Post on 22-Feb-2016
description
N-way ANOVA
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Two-factor ANOVA with equal replications
Experimental design: 2 2 (or 22) factorial with n = 5 replicate
Total number of observations: N = 2 2 5 = 20Equal replications also termed
orthogonality
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The hypothesis
H0: There is on effect of hormone treatment on the mean plasma concentration
H0: There is on difference in mean plasma concentration between sexes
H0: There is on interaction of sex and hormone treatment on the mean plasma concentration
Why not just use one-way ANOVA with for levels?
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How to do a 2-way ANOVA with equal replicationsCalculating means
Calculate cell means:
Calculate the total mean (grand mean)
Calculating treatment means
88,145,98,154,124,203,165
5
1 111
nX
egnX
X l ln
l ablab
825,211 1 1
N
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b
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nb
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How to do a 2-way ANOVA with equal replicationsCalculating general Sum of Squares
Calculate total SS:
Calculate the cell SS
Calculating treatment error SS
191DF total
7175,1762SS total2
1 1 1
N
XXa
i
b
j
n
l ijl
31DF cells
3255,1461SS cells2
1 1
ab
XXn a
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b
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161DF (error) cells-within
3920,301SS (error) cells-within2
1 1 1
nab
XXn a
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n
l ijijl
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How to do a 2-way ANOVA with equal replicationsCalculating factor Sum of Squares
Calculating factor A SS:
Calculating factor B SS
Calculating A B interaction SSA B interaction SS = cell SS – factor A SS – factor B SS = 4,9005A B DF = cell DF– factor A DF – factor B DF = 1
11DF Bfactor
3125,70SS Bfactor 2
1
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11DFA factor
1125,1386SSA factor 2
1
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How to do a 2-way ANOVA with equal replicationsSummary of calculations
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How to do a 2-way ANOVA with equal replicationsHypothesis test
H0: There is on effect of hormone treatment on the mean plasma concentration
F = hormone MS/within-cell MS = 1386,1125/18,8370 = 73,6
F0,05(1),1,16 = 4,49
H0: There is on difference in mean plasma concentration between sexes
F = sex MS/within-cell MS = 3,73F0,05(1),1,16 = 4,49
H0: There is on interaction of sex and hormone treatment on the mean plasma concentration
F = A B MS/within-cell MS = 0,260F0,05(1),1,16 = 4,49
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Visualizing 2-way ANOVA
Table 12.2 and Figure 12.1
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2-way ANOVA in SPSS
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2-way ANOVA in SPSS
Click Add
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Visualizing 2-way ANOVA without interaction
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Visualizing 2-way ANOVA with interaction
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2-way ANOVA Random or fixed factor
Random factor: Levels are selected at random…Fixed factor: The ’value’ of each levels are of interest and selected on
purpose.
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2-way ANOVA Assumptions
• Independent levels of the each factor• Normal distributed numbers in each cell• Equal variance in each cell
• Bartletts homogenicity test (Section 10.7)• s2 ~ within cell MS; ~ within cell DF
• The ANOVA test is robust to small violations of the assumptions• Data transformation is always an option (see chpter 13)• There are no non-parametric alternative to the 2-way ANOVA
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2-way ANOVA Multiple Comparisons
Multiple comparesons tests ~ post hoc tests can be used as in one-way ANOVA
Should only be performed if there is a main effect of the factor and no interaction
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2-way ANOVA Confidence limits for means
95 % confidence limits for calcium concentrations on in birds without hormone treatment
MS cellwithins DF; cellwithin
CI % 95
2
2
),2(05,01
bnstX
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2-way ANOVA With proportional but unequal replications
Proportional replications:
N jinij
col# row#
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2-way ANOVA With disproportional replications
Statistical packges as SPSS has porcedures for estimating missing values and correcting unballanced designs, eg using harmonic means
Values should not be estimated by simple cell meansSingle values can be estimated, but remember to decrease the DF
baN
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1ˆ 1 1 1
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2-way ANOVA With one replication
Get more data!
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2-way ANOVA Randomized block design
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3-way ANOVA
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3-way ANOVA
H0: The mean respiratory rate is the same for all speciesH0: The mean respiratory rate is the same for all temperaturesH0: The mean respiratory rate is the same for both sexes H0: The mean respiratory rate is the same for all speciesH0: There is no interaction between species and temperature across
both sexesH0: There is no interaction between species and sexes across
temperatureH0: There is no interaction between sexes and temperature across
both spicesH0: There is no interaction between species, temperature, and sexes
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3-way ANOVA Latin Square
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Exercises
12.1, 12.2, 14.1, 14.2