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ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Modelling State TransitionsExample of a Multinomial Logit Model Applied to Somatic
Cell Count in Dairy Cows
Aurelien Madouasse
19th April 2010
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Outline
1 BackgroundMilk RecordingDataSomatic Cell Count
2 State TransitionState DefinitionState TransitionsData
3 A Simple ModelModelWinBUGS codeResults
4 Adding ComplexitySCC VariationModelWinBUGS codeResults
5 Discussion
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Outline
1 BackgroundMilk RecordingDataSomatic Cell Count
2 State TransitionState DefinitionState TransitionsData
3 A Simple ModelModelWinBUGS codeResults
4 Adding ComplexitySCC VariationModelWinBUGS codeResults
5 Discussion
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
What is Milk Recording?
Milk recording is the regular collection of a milk samplefrom all lactating cows of a dairy herd
What is measured:
Milk yield% butterfat, % protein, % lactoseSomatic cell count
Information collected
Date of birthDate of calvingParity
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
What is Milk Recording?
Farmers pay for milk recording, in order to:
Adapt managementIdentify cows likely to have mastitisIdentify the best producers
The information is also used for
Genetic evaluationEpidemiologic studies
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
DataInitial Dataset
The National Milk Records: main provider of milkrecording in England and Wales
All the data collected by the NMR between January 2004and December 2006 were purchased:
19,893,093 recordings1,247,427 cows5,714 herds
⇒ Big!!!
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
DataData Selection
Aim: Obtain a homogeneous dataset and discard unreliabledata
Herds recording:
For the 3 complete yearsOn a monthly basisAt least 80 % of Holstein-Friesian cows
Milk samples collected on 2 consecutive milkings
Final dataset
8,211,988 recordings483,747 cows2,128 herds
⇒ Reasonably big!!!
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
DataData Selection
Aim: Obtain a homogeneous dataset and discard unreliabledata
Herds recording:
For the 3 complete yearsOn a monthly basisAt least 80 % of Holstein-Friesian cows
Milk samples collected on 2 consecutive milkings
Final dataset
8,211,988 recordings483,747 cows2,128 herds
⇒ Reasonably big!!!
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Somatic Cell CountRelation to mastitis
Mastitis
One of the biggest health problems in dairy herdsCan be clinical or subclinicalCauses an increase in milk somatic cell count (SCC)
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Somatic Cell CountRelation to mastitis
Individual Somatic Cell Count
Threshold of 200,000 cells/mL used to categorise cows asInfected/Uninfected
Bulk Milk Somatic Cell Count
Reflects herd mastitis prevalencePenalty on milk price when it is too high
Aims of the study
Can we model the transition between Low/High SCC fromindividual cow information?
Can we predict BMSCC from the predicted transitions?
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Outline
1 BackgroundMilk RecordingDataSomatic Cell Count
2 State TransitionState DefinitionState TransitionsData
3 A Simple ModelModelWinBUGS codeResults
4 Adding ComplexitySCC VariationModelWinBUGS codeResults
5 Discussion
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionDefinition of the States
First
Low/High
Low/High
Low/High
Low/High
Low/High
Last
Low/High
Low/High
Low/High
Low/High
Low/High
Low/High
Dry
Dry
Low/High
First
Low
High
Dry
Low
High
Last
Low
High
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionTransition Matrix
CurrentLow High Dry Last
Pre
viou
s Low π11 π12 π13 π14
High π21 π22 π23 π24
Dry π31 π32 π33 π34
First π41 π42 π43 π44
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionTransition Matrix
CurrentLow High Dry Last
