Modelling
   State
 Transitions

  Aur´lien
     e
 Madouasse


Background
                            Modelling State Tra...
Outline

 Modelling
   State          1   Background
 Transitions            Milk Recording
  Aur´lien
     e             ...
Outline

 Modelling
   State          1   Background
 Transitions            Milk Recording
  Aur´lien
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What is Milk Recording?

 Modelling
   State
 Transitions

  Aur´lien
     e
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                      Milk recordi...
What is Milk Recording?

 Modelling
   State
 Transitions

  Aur´lien
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Background            Farmers pa...
Data
                  Initial Dataset


 Modelling
   State
 Transitions

  Aur´lien
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Background
     ...
Data
                  Data Selection


 Modelling
   State
 Transitions       Aim: Obtain a homogeneous dataset and disca...
Data
                  Data Selection


 Modelling
   State
 Transitions       Aim: Obtain a homogeneous dataset and disca...
Somatic Cell Count
                  Relation to mastitis


 Modelling
   State
 Transitions

  Aur´lien
     e
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Somatic Cell Count
                  Relation to mastitis


 Modelling
   State
 Transitions

  Aur´lien
     e           ...
Outline

 Modelling
   State          1   Background
 Transitions            Milk Recording
  Aur´lien
     e             ...
State transition
                  Definition of the States


 Modelling
   State
 Transitions

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State transition
                  Transition Matrix


 Modelling
   State
 Transitions

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Ba...
State transition
                  Transition Matrix


 Modelling
   State
 Transitions

  Aur´lien
     e
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Ba...
Data
                  Data Used for the Study


 Modelling
   State
 Transitions

  Aur´lien
     e
 Madouasse


Backgrou...
Outline

 Modelling
   State          1   Background
 Transitions            Milk Recording
  Aur´lien
     e             ...
State transition
                  Model


 Modelling
   State
 Transitions

  Aur´lien
     e
 Madouasse


Background
Mil...
State transition
                  Model


 Modelling
   State
 Transitions

  Aur´lien
     e
 Madouasse


Background
Mil...
State transition
                  WinBUGS code


 Modelling
   State
 Transitions

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        ...
State transition
                  WinBUGS code


 Modelling
   State
 Transitions

  Aur´lien
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        ...
State transition
                  WinBUGS code


 Modelling
   State
 Transitions

  Aur´lien
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        ...
State transition
                  WinBUGS code


 Modelling
   State
 Transitions

  Aur´lien
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        ...
State transition
                  WinBUGS code


 Modelling
   State
 Transitions       p[i,1] <- 1
  Aur´lien
     e
 Ma...
State transition
                  WinBUGS code


 Modelling
   State
 Transitions       # Code for 4
  Aur´lien
     e   ...
State transition
                  Results


 Modelling
   State
 Transitions

  Aur´lien
     e
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Background
M...
State transition
                  Results


 Modelling
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Background
M...
State transition
                  Results


 Modelling
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 Transitions

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State transition
                  Results


 Modelling
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                   p1 = 1
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                  Results


 Modelling
   State
 Transitions

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State transition
                  Results


 Modelling
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State transition
                  Results


 Modelling
   State
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  Aur´lien
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State transition
                  Results


 Modelling
   State
 Transitions

  Aur´lien
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 Madouasse         p1 = 1...
State transition
                  Results


 Modelling
   State
 Transitions

  Aur´lien
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State transition
                  Results


 Modelling
   State
 Transitions

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              ...
State transition
                  Results


 Modelling
   State
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State transition
                  Results


 Modelling
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State transition
                  Results


 Modelling
   State
 Transitions
                                        Stat...
Outline

 Modelling
   State          1   Background
 Transitions            Milk Recording
  Aur´lien
     e             ...
Somatic Cell Count
                  Factors of variation


 Modelling
   State
 Transitions

  Aur´lien
     e
 Madouasse...
Multi state
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Multi state
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Multi state

