#### Hepatitis: a normal hierarchical model with measurement error

This example is taken from Spiegelhalter et al (1996) (chapter in Markov Chain Monte Carlo in Practice) and concerns 106 children whose post-vaccination anti Hb titre was measured 2 or 3 times. Both measurements and times have been transformed to a log scale. One covariate y0 = log titre at baseline, is available.

The model is essentially a random effects linear growth curve

Yij~ Normal(αi + βi (tij -tbar), τ)

αi ~ Normal(αc, τα)

βi ~ Normal(βc, τβ)

where τ represents the precision (1/variance) of a normal distribution. We note the absence of a parameter representing correlation between αi and βi unlike in Gelfand et al 1990. However, see the Birats example in Volume 2 which does explicitly model the covariance between αi and βi.

αc , τα , βc , τβ , τ are given independent ``noninformative'' priors.

Graphical model for hep example:

BUGS language for hep example:
model
{
for( i in 1 : N ) {
for( j in 1 : T ) {
Y[i , j] ~ dnorm(mu[i , j],tau)
mu[i , j] <- alpha[i] + beta[i] * (t[i,j] - 6.5) +
gamma * (y0[i] - mean(y0[]))
}
alpha[i] ~ dnorm(alpha0,tau.alpha)
beta[i] ~ dnorm(beta0,tau.beta)
}
tau ~ dgamma(0.001,0.001)
sigma <- 1 / sqrt(tau)
alpha0 ~ dnorm(0.0,1.0E-6)
tau.alpha ~ dgamma(0.001,0.001)
beta0 ~ dnorm(0.0,1.0E-6)
tau.beta ~ dgamma(0.001,0.001)
gamma ~ dnorm(0.0,1.0E-6)
}

Note the use of a very flat but conjugate prior for the population effects: a locally uniform prior could also have been used.

##### Data
``` list( N = 106, T = 3,   Y = structure(.Data =c(4.997 , 8.028 , NA, 6.830 , 4.905 , 6.295,   3.95124371858143, 4.35670882668959,    NA, 6.79794041297493, 5.27299955856375, 4.31748811353631,    4.71849887129509, 3.49650756146648, 4.29045944114839, 5.78996017089725,    3.68887945411394, 3.43398720448515, 6.13122648948314, 3.85014760171006,    NA, 6.76272950693188, 3.29583686600433, NA, 6.14846829591765,    5.25749537202778, 5.37527840768417, 10.7579879834538, 10.4925789214972,    NA, 5.63121178182137, 4.20469261939097, 3.49650756146648,    3.46573590279973, 1.79175946922805, 2.77258872223978, 7.92479591395644,    7.54750168281497, 6.40687998606931, 9.16115012778541, 8.34236350038058,    7.9483852851119, 8.74257423767064, 7.07580886397839, 6.3919171133926,    5.91079664404053, 5.11198778835654, 5.37989735354046, 6.54678541076052,    4.93447393313069, 5.86929691313377, 2.94443897916644, 3.17805383034795,    NA, 7.05272104923232, 6.25382881157547, 5.34233425196481, 0,    7.19067603433221, NA, 9.13905916997122, 7.82724090175281, NA,    8.48260174664662, 7.51697722460432, 4.45434729625351, 7.11963563801764,    5.8348107370626, 6.5424719605068, 7.64300363556072, 6.35610766069589,    6.89365635460264, 9.42294862137501, 8.57073395834427, 8.79026911147866,    8.2553088117856, 7.16394668434255, 7.46278915741245, 4.82028156560504,    3.40119738166216, 3.66356164612965, 6.54678541076052, 5.95583736946483,    4.99721227376411, 8.67470962929122, 7.32383056620232, 8.03040956213048,    7.86326672400957, 7.63385355968177, 7.16006920759613, 4.67282883446191,    3.36729582998647, 3.17805383034795, 7.41878088275079, 6.76157276880406,    NA, 6.84587987526405, 8.38845031552351, 6.31535800152233,    5.46383180502561, 9.9219166880045, NA, 3.43398720448515,    1.6094379124341, 2.77258872223978, 7.7553388128465, 7.44366368311559,    7.82803803212583, 9.04239498112674, 6.64768837356333, 7.11151211649616,    8.45212119467252, 6.85118492749374, 7.01481435127554, 7.91425227874244,    6.16961073249146, 6.00881318544259, 7.72885582385254, 9.75214127004149,    