Pre
viou
s Low π11 π12 π13 π14
High π21 π22 π23 π24
Dry π31 π32 π33 0First π41 π42 0 0
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
DataData Used for the Study
Training data
100 randomly selected herdsDataset 1: 6 consecutive test-days used for parameterestimation (70,382 lines)Dataset 2: 7th test-day for validation (11,895 lines)
Validation data (Dataset 3: 14,669 lines)
100 randomly selected herds1 test-day per herd
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Outline
1 BackgroundMilk RecordingDataSomatic Cell Count
2 State TransitionState DefinitionState TransitionsData
3 A Simple ModelModelWinBUGS codeResults
4 Adding ComplexitySCC VariationModelWinBUGS codeResults
5 Discussion
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionModel
Stateij ∼ Multinomial(πij)
ln(πij
π1j) =
4∑i ′=1
I [State i ′
i(j−1)]αi ′i
State i
Cow-recording j
Previous State i ′
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionModel
Stateij ∼ Multinomial(πij)
ln(πij
π1j) =
4∑i ′=1
I [State i ′
i(j−1)]αi ′i
State i
Cow-recording j
Previous State i ′
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionWinBUGS code
model {
for(i in 1:N) {
resp[i,1:4] ~ dmulti(pi[i,1:4],1)
for(m in 1:4){pi[i,m] <- p[i,m]/sum(p[i,])
}
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionWinBUGS code
model {
for(i in 1:N) {
resp[i,1:4] ~ dmulti(pi[i,1:4],1)
for(m in 1:4){pi[i,m] <- p[i,m]/sum(p[i,])
}
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionWinBUGS code
model {
for(i in 1:N) {
resp[i,1:4] ~ dmulti(pi[i,1:4],1)
for(m in 1:4){pi[i,m] <- p[i,m]/sum(p[i,])
}
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionWinBUGS code
model {
for(i in 1:N) {
resp[i,1:4] ~ dmulti(pi[i,1:4],1)
for(m in 1:4){pi[i,m] <- p[i,m]/sum(p[i,])
}
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionWinBUGS code
p[i,1] <- 1
# Code for 2
log(p[i,2]) <- beta[1, i] + beta[2, i] + beta[3, i] + beta[4, i]
beta[1, i] <- pstate[i, 1] * theta[1]
beta[2, i] <- pstate[i, 2] * theta[2]
beta[3, i] <- pstate[i, 3] * theta[3]
beta[4, i] <- pstate[i, 4] * theta[4]
# Code for 3
log(p[i,3]) <- beta[5, i]+beta[6, i]+ beta[7, i] + beta[8, i]
beta[5, i] <- pstate[i, 1] * theta[5]
beta[6, i] <- pstate[i, 2] * theta[6]
beta[7, i] <- pstate[i, 3] * theta[7]
beta[8, i] <- pstate[i, 4] * gamma
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionWinBUGS code
# Code for 4
log(p[i,4]) <- beta[9, i]+ beta[10, i] + beta[11, i] +
beta[12, i]
beta[9, i] <- pstate[i, 1] * theta[8]
beta[10, i] <- pstate[i, 2] * theta[9]
beta[11, i] <- pstate[i, 3] * gamma
beta[12, i] <- pstate[i, 4] * gamma
}
# Priors for fixed effects
for(k in 1:9) {
theta[k] ~ dnorm(0, .001)
}
gamma <- -2000
}
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
p1 = 1log(p2) = Ipst1 ∗ θ1 + Ipst2 ∗ θ2 + Ipst3 ∗ θ3 + Ipst4 ∗ θ4
log(p3) = Ipst1 ∗ θ5 + Ipst2 ∗ θ6 + Ipst3 ∗ θ7 + Ipst4 ∗ γlog(p4) = Ipst1 ∗ θ8 + Ipst2 ∗ θ9 + Ipste3 ∗ γ + Ipst4 ∗ γ
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
Median Ci2.5 Ci97.5
theta[1] -2.04 -2.06 -2.00theta[2] 0.80 0.76 0.83theta[3] -1.27 -1.35 -1.19theta[4] -1.52 -1.68 -1.36theta[5] -2.71 -2.75 -2.67theta[6] -0.79 -0.84 -0.73theta[7] 0.81 0.77 0.86theta[8] -3.95 -4.02 -3.88theta[9] -1.55 -1.63 -1.48
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
p1 = 1log(p2) = Ipst1∗−2.04+Ipst2∗0.80+Ipst3∗−1.27+Ipst4∗−1.52log(p3) = Ipst1∗−2.71+Ipst2∗−0.79+Ipst3∗0.81+Ipst4∗−2000log(p4) = Ipst1∗−3.95+ Ipst2∗−1.55+ Ipst3∗γ+ Ipst4∗−2000
p1 = 1log(p2) = −2.04log(p3) = −2.71log(p4) = −3.95
Σp = 1.22
π1 = 0.82 π2 = 0.11 π3 = 0.05 π4 = 0.02
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
p1 = 1log(p2) = −2.04log(p3) = −2.71log(p4) = −3.95
Σp = 1.22
π1 = 0.82 π2 = 0.11 π3 = 0.05 π4 = 0.02
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
p1 = 1log(p2) = −2.04log(p3) = −2.71log(p4) = −3.95
Σp = p1 + p2 + p3 + p4
Σp = 1.22
π1 = 0.82 π2 = 0.11 π3 = 0.05 π4 = 0.02
