One-way analysis of variance

Chapter 5

Analysis of variance (ANOVA)

Introduction

In Chapter 1 we discussed how to test whether or not the means of two populations are equal. Recall that the test involved the selection of an independent random sample from each of the populations. In this chapter we will discuss a statistical procedure for determining whether or not the means of more than two populations are equal. The technique that we will be introduced is called the analysis of variance (ANOVA) procedure.

One-way analysis of variance

This section discusses the one-way analysis of variance procedure to make tests comparing means often is called one-way analysis, because we will analyze only one factor. Sometimes we may analyze the effects of two factors. This is called two-way analysis of variance.

Suppose that we have independent samples of sizes

observations selected randomly from K populations. Assume that population means are . The one-way analysis of variance procedure designated to test the null hypothesis

(All population means are equal)

against the alternative hypothesis

At least one of the population has different mean

The application of one way analysis of variance requires that the following assumptions hold:

1. The populations from which the samples are selected are (approximately) normal.

2. The populations from which the samples are selected have the same variance (or standard deviation).

3. The samples selected from different populations are independent.

Suppose that from K populations samples of sizes are selected (Figure 5.1)

Figure 5.1

BlockPOPULATION(GROUP)

1 2 …. K

….

….

…. …. …. ….

…. …. …. …..

….

1) The first step is to calculate the sample mean for the K groups of observations. These sample means will be denoted as .

In general

where denotes the number of observations in group.

2) The second step is to find overall mean of the all sample observations, denoted , and defined as

where n denotes the total number of sample observations

An equivalent expression for overall mean is

3) In third step, we consider variability within-groups. To measure variability in the any group, we calculate the sum of squared deviations of the observations about their sample means. Within-groups variability will be denoted by SS. For example, for the first group the sum of squared deviations of the observations about their sample mean is

For the second group, whose sample mean is , we calculate

and so on.

4) In fourth step, we find total within-groups variability, denoted SSW. That is

or

5) Now we need a measure of variability between groups. It is based on the discrepancies between the individual group means and the overall mean. Total between-groups sum of squares denoted , and defined as

6) As a last step, we calculate the sum of squared discrepancies of all the sample observations about their overall mean. This is called the total sum of squares, denoted , expressed as

It can be shown, that the total sum of squares is the sum of the within-groups and between-groups sum of squares, that is

Testing the equality of population means is based on the assumption that K populations have equal variances (or standard deviations).

If

is true, each of the SSW and SSG can be used as the basis for estimate of the common population variance. To obtain these estimates, the sums of squares must be divided by the corresponding numbers of degrees of freedom.

SSW divided by results estimate called the within-groups mean square, denoted MSW, so that

SSG divided by results estimate called the between-groups mean square, denoted MSG, so that

The test of null hypothesis is based on the ratio of the mean squares

If this ratio is close to 1, there would be little cause to doubt the null hypothesis of equality of population means.

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