This discussion covers the most crucial computational background for one-way Analysis of Variance (ANOVA). You will see that this is an extension of hypothesis testing, and is related to regression. For a review of hypothesis testing, please refer to Module 1, and for regression to Modules 2, 3, 4 and 5.

The objective of this section is to give you necessary theoretical background, formulae and tools to help you understand the background and basics of one-way ANOVA. This includes formulation of the ANOVA hypothesis test and conducting the test using already familiar hypothesis test steps.

Let's look at some of the computational background.

So far, in hypothesis testing the discussion has concentrated on at most two population cases. For example, we conducted tests on the equality of population means (t-test), or on the equality of population variances (F-test). In this section we will expand the hypothesis testing to more than two populations. As an example, we would like to test the hypothesis that k population means are equal against the hypothesis that at least two population means are not equal. The hypotheses are commonly stated as follows:

The term Analysis of Variance (ANOVA) refers to a technique in which the total variability in the data is divided into components of variability each of which can be attributed to specific distinct sources of variation. In a One-Way ANOVA model there is thought to be one factor or one source of variation in addition to randomness, which may cause the population means to be unequal. The test is then to determine whether the contribution of this one factor is significant enough to make the population means significantly different.

The One Factor Model

The one factor model may be written as

Note: The term treatment is used to identify the population of concern.

The factor _i effect is a measure of the deviation of the ith treatment mean µ_i from the overall mean µ. If there is no difference between the ith treatment mean µ_i and the overall mean µ then _i=0. Hence, it is sufficient to consider the following hypothesis test:

This hypothesis test statement is equivalent to

The test is based on a comparison of two independent estimates of the overall variance ² using the same approach as in regression. It is assumed in the one-factor experiment that the variation between the treatment averages can be attributed to two components:

• Random variation within the treatment
• Variation due to differences between treatments, including systematic differences between treatments and random variation.

The random variation within the treatment or factor is measured by the error sum of squares (SSE). The variation due to differences between the treatments or factors, the so called factor effect, is measured by the sum of squares between the treatments or factors (SSA). The total variability in the data, measured by the total sum of squares (SST) is divided into these two components like in regression:

SST = SSA + SSE

where
SST = sum of squares total
SSA = sum of squares between treatments or factors
SSE = sum of squares of errors; randomness within treatments or factors

Here

It can be shown that

If the treatment effect _i=0 for all treatments i, then

is an unbiased estimator of ². The estimator s₁² is unbiased because

The second expectation, E(SSE) above, is not affected by the treatment differences, or factor effects _i. Therefore

is always an unbiased estimator of ².

Recall the above hypothesis

The hypothesis H₀ is true if the estimator s₁² is unbiased, otherwise the hypothesis is false. If the hypothesis H₀ is false, then there is at least one _i 0.

Based on the above discussion we can use an F-test to determine if the estimates from the two sources of variation are significantly different.

The estimator s₁² is called the treatment mean square, mSA, and the estimator s² is called the error mean square, mSE, because the estimators are the respective sums of squares divided by their respective degrees of freedom. These degrees of freedom, in turn, depend on the problem size, i.e. number of treatments and number of observations within treatments.

Hence, for the factor A we can write

s₁²= SSA/(k-1)= mSA if H₀ is true

For the random variation we obtain

s² = SSE/(k(n-1)) = mSE

Now, when H₀ is true the ratio

gives values of an F-random variable, which follows an f-distribution with (k-1) and k(n-1) degrees of freedom. Because s₁² overestimates ² when H₀ is false, the critical region is entirely on the right tail of the f-distribution. The decision rule to reject H₀ at the level of significance becomes

Summary of Steps

• H₀: _i=0 for all i
• H₁: _i 0 for at least one i
• Level of significance =0.05
• Critical region

  

• Determine the value of the test statistic F_calc

  

  

• Conclusion: If , then reject H₀ and conclude that at least one of the treatment effects _i 0. Otherwise, accept H₀.