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
p1 = 1log(p2) = −2.04log(p3) = −2.71log(p4) = −3.95
Σp = 1 + e−2.04 + e−2.71 + e−3.95
Σp = 1.22
π1 = 0.82 π2 = 0.11 π3 = 0.05 π4 = 0.02
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
p1 = 1log(p2) = −2.04log(p3) = −2.71log(p4) = −3.95
Σp = 1 + 0.13 + 0.07 + 0.02
Σp = 1.22
π1 = 0.82 π2 = 0.11 π3 = 0.05 π4 = 0.02
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
p1 = 1log(p2) = −2.04log(p3) = −2.71log(p4) = −3.95
Σp = 1.22
π1 = 0.82 π2 = 0.11 π3 = 0.05 π4 = 0.02
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
p1 = 1log(p2) = −2.04log(p3) = −2.71log(p4) = −3.95
Σp = 1.22
π1 = p1Σp
π2 = p2Σp
π3 = p3Σp
π4 = p4Σp
π1 = 0.82π2 = 0.11 π3 = 0.05 π4 = 0.02
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
p1 = 1log(p2) = −2.04log(p3) = −2.71log(p4) = −3.95
Σp = 1.22
π1 = 11.22 π2 = e−2.04
1.22 π3 = e−2.71
1.22 π4 = e−3.95
1.22
π1 = 0.82π2 = 0.11 π3 = 0.05 π4 = 0.02
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
p1 = 1log(p2) = −2.04log(p3) = −2.71log(p4) = −3.95
Σp = 1.22
π1 = 0.82 π2 = 0.11 π3 = 0.05 π4 = 0.02
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionResults
State Probability of transition
Pre
vio
us
Cu
rren
t CredibilityInterval
n Observed Median 2.5 % 97.5 %Low Low 37,259 0.822 0.822 0.819 0.825Low High 4,870 0.107 0.107 0.105 0.110Low dry 2,487 0.055 0.055 0.053 0.057Low culled 720 0.016 0.016 0.015 0.017High Low 3,770 0.258 0.257 0.251 0.264High High 8,349 0.570 0.570 0.563 0.579High dry 1,718 0.117 0.117 0.113 0.123High culled 798 0.055 0.054 0.051 0.058dry Low 2,647 0.283 0.283 0.274 0.292dry High 745 0.080 0.079 0.075 0.085dry dry 5,967 0.638 0.638 0.627 0.646first Low 863 0.820 0.821 0.797 0.842first High 189 0.180 0.179 0.158 0.203
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Outline
1 BackgroundMilk RecordingDataSomatic Cell Count
2 State TransitionState DefinitionState TransitionsData
3 A Simple ModelModelWinBUGS codeResults
4 Adding ComplexitySCC VariationModelWinBUGS codeResults
5 Discussion
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Somatic Cell CountFactors of variation
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050
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Days in Milk
Som
atic
Cel
l Cou
nt
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Days in Milk
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atic
Cel
l Cou
nt
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Days in Milk
Som
atic
Cel
l Cou
nt
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0 100 200 300 400
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100
150
200
250
300
Days in Milk
Som
atic
Cel
l Cou
nt
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0 100 200 300 400
050
100
150
200
250
300
Days in Milk
Som
atic
Cel
l Cou
nt SCC varies with:
Stage of lactationParity
Parity 1 vs. > 1
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Somatic Cell CountFactors of variation
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●
●
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●●
0 100 200 300 400
050
100
150
200
250
300
Days in Milk
Som
atic
Cel
l Cou
nt
●
●
●
●
●
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●
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●
●
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●●
0 100 200 300 400
050
100
150
200
250
300
Days in Milk
Som
atic
Cel
l Cou
nt
SCC varies with:
Stage of lactationParity
Parity 1 vs. > 1
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionModel
Stateijk ∼ Multinomial(πijk)
ln(πijk
π1jk) =
4∑i ′=1
I [State i ′
i(j−1)k ](αi ′i +
∑Xijkβ
i ′i + ui ′
ik)
ui ′ik ∼ MVN(0,Σu)
State i
Cow-recording j
Herd k
Previous State i ′
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionModel
Stateijk ∼ Multinomial(πijk)
ln(πijk
π1jk) =
4∑i ′=1
I [State i ′
i(j−1)k ](αi ′i +
∑Xijkβ
i ′i + ui ′
ik)
ui ′ik ∼ MVN(0,Σu)
State i
Cow-recording j
Herd k
