One-way ANOVAs, along with a number of other statistical tests, use the F distribution. Earlier in this course you learned about the \(z\) and \(t\) distributions. You computed \(z\) and \(t\) test statistics and used those values to look up p-values using statistical software. Similarly, in this lesson you are going to compute F test statistics. The F test statistic can be used to determine the p-value for a one-way ANOVA.

The video below gives a brief introduction to the F distribution and walks you through two examples of using Minitab to find the p-values for given F test statistics. The steps for creating a distribution plot to find the area under the F distribution are the same as the steps for finding the area under the \(z\) or \(t\) distribution. For the F distribution we will always be looking for a right-tailed probability. Later in this lesson we will see that this area is the p-value.

The F distribution has two different degrees of freedom: between groups and within groups. Minitab will call these the numerator and denominator degrees of freedom, respectively. Within groups is also referred to as error. 

A one-way between groups ANOVA is used to compare the means of more than two independent groups. A one-way between groups ANOVA comparing just two groups will give you the same results at the independent \(t\) test that you conducted in Lesson 8. We will use the five step hypothesis testing procedure again in this lesson.

The assumptions for a one-way between groups ANOVA are:

1. Samples are independent
2. The response variable is approximately normally distributed for each group or all group sample sizes are at least 30
3. The population variances are equal across responses for the group levels (if the largest sample standard deviation divided by the smallest sample standard deviation is not greater than two, then assume that the population variances are equal)

Given that you are comparing \(k\) independent groups, the null and alternative hypotheses are:

\(H_{0}: \mu_1 = \mu_2 = \cdots = \mu_k\)
\(H_{a}:\) Not all \(\mu_\cdot\) are equal

In other words, the null hypothesis is that at all of the groups' population means are equal. The alternative is that they are not all equal; there are at least two population means that are not equal to one another.

ANOVA uses an F test statistic. Hand calculations for ANOVAs require many steps. In this class, you will be working primarily with Minitab outputs.

Conceptually, the F statistic is a ratio: \(F=\frac{Between\;groups\;variability}{Within\;groups\;variability}\). Numerically this translates to \(F=\frac{MS_{Between}}{MS_{Within}}\). In other words how much do individuals in different groups vary from one another over how much to individuals within groups vary from one another.

Statistical software will compute the F ratio for you and produce what is known as an ANOVA source table. The ANOVA source table will give you information about the variability between groups and within groups. The table below gives you all of the formulas, but you will not be responsible for performing these calculations by hand. Minitab will do all of these calculations for you and provide you with the full ANOVA source table.

Some of the terms in the table above should look familiar, while others will be new to you. The sum of squares that appears in the ANOVA source table is similar to the sum of squares that you computed in Lesson 2 when computing variance and standard deviation. Recall, the sum of squares is the squared difference between each score and the mean. Here, there are three different sum of squares each measuring a different type of variability.

The ANOVA source table also has three different degrees of freedom: \(df_{between}\), \(df_{within}\), and \(df_{total}\). If you were to look up an F value using statistical software you would need to know two of these degrees of freedom: \(df_1 = df_{between}\) and \(df_2=df_{within}\).

When performing an  ANOVA using statistical software, you will be given the p-value in the ANOVA source table. If performing an ANOVA by hand, you would use the F distribution. Similar to the t distribution, the F distribution varies depending on degrees of freedom.

If \(p \leq \alpha\) reject the null hypothesis. If \(p>\alpha\) fail to reject the null hypothesis.

Based on your decision in Step 4, write a conclusion in terms of the original research question.

While the results of a one-way between groups ANOVA will tell you if there is what is known as a main effect of the explanatory variable, the initial results will not tell you which groups are different from one another. In order to determine which groups are different from one another, a post-hoc test is needed. Post-hoc tests are conducted after an ANOVA to determine which groups differ from one another. There are many different post-hoc analyses that could be performed following an ANOVA. Here, we will learn about one of the most common tests known as Tukey's Honestly Significant Differences (HSD) Test.

Most statistical software, including Minitab, will compute Tukey's pairwise comparisons for you. This specific post-hoc test makes all possible pairwise comparisons. In this class we will be relying on statistical software to perform these analyses, if you are interested in seeing how the calculations are performed, this information is contained in the notes for STAT 502: Analysis of Variance and Design of Experiments. This analysis takes into account the fact that multiple tests are being performed and makes the necessary adjustments to ensure that Type I error is not inflated.

In the examples later in this lesson you will see a number of Tukey post-hoc tests. Next, you will also learn how to obtain these results using Minitab.

For each pairwise comparison, \(H_0: \mu_i - \mu_j=0\) and \(H_a: \mu_i - \mu_j \ne 0\).

In one research study, 20 young pigs are assigned at random among 4 experimental groups. Each group is fed a different diet. (This design is a completely randomized design.) The data are the pigs' weights in kg after being raised on these diets for 10 months. We wish to determine if there are any differences in mean pig weights for the 4 diets.

• \(H_0: \mu_1 = \mu_2 = \mu_3 = \mu_4\)
• \(H_a:\) Not all \(\mu\) are equal

Contained in the Minitab file: ANOVA_ex.mpx

Note that in this file the data were entered so that each group is in its own column. In other words, the responses are in a separate column for each factor level. In later examples, you will see that Minitab will also conduct a one-way ANOVA if the responses are all in one column with the factor codes in another column. 

In this lesson you learned how to compare the means of three or more groups using a one-way between groups ANOVA. A one-way between groups ANOVA is used instead of multiple independent \(t\) tests in order to avoid increasing the likelihood of committing a Type I error.

An ANOVA provides information about the explanatory variable overall, but not about differences between the different levels of that variable. In order to compare the different pairs we need to conduct a post-hoc analysis such as Tukey's HSD test.