Basic Statistics and Data Analysis

Lecture notes, MCQS of Statistics

Category: Design of Experiment (DOE)

Data Collection Methods

There are many methods to collect data, but these methods can be classified in four main methods (sources) of collecting data to use in statistical inference. These are (i) Survey Method (ii) Simulation (iii) Controlled Experiments (iv) Observational Study.

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 house-holds that own more than one motor vehicle, the number of passenger 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.

Simulation

In Simulation, a computer model for the operation of a (industrial)  system is setup in which an important measurement is a percentage purity of a (chemical) product. A very large number of realizations of the model can be run in order 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.

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 a grass seed to use in sport 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 the controlled experiment as it has certain constraints such as;

i) River on right side
ii) Shadow of trees on left side
iii) There are 3 different varieties (say, v1, v2, v3) and are distributed in 12 units.

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

Controlled experiment 2There are 3 experimental units. One is close to stream and other is to trees while third one is between them which is most beneficial than others. It is now our choice where to place any one of them at any of the side.

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 the 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 a 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 Experimental Design

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

  1. Randomization

    Randomization is the corner stone 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 number of treatment = 3 (say, A, B, and C) and replication = r = 4, then the number of elements = t x r = 3 x 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, 12 are reserved for treatment A, element 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 has equal chances of its selection.

  2. Replication

    By replication we means that repetition of the basic experiments. For example, If we need to compare grain yield of two varieties of wheat then each variety is applied to more than one experimental units. The number of times these are applied on 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.
    • The 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 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 are not removed by randomization and replication, i.e. unable to control 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 are 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 a replicate.

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

 

Latin Square Design (LSD)

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 treatment must appears once and only once in each of the row 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 represents major source of variation.

Hence a Latin Square Design is an arrangement of k treatments in a k x 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 restrictional design, which provided the facility of two blocking factor which 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 other criterion (second restriction) columns are completely homogeneous blocks.

The application of Latin Square Design is mostly in animal science, agriculture and 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 observation,
$\alpha_i$ is the effect of ith row,
$\beta_j$ is the effect of jth rows,
$\tau_k$ is the effect of kth 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

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

 

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 restrictional 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 is done to create the situation of homogeneity within block. A blocking is a source of variability. An example of blocking factor might be the gender of a patient (by blocking on gender), this is 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 improve 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 block 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.

 

Completely Randomized Design (CRD)

Completely Randomized Design (CRD)

A simplest and non–restricted experimental design, in which occurrence of each treatment has equal number of chances, each treatment can be accommodate in the plan, and the replication of each treatment is unequal is known to be completely randomized design (CRD). In this regard this design is known as unrestricted (a design without any condition) design that have one primary factor. In general form it is also known as one-way analysis of variance.

Let we have three treatments names A, B, and C placed randomly in different experimental units.

C A C
B A A
B B C

 

We can see that from the table above:

  • There may or may not be repetition of treatment
  • Only source of variation is treatment
  • It is not necessary that specific treatment comes in specific unit.
  • There are three treatments such that each treatment appears three times having P(A)=P(B)=P(C)=3/9.
  • Each treatment is appearing equal number of times (it may be unequal i.e. unbalance)
  • The total number of experimental units are 9.

Some Advantages of Completely Randomized Design (CRD)

  1. The main advantage of this design is that the analysis of data is simplest even if some unit of does not response due to any reason.
  2. Another advantage of this design is that is provided maximum degree of freedom for error.
  3. This design is mostly used in laboratory experiment where all the other factors are in under control of the researcher. For example in a tube experiment CRD in best because all the factors are under control.

An assumption regarded to completely randomized design (CRD) is that the observation in each level of a factor will be independent from each other.

The general model with one factor can be defined as

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

Where$i=1,2,\cdots,t$ and $j=1,2,\cdots, r_i$ with $t$ treatments and $r$ replication. $\mu$ is the overall mean based on all observation. $eta_i$ is the effect of ith treatment response. $e_{ij}$ is the corresponding error term which is assumed to be independent and normally distributed with mean zero and constant variance for each.

Read from WikiPedia: Completely Randomized Design (CRD)


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