26 Completely Randomized Design-I

Dr. Harmanpreet Singh Kapoor

Module 32: Completely Randomized Design-I

 

  • Learning Objectives
  • Introduction
  • Analysis of Variance (ANOVA)
  • Analysis of Fixed Effect Model
  • Separation of Total Variability into Components
  • Estimation of the Parameter
  • Examples
  • Summary
  • Suggested Readings
  • Appendix 1

 

  1. Learning Objectives

 

In this module, our main objective is to give an introduction to completely randomized design or ANOVA. The two effects of completely randomized design model: fixed effect and random effect models will be discussed in detail. Statistical analysis of the model as well as the mathematical derivations for the construction of ANOVA table will be discussed. This model has played a vital role in different area of research. Examples will be discussed to understand the topic in depth.

 

  1. Introduction

 

In general, the experimenter is interested to investigate whether the observed difference between the treatment means are non-significant (occur by chance) or whether there exist an actual difference between the treatment means. With the help of experimental design that enable to compare the effect of different treatments on the response variable’s mean value. In many situations, it is possible that the effect of treatments on the mean of response variable to be interested is only due to the variability that occur from the extraneous factors that are not of interest. To take care of this situation a proper experimental design is chosen to get an appropriate result.

 

We already discussed about three principles: randomization, replication and blocking in the module “Basic Principles of Design of Experiments”. These three principles will help the experimenter to concentrate on the objective/ response variable and the statistical analysis as well as the estimation of the treatment effects and how to take care of the extraneous, uncontrolled, and non-assignable factors while building a model.

 

Basically the completely randomized design is used to split the total variability into two segments. It follows the method of separating the total variability into components that have different source of variability. These source of variability are then compared and this comparison will help to test about the significance of the null hypothesis that there is no difference exist between the treatment mean.

 

It is essential that one should have a clear idea about testing hypothesis and statistical terms involved in them. One can refer to the modules based on testing of hypothesis and some parametric tests. The topic of completely randomized design is very important as it has wide applications in all the areas. To make this module simple and understandable, we split this topic into two modules. In this module, we will discuss about the theoretical part of ANOVA including assumptions of the model, decomposition of the total sum of square, estimation of the model parameter, and statistical analysis. In the next module, we will discuss in brief about checking for the assumptions of the model and the pairwise comparison of the treatments with various tests available in the literature.

 

In the next section, the introduction to the basic model of ANOVA is given in detail.

 

  1. Analysis of Variance (ANOVA)

 

In this section we will discuss about the analysis of the variance (ANOVA) of single factor experimental design problem. This single factor design contains one type of factor that affect the result of the experimental design. This single factor design is also known as Completely Randomized Design (CRD). ANOVA is the abbreviation for the one way analysis of variance.

 

Suppose there are p treatments or different levels of a single factor. Our objective is to compare p treatments or different levels of a single factor. Observation are recorded with respect to each of p treatments and all observations are treated as realization of random variables. The recorded observation can be shown in the following table:

 

We can see that observation of response variables are represented in terms of ???, where ?, ? = 1,2, … , ?, is used for treatment or different levels of a single factor and ?,? = 1,2, … , ? represents the observation
with respect to ?

 

.

is the   ℎ treatment effect.

 

The original model is also called the means model as it involves the effect of different treatments or levels of single factor due to its mean value. Now after transforming the original model, the new model is known as effects model. Both models are linear statistical models and also both are very useful in the experimental design literature but effects model is widely used because it seems to be appropriate to incorporate constant mean and treatment effects separate in the model.

 

In the treatment effects models, we can see two types of effects. One effect is if treatments are chosen by the experimenter. In this situation, we are interested in to test about the treatment means and after applying the statistical analysis the conclusion about the significance of the hypothesis that is whether  reject the null hypothesis and do not reject the null hypothesis will be applicable only for the treatments taken in the study. The result cannot be generalized to other treatments that were not considered. Hence this type of effect is known as fixed effects model.

 

Second effect is when the treatment would be taken as a random sample from a large population of the treatments. The conclusion by the testing of the hypothesis may apply not only for this random sample but also to other random sample treatments of the population. This effect is known as random effect model. This model is also known as components of variance model because in this model treatment effects are random variable and we try to test about the variability of the treatments effect and also try to estimate the variability among them

 

In the next section, we will consider about the analysis part of different effects of model in detail.

  1. Analysis of the Fixed Effects Model

In this section we will discuss analysis of variance for the fixed effect model. Our objective for this analysis is to compare the mean effect with respect to different levels of the single factor.

 

  • ??: ?1 = ?2 = ⋯ = ?? = ? (null hypothesis)
    ?1: ?? ≠ ??
    for at least one pair (?,?) (alternative hypothesis)
    As ?? = ? + ??
    , also ∑ ??
    ?
    ? = 0, then the hypothesis in term of ? is defined as
    ??: ?1 = ?2 = ⋯ = ?? = 0
    ?1: ?? ≠ 0 for at least one ?.

Assumptions

 

There are certain assumptions for the analysis of variance. These are

 

  • (i) The population must be from normal distribution.
  • (ii) The population must have a common variance.
  • (iii) The observations must be independent.

 

One has to test these assumptions, before applying the model on the data. To test the assumption of normality and independence of observations, one has to familiar with the methods available in the literature for this purpose.

 

Now, we have to apply the following method for testing the equality of hypothesis of equality of treatment mean or to test whether the treatment effect are zero.

  1. Separation of Total Variability into Components

In this section, we will discuss about how we can split the total variability into components associated with different source of that variability. The total sum of square is defined as:

 

  • (iv) The results that we derive are based on some assumptions. The assumptions for this problem is that runs should conform to normal distribution and variance should be same for all the teams.

 

Hence in this section, examples with their solutions are discussed so that one can understand how to build ANOVA table.

 

  1. Summary

 

In this module, we discussed about the ANOVA or Completely Randomized design. In the first section, we discuss about the importance of the ANOVA in practical life. In the analysis of variance section, we define the model for the ANOVA and also discussed two effects: fixed effect model and random effect model. We start with the fixed effect model and then we discussed about the partitioning of the total variability into components. The method for the estimation of the parameters are covered in this module. Statement of Cochran theorem will help one to understand about the independence of the components of total variability. Statistical analysis of the model as well as the mathematical derivations for the construction of ANOVA table is discussed. Some examples and their step by steps analysis are discussed that will help to apply the ANOVA on the real life data sets.

 

  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.