9.59703032475801, 9.31433996199467, 8.26359043261732, NA,    6.35957386867238, 5.84643877505772, NA, 2.30258509299405, 0,    2.30258509299405, 8.29354951506035, 7.60589000105312, 10.1017644761202,    8.90082160491523, 6.25958146406492, 7.06561336359772, 8.87905466204227,    8.88391747120797, NA, 11.5129154649202, 8.54286093816481,    8.97575663051942, 8.79618763547045, 7.86095636487639, 7.08757370555797,    8.40155784781731, 6.91075078796194, 7.22256601882217, 8.89370997756851,    7.59588991771854, NA, 7.1800698743028, 7.03174125876313,    2.07944154167984, 4.82028156560504, 2.99573227355399, 2.70805020110221,    5.77765232322266, 3.91202300542815, 3.61091791264422, 6.4707995037826,    5.4510384535657, NA, 4.40671924726425, 5.12989871492307,    3.85014760171006, 8.17413934342947, 7.88419993367604, 7.22766249872866,    2.30258509299405, 2.30258509299405, 0, 9.77104128523582,    9.32892308780313, 9.61767040669386, 6.95081476844258, 10.5472081164337,    8.66905554072548, 5.97635090929793, 5.71042701737487, 5.24174701505964,    5.8805329864007, 6.50128967054039, 5.48479693349065, 5.73979291217923,    5.53338948872752, 4.04305126783455, 5.75574221358691, 4.99043258677874,    NA, 5.15329159449778, 3.61091791264422, NA, 8.11761074646623,    7.34729970074316, 6.55250788703459, 8.54636356871602, 8.03138533062553,    7.35115822643069, 7.58171964012531, 6.92264389147589, 6.52502965784346,    6.9037472575846, 6.7202201551353, 5.74939298590825, 4.66343909411207,    3.43398720448515, 3.09104245335832, 7.47986413116503, 6.19236248947487,    NA, 3.61091791264422, 3.78418963391826, NA, 4.64439089914137,    4.51085950651685, 2.99573227355399, 4.39444915467244, 3.55534806148941,    3.29583686600433, 8.23509549725836, 7.81116338502528, 7.54855597916987,    6.12905021006055, 6.19847871649231, 4.81218435537242, 8.4424696452203,    3.55534806148941, 2.94443897916644, 6.88755257166462, 5.07517381523383,    NA, 7.29233717617388, 7.25417784645652, NA, 6.38856140554563,    6.21460809842219, 5.03043792139244, 8.75998249497728, 7.19593722647557,    NA, 6.08677472691231, 4.66343909411207, 2.39789527279837,    4.02535169073515, 3.78418963391826, 2.94443897916644, 4.38202663467388,    2.484906649788, NA, 6.85540879860993, 6.08221891037645,    5.45958551414416, 8.17919979842309, 6.04500531403601, NA,    4.24849524204936, 3.52636052461616, 3.58351893845611, 8.98343977178426,    9.69990150043745, 8.74097653801779, 8.4721958254855, 7.76004068088038,    NA, 9.480214825778, 7.78113850984502, NA, 6.1463292576689,    6.289715570909, 5.87493073085203, 9.93542217106647, 9.39806397805913,    9.17294998275762, 4.27666611901606, 5.34710753071747, 3.46573590279973,    5.36597601502185, 5.88887795833288, 3.36729582998647, 4.60517018598809,    5.29831736654804, 3.58351893845611, 3.2188758248682, 4.27666611901606,    7.63867982387611, 6.81783057145415, 3.46573590279973, NA,    6.58479139238572, 7.2152399787301, 5.90808293816893, 5.91889385427315,    5.68357976733868, 5.8171111599632, 8.67641669696422, 7.72179177681754,    6.19031540585315, 2.484906649788, 1.38629436111989, 2.56494935746154,    6.73933662735717, 6.73221070646721, 4.51085950651685, 3.91202300542815,    5.4971682252932, 4.39444915467244, 4.56434819146784, 2.83321334405622,    NA, 9.15925758174687, 8.58073121222023, NA, 7.07326971745971,    7.12608727329912, 5.01063529409626, 9.08636319215647, 7.9359451033537,    NA),   .Dim = c(106,3)),   t =    structure(.Data =   c(6.541 , 6.963 , 10,    5.8406416573734, 6.52941883826223, 6.98193467715639, 6.60123011872888,    7.02553831463852, 10, 