صفحه 1:
ANOVA
Arash Mashhad
i
صفحه 2:
Why ANOVA?
* In real life things do not typically result in two
groups being compared
- Test lines on I-64 in Frankfort
* Two-sample t-tests are problematic
- Increasing the risk of a Type | error
- At .05 level of significance, with 100 comparisons, 5
will show a difference when none exists
(experimentwise error)
- So the more t-tests you run, the greater the risk of a
type | error (rejecting the null when there is no
difference)
* ANOVA allows us to see if there are differences
between means with an OMNIBUS test
حم
صفحه 3:
When ANOVA?
* Data must be experimental
* If you do not have access to statistical
software, an ANOVA can be computed by hand
* With many experimental designs, the sample
sizes must be equal for the various factor level
combinations
* Aregression analysis will accomplish the same
goal as an ANOVA.
* ANOVA formulas change from one
experimental design to another
حم
صفحه 4:
۹ سس
Variance - why do scores
vary?
° A representation of the spread of
scores
¢ What contributes to differences in
scores?
- Individual differences
- Which group you are in
صفحه 5:
a
|, Wariance to compare
Means
° We are applying the variance
concept to means
- How do means of different groups
compare to the overall mean
¢ Do the means vary so greatly from
each other that they exceed
individual differences within the
groups?
صفحه 6:
Between/Within Groups
¢ Variance can be separated into two
major components
- Within groups - variability or differences
in particular groups (individual differences)
- Between groups - differences depending
what group one is in or what treatment is
received
Formulas: page 550
حم
صفحه 7:
٠ ااا سس سس تست اس سسوم
Bottom Line
¢ We are examining the ratio of
differences (variances) from
treatment to variances from
individual differences
¢ If the ratio is large there is a
significant impact from treatment.
¢ We know if a ratio is “large enough”
by calculating the ratio of the MST to
MSE and conducting an F test.
صفحه 8:
الل ببسم
Fundamental Concepts
* You are able to compare MULTIPLE means
¢ Between-group variance reflects
differences in the way the groups were
treated
¢ Within-group variance reflects individual
differences
* Null hypothesis: no difference in means
* Alternative hypothesis: difference in
means
حم
صفحه 9:
سس سس مس تسس تمس سس سس سس سیم
Sum of Squares
¢ We are comparing “variance estimates”
- Variance = SS/df
٠ The charge is to partition the variance
into between and within group variance
* Critical factors:
- BETWEEN GROUP VARIANCE
- WITHIN GROUP VARIANCE
* How does the between group variance
compare with the within group variance?
حم
صفحه 10:
۱ Designed Experiments of
Interest
* One-factor completely randomized designs
(Formulas: p. 558)
Total SS = Treatment SS + Error SS
SS(Total) = SST + SSE
* Randomized Block Designs (Formulas: p. 575)
Total SS = Treatment SS + Block SS + Error SS
SS(Total) = SST + SSB + SSE
* Two-Factor Factorial Experiments (Formulas: p.
593)
Total SS = Main effect SS Factor A + Main effect SS
Factor B + AB Interaction SS + Error SS
SS(Total) = SS(A) + SS (B) + SS (AB) + SSE
صفحه 11:
Word check
¢ When | talk about between groups
variability, what am | talking about?
¢ What does SS between represent?
* What does MS (either within or between)
represent?
¢ What does the F ratio represent?
wit
صفحه 12:
* See MINITAB (Tukey family error rate)
Tukey's pairwise comparisons
Intervals for (column level mean) - (row level mean)
2 3
-3.320
1.854
-5.702 4.854
-0.298 0.320
1
2 - 4
1.320
3 -4.467
0.467
4 -6.854
-1.680