and replication =$r = 4$, then the number of elements = $t \times r$ = 3 \times 4 = 12 = n$. Replication means that each treatment will appear 4 times as $r = 4$. Let the design is

    A C B C
    C B A B
    A C B A
    Note from the design elements 1, 7, 9, and 12 are reserved for Treatment $A$, elements 3, 6, 8, and 11 are reserved for Treatment $B$, and elements 2, 4, 5, and 10 are reserved for Treatment $C$. $P(A)= \frac{4}{12}, P(B)= 4/12$, and $P(C)=\frac{4}{12}$, meaning that Treatment $A, B$, and $C$ have equal chances of its selection.
  2. Replication

    By replication, we mean the repetition of the basic experiments. For example, If we need to compare the grain yield of two varieties of wheat then each variety is applied to more than one experimental unit. The number of times these are applied to experimental units is called their number of replications. It has two important properties:

    • It allows the experimenter to obtain an estimate of the experimental error.
    • More replication would provide the increased precision by reducing the standard error (SE) of mean as $s_{\overline{y}}=\tfrac{s}{\sqrt{r}}$, where $s$ is sample standard deviation and $r$ is a number of replications. Note that increase in $r$ value $s_{\overline{y}}$ (standard error of $\overline{y}$).
  3. Local Control

    Local control is the last important principle among the principles of doe. It has been observed that all extraneous source of variation is not removed by randomization and replication, i.e. unable to control the extraneous source of variation.
    Thus we need to refine the experimental technique. In other words, we need to choose a design in such a way that all extraneous source of variation is brought under control. For this purpose, we make use of local control, a term referring to the amount of (i) balancing, (ii) blocking, and (iii) grouping of experimental units.

Principles of doe

Balancing: Balancing means that the treatment should be assigned to the experimental units in such a way that the result is a balanced arrangement of treatment.

Blocking: Blocking means that the like experimental units should be collected together to form relatively homogeneous groups. A block is also called a replicate.

The main objective/ purpose of local control is to increase the efficiency of experimental design by decreasing experimental error.

This is all about the Basic Principles of the Experimental Design. To learn more about DOE visit the link: Design of Experiments.

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Real Life Example

Imagine a bakery trying to improve the quality of its bread. Factors that could affect bread quality include

  • Flour type,
  • Water
  • Temperature, and
  • Yeast quantity

By using DOE, the bakery can systematically test different combinations of these factors to determine the optimal recipe.

Randomization: Randomly assign different bread batches to different combinations of flour type, water temperature, and yeast quantity.

Replication: Bake multiple loaves of bread for each combination to assess variability.

Local Control: If the oven has different temperature zones, bake similar bread batches in the same zone to reduce temperature variation.

By following the Basic Principles of Design of Experiments, the bakery can efficiently identify the best recipe for its bread, improving product quality and reducing waste.

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