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MODEL, ASSUMPTIONS, ANALYSIS OF
VARIANCE
• Factors A (with a levels) and B (b levels)
• Total of ab treatment combinations
• The same number of replications per cell, n
Nested Design
A1
1 2 3
1
2
.
n
1
2
.
n
1
2
.
n
A2
1 2 3
1
2
.
n
1
2
.
n
1
2
.
n
A3
1 2 3
1
2
.
n
1
2
.
n
1
2
.
n
Factor B
Replications
• Factor B Level 1 in A1 is completely different from Factor B Level 1 in A2 and also in A3
• The levels of Factor B are nested within Factor A and there are n replicates within each
combination.
Nested Design
A1
1 2 3
1
2
.
n
1
2
.
n
1
2
.
n
A2
1 2 3
1
2
.
n
1
2
.
n
1
2
.
n
A3
1 2 3
1
2
.
n
1
2
.
n
1
2
.
n
Factor B
Replications
Example
Hospital 1
W1 W2 W3
1
2
.
n
1
2
.
n
1
2
.
n
W1 W2 W3
1
2
.
n
1
2
.
n
1
2
.
n
W1 W2 W3
1
2
.
n
1
2
.
n
1
2
.
n
Wards
Hospital 2 Hospital 3
Patients
Beds
Locations
Times
Ward 1 in Hospital 1 is not the same level of the Ward factor as Ward 1 in Hospital 2
Ward – Random Effect
Hospital – a random effect or fixed effect depending on contect
Example
Area 1
V1 V2 V3
1
2
.
n
1
2
.
n
1
2
.
n
V1 V2 V3
1
2
.
n
1
2
.
n
1
2
.
n
V1 V2 V3
1
2
.
n
1
2
.
n
1
2
.
n
Village
Area 2 Area 3
Locations
Times
Person
Household
Village 1 in Area 1 is not the same level of the Village factor as Village 1 in Area 2
Village – Random Effect
Area – could be a Fixed Effect or Random Effect depending on context
• The levels of Factor B are completely crossed with Factor A
- n replicates within each combination
• Factor B Level 1 in A1 is exactly the same level as Factor B Level 1 in A2
- And all other levels of A
Two Factor Design Data Layout
Factor B Factor A
1 a
1 X111,X112,…, X11n Xa11,Xa12,…, Xa1n
2 X121,X122,…, X12n Xa21,Xa22,…, Xa2n
Xij1,Xij2,…, Xijn
b X1b1,X1b2,…, X1bn Xab1,Xab2,…, Xabn
Nested Design - degrees of freedom
A1
1 2 3
𝑋11
A2
1 2 3
A3
1 2 3
Factor B
• Combined factor has 𝑎𝑏 levels
• 𝑎𝑏 − 1 degrees of freedom between the means
• 𝑎 − 1 df associated with differences among the levels of A
• 𝑎 𝑏 − 1 df associated with differences among B within A
• 𝑛 replications within each combination used to estimate error
• 𝑎𝑏 𝑛 − 1 df associated with the Error
𝐴1𝐵1
𝑋12
𝐴1𝐵2
𝑋13
𝐴1𝐵3
𝑋21
𝐴2𝐵1
𝑋22
𝐴2𝐵2
𝑋23
𝐴2𝐵3
𝑋31
𝐴3𝐵1
𝑋32
𝐴3𝐵2
𝑋33
𝐴3𝐵3
Combined
levels
Means
Complete Factor
Effects Model
𝑋𝑖𝑗𝑘 = 𝜇 + 𝜏𝑖𝑗 + 𝜖𝑖𝑗𝑘 = 𝜇 + 𝛼𝑖 + 𝛽𝑗 𝑖 + 𝜖𝑖𝑗𝑘
𝑖 = 1,2, … , 𝑎
𝑗 = 1,2, … , 𝑏
𝑘 = 1,2, … , 𝑛
The combined factor model is
essentially the same model as for a
Completely Randomised Design
Differences among treatments plus
an error term 𝜖𝑖𝑗𝑘
𝛼𝑖 main effect of level i of factor A
𝛽𝑗 𝑖 effect of level j of factor B nested within level i of factor A
Treatment Effects written as a factorial
structure.