5.8636311755981, 6.51767127291227,    6.96790920180188, 5.91350300563827, 6.5694814204143, 7.00850518208228,    5.91889385427315, 6.5694814204143, 7.00760061395185, 5.87493073085203,    6.97447891102505, 10, 5.82008293035236, 6.56244409369372, 10,    5.91350300563827, 6.5510803350434, 7.00306545878646, 5.98896141688986,    7.0335064842877, 10, 5.95064255258773, 6.59987049921284,    7.00940893270864, 5.75574221358691, 6.65544035036765, 6.9555926083963,    5.92157841964382, 6.70686233660275, 6.98286275146894, 5.93489419561959,    6.63987583382654, 6.9782137426307, 5.91350300563827, 6.63987583382654,    6.9782137426307, 5.97126183979046, 6.55961523749324, 7.00124562206948,    5.97126183979046, 6.60123011872888, 7.00033446027523, 6.59987049921284,    7.00215595440362, 10, 5.92157841964382, 6.68959926917897,    6.96224346426621, 6.71780469502369, 6.98286275146894, 10,    5.92958914338989, 6.56807791141198, 10, 5.92157841964382,    6.71780469502369, 6.98286275146894, 5.92157841964382, 6.54821910276237,    6.98193467715639, 5.92157841964382, 6.54821910276237, 6.98193467715639,    5.92157841964382, 6.54821910276237, 6.98100574072173, 5.8406416573734,    6.54821910276237, 6.98100574072173, 5.89440283426485, 6.49072353450251,    6.95654544315157, 5.93489419561959, 6.70686233660275, 7.00306545878646,    5.93224518744801, 6.56103066589657, 7.0335064842877, 5.96870755998537,    6.52795791762255, 6.94215670569947, 5.8805329864007, 6.56244409369372,    6.97728134163075, 6.62671774924902, 6.94889722231331, 10,    5.87773578177964, 6.54965074223381, 6.98841318199959, 6.59030104819669,    7.00669522683704, 10, 5.92958914338989, 6.56807791141198,    6.98193467715639, 5.92958914338989, 6.56807791141198, 6.98193467715639,    5.86078622346587, 6.55819780281227, 6.97447891102505, 5.89440283426485,    6.59030104819669, 6.98378996525813, 5.95064255258773, 6.51767127291227,    6.97447891102505, 5.93224518744801, 6.70196036600254, 6.97447891102505,    5.97126183979046, 6.60934924316738, 10, 6.57088296233958,    7.01571242048723, 10, 5.93753620508243, 6.5206211275587,    6.97073007814353, 5.87493073085203, 6.56103066589657, 6.97447891102505,    5.89164421182577, 6.59030104819669, 6.98378996525813, 6.74170069465205,    6.98933526597456, 10, 5.89440283426485, 6.5410299991899,    6.98286275146894, 5.94803498918065, 6.5510803350434, 6.99668148817654,    5.77455154554441, 6.49828214947643, 6.95844839329766, 5.97126183979046,    6.54965074223381, 10, 5.96870755998537, 6.58892647753352,    7.01481435127554, 5.8805329864007, 6.59030104819669, 6.98378996525813,    5.89440283426485, 6.5410299991899, 6.98286275146894, 6.62936325343745,    6.98471632011826, 10, 5.93224518744801, 6.62007320653036,    7.03174125876313, 5.99146454710798, 6.63856778916652, 7.04053639021596,    5.88887795833288, 6.59441345974978, 7.02908756414966, 5.93489419561959,    6.61873898351722, 7.04577657687951, 5.94017125272043, 6.62140565176413,    7.05272104923232, 5.95583736946483, 6.62936325343745, 7.05272104923232,    5.93489419561959, 6.61873898351722, 7.04490511712937, 5.92692602597041,    6.61204103483309, 7.04141166379481, 6.64639051484773, 7.04403289727469,    10, 5.93224518744801, 6.62007320653036, 10, 5.96870755998537,    6.64639051484773, 7.03174125876313, 5.93224518744801, 6.64768837356333,    7.028201432058, 5.93224518744801, 6.62007320653036, 7.03174125876313,    5.93224518744801, 6.62671774924902, 7.02731451403978, 5.94542060860658,    6.62140565176413, 7.05272104923232, 5.86929691313377, 6.5875500148248,    10, 6.65801104587075, 7.05875815251866, 10, 5.89715386763674,    