Introduction to Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA)

a hypothesis-testing procedure that is used to evaluate mean differences between two or more treatments/groups/populations

F Ratio

ratio of the between-groups population variance estimate to the within-groups population variance estimate

F Table

table of cutoff scores on the F distribution

Basic Logic of ANOVA

the null hypothesis in an analysis of variance is that the population being compared all have the same mean

Step 1 • calculate the Total Sum of Squares

Step 2 • calculate the Within-Treatments Sum of Squares

Step 3 • calculate the Between-Treatments Sum of Squares

Step 4 • calculate the Total Degrees of Freedom

Step 5 • calculate the Within-Treatments Degrees of Freedom

Step 6 • calculate the Between-Treatments Degrees of Freedom

Step 7 • calculate the Between-Treatments Variance

Step 8 • calculate the Within-Treatments Variance

Step 9 • calculate the F ratio

Carrying Out an Analysis of Variance (Sum of Squares) STEPS

1. Measure your dependent variable on a continuous scale (interval or ratio).
2. Use two or more distinct groups for your independent variable.
3. Keep observations and groups independent from each other.
4. Avoid significant outliers.
5. Aim for normal distribution in your dependent variable within each group or in its residuals.
6. Ensure consistent variances (homogeneity).

Assumptions of the ANOVA

Post Hoc Comparisons

are additional hypothesis tests that are done after an ANOVA to determine exactly which mean differences are significant and which are no

Tukey’s Honestly Significant Difference (HSD) Test

is a single-step multiple comparison procedure and statistical test. It can be used on raw data or in conjunction with an ANOVA to find means that are significantly different from each other

• most commonly used for equal sample sizes

Games-Howell

a nonparametric approach in comparing combinations of groups or treatments

• it is like the Tukey’s test, but it does not assume equal variances and sample sizes

Scheffé’s Test

method of figuring the significance of post hoc comparisons that takes into account all possible comparisons that could be made

• It is customarily used for unequal sample sizes

Bonferroni Procedure

• is a multiple-comparison procedure in which the total alpha percentage is divided among the set of comparisons so that each is tested at a more stringent significance level

Kruskal-Wallis H Test

• (sometimes also called the "one-way ANOVA on ranks") is a rankbased nonparametric test that can be used to determine if there are statistically significant differences between two or more groups of an independent variable on a continuous or ordinal dependent variable

1. Measure dependent variable as ordinal or continuous.
2. Use two or more independent categorical groups.
3. Ensure observation independence.
4. Dependent variables need not be normally distributed.

Assumptions of the Kruskal-Wallis H Test

Repeated Measures ANOVA

a design with one group of individuals participating in **three (3)** or more treatment conditions

Two-Way ANOVA

an ANOVA used for a factorial design----a design **with more than one independent variable and one dependent variable**

Analysis of Covariance (ANCOVA)

• analysis of variance that controls for the effect of one or more additional variables

Covariate - variable controlled for in an analysis of variance

Multivariate Analysis of Variance (MANOVA)

• analysis of variance with more than one dependent variable

Multivariate Analysis of Covariance (MANCOVA)

analysis of covariance with more than one dependent variable