Previous State i ′
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionModel
Stateijk ∼ Multinomial(πijk)
ln(πijk
π1jk) =
4∑i ′=1
I [State i ′
i(j−1)k ](αi ′i +
∑Xijkβ
i ′i + ui ′
ik)
ui ′ik ∼ MVN(0,Σu)
State i
Cow-recording j
Herd k
Previous State i ′
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionWinBUGS code
model
{
for (i in 1:N) {
State[i, 1:4] ~ dmulti(pi[i, 1:4], 1)
for (j in 1:4) {
pi[i, j] <- p[i, j]/sum(p[i, ])
}
p[i, 1] <- 1
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionWinBUGS code
# transition to High
log(p[i, 2]) <- beta[1, i] + beta[2, i] + beta[3, i] + beta[4, i]
## from Low
beta[1, i] <- pstate[i, 1] * (
theta[1] + u1[hd_id[i], 1] + par2[i] * theta[2] +
(1 - par2[i]) * (Dim[i] * theta[3] + pow(Dim[i], 2) * theta[4]) +
par2[i] * (Dim[i] * theta[5] + pow(Dim[i], 2) * theta[6]))
## from High
beta[2, i] <- pstate[i, 2] * (
theta[7] + u1[hd_id[i], 2] + par2[i] * theta[8] +
(1 - par2[i]) * (Dim[i] * theta[9] + pow(Dim[i], 2) * theta[10]) +
par2[i] * (Dim[i] * theta[11] + pow(Dim[i], 2) * theta[12]))
## from dry
beta[3, i] <- pstate[i, 3] * (
par2[i] * (1 - day100[i]) * (theta[13] + u1[hd_id[i], 3]) +
day100[i] * gamma)
## from first
beta[4, i] <- pstate[i, 4] * (
theta[14] + u1[hd_id[i], 4] + par2[i] * gamma)
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionWinBUGS code
# transition to High
log(p[i, 2]) <- beta[1, i] + beta[2, i] + beta[3, i] + beta[4, i]
## from Low
beta[1, i] <- pstate[i, 1] * (
theta[1] + u1[hd_id[i], 1] + par2[i] * theta[2] +
(1 - par2[i]) * (Dim[i] * theta[3] + pow(Dim[i], 2) * theta[4]) +
par2[i] * (Dim[i] * theta[5] + pow(Dim[i], 2) * theta[6]))
## from High
beta[2, i] <- pstate[i, 2] * (
theta[7] + u1[hd_id[i], 2] + par2[i] * theta[8] +
(1 - par2[i]) * (Dim[i] * theta[9] + pow(Dim[i], 2) * theta[10]) +
par2[i] * (Dim[i] * theta[11] + pow(Dim[i], 2) * theta[12]))
## from dry
beta[3, i] <- pstate[i, 3] * (
par2[i] * (1 - day100[i]) * (theta[13] + u1[hd_id[i], 3]) +
day100[i] * gamma)
## from first
beta[4, i] <- pstate[i, 4] * (
theta[14] + u1[hd_id[i], 4] + par2[i] * gamma)
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionWinBUGS code
# transition to High
log(p[i, 2]) <- beta[1, i] + beta[2, i] + beta[3, i] + beta[4, i]
## from Low
beta[1, i] <- pstate[i, 1] * (
theta[1] + u1[hd_id[i], 1] + par2[i] * theta[2] +
(1 - par2[i]) * (Dim[i] * theta[3] + pow(Dim[i], 2) * theta[4]) +
par2[i] * (Dim[i] * theta[5] + pow(Dim[i], 2) * theta[6]))
## from High
beta[2, i] <- pstate[i, 2] * (
theta[7] + u1[hd_id[i], 2] + par2[i] * theta[8] +
(1 - par2[i]) * (Dim[i] * theta[9] + pow(Dim[i], 2) * theta[10]) +
par2[i] * (Dim[i] * theta[11] + pow(Dim[i], 2) * theta[12]))
## from dry
beta[3, i] <- pstate[i, 3] * (
par2[i] * (1 - day100[i]) * (theta[13] + u1[hd_id[i], 3]) +
day100[i] * gamma)
## from first
beta[4, i] <- pstate[i, 4] * (
theta[14] + u1[hd_id[i], 4] + par2[i] * gamma)
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionWinBUGS code
# transition to High
log(p[i, 2]) <- beta[1, i] + beta[2, i] + beta[3, i] + beta[4, i]
## from Low
beta[1, i] <- pstate[i, 1] * (
theta[1] + u1[hd_id[i], 1] + par2[i] * theta[2] +
(1 - par2[i]) * (Dim[i] * theta[3] + pow(Dim[i], 2) * theta[4]) +
par2[i] * (Dim[i] * theta[5] + pow(Dim[i], 2) * theta[6]))
## from High
beta[2, i] <- pstate[i, 2] * (
theta[7] + u1[hd_id[i], 2] + par2[i] * theta[8] +
(1 - par2[i]) * (Dim[i] * theta[9] + pow(Dim[i], 2) * theta[10]) +
par2[i] * (Dim[i] * theta[11] + pow(Dim[i], 2) * theta[12]))
## from dry
beta[3, i] <- pstate[i, 3] * (
par2[i] * (1 - day100[i]) * (theta[13] + u1[hd_id[i], 3]) +
day100[i] * gamma)
## from first
beta[4, i] <- pstate[i, 4] * (
theta[14] + u1[hd_id[i], 4] + par2[i] * gamma)
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionWinBUGS code
# transition to High
log(p[i, 2]) <- beta[1, i] + beta[2, i] + beta[3, i] + beta[4, i]
## from Low
beta[1, i] <- pstate[i, 1] * (
