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Multi state Multi state Presentation Transcript

  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • Somatic Cell Count Factors of variation Modelling State Transitions Aur´lien e Madouasse 300 Background Milk Recording 250 Data Somatic Cell Count qq qq qqq q qq q SCC varies with: 200 qq qq q q qq q q q qq q q q qq q q q q q qq q q q q Somatic Cell Count q qqqq qqq q qqq q qq qq qqqq q qqq State q qq q qq Stage of lactation qq q qqq q q q q q q qq qqq qq q q q q qq q qq qq q qq qqq Transition qq qq q qq 150 qq qq q qq qq q qq qq qqq Parity q q qq qq q q qq qq qq q q q State Definition q q qq qqq qq qq q qqq qq q qq qq qq q q q qq qq qq q qq qq qq qq q State q q qq qqq qq qq qq q q q q qqq q q q 100 q q q qq qq q qq q qq qq qqq q Transitions qq qqq q qq qq qq q qq q q qqq qqq q q qq q q Parity 1 vs. > 1 q q qq q q q q q qqq qq q qq q q qqq qq qqq qqq q qq q q q q q qq q q qq q q q qq q q q qq q q qq q q q q qq qq qq q q q qqq q q q qq q q q qqq q qqq qqq qq qq qq qqqq q qq q qq q qq qq qq q qq q q q qqq q qqqqq q qq Data q qq q q qq q qqq qqqq qqqq qqq qq q qq qq qq q qqq qqq q qq q qqq qq q qq qq qqq q q qq qq q q q qq qq q qqq q q q qqq qq qqq q q qq qqq q q qq qqq q q q qq q qq q q q qq q qq qqq q q qq q qq qqq q qq q q qq q q qq q q q qqqqq qqq qq q qqq q q q qqqq q q qq q qq qqqq q q qq qq qq qqqqqqq qq q q qq qqqqqq q qq q q q qqqq qqqqq q qqqqqqqq q qqqqqqqqqqq qq q qqq q q qq qq q qq q q qq qq qqq q q q qq qq q qq q q q q qq q qq qq q q q q q q qq q qq qq q q q qq qqq qq qqq q q qq q q qq qqq q qq q qq qq q q qq q q qqqqqqqqq q q 50 q q qq q qqq q q qqqq q qqqqqqqq qq q qqqqqq qqq q q qqqq qqqqq q q q q q q A Simple Model 0 Model WinBUGS code 0 100 200 300 400 Results Days in Milk Adding Complexity SCC Variation Model WinBUGS code Results
  • State transition Model Modelling State Transitions Aur´lien e Madouasse Stateijk ∼ Multinomial(πijk ) Background Milk Recording Data Somatic Cell Count State Transition State Definition State Transitions Data State i A Simple Model Cow-recording j Model WinBUGS code Results Herd k Adding Previous State i Complexity SCC Variation Model WinBUGS code Results
  • State transition Model Modelling State Transitions Aur´lien e Madouasse Stateijk ∼ Multinomial(πijk ) Background 4 Milk Recording π Data Somatic Cell ijk ln( π1jk ) = i I [Statei(j−1)k ](αii + Xijk βii + uik ) i Count i =1 State Transition State Definition State Transitions Data State i A Simple Model Cow-recording j Model WinBUGS code Results Herd k Adding Previous State i Complexity SCC Variation Model WinBUGS code Results
  • State transition Model Modelling State Transitions Aur´lien e Madouasse Stateijk ∼ Multinomial(πijk ) Background 4 Milk Recording π Data Somatic Cell ijk ln( π1jk ) = i I [Statei(j−1)k ](αii + Xijk βii + uik ) i Count i =1 State i uik ∼ MVN(0, Σu ) Transition State Definition State Transitions Data State i A Simple Model Cow-recording j Model WinBUGS code Results Herd k Adding Previous State i Complexity SCC Variation Model WinBUGS code Results
