Transcript CRIM 483 Analysis of Variance Purpose • There are times when you want to compare something across more than two groups – For instance,

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:
–
•
Between sum of squares
–
•
Between groups sum of squares/df for between groups (k-1)
Mean sum of squares for Between Groups
–
•
Sum of the between group sum of squares and the within group sum of squares
Mean sum of squares for Between Groups
–
•
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 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
–
•
Compute sum of squares for each source of variability—between groups, within groups, and
the total
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 Fvalue
– 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