Basic Statistics and Data Analysis

Statistics Lecture Notes, Online MCQs

Category: Design of Experiment (DOE)

Design of Experiment (DOE)

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Data Collection Methods

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. The Data Collection Methods are (i) Survey Method (ii) Simulation (iii) Controlled Experiments (iv) Observational Study.

(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

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 that measurement can be explained by the model 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

Controlled Experiment

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: Controlled experiment 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.

Design of Experiment (DOE)

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Basic Principles of Experimental Design

The basic principles of experimental design are (i) Randomization, (ii) Replication, and (iii) Local Control.

  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 * r = 3 * 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)= 4/12, P(B)= 4/12, and P(C)=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 replication. 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

    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 a refinement in 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.

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 far 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.

Design of Experiment (DOE)

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Latin Square Design (LSD)

In Latin Square Design 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 imposing the restriction 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 variable that influences 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 in respect of one criterion (restriction), rows are completely homogeneous blocks, and in respect of another criterion (second restriction) columns are completely homogeneous blocks.

The application of Latin Square Design 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 design. The main assumption is that there is no contact between treatments, rows, and columns effect.

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 Design 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 Design’s 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).

Design of Experiment (DOE)

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Randomized Complete Block Design

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 simpler sense groups) of the relatively homogeneous experimental material or units. The design applied in such situations is named as Randomized Complete Block Design (RCBD).

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.

Actually, RCBD is a one restriction design, used to control a variable which is influence the response variable. The main aim of the restriction is to control the variable causing the variability in response. Efforts of blocking are done to create the situation of homogeneity within the block. Blocking is a source of variability. An example of 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 a factor is fixed and 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 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 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 one of each treatment. An experimental layout for such a design using four treatments in three blocks be as follows.

Block 1 Block 2 Block 3
A B C
B C D
C D A
D A B

From RCBD layout we can see that

  • The treatments are assigned at random within blocks of adjacent subjects and each of the treatment 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 spatial effects.
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