  1. 1. Modelling State Transitions Aur´lien e Madouasse Background Modelling State Transitions Milk Recording Data Example of a Multinomial Logit Model Applied to Somatic Somatic Cell Count Cell Count in Dairy Cows State Transition State Definition State Transitions Data Aur´lien Madouasse e A Simple Model Model WinBUGS code Results 19th April 2010 Adding Complexity SCC Variation Model WinBUGS code Results
  2. 2. Outline Modelling State 1 Background Transitions Milk Recording Aur´lien e Data Madouasse Somatic Cell Count Background 2 State Transition Milk Recording State Definition Data Somatic Cell State Transitions Count Data State Transition 3 A Simple Model State Definition Model State Transitions WinBUGS code Data Results A Simple Model 4 Adding Complexity Model SCC Variation WinBUGS code Results Model Adding WinBUGS code Complexity Results SCC Variation Model 5 Discussion WinBUGS code Results
  3. 3. Outline Modelling State 1 Background Transitions Milk Recording Aur´lien e Data Madouasse Somatic Cell Count Background 2 State Transition Milk Recording State Definition Data Somatic Cell State Transitions Count Data State Transition 3 A Simple Model State Definition Model State Transitions WinBUGS code Data Results A Simple Model 4 Adding Complexity Model SCC Variation WinBUGS code Results Model Adding WinBUGS code Complexity Results SCC Variation Model 5 Discussion WinBUGS code Results
  4. 4. What is Milk Recording? Modelling State Transitions Aur´lien e Madouasse Milk recording is the regular collection of a milk sample Background from all lactating cows of a dairy herd Milk Recording Data What is measured: Somatic Cell Count Milk yield State Transition % butterfat, % protein, % lactose State Definition Somatic cell count State Transitions Data Information collected A Simple Date of birth Model Model Date of calving WinBUGS code Results Parity Adding Complexity SCC Variation Model WinBUGS code Results
  5. 5. What is Milk Recording? Modelling State Transitions Aur´lien e Madouasse Background Farmers pay for milk recording, in order to: Milk Recording Data Adapt management Somatic Cell Count Identify cows likely to have mastitis State Transition Identify the best producers State Definition State The information is also used for Transitions Data Genetic evaluation A Simple Epidemiologic studies Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  6. 6. Data Initial Dataset Modelling State Transitions Aur´lien e Madouasse Background The National Milk Records: main provider of milk Milk Recording Data recording in England and Wales Somatic Cell Count All the data collected by the NMR between January 2004 State and December 2006 were purchased: Transition State Definition 19,893,093 recordings State Transitions 1,247,427 cows Data 5,714 herds A Simple Model Model ⇒ Big!!! WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  7. 7. Data Data Selection Modelling State Transitions Aim: Obtain a homogeneous dataset and discard unreliable Aur´lien e Madouasse data Herds recording: Background Milk Recording For the 3 complete years Data Somatic Cell On a monthly basis Count State At least 80 % of Holstein-Friesian cows Transition State Definition Milk samples collected on 2 consecutive milkings State Transitions Data A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  8. 