ANOVA EXAMPLE

Miller and Vanhoutte conducted experiments in which adult ovariectomized female mongrel dogs were treated with estrogen, progesteron, or estrogen plus progesteron. Five untreated animals served as controls. A Variable of interest was concentration of progesterone in the serum of the animals 14 to 21 days after treatment. We wish to know if the treatments have different effects on the mean serum concentration of progesterone.
Solution:-

One-Way ANOVA

The simplest type of the analysis of variance is that known as one-way ANOVA, in which only one source of variation, or factor, is investigated. It is an extension to three or more samples of the t-test procedure for use with two independent samples. We can say that the t test for use with two independent samples is a special case of one-way ANOVA.
In a typical situation we want to use one-way ANOVA to test the null hypothesis that three or more treatments are equally effective. The necessary experiment is designed in such a way that the treatments of intrest are assigned completely at random to the subjects or objects on which the measurements to determine treatment effectiveness are to be made. For this reason design is called completely randomized experimental design.

ANOVA Procedure

Following are the steps of the Analysis of Variance procedure.
1- Description of the data
 in addition to describe the data in the usual way, we display the sample data in tabular form.
2- Assumptions along with the assumtions underlying the analysis, we present the model for each design we discuss above. The model consists of a symbolic representation of a typical value from the data being analyzed.
3- Hypothesis
4- Test Statistic
5- Distribution of Test Statistic
6- Decision Rule
7- Calculation of test statistic
 the results of the arithmatic calculations will be summarized in table called the analysis of variance (ANOVA) table. The entries in the table make it easy to evaluate the results of the analysis.
8- Statistical Decision
9- Conclusion

Use of Computer

The calculations required by analysis of variance are lengthier and more complicated. The computer assumes an important role in analysis of variance . All the exercises appearing in this section are suitable for computer analysis and may be used with the statistical packages. The out put of the statistical packages may vary slightly. The basic concept of the analysis of variance that we present here should provide the necessary background for understanding the description of the programs and their output in any of the statistical packages.

Assumptions of ANOVA

The valid use of analysis of variance as a tool of statistical inference are a set of fundamental assumptions. We refer to the paper of Eisenhart. Although an experimenter must not expert to find all the assumptions met to perfection, it is important that the user of analysis variance techniques be aware of the underlying assumptions and be able to recognize when they are substanially unsatisfied.The consequences of the failure to meet the assumptions are discussed by Cochran in a companion paper to that of Eisenhart. Because experiments in which all the assumptions are perfectly met are rare, Cochran suggests that ANOVA results be considered as approximate rather than exact. These assumptions are pointed out at appropriate points in the following.
We discuss ANOVA as it is used to analyze the results of two different experimental designs, the completely randomized and the randomized complete  block design. In addition to these , the concept of a factorial experiment is given through its use in a completely randomized design. These do not exhaust the possibilities.

Analysis of Variance

It is defined as a technique whereby the total variation present in a set of data is partioned into two or more components. Associated with each of these components is a  specific source of variation, so that in the analysis it is possible to ascertain the magnitude of the contributions of each of these sources to the total variation.

Applications of analysis of variance

 Analysis of variance finds its wide application in the analysis of data derived from experiments. the principles of the design of experiments are well covered in several books, including those of Cochran and Cox, Cox, Davies ,Federer, Finney, Fisher, John, Kempthorne, Li and Mendenhall.

Analysis of variance is used for two different purposes, 1- to estimate and test hypothesis about population variances and 2- to estimate and test hypothesis about population means. 

The following given example illustrates the basic ideas involved in the application of analysis of variance. 

Example:-

Addition Rule of Probability

The third property of the probability states that probability of occurrence of either one or the other of two mutually exclusive events is equal to the sum of their individual probabilities.

For Example

Multiplication Rule

A probability may be calculated from the other probabilities.For example, joint parobability may be calculated as the product of appropriate marginal probability, and also from the conditional probability.This relationship of probabilities is called the multiplication rule of probability.