ANALYSIS OF VARIANCE - MASTER OF ARTS IN EDUCATION

ANALYSIS OF VARIANCE - MASTER OF ARTS IN EDUCATION

1. Analysis of Variance

2. Experimental Design
Investigator controls one or more independent
variables
–Called treatment variables or factors
–Contain two or more levels (subcategories)
Observes effect on dependent variable
–Response to levels of independent variable
Experimental design: Plan used to test
hypotheses

3. Parametric Test Procedures
Involve population parameters
–Example: Population mean
Require interval scale or ratio scale
–Whole numbers or fractions
–Example: Height in inches: 72, 60.5, 54.7
Have stringent assumptions
Examples:
–Normal distribution
–Homogeneity of Variance
Examples: z - test, t - test

4. Nonparametric Test Procedures
Statistic does not depend on population
distribution
Data may be nominally or ordinally scaled
–Examples: Gender [female-male], Birth Order
May involve population parameters such as
median
Example: Wilcoxon rank sum test

5. Advantages
of Nonparametric Tests
Used with all scales
Easier to compute
–Developed before wide computer use
Make fewer assumptions
Need not involve population
parameters
Results may be as exact as
parametric procedures
© 1984-1994 T/Maker Co.

6. Disadvantages
of Nonparametric Tests
May waste information
–If data permit using parametric
procedures
–Example: Converting data from
ratio to ordinal scale
Difficult to compute by hand
for large samples
Tables not widely available
© 1984-1994 T/Maker Co.

7. ANOVA (one-way)
One factor,
completely randomized
design

8. Completely Randomized
Design
Experimental units (subjects) are assigned
randomly to treatments
–Subjects are assumed homogeneous
One factor or independent variable
–two or more treatment levels or classifications
Analyzed by [parametric statistics]:
–One-and Two-Way ANOVA

9. Mini-Case
After working for the Jones Graphics
Company for one year, you have the
choice of being paid by one of three
programs:
- commission only,
- fixed salary, or
- combination of the two.

10. Salary Plans
Commission only?
Fixed salary?
Combination of the
two?

11. Is the average salary under the
various plans different?
CommissionFixed SalaryCombination
425 420 430
507 448 492
450 437 470
483 437 501
466 444 ---
492 --- ---

12. Assumptions

Homogeneity of Variance
 Normality
 Additivity
 Independence

13. Homogeneity of Variance
Variances associated with each
treatment in the experiment
are equal.

14. Normality
Each treatment population is
normally distributed.

15. Additivity
The effects of the model behave in an
additive fashion [e.g. : SST = SSB + SSW].
Non-additivity may be caused by the
multiplicative effects existing in the model,
exclusion of significant interactions, or by
“outliers” - observations that are inconsistent
with major responses in the experiment.

16. Independence
Assuming the treatment populations
are normally distributed,
the errors are not correlated.

17. Compares two types of variation to test
equality of means
Ratio of variances is comparison basis
If treatment variation is significantly greater
than random variation … then means are not
equal
Variation measures are obtained by
‘partitioning’ total variation
One-Way ANOVA

18. ANOVA (one-way)
Source of
Variation
Sum of
Squares
Degrees of
Freedom
Mean
Square
Mean
Swaure
Between
Treatments
(Model)
SSB c - 1 SSB/(c - 1)
Within
Treatments
(Error)
SSW N - c SSW/(N - c)
Total SST N - 1
tests:
F = MSB/MSW
Sig. level < 0.05

19. ANOVA Partitions Total
Variation
Total variation

20. ANOVA Partitions Total
Variation
Variation due to
treatment
Total variation

21. ANOVA Partitions Total
Variation
Variation due to
treatment
Variation due to
random sampling
Total variation

22. ANOVA Partitions Total
Variation
Variation due to
treatment
Variation due to
random sampling
Total variation
Sum of squares among
Sum of squares between
Sum of squares model
Among groups variation

23. ANOVA Partitions Total
Variation
Variation due to
treatment
Variation due to
random sampling
Total variation
Sum of squares within
Sum of squares error
Within groups variation
Sum of squares among
Sum of squares between
Sum of squares model
Among groups variation

24. Hypothesis
H
0: 
1 = 
2 = 
3
H
1: Not all means are equal
tests: F -ratio = MSB / MSW
p-value < 0.05

25. One-Way ANOVA

H
0: 
1 = 
2 = 
3
–All population means are equal
–No treatment effect

H
1
: Not all means are equal
–At least one population mean is
different
–Treatment effect

1  
2  
3
– is wrongis wrong
– not correctnot correct
X
f(X)

1
= 
2
= 
3
X
f(X)

1
= 
2

3

26. StatGraphics Input
salary plan
425 1
507 1
450 1
::: ::
466 1
492 1
420 2
448 2
437 2

27. StatGraphics Results
Source of
Variation
Sum of
Squares

d.f.
Mean
Square

F-ratio

Model

3,962.68

2

1,981.34

3.001

Error

7,923.05

12

660.254

---

Total

11,885.73

14

---
p-value
0.0877

28. Diagnostic Checking

Evaluate hypothesis
H
0: 
1 = 
2 = 
3
H
1: Not all means equal
 F-ratio = 3.001 {Table value = 3.89}
 significance level [p-value] = 0.0877
 Retain null hypothesis [ H
0 ]

29. ANOVA (two-way)
Two factor factorial design

30. Mini-Case
Investigate the effect of decibel
output using four different
amplifiers and two different
popular brand speakers, and the
effect of both amplifier and
speaker operating jointly.

31. What effects decibel output?
Type of amplifier?

Type of speaker?

The interaction
between amplifier
and speaker?

32. Are the effects of amplifiers, speakers, and
interaction significant? [Data in decibel units.]
Amplifier/
Speaker
A
1 A
2 A
3 A
4
S
1
9
9
12
8
11
16
8
7
1
10
15
9
S
2
7
1
4
5
9
6
0
1
7
6
7
5

33. Hypothesis

AmplifierH
0: 
1 = 
2 = 
3 = 
4
H
1: Not all means are equal

SpeakerH
0: 
1 = 
2
H
1: Not all means are equal
InteractionH
0
: The interaction is not significant
H
1: The interaction is significant

34. StatGraphics Input
decibels amplifier speaker
9 1 1
4 1 1
12 1 1
7 1 2
1 1 2
4 1 2
8 2 1
11 2 1
16 2 1
5 2 2
::: ::: :::

35. StatGraphics Results
Source of
Variation
Sum of
Squares d.f.
Mean
Square F-ratioSig. level
Main Effects
amplifier
speaker
97.79167
135.37500
3
1
32.5972
135.3750
3.589
15.319
0.0372
0.0014
Interaction
[AB]
9.45833 3 3.152778 0.347 0.7917
Residual 145.3333 16 9.08333 --- ---
Total 387.95833 23 --- --- ---

36. Diagnostics
Amplifier p-value = 0.0372 Reject Null
Speaker p-value = 0.0014Reject Null
Interactionp-value = 0.7917Retain Null
Thus, based on the data, the type of amplifier and the
type of speaker appear to effect the mean decibel
output. However, it appears there is no significant
interaction between amplifier and speaker mean
decibel output.

37. You and StatGraphics
Specification
[Know assumptions
underlying various
models.]
Estimation
[Know mechanics of
StatGraphics Plus Win].
Diagnostic checking

38. Questions?

39. ANOVA

40. End of Chapter