8. Data Data Selection Modelling State Transitions Aim: Obtain a homogeneous dataset and discard unreliable Aur´lien e Madouasse data Herds recording: Background Milk Recording For the 3 complete years Data Somatic Cell On a monthly basis Count State At least 80 % of Holstein-Friesian cows Transition State Definition Milk samples collected on 2 consecutive milkings State Transitions Data A Simple Final dataset Model Model 8,211,988 recordings WinBUGS code 483,747 cows Results Adding 2,128 herds Complexity SCC Variation ⇒ Reasonably big!!! Model WinBUGS code Results
  9. 9. Somatic Cell Count Relation to mastitis Modelling State Transitions Aur´lien e Madouasse Background Milk Recording Data Somatic Cell Mastitis Count One of the biggest health problems in dairy herds State Transition Can be clinical or subclinical State Definition State Causes an increase in milk somatic cell count (SCC) Transitions Data A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  10. 10. Somatic Cell Count Relation to mastitis Modelling State Transitions Aur´lien e Individual Somatic Cell Count Madouasse Threshold of 200,000 cells/mL used to categorise cows as Background Infected/Uninfected Milk Recording Data Somatic Cell Count Bulk Milk Somatic Cell Count State Reflects herd mastitis prevalence Transition State Definition Penalty on milk price when it is too high State Transitions Data Aims of the study A Simple Model Model Can we model the transition between Low/High SCC from WinBUGS code Results individual cow information? Adding Complexity Can we predict BMSCC from the predicted transitions? SCC Variation Model WinBUGS code Results
  11. 11. Outline Modelling State 1 Background Transitions Milk Recording Aur´lien e Data Madouasse Somatic Cell Count Background 2 State Transition Milk Recording State Definition Data Somatic Cell State Transitions Count Data State Transition 3 A Simple Model State Definition Model State Transitions WinBUGS code Data Results A Simple Model 4 Adding Complexity Model SCC Variation WinBUGS code Results Model Adding WinBUGS code Complexity Results SCC Variation Model 5 Discussion WinBUGS code Results
  12. 12. State transition Definition of the States Modelling State Transitions Aur´lien e Madouasse Background Milk Recording Data Somatic Cell Count State Transition State Definition First Low/High Low/High Low/High Low/High Low/High Low/High Low/High Low/High Low/High Low/High Low/High Low/High Last Dry Dry State Transitions Data A Simple Model Low Low Low Model WinBUGS code First Dry Last Results High High High Adding Complexity SCC Variation Model WinBUGS code Results
  13. 13. State transition Transition Matrix Modelling State Transitions Aur´lien e Madouasse Background Milk Recording Current Data Somatic Cell Low High Dry Last Count Low π11 π12 π13 π14 Previous State Transition High π21 π22 π23 π24 State Definition State Dry π31 π32 π33 π34 Transitions Data First π41 π42 π43 π44 A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  14. 14. State transition Transition Matrix Modelling State Transitions Aur´lien e Madouasse Background Milk Recording Current Data Somatic Cell Low High Dry Last Count Low π11 π12 π13 π14 Previous State Transition High π21 π22 π23 π24 State Definition State Dry π31 π32 π33 0 Transitions Data First π41 π42 0 0 A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  15. 