Effect of A
Additional effect of B nested within A
• Factor B – the nested factor - generally defines a random effect,
- e.g., randomly selected subjects or units
- Assume 𝛽𝑗 𝑖 ~𝑁 0, 𝜎𝛽
2
independently of the errors, 𝜖𝑖𝑗𝑘.
• Factor A
- may be either a fixed effect with 𝛼𝑖 = 0 as constraint
- or random effect with 𝛼𝑖~𝑁 0, 𝜎𝛼
2
assumed.
• Therefore, ND based experiments will be mostly mixed effects (type III) or random effects (type
II) models
• Errors, 𝜖𝑖𝑗𝑘, normally distributed and independent with constant variance; additive terms
Assumptions
Hypotheses
• Fixed Effect
• Factor A
• 𝐻0: 𝛼𝑖 = 0, for all i
• 𝐻1: 𝛼𝑖 ≠ 0, for at least one i
• Factor B
• 𝐻0: 𝛽𝑗 𝑖 = 0, for all i, j
• 𝐻1: 𝛽𝑗 𝑖 ≠ 0, for at least one i,j
• Random Effect
•
• Factor A
• 𝐻0: 𝜎𝛼
2 = 0
• 𝐻1: 𝜎𝛼
2
> 0
• Factor B
• 𝐻0: 𝜎𝛽
2
= 0
• 𝐻1: 𝜎𝛽
2
> 0
Source df SS MS
Factor A 𝑎 − 1 SSA
𝑀𝑆𝐴 =
𝑆𝑆𝐴
𝑎 − 1
Factor B
(within A)
𝑎 𝑏 − 1 SSB 𝐴
𝑀𝑆𝐵 𝐴 =
SSB 𝐴
𝑎 𝑏 − 1
Error
(Residual)
𝑎𝑏 𝑛 − 1 SSE = 𝑆𝑆𝑇 − 𝑆𝑆𝐴 − SSB 𝐴
𝑀𝑆𝐸 =
𝑆𝑆𝐸
𝑎𝑏 𝑛 − 1
Total 𝑎𝑏𝑛 − 1 𝑆𝑆𝑇
Anova for 2 factor Nested model
Source df MS F EMS
Factor A 𝑎 − 1
𝑀𝑆𝐴 =
𝑆𝑆𝐴
𝑎 − 1
𝑀𝑆𝐴
𝑀𝑆𝐵 𝐴
𝜎2
+ 𝑛𝜎𝛽
2
+ 𝑏𝑛𝜎𝛼
2
Factor B
(within A)
𝑎 𝑏 − 1
𝑀𝑆𝐵 𝐴 =
SSB 𝐴
𝑎 𝑏 − 1
𝑀𝑆𝐵 𝐴
𝑀𝑆𝐸
𝜎2 + 𝑛𝜎𝛽
2
Error
(Residual)
𝑎𝑏 𝑛 − 1
𝑀𝑆𝐸 =
𝑆𝑆𝐸
𝑎𝑏 𝑛 − 1
𝜎2
Total 𝑎𝑏𝑛 − 1
F tests B Random Effect
Factor A can be Fixed or Random
Source df MS F EMS
Factor A 𝑎 − 1
𝑀𝑆𝐴 =
𝑆𝑆𝐴
𝑎 − 1
𝑀𝑆𝐴
𝑀𝑆𝐸
𝜎2 + 𝑏𝑛𝜎𝛼
2
Factor B
(within A)
𝑎 𝑏 − 1
𝑀𝑆𝐵 𝐴 =
SSB 𝐴
𝑎 𝑏 − 1
𝑀𝑆𝐵 𝐴
𝑀𝑆𝐸
𝜎2
+ 𝑛𝜎𝛽
2
Error
(Residual)
𝑎𝑏 𝑛 − 1
𝑀𝑆𝐸 =
𝑆𝑆𝐸
𝑎𝑏 𝑛 − 1
𝜎2
Total 𝑎𝑏𝑛 − 1
F tests B Fixed Effect
𝜎𝛽
2
=
𝑖𝑗
𝛽𝑗 𝑖
2
Factor A can be Fixed or Random

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4_4_WP_4_06_ND_Model.pptx

  • 2. • Factors A (with a levels) and B (b levels) • Total of ab treatment combinations • The same number of replications per cell, n Nested Design A1 1 2 3 1 2 . n 1 2 . n 1 2 . n A2 1 2 3 1 2 . n 1 2 . n 1 2 . n A3 1 2 3 1 2 . n 1 2 . n 1 2 . n Factor B Replications
