N-way ANOVA

N-way ANOVA

2

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

3

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?

4

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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a

i

b

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5,1311 1

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nb

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5

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

j ij

161DF (error) cells-within

3920,301SS (error) cells-within2

1 1 1

nab

XXn a

i

b

j

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

b

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j j

11DFA factor

1125,1386SSA factor 2

1

a

XXbn a

i i

7

How to do a 2-way ANOVA with equal replicationsSummary of calculations

8

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

11

2-way ANOVA in SPSS

Click Add

12

Visualizing 2-way ANOVA without interaction

13

Visualizing 2-way ANOVA with interaction

14

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.

15

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

16

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

18

2-way ANOVA With proportional but unequal replications

Proportional replications:

N jinij

col# row#

19

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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a

i

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1ˆ 1 1 1

20

2-way ANOVA With one replication

Get more data!

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2-way ANOVA Randomized block design

22

3-way ANOVA

23

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