1. PS 225 Lecture 15 Analysis of Variance ANOVA Tables

2. Analysis of Variance  Compare the values of a variable when grouped according to the values of another variable  Grouping variable called a ‘factor’

3. Differences Between Groups  Observed sample differences caused by: True population differences Variation  Different Types of Variance: Variation within groups Variation between group means

4. Two Types of Variability  With-in group variability Similar in concept to standard deviation Variability between the data in a sample  Between-group variability Similar to the standard deviation of sample means Variation between sample means

5. Box Plot MiddleHigh SchoolCollege Hours Spent at School Per Day 4 6 8 10 12

6. Assumptions Needed for ANOVA  Independent random samples taken from each population  Normality  Equality of Variance Levine Test Visual Examination of Box plot

7. Comparing the Variation Types Between-Groups Variation Between-Groups Mean Square Error Within-Groups Variation Within-Groups Mean Square Error F == Division creates a ratio of the different types of variation

8. Mean Square Error  A measure of variation  Takes into account number of samples

9. Within-Groups Sum of Squares  Take standard deviation for all groups and square them to obtain variance  Multiply each variance by the degree of freedom for each group (n-1)

10. Between-Groups Sum of Squares  Subtract the overall mean from each group mean and square the difference. Multiply the square by the number of observations in the group. Add all of the results to get the mean square  Calculate the degrees of freedom- the number of groups minus 1  Divide the mean square by the degrees of freedom

11. F-distribution  Probability Distribution of the ratio of mean squares  Small significance means reject Ho  Ho: The mean is the same for all groups

12. Why The F-Distribution?  The more individual mean difference tests conducted, the greater the probability of observing a mean difference when there is none  Conduct f-test to determine if there are differences  Conduct Bonferroni multiple comparisons test to determine which means are different

13. SPSS Anova Table Salary divided by Education Level

14. SPSS Bonferroni Comparison

15. Assignment  What determines the age at which an individual is first married? Sex? Education (highest degree)? Determine if there are mean differences using an Anova table or Independent Sample T-test Determine which means are significantly different using the Boniferri Comparison when applicable Explain the relationship between all variable pairs, can you determine a specific cause of early marriage?