  • 3. • Factor B Level 1 in A1 is completely different from Factor B Level 1 in A2 and also in A3 • The levels of Factor B are nested within Factor A and there are n replicates within each combination. Nested Design A1 1 2 3 1 2 . n 1 2 . n 1 2 . n A2 1 2 3 1 2 . n 1 2 . n 1 2 . n A3 1 2 3 1 2 . n 1 2 . n 1 2 . n Factor B Replications
  • 4. Example Hospital 1 W1 W2 W3 1 2 . n 1 2 . n 1 2 . n W1 W2 W3 1 2 . n 1 2 . n 1 2 . n W1 W2 W3 1 2 . n 1 2 . n 1 2 . n Wards Hospital 2 Hospital 3 Patients Beds Locations Times Ward 1 in Hospital 1 is not the same level of the Ward factor as Ward 1 in Hospital 2 Ward – Random Effect Hospital – a random effect or fixed effect depending on contect
  • 5. Example Area 1 V1 V2 V3 1 2 . n 1 2 . n 1 2 . n V1 V2 V3 1 2 . n 1 2 . n 1 2 . n V1 V2 V3 1 2 . n 1 2 . n 1 2 . n Village Area 2 Area 3 Locations Times Person Household Village 1 in Area 1 is not the same level of the Village factor as Village 1 in Area 2 Village – Random Effect Area – could be a Fixed Effect or Random Effect depending on context
  • 6. • The levels of Factor B are completely crossed with Factor A - n replicates within each combination • Factor B Level 1 in A1 is exactly the same level as Factor B Level 1 in A2 - And all other levels of A Two Factor Design Data Layout Factor B Factor A 1 a 1 X111,X112,…, X11n Xa11,Xa12,…, Xa1n 2 X121,X122,…, X12n Xa21,Xa22,…, Xa2n Xij1,Xij2,…, Xijn b X1b1,X1b2,…, X1bn Xab1,Xab2,…, Xabn
  • 7. Nested Design - degrees of freedom A1 1 2 3 𝑋11 A2 1 2 3 A3 1 2 3 Factor B • Combined factor has 𝑎𝑏 levels • 𝑎𝑏 − 1 degrees of freedom between the means • 𝑎 − 1 df associated with differences among the levels of A • 𝑎 𝑏 − 1 df associated with differences among B within A • 𝑛 replications within each combination used to estimate error • 𝑎𝑏 𝑛 − 1 df associated with the Error 𝐴1𝐵1 𝑋12 𝐴1𝐵2 𝑋13 𝐴1𝐵3 𝑋21 𝐴2𝐵1 𝑋22 𝐴2𝐵2 𝑋23 𝐴2𝐵3 𝑋31 𝐴3𝐵1 𝑋32 𝐴3𝐵2 𝑋33 𝐴3𝐵3 Combined levels Means Complete Factor