  • State transition WinBUGS code Modelling State Transitions Aur´lien e Madouasse model Background { Milk Recording Data for (i in 1:N) { Somatic Cell Count State State[i, 1:4] ~ dmulti(pi[i, 1:4], 1) Transition State Definition State for (j in 1:4) { Transitions Data pi[i, j] <- p[i, j]/sum(p[i, ]) A Simple } Model Model WinBUGS code p[i, 1] <- 1 Results Adding Complexity SCC Variation Model WinBUGS code Results
  • State transition WinBUGS code Modelling State # transition to High Transitions log(p[i, 2]) <- beta[1, i] + beta[2, i] + beta[3, i] + beta[4, i] 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
  • State transition WinBUGS code Modelling State # transition to High Transitions log(p[i, 2]) <- beta[1, i] + beta[2, i] + beta[3, i] + beta[4, i] Aur´lien e Madouasse ## from Low beta[1, i] <- pstate[i, 1] * ( Background theta[1] + u1[hd_id[i], 1] + par2[i] * theta[2] + Milk Recording (1 - par2[i]) * (Dim[i] * theta[3] + pow(Dim[i], 2) * theta[4]) Data Somatic Cell par2[i] * (Dim[i] * theta[5] + pow(Dim[i], 2) * theta[6])) Count State Transition State Definition State Transitions Data A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  • State transition WinBUGS code Modelling State # transition to High Transitions log(p[i, 2]) <- beta[1, i] + beta[2, i] + beta[3, i] + beta[4, i] Aur´lien e Madouasse Background Milk Recording Data Somatic Cell Count State ## from High Transition beta[2, i] <- pstate[i, 2] * ( State Definition State theta[7] + u1[hd_id[i], 2] + par2[i] * theta[8] + Transitions Data (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])) A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  • State transition WinBUGS code Modelling State # transition to High Transitions log(p[i, 2]) <- beta[1, i] + beta[2, i] + beta[3, i] + beta[4, i] Aur´lien e Madouasse Background Milk Recording Data Somatic Cell Count State Transition State Definition State Transitions Data A Simple Model Model ## from dry WinBUGS code beta[3, i] <- pstate[i, 3] * ( Results par2[i] * (1 - day100[i]) * (theta[13] + u1[hd_id[i], 3]) + Adding day100[i] * gamma) Complexity SCC Variation Model WinBUGS code Results
  • State transition WinBUGS code Modelling State # transition to High Transitions log(p[i, 2]) <- beta[1, i] + beta[2, i] + beta[3, i] + beta[4, i] 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 ## from first WinBUGS code Results beta[4, i] <- pstate[i, 4] * ( theta[14] + u1[hd_id[i], 4] + par2[i] * gamma)