theta[1] + u1[hd_id[i], 1] + par2[i] * theta[2] +
(1 - par2[i]) * (Dim[i] * theta[3] + pow(Dim[i], 2) * theta[4]) +
par2[i] * (Dim[i] * theta[5] + pow(Dim[i], 2) * theta[6]))
## from High
beta[2, i] <- pstate[i, 2] * (
theta[7] + u1[hd_id[i], 2] + par2[i] * theta[8] +
(1 - par2[i]) * (Dim[i] * theta[9] + pow(Dim[i], 2) * theta[10]) +
par2[i] * (Dim[i] * theta[11] + pow(Dim[i], 2) * theta[12]))
## from dry
beta[3, i] <- pstate[i, 3] * (
par2[i] * (1 - day100[i]) * (theta[13] + u1[hd_id[i], 3]) +
day100[i] * gamma)
## from first
beta[4, i] <- pstate[i, 4] * (
theta[14] + u1[hd_id[i], 4] + par2[i] * gamma)
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionTransition Matrix for Primiparous Cows
●●●●●
●
●
●●●
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●
●
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●●
●
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●
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●
●
●●
●
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●
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●
●●
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●
●
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●
●●●
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●
●●
●
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●●
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●
●
●
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●
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●
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●
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●
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●
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●
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●
●
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●
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●
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●
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●
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●
●
●
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●
●
●
●
●
●
●
●●
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●
●●
●●
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●
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●
●
●
●
●
●
●
●
●●
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●
●
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●
●
●
●
●
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●
●
●
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●
●
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●
●●
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●
●●
●
●●
●
●
●
●
●
●
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●
●
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●
●
●
●
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●
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●
●
●
●●
●
●
●
●
●
●
●
●
●
●
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●
●
●●
●
●
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●
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●
●
●
●
●
●
●
●
●
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●
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●
●
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●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
Dim
Obs
Dim
Med
Low
Low
0 100 300 500
0.0
0.2
0.4
0.6
0.8
1.0
●●●●●●●
●●●
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●
●
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●
●
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●
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●
●●●●●●●●●●●
●
●●●
●
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●
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1.0
Current State
Pre
viou
s S
tate
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionTransition Matrix for Multiparous Cows
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Med
