analysis of variance. experimental design u investigator controls one or more independent variables...
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Analysis of Variance
Experimental DesignExperimental 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
Parametric Test ProceduresParametric 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 assumptionsExamples:
– Normal distribution
– Homogeneity of Variance
Examples: z - test, t - test
Nonparametric Test ProceduresNonparametric 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
Advantages of Nonparametric Tests
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.
Disadvantages of Nonparametric Tests
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.
ANOVA (one-way)
One factor,
completely randomized
design
Completely Randomized Design
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
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.
Salary Plans
Commission only?
Fixed salary?
Combination of the two?
Is the average salary under the various plans different?
Commission Fixed Salary Combination425 420 430507 448 492450 437 470483 437 501466 444 ---492 --- ---
Assumptions
Homogeneity of Variance Normality Additivity Independence
Homogeneity of Variance
Variances associated with each treatment in the experiment
are equal.
Normality
Each treatment population is normally distributed.
AdditivityThe 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.
Independence
Assuming the treatment populations are normally distributed,
the errors are not correlated.
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 ANOVAOne-Way ANOVA
ANOVA (one-way)
Source ofVariation
Sum ofSquares
Degrees ofFreedom
MeanSquare
MeanSwaure
BetweenTreatments(Model)
SSB c - 1 SSB/(c - 1)
WithinTreatments(Error)
SSW N - c SSW/(N - c)
Total SST N - 1tests: F = MSB/MSWSig. level < 0.05
ANOVA Partitions Total Variation
ANOVA Partitions Total Variation
Total variationTotal variation
ANOVA Partitions Total Variation
ANOVA Partitions Total Variation
Variation due to treatment
Variation due to treatment
Total variationTotal variation
ANOVA Partitions Total Variation
ANOVA Partitions Total Variation
Variation due to treatment
Variation due to treatment
Variation due to random samplingVariation due to
random sampling
Total variationTotal variation
ANOVA Partitions Total Variation
ANOVA Partitions Total Variation
Variation due to treatment
Variation due to treatment
Variation due to random samplingVariation due to
random sampling
Total variationTotal variation
Sum of squares among Sum of squares between Sum of squares model Among groups variation
ANOVA Partitions Total Variation
ANOVA Partitions Total Variation
Variation due to treatment
Variation due to treatment
Variation due to random samplingVariation due to
random sampling
Total variationTotal 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
Hypothesis
H0: 1 = 2 = 3
H1: Not all means are equal
tests: F -ratio = MSB / MSW
p-value < 0.05
One-Way ANOVA One-Way ANOVA
H0: 1 = 2 = 3 – All population means are equal– No treatment effect
H1: 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 = 3X
f(X)
1 = 2 = 3
X
f(X)
1 = 2 3X
f(X)
1 = 2 3
StatGraphics Inputsalary plan
425 1507 1450 1::: ::
466 1492 1420 2448 2437 2
StatGraphics ResultsSource 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
Diagnostic Checking Evaluate hypothesis
H0: 1 = 2 = 3
H1: Not all means equal
F-ratio = 3.001 {Table value = 3.89}
significance level [p-value] = 0.0877
Retain null hypothesis [ H0 ]
ANOVA (two-way)
Two factor factorial design
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.
What effects decibel output?
Type of amplifier?
Type of speaker?
The interaction
between amplifier and speaker?
Are the effects of amplifiers, speakers, and interaction significant? [Data in decibel units.]
Amplifier/Speaker
A1 A2 A3 A4
S1
9
9
12
8
11
16
8
7
1
10
15
9
S2
7
1
4
5
9
6
0
1
7
6
7
5
Hypothesis Amplifier H0: 1 = 2 = 3 = 4
H1: Not all means are equal
Speaker H0: 1 = 2
H1: Not all means are equal
Interaction H0: The interaction is not significant
H1: The interaction is significant
StatGraphics Inputdecibels amplifier speaker
9 1 14 1 112 1 17 1 21 1 24 1 28 2 111 2 116 2 15 2 2::: ::: :::
StatGraphics ResultsSource ofVariation
Sum ofSquares d.f.
MeanSquare F-ratio Sig. 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 --- --- ---
Diagnostics Amplifier p-value = 0.0372 Reject Null
Speaker p-value = 0.0014 Reject Null
Interaction p-value = 0.7917 Retain 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.
You and StatGraphics
Specification[Know assumptions underlying various models.]
Estimation
[Know mechanics of StatGraphics Plus Win].
Diagnostic checking
Questions?
ANOVA
End of Chapter