  • State transition Transition Matrix for Primiparous Cows Modelling Current State State Transitions Low High dry culled 0 100 300 500 0 100 300 500 0 100 300 500 0 100 300 500 1.0 qqqqqqqq qqq qqqqq qq qq q qqqq qqq qqq qq q qq qq q q q q q qqqq qqq qqqq qq q 1.0 Aur´lien e q qq q q q q qqqqqqqqqqq qqqqqqq qqqq qq q q q qqqqqqqqqq q q q q q q q qq q qqq q qqq q qq q qq qq q qq qqqqq qqqqqqqq q q qq qqqq q qqqqqqqq q qq qq q qq q qq qqqq qqq q q q qq q q q q q q qqq q q qq qqq q qq 0.8 q q q q qq q q qqq qqq q qq q qq q 0.8 Madouasse q qq q q q qq q q qq qqq q qqq qqq q qq q q Probability q q qq q qqq q q q qq qqqq qq q qqq q qqq qq qq q qqqq q qq q qq q q q q q Low 0.6 qqqqq q q qqq qq q q q qq q q q q q q qq q qq 0.6 Med Med Med Med q Obs Obs Obs Obs q qq q q qq q qq q q q q q q q qq q qq q qq 0.4 q qq qq q qq q q qq qq qqq qq q q q q q q qq qq q 0.4 q q q qq q q qqqqq q q q qqq q q qqqqqq qq q q qq qq q q q q q qqqqqq q qq qqqq q qq qq q Background 0.2 q q qq q q q q q q q qq q qq qqqqq qq q q q q q q qq q q q q q qq q qqqq q qq qqqq qqqqqqqq qqqq q qqqqqqqqqqqqq qq q q q qq qq q q qq q q qqqq qq q qqqqqqq q qq q qqq qqq q q q q qqq qqqq q qq q q q q q qq q q q q 0.2 qqqqqqqqq qqqqq qq qq q qq qqqqqqqq qq qqq qqqqqq q qq q qqq q q q qq q q qqq q q q q qq q q q qqq qqqq qqqqqqqqqqqqq q q qqqq qqqqqqq qqq qq q qq qqqqqqq q q q q q q qq qq q qqqqq q q q q q qq qqqqqqqqqqqqqqq qq qq q q q qq q q qq q q q qqqq q qqq q q q q q qqq qqqqqqqqqqqq q qqqqqqqqqqq qq q qqqqqqq qq qqqqqqqqqqqq qqq qqqqq q qqqq qq qq qqq qqqq q qqqqq qqqqqqqqqqqq qqqqqqqqqqqqq qq qq q qq q q q q q qq q qq q q qq q q q q q q q qqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqq qqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqq qq qqqqqq qqqq qqq qqqqq q qq qqqq q qqq Milk Recording 0.0 qq qq qq q q q qq q q qq qqqq q qq q q q q q qqq q q qqqqqqqqqqqqq qqqqqqqqqqqqq qqqqqqqqqqqq q q q q qqqqq q qqq qq qq q q q qq q q 0.0 Data 1.0 q q q qqqqq q q q q q qqq q q q q q q q q q qq q qq q q q qqq q q qq q q qqqqqq qq qq q q qq q q q qq qqq qq qqq q q q q q qq q qqq q q q qq 1.0 Somatic Cell q q qq Dim q q Dim q Dim Dim Count 0.8 q q q q q q q qq q q q 0.8 q q qqq q q q q q q q q q qq q q q q q q q qq q q qq q q Probability q q q q qq qqq qq q q q q q q q q q qq qq q qq q q qq q qqqqqqq qq q q qq q q q q q q qqq q q q q q q qq q q q qq qq q q q q q High 0.6 qq q q q q q qq q q qq q qq q q q q qqq qq q q q q q q q q q q qq 0.6 Med Med Med Med Obs Obs Obs Obs Previous State qqqqq q q qqqqqqqqq qq q q qq qq q q qq q q q q qq q q qq q q qq q q qq q q q q q q qqq q q qq qqqqqqqqq qq q qqqq q q qqq qqqqqq q q qq q q qqq qqq q q q q qq qq q q q q qq q qqq qqq q qqq qq qq qq q qq q q q qq q q qq State 0.4 q qqq qq q q qq q qq q q qq q q qq q q q q q q q qq q q qq q qq qq qq q qqqqqq qq qq q q q q q q q qq q q q q q q qq q q q q q q q q q qq q q q qq q q qq q qqq q qq q q q q qq qqqq qq qq q q q qq q q q q q q q q q q q qq q qq q q q q q qq q q qq q q qq q q q q q q q q qq q q q qq 0.4 q q qqq q q q q qqq q q Transition 0.2 q q qq q q q q qqq q q qq q qqq q q q q qqqq q qq q q q q q qq q q q q qq q q q q q q qqq qqq qq q q q q q q q q q qq q q q q q q q qq q q q q q q qqqq q qq q q qq q q qqq q q q qq q q q q qq q q