6.59714570188665, 7.03174125876313, 6.00881318544259, 6.62936325343745,    7.03966034986208, 5.97126183979046, 6.66568371778241, 7.03878354138854,    5.97126183979046, 6.66440902035041, 7.04403289727469, 5.97126183979046,    6.66440902035041, 7.04403289727469, 6.00881318544259, 7.05185562295589,    10, 6.64639051484773, 7.04403289727469, 10, 5.88610403145016,    6.59167373200866, 7.028201432058, 5.75257263882563, 6.96129604591017,    10, 5.68017260901707, 6.50876913697168, 6.97541392745595,    5.77144112313002, 6.56244409369372, 6.99759598298193, 5.96100533962327,    6.66695679242921, 10, 5.75257263882563, 6.49828214947643,    6.99025650049388, 5.81413053182507, 7.02108396428914, 10,    5.63478960316925, 6.46769872610435, 6.95081476844258, 5.77455154554441,    6.48768401848461, 6.92067150424868, 5.77765232322266, 6.52209279817015,    10, 5.63121178182137, 6.55535689181067, 10, 5.70711026474888,    6.45833828334479, 6.91473089271856, 5.89164421182577, 6.48920493132532,    6.92165818415113, 5.77455154554441, 6.48920493132532, 6.92165818415113,    6.0591231955818, 6.62671774924902, 7.01391547481053, 5.8171111599632,    6.49223983502047, 6.94119005506837, 5.65248918026865, 6.42162226780652,    6.86380339145295, 5.60947179518496, 6.44413125670044, 10,    5.83773044716594, 6.48616078894409, 6.92165818415113, 5.91350300563827,    6.52649485957079, 6.97447891102505, 5.77455154554441, 6.45047042214418,    6.92755790627832, 5.83773044716594, 6.48463523563525, 6.91869521902047,    5.98645200528444, 6.51025834052315, 6.96413561241824, 5.8171111599632,    6.48463523563525, 6.94793706861497, 6.49526555593701, 6.94215670569947,    10, 5.68697535633982, 6.47850964220857, 10, 5.8171111599632,    6.51174532964473, 6.93634273583405, 5.85793315448346, 6.92264389147589,    10),    .Dim = c(106,3)),   y0 = c(8.613 , 7.105 , 6.896 ,    5.63835466933375, 6.35088571671474, 5.03043792139244, 5.8664680569333,    5.73979291217923, 7.07834157955767, 8.56655462095396, 6.86171134048073,    6.97354301952014, 7.34213173058472, 8.82629423124132, 6.48768401848461,    6.66440902035041, 7.0343879299155, 4.49980967033027, 6.85118492749374,    3.04452243772342, 8.63905677917308, 1.79175946922805, 7.01660968389422,    7.09090982207998, 7.42356844425917, 8.09620827165004, 5.70711026474888,    7.01031186730723, 7.18311170174328, 8.15219801586179, 5.99645208861902,    7.0825485693553, 5.06890420222023, 7.04228617193974, 4.0943445622221,    7.32052696227274, 7.12689080889881, 7.24136628332232, 6.62007320653036,    7.36897040219479, 8.47720418319987, 7.20340552108309, 2.89037175789616,    7.39141523467536, 6.25190388316589, 8.38160253710989, 11.5129154649202,    6.60258789218934, 7.61283103040736, 7.42476176182321, 6.71174039505618,    5.56068163101553, 6.0282785202307, 7.10085190894405, 7.52671756135271,    7.92948652331429, 1.6094379124341, 6.85856503479136, 8.05006542291597,    5.83188247728352, 7.15851399732932, 7.01481435127554, 7.70120018085745,    7.16239749735572, 5.99893656194668, 5.41610040220442, 8.40782465436087,    7.18462915271731, 3.25809653802148, 8.14380797677148, 5.24702407216049,    5.73979291217923, 4.55387689160054, 7.72444664563354, 7.91425227874244,    7.11476944836646, 8.24064886337491, 8.56044423341055, 6.78558764500793,    10.8861838116069, 2.07944154167984, 5.82894561761021, 4.0943445622221,    7.53208814354172, 5.66988092298052, 3.46573590279973, 10.7229148990064,    8.02910705461974, 9.23092700558457, 6.42971947803914, 11.5129154649202,    7.25700270709207, 6.62007320653036, 9.41995278900604, 5.28826703069454,    8.04654935728308, 7.58984151218266, 6.39024066706535, 6.04025471127741,    2.77258872223978, 7.62948991639399, 7.85476918349913, 7.09007683577609,    5.50125821054473, 6.98100574072173, 5.50533153593236)) ```
