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### CRIM 483

Analysis of Variance

Purpose

- There are times when you want to compare something across more than two groups
- For instance, level of education, SES, age groups, etc.
- Book example related to sports performance—examining the difference in coping skills across different levels of experience
- Group 1: 6 years or less of experience
- Group 2: 7-10 years of experience
- Group 3: More than 10 years of experience

Description & Use

- Simple analysis of variance: There is one factor or one treatment variable (e.g., group membership).
- The variance due to differences is separated into:
- Variance that is due to differences between individuals within groups
- Variance due to differences between groups
- In an ANOVA procedure, the two types of variance are compared to one another to determine if there is a significant difference between the tested groups
- Use ANOVA when:
- There is only one dimension or treatment
- There are more than two levels of the grouping factor
- You are looking at differences across groups in average scores

Testing the ANOVA

- The test statistic for significance with ANOVA is the F test
- F=MSbetween/MSwithin
- Thus, the ANOVA is a ratio that compares the amount of variability between groups to the amount of variability within groups
- Variability between groups=the variability due to the grouping factor
- Variability within groups=the variability due to chance
- If the ratio is 1, than the two types of variability is equal; hence, no group differences on the factor you are comparing (e.g., coping skills)

Determining Significance

- As the average difference between groups (numerator) gets larger, so does the F-value; the larger the difference, the more likely that the difference will obtain statistical significance
- As the F-value increases, it becomes more extreme in relation to the distribution of all F values and is more likely due to something other than chance
- .25/.25=1.00—no difference b/t groups
- .50/.25=2.00—possible difference b/t groups
- .50/.75=.67—no difference b/t groups
- F-value works in only one direction because the ANOVA can only test a non-directional hypothesis

An Example

- Null and Research Hypothesis:
- There will be no difference between the means for the three different groups of preschoolers.
- There will be a difference between groups of preschoolers on these scores.
- Level of Risk=.05
- Appropriate test statistic=ANOVA

To calculate the F-statistic, you must first:

Compute sum of squares for each source of variability—between groups, within groups, and the total

Between sum of squares

Sum of the differences between the mean of all scores and the mean of each group’s score…squared (how different is each group’s mean from the overall mean).

Within sum of squares

Sum of the differences between each individual score in the a group and the mean of each group…squared (how different is each score in a group is from the group’s mean).

Total

Sum of the between group sum of squares and the within group sum of squares

Mean sum of squares for Between Groups

Between groups sum of squares/df for between groups (k-1)

Mean sum of squares for Between Groups

Within groups sum of squares/df for within groups (N-k)

F-value= Mean Sum of Squares for Between Groups

Mean Sum of Squares for Within Groups

4. Compute the test statistic value (obtained value):Computations

- Between sum of squares=

∑(∑X)2/n-(∑∑X)2/N

215,171.60-214,038.53=1,133.07

- Within sum of squares=

∑∑(X2)-∑(∑X)2/n

216,910-215171.60=1,738.40

- Total sum of squares=

∑∑(X2)-(∑∑X)2/N

216,910-214,038.53=2,871.47

5. Determine critical value to determine significance of F-value

Like the t-test, you will need degrees of freedom to find a critical value for the F-value. This time, you will need a DF for between groups and a DF for within groups

DF (between groups)=k-1, where k=# of groups

3 groups-1=2

DF (within groups)=N-k, where N=# of cases and k=# of groups

30 cases-3 groups=27

The obtained, computed F-value is 8.80 with DF (2, 27)

Using Table B3 in Appendix B, you can now obtain the critical value at which any F-value that is greater will be significant at the p<.05 level

@ .05 threshold, the critical value is 3.36

@ .01 threshold, the critical value is 5.49

Compare obtained value to critical value

@ .05: 8.80 ___ 3.36

@ .01: 8.80 ___ 5.49

Is the difference between groups on this score significant?

If obtained F-value is less than critical value, difference between groups is statistically significant

Accept research hypothesis/reject null

If obtained F-value is greater than critical value, difference between groups is not statistically significant

Accept null/reject research hypothesis

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