# Advanced ANOVA/One-way ANOVA

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This tutorial teaches use of one-way ANOVA, a statistical technique for testing mean differences betweeen three or more independent groups on a single dependent variable. Practical exercises are based on using SPSS.

## Purpose

• Assesses the statistical significance of differences between three or more group means for a single dependent variable
• Extension of a t-test
• Use one-way ANOVA in preference to multiple pairwise comparisons (t-tests) because:
• Computationally easier
• Limits the probability of type I and type II errors.
• With multiple comparisons, if they are all independent (which is unlikley) in a series of 100 tests we would expect to get five Type I errors with a .05 level of significance
• By simultaneously computing all possible comparisons in a single significance test, the ANOVA avoids these inflated error rates
• However, use of one-way ANOVA limits error rates at the expense of specificity - statistic tells us that there is a significant difference somewhere among the sample means, but does not tell us which means differ significantly (have to use post-hoc and a priori comparison procedures)

## Examples

• Experimental study: Examine reaction time under different levels of alcohol consumption by randomly assigning participants to four conditions (none, low, medium, and high alcohol)
• Quasi-experimental study: Examine whether students with behaviour problems behave better in classrooms where teachers have a humanistic philosophy and have control of their classrooms. Classify teachers as (1) humanists with control, (2) strict disciplinarians, and (3) laissez-faire.

## General steps

1. Establish hypothesis/hypotheses
2. Examine assumptions - If assumptions are not met, use the Kruskal-Wallis non-parametric procedure
3. Examine descriptive statistics, particularly the four moments (M, SD, Skewness, Kurtosis) overall, and also for each group
4. Examine graphs, e.g.,:
• Histograms
• Normal probability plot
• Error-bar graph
5. Conduct inferential test (ANOVA) and interpret significance of F
6. Conduct follow-up tests (planned contrasts or post-hoc tests) if F is significant
7. Calculate and interpret effect sizes
• Eta-square (omnibus - equivalent to R2)
• Standardised mean effect size (difference b/w two means) - e.g., Cohen's d

## Error bar graphs

• Use any dataset
• Conduct a one-way ANOVA and graphically present the means and confidence intervals using an Error Bar Graph - is this error bar chart consistent with the statistical results?
• Why?
• Why not?