Analysis of Variance Notes
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Experiments versus Studies
Types of Experiments
Assumptions & Assumption Checks
Types of Analysis
NCSS
1. Experiments versus Studies
1.1 Terminology
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Factors versus Independent Variables
Example: hours studied and major are two factors affecting Grade
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Treatments –
Example: specific combinations of hours studied and teaching method
1.2 Purpose
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Observational Study –
o Correlational –
o Observe values of X
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Experiment –
o Cause-Effect
o Control values of X
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Designs
o Balanced
o unbalanced
2. Types of Experimental Designs
2.1 Randomized Design
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one factor
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two factor
2.2 Randomized Block Design
2.3 Examples
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Teaching Method only
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Teaching method and hours studied
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Teaching method within major
3. Assumptions & Assumption Checks
3.1 Assumptions
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Same Variance
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Independence
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Normality
3.2 Assumption checks
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Modified Levine – comparing differences to center
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Normality Tests and Box Plots
4. Analysis
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Sources of Variability and degrees of freedom
Tests of effects of
o One factor designs
o Each factor in two factor designs
o Combination effects in two factor designs
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Tests of Assumptions
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Tests and estimation of differences in averages
4.1 Sources of variability and degrees of freedom
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 Total: Values around overall average: divisor of (n-1)
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Factor: Factor averages variation : divisor of (# averages – 1)
Interaction: Combination effects: divisor of (product of factor divisors)
Error: Randomness: divisor of (n - # of averages or combination of averages)
4.2 Tests of effects of
4.2.1 One Factor – Completely Randomized Design or Independent Sample Study
4.2.1.1 Test Template:
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Null hypothesis: average value of Y is the same for all levels of the factor
Alternative: at least two are different
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Test Statistic: Compares variation of factor averages to variation of
random data
Among-Group variation to within-group variation
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Rejection Region: Above ratio is large (F ratio) > F table
Two degrees of freedom: numerator degrees of freedom and divisor
degrees of freedom
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Conclusion: We can (not) say the average value of Y differs for at least
two levels of the factor.
4.2.1.2 Example: Y = tensile strength of a product
Factor = 4 Suppliers
Obtain samples of size 5 from each supplier (n = ____ )
MSA = sample factor variability = 21.095
MSW = sample error variability = 6.094
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Null hypothesis: 1=2=3=4 (average value of ______________ is the
Same for all ________________)
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Alternative: at least two are different
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Test Statistic: MSA/MSW =
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Rejection Region: Reject Ho if F > F table with
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Numerator degrees of freedom = ______ and denomination d.f. = _____
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F-Table = _______
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Conclusion: We can (not) say that the average _________________
differs for at least two ________________________
4.2.2 One Factor – Randomized Block
4.2.2.1 Test Template:
Same as in 4.2.1.1 but divisor degrees of freedom =
(# of factor means-1)*( # of block means-1)
4.2.1.2 Example: Y = Rating of a restaurant’s service
Factor = 4 Restaurants
Block = all restaurants reviewed by same 6 raters (n = ____ )
MSA = sample factor variability = 595.8
MSE = sample error variability = 14.986
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Null hypothesis: 1=2=3=4 (average value of ______________ is the
Same for all ________________)
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Alternative: at least two are different
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Test Statistic: MSA/MSW =
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Rejection Region: Reject Ho if F > F table with
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Numerator degrees of freedom = ______ and denomination d.f. = _____
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F-Table = _______
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Conclusion: We can (not) say that the average _________________
differs for at least two ________________________
4.2.3 Two Factors – Interaction or combination effects
4.2.3.1 Test Template:
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Null hypothesis: (no interaction) difference in average value of Y between
any two levels of factor one does not depend on the level of factor two
Alternative: (interaction) difference in average value of Y between any
two levels of factor one does depend on the level of factor two
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Test Statistic: Compares variation of interaction to variation of random
data
Among-Group variation to within-group variation
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Rejection Region: Above ratio is large (F ratio) > F table
Two degrees of freedom:
numerator d.f. = product of factor 1 and 2 d.f.
denominator = n – number of combination of factor 1 and 2
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Conclusion: we can (not) say that the difference in average value of Y
between any two levels of factor one does depend on the level of factor
two.
4.2.1.2 Example: Y = length of a ball-bearings life
Factor 1 = heat treatment (high or low)
Factor 2 = ring osculation (high or low)
Obtain samples of size 2 from each combination (n = ____ )
MSAB = sample interaction variability = 3280.5
MSE = sample error variability = 61
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Null hypothesis: (no interaction) difference in average value of
_______________ between any two levels of ________________ does
not depend on the level of ___________________
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Alternative: (interaction) difference in average value of _______________
between any two levels of ________________ does depend on the level of
___________________
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Test Statistic: Compares variation of interaction to variation of random
data
F= MSAB / MSE =
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Rejection Region: Above ratio is large (F ratio) > F table
Two degrees of freedom:
numerator d.f. = product of factor 1 and 2 d.f = .