qqqq q q qq q q qq q qq q q q q 0.2 q qq q State Definition 0.0 q qqqq qqq q qqqqqqqqq qq qq q qqq qqqqq qqqq q q q qq qqqqqqq q q q qqqqqq qq qq qq qqqqq qqqq q q qqqqqqq qqq q qqq q q qq qq qqqqq qqq q q qq q q q q qqqq q qq q q qq q qqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqq qq qqqqqqqqqqqqqqqqqq qqqq qqqqqqqqqqqqqqq qqqqqqq qqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq q 0.0 State Transitions 1.0 qqqqqqqqqqqqqqqq q qqqqqqqqqqqqq qqqqqqqqqqqqq q qqqqqqqqqqq qqqqqqqqqq 1.0 Dim Dim Dim Dim Data 0.8 0.8 Probability 0.6 0.6 dry Med Med Med Med Obs Obs Obs Obs A Simple 0.4 0.4 Model 0.2 0.2 Model 0.0 qqqqqqqqqqqqqqqq q qqqqqqqqqqqqq qqqqqqqqqqqqq q qqqqqqqqqqq qqqqqqqqqq qqqqqqqqqqqqqqqq q qqqqqqqqqqqqq qqqqqqqqqqqqq q qqqqqqqqqqq qqqqqqqqqq qqqqqqqqqqqqqqqq q qqqqqqqqqqqqq qqqqqqqqqqqqq q qqqqqqqqqqq qqqqqqqqqq 0.0 WinBUGS code 1.0 qq 1.0 Results q q q q q q q Dim Dim Dim Dim q qq 0.8 q qq q q 0.8 q qq Probability q Adding first 0.6 0.6 Med Med Med Med Obs Obs Obs Obs q q Complexity 0.4 q 0.4 qq qq SCC Variation 0.2 q q q q q q q q q 0.2 qq qq q qq Model 0.0 qq qq qq q qq qq qq q 0.0 WinBUGS code 0 100 300 500 0 100 300 500 0 100 300 500 0 100 300 500 Results Days in Milk Dim Days in Milk Dim Days in Milk Dim Days in Milk Dim
  • State transition Transition Matrix for Multiparous Cows Modelling Current State State Transitions Low High dry culled 0 100 300 500 0 100 300 500 0 100 300 500 0 100 300 500 1.0 qq qq q q qq q q q q q 1.0 Aur´lien e q q q q qqq q qqqq qq qqq qq q qqqqqqqq qq qqqqqqq q qq q q qqqqqqqqq qqqqqqqqqqq q qqqqqqqqqq q q q qq q qqqqqqqqqqqq qqqqqqqqqqqq q qqq q qqqqqq q qqqq qq q qqq q qq qq q q q q q q qqqqq q 0.8 qq q qqqq qqq q 0.8 Madouasse q qq qqq q qqq q q q q q qq q qq q q q q q Probability q q q qq q q q q q q q q q q qq qqqq q q q q q q qq q q q qqq qq qq q q Low 0.6 qq qqq qq qqq qq q q 0.6 qqqqq q q q q qq qq Med qq qq q Med Med Med qqqqqqqq Obs Obs Obs Obs q q q qqq qq q q q q q qq qq qqq qq q qq qq q qq q q q qq qq qq q q q q qqq q qqq q q q q q q qqqq q q q q 0.4 qqq q q q q q q qq q q q q 0.4 q q q qq q qq q qq q q qqq qq q q q q q q q qqqq qq q q q q q q q qq q q q q q qqqqqqqqq q qqqq q q qq q q q qq q q qqq q q qq q q q q qqqqqqq q qq q qqq q qqq qq q q q q qq q Background 0.2 qq qq q q qqq qqq q q q q q qq q q qq qq qqqq q q q q q qqqqq q q q qqqqq qqqqq q q q q q q qqqqqqqqqqq qqq qq q q q qq q q qqqq q qq q qqq q qqq q q qqqqqqq qqqq q q qqqq qqqqqqqqqqqqqqqqq q qqqq qqqqqqqqqqq qqqq qqqq qqqqqqqqq qqq q q q q q q q qq q qqqq q q qqqqq qq q qq q q qq q qq q qqq q qq q qqq qq q qqq qq qqqqq q q q q qq qqq q q q q q q q qq qq q qq q q q qq q q qq q q q 0.2 q qq q q qq q qqqqq qqqq qqqqqqqqqqq qq q q qqq q q qqqqqq q q q q qq q q q q q qqqq qqqqq qq q q q q q qq qq qq q q qq qq q qq qqq q qqqq qqq qq qq q q q q q qq qq q qq q q qq qq qq q q qq q q qq q q qqq q q q q qq qqqqq q qq qq q q q q qq qqq q qq qq q qq qq qqqqq