Nested ANOVA: Example Nested ANOVA: models

Nested ANOVA: Example • 3 cages • 4 mosquitoes within each cage • 2 independent measurements per mosquito Cage I 1

2

Cage II

3

4

58.5 77.8 84.0 70.1 59.5 80.9 83.6 68.3

1

2

3

Cage III 4

1

69.8 56.0 50.7 63.8 69.8 54.5 49.3 65.8

2

3

4

56.6 77.8 69.9 62.1 57.5 79.2 69.2 64.5

Nested ANOVA: models Yijk = µ + αi + βij + ǫijk µ = overall mean αi = “effect” for ith cage βij = “effect” for jth mosquito within ith cage ǫijk = random error

Random effects model αi ∼ Normal(0, σA2 )

Mixed effects model αi fixed;

P

αi = 0

βij ∼ Normal(0, σB2 |A)

βij ∼ Normal(0, σB2 |A)

ǫijk ∼ Normal(0, σ 2)

ǫijk ∼ Normal(0, σ 2)

The model

40

50

60

70

80

90

40

50

60

Cages

−30

−20

−10

0

10

20

30

−30

−20

−10

Individuals

−30

−20

−10

0

70

80

90

100

Cages

0

10

20

30

10

20

30

Individuals

10

20

30

−30

−20

Residuals

−10

0

Residuals

Example: sample means Cage I 1

¯ ij· Y ¯ i·· Y ¯ ··· Y

2

3

Cage II 4

1

2

3

Cage III 4

1

2

3

4

58.5 77.8 84.0 70.1 59.5 80.9 83.6 68.3

69.8 56.0 50.7 63.8 69.8 54.5 49.3 65.8

56.6 77.8 69.9 62.1 57.5 79.2 69.2 64.5

59.00

69.80

57.05

79.35

83.80

72.84

69.20

55.25

50.00

59.96 66.63

64.80

78.50

69.55

67.10

63.30

Calculations (equal sample sizes) Source

Sum of squares

among groups

SSamong = bn

subgroups within groups

SSsubgr = n

within subgroups

SSwithin =

TOTAL

PPP i

j

df

P ¯ ¯ 2 i (Yi·· − Y··· )

PP ¯ ¯ 2 i j (Yij· − Yi·· )

PPP i

j

k (Yijk

k (Yijk

¯ ij·)2 −Y

¯ ···)2 −Y

a–1

a (b – 1)

a b (n – 1)

abn–1

ANOVA table SS

df

MS

F

expected MS

SSamong

a–1

SSamong a–1

MSamong MSsubgr

σ 2 + n σB2 |A + n b σA2

SSsubgr

a (b – 1)

SSsubgr a(b – 1)

MSsubgr MSwithin

σ 2 + n σB2 |A

SSwithin

a b (n – 1)

SSwithin ab(n – 1)

SStotal

abn–1

σ2

Example source

df

SS

MS

F

P-value

among groups

2

665.68

332.84

1.74

0.23

among subgroups within groups

9

1720.68

191.19

146.88

< 0.001

within subgroups

12

15.62

1.30

TOTAL

23

2401.97

aov.out