##### Inits for chain 1
``` list(alpha0 = 4, beta0 = 0, gamma = 0, tau.alpha = 1, tau.beta = 1, tau = 1 ) ```

##### Inits for chain 2
``` list(alpha0 = 1, beta0 = 1, gamma = 1, tau.alpha = 0.1, tau.beta = 1, tau = 0.1 ) ```

Results

With measurement error

model
{
tau.alpha ~ dgamma(0.001,0.001)
alpha0 ~ dnorm( 0.0,1.0E-6)
beta0 ~ dnorm( 0.0,1.0E-6)
tau.beta ~ dgamma(0.001,0.001)
for( i in 1 : N ) {
alpha[i] ~ dnorm(alpha0,tau.alpha)
beta[i] ~ dnorm(beta0,tau.beta)
y0[i] ~ dnorm(mu0[i],tau)
mu0[i] ~ dnorm(theta,psi)
}
for( j in 1 : T ) {
for( i in 1 : N ) {
Y[i , j] ~ dnorm(mu[i , j],tau)
mu[i , j] <- alpha[i] + beta[i] * (t[i , j] - 6.5) +
gamma * (mu0[i] - mean(y0[]))
}
}
tau ~ dgamma(0.001,0.001)
sigma <- 1 / sqrt(tau)
gamma ~ dnorm( 0.0,1.0E-6)
theta ~ dnorm( 0.0,1.0E-6)
psi ~ dgamma(0.001,0.001)
}
##### Data
``` list( N = 106, T = 3,   Y = structure(.Data =c(4.997 , 8.028 , NA, 6.830 , 4.905 , 6.295,   3.95124371858143, 4.35670882668959,    NA, 6.79794041297493, 5.27299955856375, 4.31748811353631,    4.71849887129509, 3.49650756146648, 4.29045944114839, 5.78996017089725,    3.68887945411394, 3.43398720448515, 6.13122648948314, 3.85014760171006,    NA, 6.76272950693188, 3.29583686600433, NA, 6.14846829591765,    5.25749537202778, 5.37527840768417, 10.7579879834538, 10.4925789214972,    NA, 5.63121178182137, 4.20469261939097, 3.49650756146648,    3.46573590279973, 1.79175946922805, 2.77258872223978, 7.92479591395644,    7.54750168281497, 6.40687998606931, 9.16115012778541, 8.34236350038058,    7.9483852851119, 8.74257423767064, 7.07580886397839, 6.3919171133926,    5.91079664404053, 5.11198778835654, 5.37989735354046, 6.54678541076052,    4.93447393313069, 5.86929691313377, 2.94443897916644, 3.17805383034795,    NA, 7.05272104923232, 6.25382881157547, 5.34233425196481, 0,    7.19067603433221, NA, 9.13905916997122, 7.82724090175281, NA,    8.48260174664662, 7.51697722460432, 4.45434729625351, 7.11963563801764,    5.8348107370626, 6.5424719605068, 7.64300363556072, 6.35610766069589,    6.89365635460264, 9.42294862137501, 8.57073395834427, 8.79026911147866,    8.2553088117856, 7.16394668434255, 7.46278915741245, 4.82028156560504,    3.40119738166216, 3.66356164612965, 6.54678541076052, 5.95583736946483,    4.99721227376411, 8.67470962929122, 7.32383056620232, 8.03040956213048,    7.86326672400957, 7.63385355968177, 7.16006920759613, 4.67282883446191,    3.36729582998647, 3.17805383034795, 7.41878088275079, 6.76157276880406,    NA, 6.84587987526405, 8.38845031552351, 6.31535800152233,    5.46383180502561, 9.9219166880045, NA, 3.43398720448515,    1.6094379124341, 2.77258872223978, 7.7553388128465, 7.44366368311559,    7.82803803212583, 9.04239498112674, 6.64768837356333, 7.11151211649616,    8.45212119467252, 6.85118492749374, 7.01481435127554, 7.91425227874244,    6.16961073249146, 6.00881318544259, 7.72885582385254, 9.75214127004149,    9.59703032475801, 9.31433996199467, 8.26359043261732, NA,    6.35957386867238, 5.84643877505772, NA, 2.30258509299405, 0,    2.30258509299405, 8.29354951506035, 7.60589000105312, 10.1017644761202,    8.90082160491523, 6.25958146406492, 7.06561336359772, 8.87905466204227,    8.88391747120797, NA, 11.5129154649202, 8.54286093816481,    8.97575663051942, 8.79618763547045, 7.86095636487639, 7.08757370555797,    8.40155784781731, 6.91075078796194, 7.22256601882217, 