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Between-Groups ANOVA. Chapter 12. When to use an F distribution Working with more than two samples ANOVA Used with two or more nominal independent variables and an interval dependent variable. Why not use multiple t -tests?. The problem of too many t tests Fishing for a finding - PowerPoint PPT Presentation
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Between-Groups ANOVA
Chapter 12
> When to use an F distribution•Working with more than two samples
> ANOVA•Used with two or more nominal
independent variables and an interval dependent variable
> The problem of too many t tests• Fishing for a finding• Problem of Type I error
Why not use multiple t-tests?
> Analyzing variability to compare means
• F = variance between groups variance within groups
> That is, the difference among the sample means divided by the average of the sample variances
The F Distribution
Types of Variance
> Between groups: estimate of the population variance based on differences among group means
> Within groups: estimate of population variance based on differences within (3 or more) sample distributions
Check Your Learning
> If between-groups variance is 8 and within-groups variance is 2, what would F be?
Types of ANOVA
> One-Way: hypothesis test including one nominal variable with more than two levels and a scale DV
> Within-Groups: more than two samples, with the same participants; also called repeated-measures
> Between-Groups: more than two samples, with different participants in each sample
Assumptions of ANOVAs
> Random selection of samples> Normally distributed sample> Homoscedasticity: samples come from
populations with the same variance
One-Way Between-Groups ANOVA
> Everything about ANOVA but the calculations> 1. Identify the populations, distribution, and
assumptions.> 2. State the null and research hypotheses.> 3. Determine the characteristics of the
comparison distribution.> 4. Determine the critical value, or cutoff.> 5. Calculate the test statistic.> 6. Make a decision.
Step 3. Characteristics
•What are the degrees of freedom?> If there are three levels of the independent
variable?> If there are a total of 20 participants in each of
the three levels?
1 groupsbetween Ndf
lastwithin dfdfdfdfdf ...321
111 ndf
> Step 4: Critical Values
Determine Cutoffs for an F Distribution (Step 4)
Formulae
2)( GMXSStotal
2)( MXSSwithin
2)( GMXSSbetween
betweenwithintotal SSSSSS
between
betweenbetween df
SSMS
within
withinwithin df
SSMS
within
between
MSMSF
> Quantifies overlap> Two ways to estimate population
variance• Between-groups variability•Within-groups variability
Logic behind the F Statistic
The Logic of ANOVA
> Presents important calculations and final results in a consistent, easy-to-read format
The Source Table
> What is the ANOVA telling us to do about the null hypothesis?
> Do we reject or accept the null hypothesis?
Bringing it All Together
An F Distribution
Here the F statistic is 8.27 while the cutoff is 3.86. Do we reject the null hypothesis?
Making a Decision
> Step 1. Compare the variance (MS) by diving the sum squares by the degrees of freedom.
> Step 2. Divide the between-groups MS by the within-groups MS value.
> Step 3. Compare the calculated F to the critical F (in Appendix B).• If calculated is bigger than critical, we have
a significant difference between means
Calculating Effect Size
> R2 is a common measure of effect size for ANOVAs.
total
between
SSSSR 2
Post-Hoc Tests to Determine Which Groups Are Different
> When you have three groups, and F is significant, how do you know where the difference(s) are?• Tukey HSD• Bonferonni
> A priori (planned) comparisons
Tukey HSD Test
> Widely used post hoc test that uses means and standard error
MsMMHSD 21
NMSs within
M
The Bonferroni Test
> A post-hoc test that provides a more strict critical value for every comparison of means.
> We use a smaller critical region to make it more difficult to reject the null hypothesis. •Determine the number of comparisons we
plan to make. > Divide the p level by the number of
comparisons.