ANOVA

One-factor ANOVA by example

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One-factor ANOVA by visual inspection

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One-factor ANOVA H0

H0: µ1 = µ2 = µ3 = …

HA : not all means are equal

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One-factor ANOVA but why not t-tests

• t-tests?

• 3+2+1 tests -> multiple comparisons

• The variance is correctly estimated

• We need a method that uses the full dataset

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One-factor ANOVA the cook book I

• Find the Within groups SS

Fx: 𝑆𝑆1 = 𝑥𝑖 − 𝑥 2

𝑖 = 8.2 − 6 2 +8.2 − 7 2 + 8.2 − 8 2 + 8.2 − 8 2 +8.2 − 9 2 + 8.2 − 11 2 = 14.4

Sum the sum of squares from each group:

SS1+SS2+SS3+SS4 = 14.4+8.8+20.8+13.3

=57.8

df = 20

• Within group variance

• =𝑤𝑖𝑡𝑕𝑖𝑛 𝑔𝑟𝑜𝑢𝑝 𝑆𝑆

𝑑𝑓=57.8

20= 2.9

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One-factor ANOVA the cook book II

• Find the total SS

𝑆𝑆𝑡𝑜𝑡 = 𝑥𝑖 − 𝑥 2

𝑖

= 140.0

df = 23

• Find the between group SS

𝑆𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛 = 𝑛 𝑥 − 𝑥 2

𝑚

= 6( 8.2 − 7.5 2 + 5.8 − 7.5 2

+ 10.2 − 7.5 2 + 5.7 − 7.5 2) = 82.1

df = 3

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The ANOVA table

ANOVA

Outcome

Sum of Squares df Mean Square F Sig.

Between Groups 82,125 3 27,375 9,467 ,000

Within Groups 57,833 20 2,892

Total 139,958 23

Variance aka mean square aka s2 is simply SS/df

F is the Between SS devided by the Within SS

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Assumptions

• The data needs to be normal distributed in the

groups

• The variance needs to be equal in all groups:

homoscedasticity

• The groups needs to be independent

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Multiple comparisons procedures aka post hoc analysis

• Rejecting H0 only states that one or more

pairs of means are different, but not which.

• Tukeys multiple comparisons test as an

example.

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Tukeys multiple comparisons

Rank the sample means:

q > 3,958

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Rank 1 2 3 4

Group 3 1 2 4

µ 10.2 8.2 5.8 5.7

𝑆𝐸 =𝑠2

𝑛=

2,892

6= 0,67

ANOVA

Outcome

Sum of

Squares df

Mean

Square F Sig.

Between Groups 82,125 3 27,375 9,467 ,000

Within Groups 57,833 20 2,892

Total 139,958 23

pair difference q H0

3vs4 4.5 6,7 reject

3vs2 4,4 6,6 reject

3vs1 2 3,0 Do not reject

1vs4 2,5 3,7 Do not reject

1vs2 Don not test Do not reject

2vs4 Don not test Do not reject

1-way ANOVA in SPSS

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Comparison between sevreal medians

Kruskal-Wallis test

H0: The distribution of the groups are equal

1-Way ANOVA for non-normal data

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Kruskal-Wallis test

A few definitions:

k is the number of groups

ni: : the numner of observations in the i’th group.

N : total numner of observations

Ri : the sum of ranks in the i’th group

How to:

Rank all observations

Calculate the rank sum for each group

Calculate H

H is chi-square distributed with k-1 degrees of redom

Look up the p-value in a table

14

13

1

122

NNN

H i

in

T

Kruskal-Wallis test – An example

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Kruskal-Wallis test – An example

The data is ranked

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Kruskal-Wallis test – An example

The data is ranked

H is calculated

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2,6632,692132120

242212

120312020

579365342

122222

H

Kruskal-Wallis test – An example

The data is ranked

H = 6,2

# d.f. = k-1 = 3

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Kruskal-Wallis test – in SPSS

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Kruskal-Wallis test – i SPSS

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Kruskal-Wallis test – i SPSS

Ranks

group N Mean Rank

count 1,00 5 8,40

2,00 5 10,60

3,00 5 7,20

4,00 5 15,80

Total 20

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Test Statisticsa,b

count

Chi-Square 6,205

df 3

Asymp. Sig. ,102

a. Kruskal Wallis Test

b. Grouping Variable:

group

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 = 20

Equal 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 replications

Calculating means

Calculate cell means:

Calculate the total mean (grand mean)

Calculating treatment means

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88,145,98,154,124,203,16

5

5

1 111

n

Xeg

n

XX l l

n

l abl

ab

825,211 1 1

N

XX

a

i

b

j

n

l ijl

5,131

1 1

Xeg

nb

XX

b

j

n

l ijl

i

How to do a 2-way ANOVA with equal replications

Calculating general Sum of Squares

Calculate total SS:

Calculate the cell SS

Calculating treatment error SS

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191DF total

7175,1762SS total2

1 1 1

N

XXa

i

b

j

n

l ijl

31DF cells

3255,1461SS cells2

1 1

ab

XXna

i

b

j ij

161DF (error) cells-within

3920,301SS (error) cells-within2

1 1 1

nab

XXna

i

b

j

n

l ijijl

How to do a 2-way ANOVA with equal replications

Calculating factor Sum of Squares

Calculating factor A SS:

Calculating factor B SS

Calculating A B interaction SS

A B interaction SS = cell SS – factor A SS – factor B SS = 4,9005

A B DF = cell DF– factor A DF – factor B DF = 1

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11DF Bfactor

3125,70SS Bfactor 2

1

b

XXanb

j j

11DFA factor

1125,1386SSA factor 2

1

a

XXbna

i i

How to do a 2-way ANOVA with equal replications

Summary of calculations

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How to do a 2-way ANOVA with equal replications

Hypothesis 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 = 3,73

F0,05(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 = 0,260

F0,05(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

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Click Add

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

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MS cellwithins DF; cellwithin

CI % 95

2

2

),2(05,01

bn

stX

2-way ANOVA

With proportional but unequal replications

Proportional replications:

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N

jinij

col# row#

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 means

Single values can be estimated, but remember to decrease the DF

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baN

XbBaAX

a

i

b

j

n

l ijlji

ijl

ij

1ˆ 1 1 1

2-way ANOVA

With one replication

Get more data!

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2-way ANOVA

Randomized block design

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2-way ANOVA

Repeated measures

• Repeating measurements in the same

‘subject’, like a paired t-test

• An additional assumption is that the

correlation between pairs of groups is

equal: compound symmetry

• if this is not the case, try multivariate

ANOVA or linear mixed model

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