8.89370997756851,    7.59588991771854, NA, 7.1800698743028, 7.03174125876313,    2.07944154167984, 4.82028156560504, 2.99573227355399, 2.70805020110221,    5.77765232322266, 3.91202300542815, 3.61091791264422, 6.4707995037826,    5.4510384535657, NA, 4.40671924726425, 5.12989871492307,    3.85014760171006, 8.17413934342947, 7.88419993367604, 7.22766249872866,    2.30258509299405, 2.30258509299405, 0, 9.77104128523582,    9.32892308780313, 9.61767040669386, 6.95081476844258, 10.5472081164337,    8.66905554072548, 5.97635090929793, 5.71042701737487, 5.24174701505964,    5.8805329864007, 6.50128967054039, 5.48479693349065, 5.73979291217923,    5.53338948872752, 4.04305126783455, 5.75574221358691, 4.99043258677874,    NA, 5.15329159449778, 3.61091791264422, NA, 8.11761074646623,    7.34729970074316, 6.55250788703459, 8.54636356871602, 8.03138533062553,    7.35115822643069, 7.58171964012531, 6.92264389147589, 6.52502965784346,    6.9037472575846, 6.7202201551353, 5.74939298590825, 4.66343909411207,    3.43398720448515, 3.09104245335832, 7.47986413116503, 6.19236248947487,    NA, 3.61091791264422, 3.78418963391826, NA, 4.64439089914137,    4.51085950651685, 2.99573227355399, 4.39444915467244, 3.55534806148941,    3.29583686600433, 8.23509549725836, 7.81116338502528, 7.54855597916987,    6.12905021006055, 6.19847871649231, 4.81218435537242, 8.4424696452203,    3.55534806148941, 2.94443897916644, 6.88755257166462, 5.07517381523383,    NA, 7.29233717617388, 7.25417784645652, NA, 6.38856140554563,    6.21460809842219, 5.03043792139244, 8.75998249497728, 7.19593722647557,    NA, 6.08677472691231, 4.66343909411207, 2.39789527279837,    4.02535169073515, 3.78418963391826, 2.94443897916644, 4.38202663467388,    2.484906649788, NA, 6.85540879860993, 6.08221891037645,    5.45958551414416, 8.17919979842309, 6.04500531403601, NA,    4.24849524204936, 3.52636052461616, 3.58351893845611, 8.98343977178426,    9.69990150043745, 8.74097653801779, 8.4721958254855, 7.76004068088038,    NA, 9.480214825778, 7.78113850984502, NA, 6.1463292576689,    6.289715570909, 5.87493073085203, 9.93542217106647, 9.39806397805913,    9.17294998275762, 4.27666611901606, 5.34710753071747, 3.46573590279973,    5.36597601502185, 5.88887795833288, 3.36729582998647, 4.60517018598809,    5.29831736654804, 3.58351893845611, 3.2188758248682, 4.27666611901606,    7.63867982387611, 6.81783057145415, 3.46573590279973, NA,    6.58479139238572, 7.2152399787301, 5.90808293816893, 5.91889385427315,    5.68357976733868, 5.8171111599632, 8.67641669696422, 7.72179177681754,    6.19031540585315, 2.484906649788, 1.38629436111989, 2.56494935746154,    6.73933662735717, 6.73221070646721, 4.51085950651685, 3.91202300542815,    5.4971682252932, 4.39444915467244, 4.56434819146784, 2.83321334405622,    NA, 9.15925758174687, 8.58073121222023, NA, 7.07326971745971,    7.12608727329912, 5.01063529409626, 9.08636319215647, 7.9359451033537,    NA),   .Dim = c(106,3)),   t =    structure(.Data =   c(6.541 , 6.963 , 10,    5.8406416573734, 6.52941883826223, 6.98193467715639, 6.60123011872888,    7.02553831463852, 10, 5.8636311755981, 6.51767127291227,    6.96790920180188, 5.91350300563827, 6.5694814204143, 7.00850518208228,    5.91889385427315, 6.5694814204143, 7.00760061395185, 5.87493073085203,    6.97447891102505, 10, 5.82008293035236, 6.56244409369372, 10,    5.91350300563827, 6.5510803350434, 7.00306545878646, 5.98896141688986,    7.0335064842877, 10, 5.95064255258773, 6.59987049921284,    7.00940893270864, 5.75574221358691, 6.65544035036765, 6.9555926083963,    5.92157841964382, 