15. Data Data Used for the Study Modelling State Transitions Aur´lien e Madouasse Background Training data Milk Recording Data 100 randomly selected herds Somatic Cell Count Dataset 1: 6 consecutive test-days used for parameter State estimation (70,382 lines) Transition Dataset 2: 7th test-day for validation (11,895 lines) State Definition State Transitions Validation data (Dataset 3: 14,669 lines) Data 100 randomly selected herds A Simple Model 1 test-day per herd Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  16. 16. Outline Modelling State 1 Background Transitions Milk Recording Aur´lien e Data Madouasse Somatic Cell Count Background 2 State Transition Milk Recording State Definition Data Somatic Cell State Transitions Count Data State Transition 3 A Simple Model State Definition Model State Transitions WinBUGS code Data Results A Simple Model 4 Adding Complexity Model SCC Variation WinBUGS code Results Model Adding WinBUGS code Complexity Results SCC Variation Model 5 Discussion WinBUGS code Results
  17. 17. State transition Model Modelling State Transitions Aur´lien e Madouasse Background Milk Recording Data Somatic Cell Count Stateij ∼ Multinomial(πij ) State i State Transition Cow-recording j State Definition State Transitions Data A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  18. 18. State transition Model Modelling State Transitions Aur´lien e Madouasse Background Milk Recording Data Somatic Cell Count Stateij ∼ Multinomial(πij ) State i 4 State π Cow-recording j Transition State Definition ij ln( π1j ) = I [Statei(j−1) ]αii i State i =1 Previous State i Transitions Data A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  19. 19. State transition WinBUGS code Modelling State Transitions Aur´lien e Madouasse model { Background Milk Recording Data Somatic Cell Count State Transition State Definition State Transitions Data A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  20. 20. State transition WinBUGS code Modelling State Transitions Aur´lien e Madouasse model { Background Milk Recording Data Somatic Cell for(i in 1:N) { Count State Transition State Definition State Transitions Data A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  21. 21. State transition WinBUGS code Modelling State Transitions Aur´lien e Madouasse model { Background Milk Recording Data Somatic Cell for(i in 1:N) { Count State Transition resp[i,1:4] ~ dmulti(pi[i,1:4],1) State Definition State Transitions Data A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  22. 22. State transition WinBUGS code Modelling State Transitions Aur´lien e Madouasse model { Background Milk Recording Data Somatic Cell for(i in 1:N) { Count State Transition resp[i,1:4] ~ dmulti(pi[i,1:4],1) State Definition State Transitions Data for(m in 1:4){ A Simple Model pi[i,m] <- p[i,m]/sum(p[i,]) Model WinBUGS code } Results Adding Complexity SCC Variation Model WinBUGS code Results
  23. 23. State transition WinBUGS code Modelling State Transitions p[i,1] <- 1 Aur´lien e Madouasse # Code for 2 Background log(p[i,2]) <- beta[1, i] + beta[2, i] + beta[3, i] + beta[4, i] Milk Recording Data Somatic Cell beta[1, i] <- pstate[i, 1] * theta[1] Count beta[2, i] <- pstate[i, 2] * theta[2] State Transition beta[3, i] <- pstate[i, 3] * theta[3] State Definition beta[4, i] <- pstate[i, 4] * theta[4] State Transitions Data # Code for 3 A Simple log(p[i,3]) <- beta[5, i]+beta[6, i]+ beta[7, i] + beta[8, i] Model Model WinBUGS code Results beta[5, i] <- pstate[i, 1] * theta[5] Adding beta[6, i] <- pstate[i, 2] * theta[6] Complexity beta[7, i] <- pstate[i, 3] * theta[7] SCC Variation Model beta[8, i] <- pstate[i, 4] * gamma WinBUGS code Results
