Design of Experiments (DOE)

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

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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Latin Square Designs (LSD) Definition and Introduction

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Introduction to Latin Square Designs

In Latin Square Designs the treatments are grouped into replicates in two different ways, such that each row and each column is a complete block, and the grouping for balanced arrangement is performed by restricting that each of the treatments must appear once and only once in each of the rows and only once in each of the column. The experimental material should be arranged and the experiment conducted in such a way that the differences among the rows and columns represent a major source of variation.

Hence a Latin Square Design is an arrangement of $k$ treatments in a $k\times k$ squares, where the treatments are grouped in blocks in two directions. It should be noted that in a Latin Square Design the number of rows, the number of columns, and the number of treatments must be equal.

In other words unlike Randomized Completely Block Design (RCBD) and Completely Randomized Design (CRD) a Latin Square Design is a two-restriction design, which provides the facility of two blocking factors that are used to control the effect of two variables that influence the response variable. Latin Square Design is called Latin Square because each Latin letter represents the treatment that occurs once in a row and once in a column in such a way that for one criterion (restriction), rows are completely homogeneous blocks, and concerning another criterion (second restriction) columns are completely homogeneous blocks.

Application of Latin Square Designs

The application of Latin Square Designs is mostly in animal science, agriculture, industrial research, etc. A daily life example can be a simple game called Sudoku puzzle is also a special case of Latin square designs. The main assumption is that there is no contact between treatments, rows, and columns effect.

Latin Square Designs

The general model is defined as
\[Y_{ijk}=\mu+\alpha_i+\beta_j+\tau_k +\varepsilon_{ijk}\]

where $i=1,2,\cdots,t; j=1,2,\cdots,t$ and $k=1,2,\cdots,t$ with $t$ treatments, $t$ rows and $t$ columns,
$\mu$ is the overall mean (general mean) based on all of the observations,
$\alpha_i$ is the effect of the $i$th row,
$\beta_j$ is the effect of $j$th rows,
$\tau_k$ is the effect of the $k$th column.
$\varepsilon_{ijk}$ is the corresponding error term which is assumed to be independent and normally distributed with mean zero and constant variance i.e $\varepsilon_{ijk}\sim N(0, \sigma^2)$.

Latin Square Designs Experimental Layout

Suppose we have 4 treatments (namely: $A, B, C$, and $D$), then it means that we have

Number of Treatments = Number of Rows = Number of Columns =4

The Latin Square Designs Layout can be for example

A
$Y_{111}$		B
$Y_{122}$		C
$Y_{133}$		D
$Y_{144}$
B
$Y_{212}$		C
$Y_{223}$		D
$Y_{234}$		A
$Y_{241}$
C
$Y_{313}$		D
$Y_{324}$		A
$Y_{331}$		B
$Y_{342}$
D
$Y_{414}$		A
$Y_{421}$		B
$Y_{432}$		C
$Y_{443}$

The number in subscript represents a row, block, and treatment number respectively. For example, $Y_{421}$ means the first treatment in the 4th row, the second block (column).

Latin Square Designs

Benefits of using Latin Square Designs

  • Efficiency: It allows to examination of multiple factors (treatments) within a single experiment, reducing the time and resources needed.
  • Controlling Variability: By ensuring a balanced distribution of treatments across rows and columns, one can effectively control for two sources of variation that might otherwise influence the results.

Limitations

The following limitations need to be considered:

  • Number of Treatments: The number of rows and columns in the Latin square must be equal to the number of treatments. This means it works best with a small to moderate number of treatments.
  • Interaction Effects: Latin squares are good for analyzing the main effects of different factors, but they cannot account for interaction effects between those factors.

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Randomized Complete Block Design (RCBD)

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The Randomized Complete Block Design may be defined as the design in which the experimental material is divided into blocks/groups of homogeneous experimental units (experimental units have same characteristics) and each block/group contains a complete set of treatments which are assigned at random to the experimental units.

In Randomized Complete Design (CRD), there is no restriction on the allocation of the treatments to experimental units. But in practical life there are situations where there is relatively large variability in the experimental material, it is possible to make blocks (in a simpler sense groups) of the relatively homogeneous experimental material or units. The design applied in such situations is called a Randomized Complete Block Design (RCBD).

Randomized Complete Block Design

RCBD is a one-restriction design, used to control a variable that influences the response variable. The main aim of the restriction is to control the variable causing the variability in response. Efforts of blocking are made to create a situation of homogeneity within the block. Blocking is a source of variability. An example of a blocking factor might be the gender of a patient (by blocking on gender), this is a source of variability controlled for, leading to greater accuracy. RCBD is a mixed model in which one factor is fixed and the other is random. The main assumption of the design is that there is no contact between the treatment and block effect.

Randomized Complete Block design is said to be a complete design because in this design the experimental units and number of treatments are equal. Each treatment occurs in each block.

The general model is defined as

\[Y_{ij}=\mu+\eta_i+\xi_j+e_{ij}\]

where $i=1,2,3\cdots, t$ and $j=1,2,\cdots, b$ with $t$ treatments and $b$ blocks. $\mu$ is the overall mean based on all observations, $\eta_i$ is the effect of the ith treatment response, $\xi$ is the effect of the jth block, and $e_{ij}$ is the corresponding error term which is assumed to be independent and normally distributed with mean zero and constant variance.

The main objective of blocking is to reduce the variability among experimental units within a block as much as possible and to maximize the variation among blocks; the design would not contribute to improving the precision in detecting treatment differences.

Randomized Complete Block Design Experimental Layout

Suppose there are $t$ treatments and $r$ blocks in a randomized complete block design, then each block contains homogeneous plots for one of each treatment. An experimental layout for such a design using four treatments in three blocks is as follows.

Block 1Block 2Block 3
ABC
BCD
CDA
DAB
Randomized Complete Block Design (RCBD)

From the RCBD layout, we can see that

  • The treatments are assigned at random within blocks of adjacent subjects and each of the treatments appears once in a block.
  • The number of blocks represents the number of replications
  • Any treatment can be adjacent to any other treatment, but not to the same treatment within the block.
  • Variation in an experiment is controlled by accounting for spatial effects.

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