Analysis of Variance (ANOVA)

• Analysis of Variance (ANOVA) allows statistical comparison among samples taken from many populations.
• The comparison is typically the result of an experiment, such as trying the same test at many different locations.
• The factor is the major thing that differs between each experiment (e.g., location).
• The specific values of a given factor are called levels.
• Levels cause the variable under study to be divided into groups.
• Examples of experiments that can be conducted when performing ANOVA analysis:
  • Completely Randomized Design: an experiment with only one factor.
  • Factorial Design: more than one factor is considered.
  • Randomized Block Design: where groups are also divided into subgroups.
• The purpose of ANOVA is to reach conclusions about possible differences among the means of each group.

Completely Randomized Design

• Analyzes a single factor.
• Involves a two-step process:
  • Step 1: Determine if there is a significant difference among the group means; the null hypothesis is that there is not.
  • Step 2: Determine which groups contain means that are significantly different from the other group means.
• For analyzing variation, the goal is to separate the total variation into variation due to differences among groups and variation due to differences within the groups.

Completely Randomized Design - One Way ANOVA

• Sum of Squares Total (SST) is the total variation.
• Within-Group Variation (SSW) is the variation measured within each group.
• Among-Group Variation (SSA) is the variation measured among the groups.
• SST = SSA + SSW
• The grand mean is the mean of the means of each group.
• Sum of squares among groups has c-1 degrees of freedom where c = number of groups
• Sum of squares within groups has n-c degrees of freedom where n = number of items in all groups

MSA / MSW

• MSW = Mean Square Within
• MSA = Mean Square Among
• MSW = SSW /(n-c)
• MSA = SSA /(c-1)

F test for Differences Among More Than Two Means

• F test is used to determine if there is a significant difference among the group means.
• The F test is the ratio of MSA divided by MSW.
• The null hypothesis is that there is no significant difference among the means.
• Reject the null hypothesis if the F test result is above the upper tail critical value.
• Critical value can be looked up in table A.6 (page 877)
  • c - 1 degrees of freedom in the numerator
  • n - c degrees of freedom in the denominator

ANOVA Summary Table

• Used to summarize the results of a one-way ANOVA.

F test assumptions

• Randomness and Independence:
  • Random samples were selected from the c groups
• Normality:
  • Normality of the c groups from which the samples are selected is assumed
• Homogeneity of variance:
  • The variances of the c groups are equal