6.70686233660275, 6.98286275146894, 5.93489419561959,    6.63987583382654, 6.9782137426307, 5.91350300563827, 6.63987583382654,    6.9782137426307, 5.97126183979046, 6.55961523749324, 7.00124562206948,    5.97126183979046, 6.60123011872888, 7.00033446027523, 6.59987049921284,    7.00215595440362, 10, 5.92157841964382, 6.68959926917897,    6.96224346426621, 6.71780469502369, 6.98286275146894, 10,    5.92958914338989, 6.56807791141198, 10, 5.92157841964382,    6.71780469502369, 6.98286275146894, 5.92157841964382, 6.54821910276237,    6.98193467715639, 5.92157841964382, 6.54821910276237, 6.98193467715639,    5.92157841964382, 6.54821910276237, 6.98100574072173, 5.8406416573734,    6.54821910276237, 6.98100574072173, 5.89440283426485, 6.49072353450251,    6.95654544315157, 5.93489419561959, 6.70686233660275, 7.00306545878646,    5.93224518744801, 6.56103066589657, 7.0335064842877, 5.96870755998537,    6.52795791762255, 6.94215670569947, 5.8805329864007, 6.56244409369372,    6.97728134163075, 6.62671774924902, 6.94889722231331, 10,    5.87773578177964, 6.54965074223381, 6.98841318199959, 6.59030104819669,    7.00669522683704, 10, 5.92958914338989, 6.56807791141198,    6.98193467715639, 5.92958914338989, 6.56807791141198, 6.98193467715639,    5.86078622346587, 6.55819780281227, 6.97447891102505, 5.89440283426485,    6.59030104819669, 6.98378996525813, 5.95064255258773, 6.51767127291227,    6.97447891102505, 5.93224518744801, 6.70196036600254, 6.97447891102505,    5.97126183979046, 6.60934924316738, 10, 6.57088296233958,    7.01571242048723, 10, 5.93753620508243, 6.5206211275587,    6.97073007814353, 5.87493073085203, 6.56103066589657, 6.97447891102505,    5.89164421182577, 6.59030104819669, 6.98378996525813, 6.74170069465205,    6.98933526597456, 10, 5.89440283426485, 6.5410299991899,    6.98286275146894, 5.94803498918065, 6.5510803350434, 6.99668148817654,    5.77455154554441, 6.49828214947643, 6.95844839329766, 5.97126183979046,    6.54965074223381, 10, 5.96870755998537, 6.58892647753352,    7.01481435127554, 5.8805329864007, 6.59030104819669, 6.98378996525813,    5.89440283426485, 6.5410299991899, 6.98286275146894, 6.62936325343745,    6.98471632011826, 10, 5.93224518744801, 6.62007320653036,    7.03174125876313, 5.99146454710798, 6.63856778916652, 7.04053639021596,    5.88887795833288, 6.59441345974978, 7.02908756414966, 5.93489419561959,    6.61873898351722, 7.04577657687951, 5.94017125272043, 6.62140565176413,    7.05272104923232, 5.95583736946483, 6.62936325343745, 7.05272104923232,    5.93489419561959, 6.61873898351722, 7.04490511712937, 5.92692602597041,    6.61204103483309, 7.04141166379481, 6.64639051484773, 7.04403289727469,    10, 5.93224518744801, 6.62007320653036, 10, 5.96870755998537,    6.64639051484773, 7.03174125876313, 5.93224518744801, 6.64768837356333,    7.028201432058, 5.93224518744801, 6.62007320653036, 7.03174125876313,    5.93224518744801, 6.62671774924902, 7.02731451403978, 5.94542060860658,    6.62140565176413, 7.05272104923232, 5.86929691313377, 6.5875500148248,    10, 6.65801104587075, 7.05875815251866, 10, 5.89715386763674,    6.59714570188665, 7.03174125876313, 6.00881318544259, 6.62936325343745,    7.03966034986208, 5.97126183979046, 6.66568371778241, 7.03878354138854,    5.97126183979046, 6.66440902035041, 7.04403289727469, 5.97126183979046,    6.66440902035041, 7.04403289727469, 6.00881318544259, 7.05185562295589,    10, 6.64639051484773, 7.04403289727469, 10, 5.88610403145016,    6.59167373200866, 7.028201432058, 5.75257263882563, 6.96129604591017,    10, 5.68017260901707, 6.50876913697168, 