denominator = n – number of combination of factor 1 and 2 =
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Conclusion: we can (not) say that the difference in average value of
_____________________ between any two levels of
____________________ does depend on the level of _______________.
4.2.4 One of the two factors – Completely Randomized Design or Independent
Sample Study – NO SIGNIFICANT INTERACTION
4.2.4.1 Test Template:
same as in the one-factor test but
divisor d.f. = n – (#of levels of factor 1)*(# in factor 2)
4.2.4.2 Example: Y = rating of a photographic plate
Factor A = 2 levels of development strength,
Factor B = 2 levels of development time (10 and 14 minutes)
Randomly assign 4 plates to each of the 4 combinations
MSA = sample variability of factor A (time) = 1.5625
MSB = sample variability of factor B (strength) = 56.5625
MSE = sample error variability = 2.229
(no interaction was found – testing time effect)
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Null hypothesis: 1=2 (average value of ______________ is the
Same for all ________________)
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Alternative: at least two are different
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Test Statistic: MSB/MSE =
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Rejection Region: Reject Ho if F > F table with
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Numerator degrees of freedom = ______ and denomination d.f. = _____
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F-Table = _______
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Conclusion: We can (not) say that the average _________________
differs for at least two ________________________
4.3 Tests of Assumptions
4.3.1 Equal Variance –
4.3.1.1 Test Template:
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Null hypothesis: variation of Y is the same for all levels of the factor
Alternative: at least two are different
Compute the absolute difference between each value in a group and the
median of the group
Test Statistic and rejection region: same as for the factor tests
Conclusion: We can (not) say the variation of Y differs for at least two
levels of the factor.
4.3.1.2 Example: Y = tensile strength of a product
Factor = 4 Suppliers
Obtain samples of size 5 from each supplier (n = ____ )
MSDifference = sample factor variability = 0.59
MSE = sample error variability = 2.2853
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Null hypothesis: 1=2=3=4 (variability of __________ is the
Same for all ________________)
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Alternative: at least two are different
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Test Statistic: MSDiff/MSE =
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Rejection Region: Reject Ho if F > F table with
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Numerator degrees of freedom = ______ and denomination d.f. = _____
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F-Table = _______
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Conclusion: We can (not) say that the variability of _________________
differs for at least two ________________________
4.3.2 Normality –
4.3.2.1 Test Template:
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Null hypothesis: distribution of Y is the normal for all levels of the factor
Alternative: at least one is not normal
Test Statistic and rejection region: use tests on NCSS and p-value is less
than alpha reject normality.
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Conclusion: We can (not) say the distribution of Y is not normal for at
least two levels of the factor.
4.3.2.2 Example: Y = tensile strength of a product
Factor = 4 Suppliers
Obtain samples of size 5 from each supplier (n = ____ )
Assumption Test
Skewness Normality of Residuals
Kurtosis Normality of Residuals
Omnibus Normality of Residuals
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Prob -Level
0.605780
0.548522
0.731126
Null hypothesis: distribution of __________ is normality distributed for
all ________________)
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Alternative: distribution of __________ is non-normally distributed for at
least one level of ________________)
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Test Statistic: p-value
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Rejection region: p-value < alpha
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Conclusion: We can (not) say that the distribution of __________ is nonnormally distributed for at least one level of ________________)
4.4 Testing the difference in means
4.4.1 Expermentwise error versus comparison error
4.4.2 Testing one factor
Use NCSS. The output will tell you which means are statistically different
Example: Y = tensile strength of a product
Factor = 4 Suppliers
Obtain samples of size 5 from each supplier
Tukey-Kramer Multiple-Comparison Test
Response: strength
Term A: supplier
Alpha=0.050 Error Term=S(A) DF=16 MSE=6.094 Critical
Value=4.046122
Different
Group
Count Mean
From Groups
1
5
19.52
2
4
5
21.16
3
5
22.84
2
5
24.26
1
Conclusions: We can say that the average value of (Y) _________ for (factor
level) ________ differs from (factor level).
<Repeat for each difference>
The average (Y) for the other (factor levels) ______________ are not
significantly different.
4.4.3 Same procedure works for Randomized Block and Two-factor studies
without interaction.
4.5 Nonparametric tests
4.5.1 Kruskal-Wallis test
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One-factor designs
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Compares medians instead of means
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Test similar to ANOVA but does not require normality
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Using NCSS: p-value < alpha reject equality of medians
4.5.2 Friedman’s Test
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Randomized block designs
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Compares medians instead of means
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Test similar to ANOVA but does not require normality
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Using NCSS p-value < alpha reject equality of medians
5. NCSS
5.1 data format: place all the values of Y in one column and let the next column(s) be the
values of the factor(s).
5.2 Approach
5.2.1 One factor designs
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Click on Analysis, ANOVA, one-way anova
Choose the dependent variable and factor
In reports, uncheck EMS report and check Tukey-Kramer Test
5.2.2 Randomized Block Designs
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Click on Analysis, ANOVA, Analysis of Variance
Choose
o First, the dependent variable
o Second, for factor 1 the block and choose Random from Type-list
o Third, for factor 2 the factor of interest, (fixed type)
In reports, uncheck EMS report and check Tukey-Kramer Test
5.2.3 Two-factor designs
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Click on Analysis, ANOVA, Analysis of Variance
Choose
o First, the dependent variable
o Second, factor 1 Type Fixed
o Third, factor 2 Type fixed
o If interaction exists, tests for two-factor interaction
In reports, uncheck EMS report and check Tukey-Kramer Test