  24. 24. State transition WinBUGS code Modelling State Transitions # Code for 4 Aur´lien e log(p[i,4]) <- beta[9, i]+ beta[10, i] + beta[11, i] + Madouasse beta[12, i] Background Milk Recording beta[9, i] <- pstate[i, 1] * theta[8] Data Somatic Cell beta[10, i] <- pstate[i, 2] * theta[9] Count beta[11, i] <- pstate[i, 3] * gamma State Transition beta[12, i] <- pstate[i, 4] * gamma State Definition } State Transitions Data # Priors for fixed effects A Simple Model for(k in 1:9) { Model theta[k] ~ dnorm(0, .001) WinBUGS code Results } Adding Complexity gamma <- -2000 SCC Variation Model } WinBUGS code Results
  25. 25. State transition Results Modelling State Transitions Aur´lien e Madouasse Background Milk Recording Data Somatic Cell Count State Transition State Definition State Transitions Data A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  26. 26. State transition Results Modelling State Transitions Aur´lien e Madouasse Background Milk Recording Data Somatic Cell p1 = 1 Count log (p2 ) = Ipst1 ∗ θ1 + Ipst2 ∗ θ2 + Ipst3 ∗ θ3 + Ipst4 ∗ θ4 State Transition log (p3 ) = Ipst1 ∗ θ5 + Ipst2 ∗ θ6 + Ipst3 ∗ θ7 + Ipst4 ∗ γ State Definition State log (p4 ) = Ipst1 ∗ θ8 + Ipst2 ∗ θ9 + Ipste3 ∗ γ + Ipst4 ∗ γ Transitions Data A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  27. 27. State transition Results Modelling State Transitions Aur´lien e Madouasse Median Ci2.5 Ci97.5 Background theta[1] -2.04 -2.06 -2.00 Milk Recording Data theta[2] 0.80 0.76 0.83 Somatic Cell Count theta[3] -1.27 -1.35 -1.19 State theta[4] -1.52 -1.68 -1.36 Transition State Definition theta[5] -2.71 -2.75 -2.67 State Transitions Data theta[6] -0.79 -0.84 -0.73 A Simple theta[7] 0.81 0.77 0.86 Model Model theta[8] -3.95 -4.02 -3.88 WinBUGS code Results theta[9] -1.55 -1.63 -1.48 Adding Complexity SCC Variation Model WinBUGS code Results
  28. 28. State transition Results Modelling State Transitions p1 = 1 Aur´lien e Madouasse log (p2 ) = Ipst1 ∗−2.04+Ipst2 ∗0.80+Ipst3 ∗−1.27+Ipst4 ∗−1.52 Background log (p3 ) = Ipst1 ∗−2.71+Ipst2 ∗−0.79+Ipst3 ∗0.81+Ipst4 ∗−2000 Milk Recording Data log (p4 ) = Ipst1 ∗ −3.95 + Ipst2 ∗ −1.55 + Ipst3 ∗ γ + Ipst4 ∗ −2000 Somatic Cell Count State Transition State Definition State Transitions Data A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  29. 29. State transition Results Modelling State Transitions Aur´lien e Madouasse p1 = 1 Background log (p2 ) = −2.04 Milk Recording log (p3 ) = −2.71 Data Somatic Cell Count log (p4 ) = −3.95 State Transition State Definition State Transitions Data A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  30. 30. State transition Results Modelling State Transitions Aur´lien e Madouasse p1 = 1 Background log (p2 ) = −2.04 Milk Recording log (p3 ) = −2.71 Data Somatic Cell Count log (p4 ) = −3.95 State Transition State Definition State Transitions Data Σ p = p1 + p2 + p3 + p4 A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  31. 31. State transition Results Modelling State Transitions Aur´lien e Madouasse p1 = 1 Background log (p2 ) = −2.04 Milk Recording log (p3 ) = −2.71 Data Somatic Cell Count log (p4 ) = −3.95 State Transition State Definition State Transitions Data Σp = 1 + e −2.04 + e −2.71 + e −3.95 A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  32. 