28 Randomized Block Design and Latin Square Design

Dr. Harmanpreet Singh Kapoor

Module 34: Randomized Block Design and Latin Square Design

 

  • Learning Objectives
  • Introduction
  • Randomized Block Design
  • Latin Square Design
  • Examples
  • Summary
  • Suggested Readings
  • Appendix 1

 

  1. Learning Objectives

 

In this module, we will discuss about the second most widely used design to solve practical problems, this design is called Randomized Block Design (RBD). The construction of the RBD, its mathematical derivation as well as its statistical analysis will be discussed. Some other aspects will also be discussed in brief. We will also discuss a briefly about the random treatments and blocks. We will also discuss another important design, which is called the Latin square design. Latin square design is used in those situation where we have two nuisance factors. In this module, introduction to this design and its mathematical derivation will be discussed. A few brief introduction about the replication of the Latin square design will be discussed.

 

  1. Introduction

 

This design is an extension over the CRD design, where the effect the effect of nuisance factor needs to be control. Nuisance factor is that factor which has an effect on the response variable in an indirect manner. It is possible that this nuisance factor cannot be determined and uncontrollable, but we just have the information about the existence of this factor. As we already discussed about the three principle of experimental design: randomization, replication and local control. Here randomization is used to remove the effects that occur due to nuisance factor. It is also possible that we have some information about the nuisance factor and it can be controllable. In this situation, we have to introduce the concept of blocking or local control in our model. It is basically used to eliminate the effect of nuisance factor while the testing of treatment means. Hence due to involvement of the blocking term as well as randomization in the model. The design is named as Randomized Block Design. While building the  model for the observation, we want to keep the experimental error term to be small as possible as we are more interested in removing the variability that arise due to the effect of experimental error. Hence the randomized block design helps the experimenter to take care of this variation.

 

Suppose the experimental area is heterogeneous and we want to control the variability of the experimental material with respect to effect of randomization in one direction. Randomized Block Design is used to control the variability of the experimental material that are divided in groups into homogeneous sub-group or strata or blocks perpendicular to the effect that arise in the experimental units. Hence the main purpose for blocking is to provide same platform to the treatments. So that one can test the treatment mean in more appropriate manner and the variation that appear in the treatment’s values on the experimental units are not influenced by any external factor rather they only arise from the variation in the features of the treatments.

 

  1. Randomized Block Design

 

In randomized block design, treatment are randomly applied to the homogeneous units within each block and replicate over all the blocks. In this module, we discuss only about the model of the RBD. In literature, one can read about how to estimate the parameters of the models and what are the methods that are available in the literature for estimation. While estimation, it is also essential to take account of some conditions. In practical life, there are situations when data from some individual units is/are missing. To solve this type of problem the missing plot techniques is used. One can refer to the literature for in-depth study of RBD. Nowadays, statistical softwares like SPSS and programming language like R, python are used to do the analysis part of the experimental design like CRD, RBD and Latin square design etc.

 

3.1 Statistical Analysis of the RBD:

 

Suppose we have treatments that are to be compared within blocks. The structure of the RBD will be shown as:

 

 

From Table 2, we see that the last column will give you the F statistic value. This F statistic value is used to draw conclusion about the null hypothesis. One can also test for the equality of blocks by making statement about the null hypothesis and evaluate statistic for the blocks. After comparing the calculated statistic value with the tabulated value one can make a decision about the equality of the blocks.

 

3.2 Other Aspects of Randomized Block Design

 

There are situations when increment in the expected response variable by few units due to any treatment or due to any block then how to take care of this increment in the model. In general theory, it is considered that in RBD the principle of additive theory is used. As we have already defined the RBD model as the sum of mean effect, block effect, treatment effect and error term. Hence it is assumed that if treatment or block increase the value of the response variable in units then that incremental value will be reflected in the total value but this is not considered as a relevant situation in general. The reason behind this is that if change occurs due to any treatment or block in response variable then there is a chance of interaction that occur between treatment and block. So there is a need to incorporate interaction term in the model but it not recommended to use interaction term as it inflates the residual mean square that adversely influenced the treatment mean comparison. To tackle this problem, we have to use another design like factorial designs.