6.97541392745595,    5.77144112313002, 6.56244409369372, 6.99759598298193, 5.96100533962327,    6.66695679242921, 10, 5.75257263882563, 6.49828214947643,    6.99025650049388, 5.81413053182507, 7.02108396428914, 10,    5.63478960316925, 6.46769872610435, 6.95081476844258, 5.77455154554441,    6.48768401848461, 6.92067150424868, 5.77765232322266, 6.52209279817015,    10, 5.63121178182137, 6.55535689181067, 10, 5.70711026474888,    6.45833828334479, 6.91473089271856, 5.89164421182577, 6.48920493132532,    6.92165818415113, 5.77455154554441, 6.48920493132532, 6.92165818415113,    6.0591231955818, 6.62671774924902, 7.01391547481053, 5.8171111599632,    6.49223983502047, 6.94119005506837, 5.65248918026865, 6.42162226780652,    6.86380339145295, 5.60947179518496, 6.44413125670044, 10,    5.83773044716594, 6.48616078894409, 6.92165818415113, 5.91350300563827,    6.52649485957079, 6.97447891102505, 5.77455154554441, 6.45047042214418,    6.92755790627832, 5.83773044716594, 6.48463523563525, 6.91869521902047,    5.98645200528444, 6.51025834052315, 6.96413561241824, 5.8171111599632,    6.48463523563525, 6.94793706861497, 6.49526555593701, 6.94215670569947,    10, 5.68697535633982, 6.47850964220857, 10, 5.8171111599632,    6.51174532964473, 6.93634273583405, 5.85793315448346, 6.92264389147589,    10),    .Dim = c(106,3)),   y0 = c(8.613 , 7.105 , 6.896 ,    5.63835466933375, 6.35088571671474, 5.03043792139244, 5.8664680569333,    5.73979291217923, 7.07834157955767, 8.56655462095396, 6.86171134048073,    6.97354301952014, 7.34213173058472, 8.82629423124132, 6.48768401848461,    6.66440902035041, 7.0343879299155, 4.49980967033027, 6.85118492749374,    3.04452243772342, 8.63905677917308, 1.79175946922805, 7.01660968389422,    7.09090982207998, 7.42356844425917, 8.09620827165004, 5.70711026474888,    7.01031186730723, 7.18311170174328, 8.15219801586179, 5.99645208861902,    7.0825485693553, 5.06890420222023, 7.04228617193974, 4.0943445622221,    7.32052696227274, 7.12689080889881, 7.24136628332232, 6.62007320653036,    7.36897040219479, 8.47720418319987, 7.20340552108309, 2.89037175789616,    7.39141523467536, 6.25190388316589, 8.38160253710989, 11.5129154649202,    6.60258789218934, 7.61283103040736, 7.42476176182321, 6.71174039505618,    5.56068163101553, 6.0282785202307, 7.10085190894405, 7.52671756135271,    7.92948652331429, 1.6094379124341, 6.85856503479136, 8.05006542291597,    5.83188247728352, 7.15851399732932, 7.01481435127554, 7.70120018085745,    7.16239749735572, 5.99893656194668, 5.41610040220442, 8.40782465436087,    7.18462915271731, 3.25809653802148, 8.14380797677148, 5.24702407216049,    5.73979291217923, 4.55387689160054, 7.72444664563354, 7.91425227874244,    7.11476944836646, 8.24064886337491, 8.56044423341055, 6.78558764500793,    10.8861838116069, 2.07944154167984, 5.82894561761021, 4.0943445622221,    7.53208814354172, 5.66988092298052, 3.46573590279973, 10.7229148990064,    8.02910705461974, 9.23092700558457, 6.42971947803914, 11.5129154649202,    7.25700270709207, 6.62007320653036, 9.41995278900604, 5.28826703069454,    8.04654935728308, 7.58984151218266, 6.39024066706535, 6.04025471127741,    2.77258872223978, 7.62948991639399, 7.85476918349913, 7.09007683577609,    5.50125821054473, 6.98100574072173, 5.50533153593236)) ```
##### Inits for chain 1
``` list(alpha0 = 4, beta0 = 0, gamma = 0, tau.alpha = 1, tau.beta = 1, tau = 1, theta = 6, psi = 1) ```

##### Inits for chain 2
``` list(alpha0 = 1, beta0 = 1, gamma = 0, tau.alpha = 0.1, tau.beta = 0.1, tau = 0.1, theta = 1, psi = 1) ```

Results