Pro
babi
lity
0.0
0.2
0.4
0.6
0.8
1.0
Dim
Obs
Dim
Med
first
Days in Milk
0 100 300 500
0.0
0.2
0.4
0.6
0.8
1.0
Dim
Obs
Dim
Med
Days in Milk
0 100 300 500
Dim
Obs
Dim
Med
Days in Milk
0 100 300 500
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Obs
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Med
Days in Milk
Pro
babi
lity
0 100 300 500
0.0
0.2
0.4
0.6
0.8
1.0
Current State
Pre
viou
s S
tate
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
State transitionTransition Between Low and High SCC
0.0
0.2
0.4
0.6
0.8
1.0
Previous State: SCC < 200,000 cells/mL
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Previous State: SCC < 200,000 cells/mL
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Previous State: SCC < 200,000 cells/mL
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Previous State: SCC < 200,000 cells/mL
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Days in Milk
Pro
babi
lity
of T
rans
ition
30 100
150
200
250
300
350
400
450
500
0.0
0.2
0.4
0.6
0.8
1.0
Previous State: SCC > 200,000 cells/mL
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Previous State: SCC > 200,000 cells/mL
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Previous State: SCC > 200,000 cells/mL
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Previous State: SCC > 200,000 cells/mL
Days in Milk
Pro
babi
lity
of T
rans
ition
0.0
0.2
0.4
0.6
0.8
1.0
Days in Milk
Pro
babi
lity
of T
rans
ition
30 100
150
200
250
300
350
400
450
500
Current State − Parity = 1:
Current State − Parity > 1:
< 200,000 > 200,000 dry culled
< 200,000 > 200,000 dry culled
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Prediction of BMSCCFrom Individual Cows to Bulk Milk
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Dataset 1B
MS
CC
(/1
,000
cel
ls/m
L)
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200
400
600
800
1000
BM
SC
C (
/1,0
00 c
ells
/mL)
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200
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800
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SC
C (
/1,0
00 c
ells
/mL)
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200
400
600
800
1000
Dataset 2
BM
SC
C (
/1,0
00 c
ells
/mL)
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200
400
600
800
1000
BM
SC
C (
/1,0
00 c
ells
/mL)
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200
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600
800
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Dataset 3
BM
SC
C (
/1,0
00 c
ells
/mL)
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Outline
1 BackgroundMilk RecordingDataSomatic Cell Count
2 State TransitionState DefinitionState TransitionsData
3 A Simple ModelModelWinBUGS codeResults
4 Adding ComplexitySCC VariationModelWinBUGS codeResults
5 Discussion
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Discussion
The model described the data well
Takes a long time to run in WinBUGS (∼ 30seconds/iteration)
Coefficients can be interpreted as odds-ratios for simplemodels
Model results must be interpreted by generatingpredictions in more complex cases
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Discussion
This type of model could be applied to a wide range ofsituations in Veterinary Epidemiology
e.g. locomotion scores, SIR models . . .
MCMC as implemented in WinBUGS converge slowly,even for simple models
ModellingState
Transitions
AurelienMadouasse
Background
Milk Recording
Data
Somatic CellCount
StateTransition
State Definition
StateTransitions
Data
A SimpleModel
Model
WinBUGS code
Results
AddingComplexity
SCC Variation
Model
WinBUGS code
Results
Discussion
Acknowledgments
Prof. Martin Green
Dr Jon HuxleyDr Andrew Bradley
School of Vetrinary Medicine and Science
University of Nottingham
Prof. William BrowneSchool of Clinical Veterinary Sciences
University of Bristol
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