qq qqqq q q q qqq qqq qqq qqqqqqqqqqqqq qqqqqqqqqqqq q q qq q q q qqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqqqq q qqqqqqqqqqqqqqqqq q qq qq q qqqq qqqqq q qq qq q q qq q q q q qq q q qq Milk Recording 0.0 q q q q qq qq q q q qq qqq qqq qqq qqqqqqqqqqqq q qqqqqqqqqqqq q qqqqqqqqqqqq qq q qq qq q qq q qq q q 0.0 Data 1.0 qq qq qq q qq q 1.0 Somatic Cell Dim q qq q q qq q q qDim q q q qq Dim Dim 0.8 q qqq qqq q q q 0.8 Count q q q qqq q qq qqqqqq q q qq q q qq qq q qqq qqqq q q q qq q q qq Probability q q q qqqqqqq qq qq q qq qq q q q qq q qqqqqq qqq qqq q qqqq qq q q q q qq qq q q q q q q q qq qqq q qq q qq qq qqq q q qqq qq q qqqq q q q q q High q q q q qq q qqq q q q q qq q q q q 0.6 q q qq q qq qq q q qqqq qqqq qqqqqqq q q q q qq q q qqq q q q q qq qq qq qqq q q qqqqq qq q q q q q q qqq q q q q 0.6 qqqqqqqq q Med Med Med Med qq q Obs Obs q Obs Obs q q q q qq qqq qq qqqqqqqqq qq q q q qq q q qqqq qqqqq q qqqq q q qq q q Previous State qq q qq q qq q qqq q qq q qq q q q q qq qqq qq q qq q q q q q qq State 0.4 q qqq q q q q qq qq q q qqqqq q q qq qq q q q q qqqqq qq q qq q qq q q q qq qq qqqqqq q q q qqqq q qqqqqq q qq qq qq q qq qq q qq qq q q q q qq q qqqqq q q qq q q qqq qq q qq q qq q q qq qq q q q q qq q q q q qq q q qq q q qq q q q qqq q q q qqq q q q q q qq qq qq q qq q qq qqq q qqqq qq q q q qqq qq q q qqqq qq q q 0.4 qqqqqqqqq q q q q qqq qqqqqqqqq q q qq qqqq q q q q q q qq q q qq q qqq qq q q q qqqqqqqqq qqq qq q q q qqqq qqqqqqqq q q q q q q q qqq qqqqqqqqq q qq q q q q Transition 0.2 q q qqqqqqq qq q q qq q qqqq q q q qqqqq qq q qq q q q q q q q q qq qq qqq qqqqq q qqqq q q q qqqqq q q qq qqq qqqq qqq qq q qq qq q qqqqqqqqq qqq q q q qq qq q q q q q qq q q q q q qq qqq qqqqqq qq q qq q q qq q qqq q qq q qq qq q qq q q q q qq qq q qq qq qq q q q qq qq q qq q q q q qqq qqq q qq q q qqq q q q qq q qq qqq qq q qqq q q q q qqq q qqqqqqqqq q q qq q q qqq qqqq q qqqq q q q q q 0.2 q qqqqqqqqqq q q q q q qqqq qqqq q q qqqq qqq qqqqqq q qq q qqqqq qq qq q qq q q q q qq q qqq qq qq q qqqqqqqq qqqqqqqqqqqqqqqqqq qqqqq q q q q q q qqq q qqqqqqqqqqqqq qq qqq q qq q q q qq q q qq qq q qq q q q qqq q qqqqq q qqqq q qq q qqqq q q qq qq q q q qq qqqqqq qqqqqqqqq qqq q q qq q q qq qq qqqqqq qq qq qqqqqqqqqqqqq qqqqqqqqqqqqq qqqqqqqqqqqqq q qqqqqqq qqqqqq q q q q q q q qqqqqqqqqq qqq qqq q qqqqq qqqqq qqqqq qqq qq q q qqqq qqqqq qq qq q qq q q q q qqq qqqq q qq q q qq qq q qq State Definition 0.0 q qq qqqqqqq q qqq q qq qq qq qq q qqqqqqqqqqqqqq qqqqqqqqqqqqq qq qq q qq q q q q qq q q qq 0.0 State Transitions 1.0 q qqq q qq q qq q qq q qq qqq qq q q q q qqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq q qqqqqqqqqqqqqq q qqqqqqqqqqqqq q qqqqqqqqqqq 1.0 qq q q q q Dim Dim Dim Dim Data 0.8 q q q q q q q q qq 0.8 Probability q q q qq q q q q 0.6 0.6 dry Med Med Med Med Obs Obs Obs Obs qq q qq q q q A Simple 0.4 q 0.4 q q qq qq q q q Model 0.2 q q qq qq q q qq q q q