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

1. 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.

2. Level of Risk=.05

3. Appropriate test statistic=ANOVA

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

• 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/N215,171.60-214,038.53=1,133.07

• Within sum of squares=∑∑(X2)-∑(∑X)2/n216,910-215171.60=1,738.40

• Total sum of squares=∑∑(X2)-(∑∑X)2/N216,910-214,038.53=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

6. Compare obtained value to critical value• @ .05: 8.80 ___ 3.36• @ .01: 8.80 ___ 5.49

7. Is the difference between groups on this score significant?

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

• Accept research hypothesis/reject null

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

• Accept null/reject research hypothesis

Computer Example: Chapter 11 Dataset 1