Analysis of Variance

Model II - example

Width (in μm) of scutum (dorsal shield) of

larvae of the tick Haemaphysalis

leporispalustris in samples from 4

cottontail rabbits

Source:

Sokal and Rohlf, Biometry

host 1

host 2

host 3

host 4

380

350

354

376

376

356

360

344

360

358

362

342

368

376

352

372

372

338

366

374

366

342

372

360

374

366

362

382

350

344

344

342

364

358
351
348
348

host 1

host

2

host 3 host 4

2978

3544

4619

2168

13 309

8

10

13

6

37

372.25

354.4

0

355.31 361.33 359.70

379.50 1278.

4

954.77 1165.3 3778.0

Step 1 – Preliminary
computations





a

1

i





i

n

1

j

ij

Y

i

Y

i

n











i

n

1

j

2

i

ij

Y

Y

37

13309



Step 2 – Sum of squares





73

.

1807

)

7

.

359

33

.

361

(

6

)

7

.

359

31

.

355

(

13

...

)

7

.

359

4

.

354

(

10

)

7

.

359

25

.

372

(

8

Y

n

Y

n

Y

Y

n

groups

among

squares

of

sum

,

SS

2

2

2

2

2

i

n

1

j

ij

i

a

1

i

a

1

i

2

i

i

among

i

























































Step 2 – Sum of squares





0

.

3778

33

.

1165

77

.

954

40

.

1278

50

.

379

Y

Y

groups

within

squares

of

sum

,

SS

a

1

i

n

1

j

2

i

ij

within

i





















Step 2 – Sum of squares

73

.

5585

0

.

3778

73

.

1807

SS

SS

squares

of

sum

total

,

SS

within

among

total











Step 3 – ANOVA table

Source of

variation

df

SS

MS

F

S

Expected

MS

a-1

SS

among

SS

within

SS

total

Y

Y 

among groups

1

a

SS

among



within

among

MS

MS

2
A

0

2

n 





Y

Y 

within groups

a

n

a

1

i

i







a

n

SS

a

1

i

i

within







2



Y

Y 

total

1

n

a

1

i

i







Step 3 – ANOVA table

Source of

variation

df

SS

MS

F

S

3

1807.7

3

602.58

5.26

33

3778.0

0

114.48

36

5585.7

3

Y

Y 

among groups

Y

Y 

within groups

Y

Y 

total

Step 4 – Conclusion



The corresponding p-value for calculated F
statistic with [df

among

, df

within

] degrees of

freedom is equal to 0.0045 < 0.05.



We reject the H

0

hypothesis at p < 0.005.



There is a significant added variance
component among hosts for width of scutum
in larval ticks

Step 5 – Estimation of variance
components (model II)

Since sample size differs among groups, no
single value of n is appropriate. We therefore
use an „average” n – this is not the arithmetic
mean of the n

i

’s, but is





009

.

9

6

13

10

8

6

13

10

8

6

13

10

8

1

4

1

n

n

n

1

a

1

n

2

2

2

2

a

1

i

a

1

i

i

a

1

i

2
i

i

0







































































Step 5 – Estimation of variance
components (model II)

We may estimate the variance component as
follows









2

within

2
A

0

2

among

2

within

2
A

0

2

among

0

within

among

2
A

s

MS

s

n

s

MS

so

MS

E

n

MS

E

because

179

.

54

009

.

9

48

.

114

58

.

602

n

MS

MS

s





























Step 5 – Estimation of variance
components (model II)

Percent of variation among groups

%

1

.

32

100

179

.

54

48

.

114

179

.

54

100

s

s

s

2
A

2

2
A













The coefficient of interclass correlation
r

I

=0.321