q q q 0.2 q Model 0.0 qqqq qqqqqqqqqqqqqqqqqq qq qqqqqqqqqqqqqqqqq qq qqqqqqqqqqqqqq q qqqqqqqqqqqqq q qqqqqqqqqqq q qqq q q qqqqqqqqqqqqqqqqq q qq q qq q qq q qq qqqqqqqqqqqqqqqq q qqqqqqqqqqqqqq q qqqqqqqqqqqqq q qqqqqqqqqqq qqqqq qqqqq qqqqq qqqq qqqq q q qqqqqq qqqqqqqqqqqqqqqqqq qqqqq qqqqq qqqq qqqq qqqqqqqqqqqqqqqqq qq qqqqqqqqqqqqqq q qqqqqqqqqqqqq q qqqqqqqqqqq 0.0 WinBUGS code 1.0 1.0 Results Dim Dim Dim Dim 0.8 0.8 Probability Adding first 0.6 0.6 Med Med Med Med Obs Obs Obs Obs Complexity 0.4 0.4 SCC Variation 0.2 0.2 Model 0.0 0.0 WinBUGS code 0 100 300 500 0 100 300 500 0 100 300 500 0 100 300 500 Results Days in Milk Dim Days in Milk Dim Days in Milk Dim Days in Milk Dim
  • State transition Transition Between Low and High SCC Modelling State Transitions Previous State: Previous State: SCC < 200,000 cells/mL SCC > 200,000 cells/mL Aur´lien e Madouasse 1.0 1.0 Background 0.8 0.8 Milk Recording Data Probability of Transition Probability of Transition Somatic Cell Count 0.6 0.6 State Transition 0.4 0.4 State Definition State Transitions Data 0.2 0.2 A Simple Model Model 0.0 0.0 WinBUGS code 30 100 150 200 250 300 350 400 450 500 30 100 150 200 250 300 350 400 450 500 Results Days in Milk Days in Milk Adding Complexity Current State − Parity = 1: < 200,000 > 200,000 dry culled SCC Variation Current State − Parity > 1: < 200,000 > 200,000 dry culled Model WinBUGS code Results
  • Prediction of BMSCC From Individual Cows to Bulk Milk Modelling Dataset 1 State Transitions 1000 q Aur´lien e 800 q Madouasse BMSCC (/1,000 cells/mL) q q 600 qq qq q q Background q qq q q qq qq q q q q q q q q q 400 q qq qq qqq q Milk Recording q q q q q q q q q qq q qqqqq q qqq qqq qqq qqq qq qqq q qq q qq qqq q q q qqq q q q q qq qq q q q q qq q q q q q q qq q q q q qqqqqq qqqqqqq qq qqqq q qqqqqq q q q q q q q q q Data q qq q qq q q q qq qq q q q qq q qqqqq q q q qqqqqq qq qq q q q q q q q q q q q qq qq q q qq q qqq qq q q q q q qq q q qq q q q q q qq qqqqq qqqq qqqq q qq qq q qq q qq q q qqq q q qqq q qq q qq qq q qq q q qqq q qq q q qqq q q q qq qqq q qq q q q q q q qq qq q q q qqqq qq qqqqqqq q q qq q qq qq q q q qqqqqqqqqqqq q q q q q q q q q q q q q q q q q qq qq q qqqq q qqqq qqqq q q q qqqqqqqqq qqqq q q qq q q q q q qq q qq q q q qqq q q q q qq q qqqqqqqqqqq q q q qqqqqqqqq q qqqqqqqqq q q q q q qq qq q q q q q q qqqqqqqqqqqqqqq q q qq q q q q q q q q q q q q q q q q q q q q qq q qq q qqqqq qqq q q qq qqqqq qq q q q q q q qqqqqqqqqqq q qq qqqqqq qqqqqqq q q Somatic Cell 200 q q q q qq q q qq q q q q q qq q qq q q q q q q q q q q q q qq q q q q q q q q qq q qq q qqq qq q q qqq q q q q q q q q qq q q q q q q q q q q q q q q qq qq qq q q q q q q q qq q q q q qq q q q qq q qq q q qq q q q q qqqqqqqqqqqqqqqqqqqqqqqq q q q qq qq q q q qq qqqqqqqqqqqqqqqq q q qqqq qqqqqq q qqq q qqqqqqq qqq q q q qq q qqqqqqqqqq qq q qq q qq qqq qq q qq q qq q qqqq q q qqq qqq qqqq qqqqqq q qqqqqqqqqqq q