 

3.3 Random Treatment and Block

 

The design that we discussed in the previous section consider both the treatments and block as a fixed factors. There are many situations where both the treatments and blocks are random. In most of the situations, blocks are considered to be a random quantity. It will be beneficial for the experimenter to check the validity of the experiment across the random sample of the blocks.

 

  1. Latin Square Design

 

Latin square design is another design that utilizes the principle of blocking. It is possible that in the experimentation of the system we have more than one nuisance factors. If we have only two nuisance factors then the design that consists of taking account of these factors is called Latin square design. The most important feature of Latin square design is that it is in square arrangement and usually denoted by the Latin letters. Hence the designed is named after the Latin letters. The main purpose of Latin square design is to remove the variability that arises from two nuisance factors. In Latin square design we implement the principle of blocking in two directions. Hence rows and columns are generally considered as the two way factors that restrict the implementation of randomization in the testing of treatment mean. In this we have 2 cells and treatments are represented in each cell with Latin letter and each letter occurs one and only one time in each column and row.

 

Here, we can see that           = 9.55 <  0 = 40.711.

 

So that we can see say there is significant difference in the mean value of all the treatments. The three research project differ in terms of strength of cloth. Hence the strength of cloth of research projects differ in different blocks.

Example 2

Suppose a manufacturer wants to study the effects of the six different formulations to manufacture the part of the swing machine. Each formulation mixed from a batch of raw material that is only large enough for six formulations to be tested. The formulation of the parts of the swing machine are prepared by several operators. Thus, it would seem that there are two nuisance factor to be controlled that are: batches of raw material and operators.

Solution

 

The design that will be appropriate for this problem is Latin square design. LSD is a square arrangement design and six formulations represented by Latin letters A, B, C, D, E, and F.

We can see that table is a 6*6 table. After coding the above table by subtracting 6, the new table is given as:

 

To make a decision about the hypothesis, we have to compare the value of the F statistics with the tabulated value of the F distribution. If the value of the test statistics is greater than the tabulated value i.e. 5,20 at 5% level of significance then we reject our null hypothesis i.e. there is a significance difference between the means of treatments. Otherwise we do not reject our null hypothesis and conclude that there is no significant difference in the mean values of the treatments. Here, we can see that = 4.56 > 0 = 0.117.

So that we can see say there is no significant difference in the (treatments) generated by the combination of operators and raw material.

 

  1. Summary

 

In this module, we discussed about how to take into account of nuisance factor when this factor is known and controllable. The principle of local control or blocking is used for this purpose. The design that take care of the concept of blocking i.e. randomized block design is discussed here. This design is also most widely used due to its feature of blocking to solve practical problems. In this module, we discussed the theory of construction of RBD, it mathematical derivation and its statistical analysis are discussed. Example with solution for RBD is also discussed that helps in understanding the concept in depth. Latin square design that is the extension of the randomized block design is also discussed. The mathematical derivation of Latin square and steps for the construction of ANOVA table are also discussed in detail.Example with solution of Latin square is also discussed.

 

  1. Suggested Readings

 

  • Chakarbarti, M.C., Mathematics of Design and Analysis of Experiments, Asia Publishing House, 1970.
  • Cochran W. G. and G. M. Cox, Design of Experiments, Wiley, 1992.
  • Dass, M. N. and N. C. Giri, Design and Analysis of Experiments, New Age International Publishers, 1986.
  • Kempthorne, O., Design and Analysis of Experiments Vol I-II, Wiley, 2007.
  • Montgomery, D. C., Design and Analysis of Experiment, Wiley, 2004.
  • Raghavarao, D.,Construction and Combinatorial Problems in Design of Experiments, Wiley, 1971.