32. State transition Results Modelling State Transitions Aur´lien e Madouasse p1 = 1 Background log (p2 ) = −2.04 Milk Recording log (p3 ) = −2.71 Data Somatic Cell Count log (p4 ) = −3.95 State Transition State Definition State Transitions Data Σp = 1 + 0.13 + 0.07 + 0.02 A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  33. 33. State transition Results Modelling State Transitions Aur´lien e Madouasse p1 = 1 Background log (p2 ) = −2.04 Milk Recording log (p3 ) = −2.71 Data Somatic Cell Count log (p4 ) = −3.95 State Transition State Definition State Transitions Data Σp = 1.22 A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  34. 34. State transition Results Modelling State Transitions Aur´lien e Madouasse p1 = 1 log (p2 ) = −2.04 Background Milk Recording log (p3 ) = −2.71 Data Somatic Cell log (p4 ) = −3.95 Count State Transition State Definition State Transitions Σp = 1.22 Data A Simple Model Model p1 p2 p3 p4 WinBUGS code π1 = Σp π2 = Σp π3 = Σp π4 = Σp Results Adding Complexity SCC Variation Model WinBUGS code Results
  35. 35. State transition Results Modelling State Transitions Aur´lien e Madouasse p1 = 1 log (p2 ) = −2.04 Background Milk Recording log (p3 ) = −2.71 Data Somatic Cell log (p4 ) = −3.95 Count State Transition State Definition State Transitions Σp = 1.22 Data A Simple Model Model 1 e −2.04 e −2.71 e −3.95 WinBUGS code π1 = 1.22 π2 = 1.22 π3 = 1.22 π4 = 1.22 Results Adding Complexity SCC Variation Model WinBUGS code Results
  36. 36. State transition Results Modelling State Transitions Aur´lien e Madouasse p1 = 1 Background log (p2 ) = −2.04 Milk Recording log (p3 ) = −2.71 Data Somatic Cell Count log (p4 ) = −3.95 State Transition State Definition State Transitions Data Σp = 1.22 A Simple Model Model WinBUGS code Results π1 = 0.82 π2 = 0.11 π3 = 0.05 π4 = 0.02 Adding Complexity SCC Variation Model WinBUGS code Results
  37. 37. State transition Results Modelling State Transitions State Probability of transition Aur´lien e Previous Current Madouasse Credibility Interval Background n Observed Median 2.5 % 97.5 % Milk Recording Low Low 37,259 0.822 0.822 0.819 0.825 Data Somatic Cell Low High 4,870 0.107 0.107 0.105 0.110 Count Low dry 2,487 0.055 0.055 0.053 0.057 State Low culled 720 0.016 0.016 0.015 0.017 Transition High Low 3,770 0.258 0.257 0.251 0.264 State Definition State High High 8,349 0.570 0.570 0.563 0.579 Transitions High dry 1,718 0.117 0.117 0.113 0.123 Data High culled 798 0.055 0.054 0.051 0.058 A Simple Model dry Low 2,647 0.283 0.283 0.274 0.292 Model dry High 745 0.080 0.079 0.075 0.085 WinBUGS code dry dry 5,967 0.638 0.638 0.627 0.646 Results first Low 863 0.820 0.821 0.797 0.842 Adding first High 189 0.180 0.179 0.158 0.203 Complexity SCC Variation Model WinBUGS code Results
  38. 38. Outline Modelling State 1 Background Transitions Milk Recording Aur´lien e Data Madouasse Somatic Cell Count Background 2 State Transition Milk Recording State Definition Data Somatic Cell State Transitions Count Data State Transition 3 A Simple Model State Definition Model State Transitions WinBUGS code Data Results A Simple Model 4 Adding Complexity Model SCC Variation WinBUGS code Results Model Adding WinBUGS code Complexity Results SCC Variation Model 5 Discussion WinBUGS code Results
