Design of Experiments (DOE)

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

Block Design, Incidence, and Concurrence Matrix (2018)

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Block Design Properties

The necessary conditions that the parameters of a Balanced Incomplete Block Design (BIB design) must satisfy are

  • $bk = vr$, where $r=\frac{bk}{v}$ each treatment has $r$ replications
  • no treatment appears more than once in any block
  • all unordered pairs of treatments appear exactly in $\lambda$ blocks (equi-concurrence)
    where $\lambda=\frac{r(k-1)}{v-1}=\frac{bk(k-1}{v(v-1)}$ is often referred to as the concurrence parameter of a BIB design.

A design say $d$ with parameters $(v, b, r, k, \lambda)$ can be represented as a $v \times b$ treatment block incidence matrix (having $v$ rows and $b$ columns). Let denote it by $N=n_{ij}$ whose elements $n_{ij}$ signify the number of units in block $j$ allocated to treatment $i$. The rows of the incidence matrix are labeled with varieties (treatments) of the design and the columns with the blocks.

We have to put 1 in the ($i$, $j$)th cell of the matrix if variety $i$ is contained in block $j$ and 0 otherwise. Each row of the incidence matrix has $r$ 1’s, each column has $k$ 1’s, and each pair of distinct rows has $\lambda$ column 1’s, leading to a useful identity matrix.
The matrix $NN’$ has $v$ rows and $v$ columns, referred to as the concurrence matrix of design $d$, and its entries, the concurrence parameters are denoted by $\lambda_{dij}$. For a BIBD, $n_{ij}$ is either one or zero, and $n_{ij}^2= n_{ij}$.

Theorem: If $N$ is the incidence matrix of a $(v, b, r, k, \lambda)$-design then $NN’=(r-\lambda)I+\lambda J$ where $I$ is $v\times v$ identity matrix and $J$ is the $v\times v$ matrix of all 1’s.

Example: For Block Design {1,2,3}, {2,3,4}, {3,4,1}, {4,1,2} construct incidence matrix

Block Design: incidence matrix
Incidence and Concurrence matrix


Denoting the elements of $NN’$ by $q_{ih}$, we see that $q_{ii}=\sum_j n_{ij}^2$ and $q_{ih}=\sum_j n_{ij} n_{hj}, (i \ne h)$. For any block design $NN’$, the treatment concurrence with diagonal elements equal to $q_{ii}=r$ and off-diagonal elements are $q_{ih}=\lambda, (i\ne h)$ equal to the number of times any pairs of treatment occur together within the block. In a balanced design, the off-diagonal entries in $NN’$ are all equal to a constant $\lambda$ i.e., the common replication for a BIBD is $r$, and the common pairwise treatment concurrence is $\lambda$.

$N$ is a matrix of $v$ rows and $b$ columns that $r(N)\le min(b, c)$. Hence, $t\le min(b, v)$. If design is symmetric $b=v$ and $N$ is square the $|NN’|=|N|^2$, so $(r-\lambda)^{v-1}r^2$ is a perfect square.

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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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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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