qqqqq qqqqqq q qq q qqqqqq q qqqqqqqqqqqqq q q qqq q q q qq q qqqqqqqq q q q q q q q q q q qq q q q q q q q q q q qq q q q q qq qq q q qqqqqqqqqqq q q qq qq q qqqqqq qqq q q qqqqqqqqqq q q q qq q q q q q qq qqq qqqqqqqqq q qq q q q qq q q q qq q qq q q q q q q qqq q qqqq qq q q q qqq q qqqq q qqqq qqqqqqqqqqqqq q q q q q qq Count q q qq q q qq qqqqqq q q q qqqq qqq qqqqqqqq qqqqq qqqqqq qqqqqq q q qq qqqqqqq qqqqqqqqq q q qqqqqqq qqqqqqqq q qqqqqqqq q q q q q q q q qq q q qqqqq qqqq qqqq qqqqq qqq 0 State Transition State Definition State Dataset 2 Dataset 3 Transitions Data 1000 1000 A Simple 800 800 Model BMSCC (/1,000 cells/mL) Model q BMSCC (/1,000 cells/mL) 600 600 WinBUGS code q Results q q q q qq q 400 q q 400 q q q q q q q q q q q q Adding q q q q q q q q qq q q qq qq q q q qqq q qqqq qq q q q q qq qq q q q q q q qq q qq q q q q q q q q q qqq qqqq q q qq qq q qq q q q q q q qq q q q q q qq q qq q q qq q q q q q q qqq q q q q q q qq qq qq q q qq q q qqq qq qqqqqq q q qq qq qqq q q q q qqq Complexity 200 q q q q q q q q q q q q qq q q qq q q q q q qq q qqqq qqqqq q q qqqqqq q q qq q q q q q q q q q qq q q qqq q q qqqqq q q qqqq q q q q q q q q q q 200 q q q qq q q q q q qq q q q q qq q qqq q qqq q q q q qqqq qq q qq q q q qq qq qqq q q qqq q qqq qq q q q q q q q qq q q q q q qqq q q qq q q qq q q q q qq q qqqqq qq qqq q q q qqqqqq qqqq qq qqqqq qqq qqq qqq qq SCC Variation qqqq q qq Model 0 0 WinBUGS code Results
  • 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
  • Discussion Modelling State Transitions Aur´lien e Madouasse Background The model described the data well Milk Recording Data Takes a long time to run in WinBUGS (∼ 30 Somatic Cell Count seconds/iteration) State Transition Coefficients can be interpreted as odds-ratios for simple State Definition State Transitions models Data Model results must be interpreted by generating A Simple Model predictions in more complex cases Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  • Discussion Modelling State Transitions Aur´lien e Madouasse Background Milk Recording Data This type of model could be applied to a wide range of Somatic Cell Count situations in Veterinary Epidemiology State e.g. locomotion scores, SIR models . . . Transition State Definition State MCMC as implemented in WinBUGS converge slowly, Transitions Data even for simple models A Simple Model Model WinBUGS code Results Adding Complexity SCC Variation Model WinBUGS code Results
  • Acknowledgments Modelling State Transitions Aur´lien e Madouasse Prof. Martin Green Background Dr Jon Huxley Milk Recording Data Somatic Cell Dr Andrew Bradley Count School of Vetrinary Medicine and Science State University of Nottingham Transition State Definition State Transitions Data A Simple Model Prof. William Browne Model School of Clinical Veterinary Sciences WinBUGS code Results University of Bristol Adding Complexity SCC Variation Model WinBUGS code Results