  39. 39. Somatic Cell Count Factors of variation Modelling State Transitions Aur´lien e Madouasse 300 q q q qq qq q q q q q q q q q q qq q q q q qq q q q q q qq q qq q q q qqq qqq q Background q qq q qq q q qq q qq q q 250 qqq q q q q q qq q q q q qq q q qqq q qq q qqq Milk Recording qqq qq q qqq q q q q qq qq q q q qq q qq q qq q q qq q qq q q q q qq q q qq q qq q qq Data qqq q q qqq qqq qq qqqqqq qq qq q q q qqq q q qq qq q q q q qqq q q q qq q q qqqq Somatic Cell qqq qqq qq 200 q q qqq q qq q SCC varies with: qq qqq q q q qq q q q q q qq q q q q qq qq q q q qq qq q qqqq Count qqq qqq q q qq q q q qq q q q qqqqq q q q q q q q qq qq q qqq Somatic Cell Count q qq qq q qq qqq q q q q qqqq q q qq q qqqq q q qqqqq q qqq q q q q q q qq q qqq q qqq q q q q q q q q q qqq q q qqqqq q q q q qqq q q qq qq qqq q q qq q q q State qqq q qqq q qq q qq qq Stage of lactation q q q qq qqq q q q q qq qq q qq qq qqq qq q q 150 q qq qqq q q qq q q qqq q q q qqq q qq qq qq qqqqqq qq q q q qq q qq qq q qq qq qq q Transition q qq q q qq qq qq q qqq q qq q qq qq qqqq qqq qq qq q q q qqq qq q qqq qqqqq qqq qq q q qq q q qqq qqq q q q qq qqq q q q q qq q q q qqq qq q qqqq q qqq qqq q qq qq q qq q q q q qq q q q qq q q q q Parity q qqq q qqq q q qqqqq qq q qqq qqq q qq q q q q q qq q qq qqqqqq q q qq qqq q q q qqq q q qqq qq q q qqqqq q State Definition q q qqq q qqqq q q qqq qq q qqq qqq q qqqqqq q q q qqqq q qqqq qqqqq q qq q q qq qqqq qqqq qq qq q qq q q qqqqq q q q q qqqqq qqq q qq qq qqqq qqqq q q q qq q qqqq q qq q qq qqqqq qq qqqq qqq qq q qq qqq q qqq qq qqqqq qq q qq q qqq q qqq 100 State qqq qq q q q q q q qq qqqq q qqqq q qq qqqqqq qq q q q q qqqq q q q qqqq q qqqqq qq q q q qq qq q qq q qqq q q q q qqqqqq q q qqqqq q q qq q q q qqq qqqqqq q q qq qqq q qq q q qq qq qq qqq q qqqq qqq q q q q q qq q q q q q q q q q qq qqqq q q qqqq qqqq qqq q qq q q qqq qqqq q qqqq q Transitions qq q q q qq qq qqq qq q qqq q q qqqqqq qqqqqqqq qqqq q qq qq qqq qqqqqq qqqq q q qq q q q q q qq qqq q qqqq qq qqq q qq qq qq qqqqq q qqqqq q qq q q qq q qq q q q q q q q q qq q qqq q qq q qqqqq q qqq q qqqqq qq qqqqq q qq q q q q qqq qq qq q qqqq q q q qqqqqq q q qqq q qqq qq qqqqq qq qq qqqq q qq q q q q q qqq qqq qqq q qq qq qq q q q qq qq qq q q q q q qqqq q qqqqq qqqqq qq q qqqqqq q q q q q q qq q q q qqq qq qqq q qq q q qq q q qqqqqqqq qq qqqqq qqqqq qqqqq q qq q qqqqq qqqq q q Data q q qqqq q q qq qqq qqqqq q qq qqq q qqq qqqqqqqqq q q qqqqqqqq qqq q q qq qq qqqq q q qqqqqq qqqq q q q qq qq q q q q qqq qqqqq q qqq qq qq qq qqqqq q qq qq q q q qqq q qqqq q qqqqqqq qqq q qq qqqq q qqqqqqqq qq q qqq qq qqq qq qq qqqqq qqqqq q q qqqqqqqqq q q q qq qqq q q qq qqqqqqqq qq q qqqqqqqqqq q q qqq qq qq q qqqqqq q qqqqq qq qqqqqqqqqqq qqq qqqqq q qqq q qq qq q q qq q q q qq qq qqqqq q qqq q qqqq q q qqqqqqqqqqqqq qqq q q q qqqqqqqqqq qqqqq qqq q qq qq qqqqqq qqqqqqqqqqqqqqqqqqqqq qq qqq qqqq qqqqq q q qqqqqqq qqqqqqq qqqqqqq q qqq qqq qqqqqqqqq qqqqq qq q qqqq q qqqq q q q q qqqq q qqqq qqq qqqqqq q qqq q qqqqqqqqqqq qq q q 50 q qqqqqq qqqqqqqq q qqqqqqqqqqqqqqqqqqqqq q qqqq q qqq qqqqqq q qqqqqqqqqqqqqqqqqqqq q q qqqq qqqqqq qqqqqqqqqq q qq qqqqqq q qq qqqqqqqqqqqq qqqq qq q qqq q A Simple q q Model Model 0 WinBUGS code 0 100 200 300 400 Results Days in Milk Adding Complexity SCC Variation Model WinBUGS code Results

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