  • 8. Effects Model 𝑋𝑖𝑗𝑘 = 𝜇 + 𝜏𝑖𝑗 + 𝜖𝑖𝑗𝑘 = 𝜇 + 𝛼𝑖 + 𝛽𝑗 𝑖 + 𝜖𝑖𝑗𝑘 𝑖 = 1,2, … , 𝑎 𝑗 = 1,2, … , 𝑏 𝑘 = 1,2, … , 𝑛 The combined factor model is essentially the same model as for a Completely Randomised Design Differences among treatments plus an error term 𝜖𝑖𝑗𝑘 𝛼𝑖 main effect of level i of factor A 𝛽𝑗 𝑖 effect of level j of factor B nested within level i of factor A Treatment Effects written as a factorial structure. Effect of A Additional effect of B nested within A
  • 9. • Factor B – the nested factor - generally defines a random effect, - e.g., randomly selected subjects or units - Assume 𝛽𝑗 𝑖 ~𝑁 0, 𝜎𝛽 2 independently of the errors, 𝜖𝑖𝑗𝑘. • Factor A - may be either a fixed effect with 𝛼𝑖 = 0 as constraint - or random effect with 𝛼𝑖~𝑁 0, 𝜎𝛼 2 assumed. • Therefore, ND based experiments will be mostly mixed effects (type III) or random effects (type II) models • Errors, 𝜖𝑖𝑗𝑘, normally distributed and independent with constant variance; additive terms Assumptions
  • 10. Hypotheses • Fixed Effect • Factor A • 𝐻0: 𝛼𝑖 = 0, for all i • 𝐻1: 𝛼𝑖 ≠ 0, for at least one i • Factor B • 𝐻0: 𝛽𝑗 𝑖 = 0, for all i, j • 𝐻1: 𝛽𝑗 𝑖 ≠ 0, for at least one i,j • Random Effect • • Factor A • 𝐻0: 𝜎𝛼 2 = 0 • 𝐻1: 𝜎𝛼 2 > 0 • Factor B • 𝐻0: 𝜎𝛽 2 = 0 • 𝐻1: 𝜎𝛽 2 > 0
  • 11. Source df SS MS Factor A 𝑎 − 1 SSA 𝑀𝑆𝐴 = 𝑆𝑆𝐴 𝑎 − 1 Factor B (within A) 𝑎 𝑏 − 1 SSB 𝐴 𝑀𝑆𝐵 𝐴 = SSB 𝐴 𝑎 𝑏 − 1 Error (Residual) 𝑎𝑏 𝑛 − 1 SSE = 𝑆𝑆𝑇 − 𝑆𝑆𝐴 − SSB 𝐴 𝑀𝑆𝐸 = 𝑆𝑆𝐸 𝑎𝑏 𝑛 − 1 Total 𝑎𝑏𝑛 − 1 𝑆𝑆𝑇 Anova for 2 factor Nested model
  • 12. Source df MS F EMS Factor A 𝑎 − 1 𝑀𝑆𝐴 = 𝑆𝑆𝐴 𝑎 − 1 𝑀𝑆𝐴 𝑀𝑆𝐵 𝐴 𝜎2 + 𝑛𝜎𝛽 2 + 𝑏𝑛𝜎𝛼 2 Factor B (within A) 𝑎 𝑏 − 1 𝑀𝑆𝐵 𝐴 = SSB 𝐴 𝑎 𝑏 − 1 𝑀𝑆𝐵 𝐴 𝑀𝑆𝐸 𝜎2 + 𝑛𝜎𝛽 2 Error (Residual) 𝑎𝑏 𝑛 − 1 𝑀𝑆𝐸 = 𝑆𝑆𝐸 𝑎𝑏 𝑛 − 1 𝜎2 Total 𝑎𝑏𝑛 − 1 F tests B Random Effect Factor A can be Fixed or Random
  • 13. Source df MS F EMS Factor A 𝑎 − 1 𝑀𝑆𝐴 = 𝑆𝑆𝐴 𝑎 − 1 𝑀𝑆𝐴 𝑀𝑆𝐸 𝜎2 + 𝑏𝑛𝜎𝛼 2 Factor B (within A) 𝑎 𝑏 − 1 𝑀𝑆𝐵 𝐴 = SSB 𝐴 𝑎 𝑏 − 1 𝑀𝑆𝐵 𝐴 𝑀𝑆𝐸 𝜎2 + 𝑛𝜎𝛽 2 Error (Residual) 𝑎𝑏 𝑛 − 1 𝑀𝑆𝐸 = 𝑆𝑆𝐸 𝑎𝑏 𝑛 − 1 𝜎2 Total 𝑎𝑏𝑛 − 1 F tests B Fixed Effect 𝜎𝛽 2 = 𝑖𝑗 𝛽𝑗 𝑖 2 Factor A can be Fixed or Random