Dear BUGS users,
Thank your for your response to my post of May 9.
Here is a summary of the answers.
1. Several people pointed out that in the generated data (see below),
the ratio of the measurement error variability to variability of the true
response was very high. This was true.
However the problem persists even when the measurement error is very small
in comparison to the variability of the true response.
If anyone wants more data, I can supply.
2. E.G. Tsionas pointed out that
" an obvious answer to that would be that the posterior is not proper when a
flat prior is used. One would need to prove analytically that the likelihood is
integrable in that case."
3. Samuel Mwalili said
Your results may be correct since you are sampling from the naïve (the
error corrupted) model. The 'poor' results are due to the attenuation
effect of measurement error.
Reference (Carroll et al 1995, Measurement error in nonlinear models).
I'm still trying to determine whether 2 or 3 is to blame.
Thanks again,
Julie Horrocks
Quoting Julie Horrocks <[log in to unmask]>:
> Dear BUGS users,
> I wonder if anyone else has tried this:
> I am trying to fit a joint model for longitudinal and binary data.
> (as opposed to the usual joint models for longitudinal and survival data.)
> The longitudinal data are measured with error.
>
> In the generated data below, there are 100 individuals, and
> a continuous variable (Y) is measured at 8 different times (longitudinal
> data)
> while a binary variable (R) is measured once.
> The continuous measurements include true measure (U) and
> measurement error (e): Y=U+e
> The binary variable follows a logit model: logit(R) = G0 + G1*U.
>
> The model is correct, since the data are generated, and the
> true value of G1 is 1.6.
>
> When I put a non-informative prior on the parameters,
> the posterior distribution of the parameter of interest (G1)
> is so variable as to be practically useless. Here is the output
> from a not particularly vague prior on G1 (G1~dnorm(0, .02)).
>
> mean sd MC_error val2.5pc median val97.5pc start sample
> G0 -0.08904 0.3394 0.008596 -0.789 -0.08988 0.6018 10000 10001
> G1 -0.439 6.725 0.229 -13.18 -0.7294 13.5 10000 10001
>
> It gets worse the lower the precision of the prior on G1.
> Is there any obvious reason for this? Is this model just doomed,
> or have I done something silly?
>
> Thanks,
>
> Julie Horrocks
> University of Guelph
> Guelph, Ontario, Canada
>
> xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> model{
> for (i in 1:N) {
> for (j in 1:M) {
> Y[i,j] ~ dnorm(muy[i, j], tauz)
> muy[i, j]<-U1[i]
> }
> U1[i] ~ dnorm(muu,tau1)
> r[i]~dbin(p[i],1)
>
> logit(p[i])<-G0+G1*U1[i]
> }
>
> sigmaz<- sqrt(1/tauz )
>
> #priors
>
> tauz~dgamma(0.1, 0.1)
> sigma1~dunif(0,1)
> G0~dnorm(0,0.000001)
> G1~dnorm(1.6, .02)
> muu~dnorm(0,1.0E-6)
> }
>
> list(N = 100, M = 8, Y = structure(.Data=c(-0.262313843855381,
> -0.0829926192505581, 0.470176890117265, -0.899980830698257,
> 0.00657683988293917,
> 0.108985362617801, 0.94320290019508, 0.0793846298459693, 1.08080717460911,
> -0.536724926512684, -0.859525603764751, 1.03465788624157, 0.291866101942854,
> -0.78205891349431, 0.890040456790485, -1.34238948129829, -0.285357264120757,
> -0.264662266394864, 0.492223883893569, 0.0551348373728768, 1.91005928384055,
> -0.566341302158937, 1.28394586810013, 1.18665003359530, 0.178659798447664,
> -2.21284913124418, -0.0566859936326375, 0.175730258764369, 0.947640582071549,
> 0.636540067518711, -0.129009761313290, -0.783013817334896, 0.473276518546235,
> -0.51145302291504, -0.555167927968795, -0.182915323655587,
> -0.361489617986338,
> -1.10770530377032, -1.29048206702299, -0.63535156894623, -0.98214606965787,
> -0.7265414729711, -0.296085918095656, 1.17368226042249, 0.129501072338256,
> 0.956177306929572, -0.263667970562866, -0.275316829867991,
> -0.620513482010366,
> -0.281528194582710, -0.590363917465252, 0.980843770841761,
> 0.0940104283677036,
> 0.906369114937596, -0.177347963079963, -0.431977029430145,
> -0.373065444174461,
> 0.305169949036871, 1.00188092393959, 0.388498697580012, -0.292361180751258,
> 1.15719791066870, 0.0392095305692937, 1.11049908258761, 0.0995255028876407,
> -0.934415471154188, -1.05694964977726, -1.00521530978515, -0.829533959525446,
> -0.593711408561849, -0.512109510046093, 0.200011936381755, 0.322091135692809,
> 0.404908862838225, 1.05026538462093, 0.492047013847931, 0.635850081360298,
> 0.437668752092144, 0.806098859834227, -0.993813559617363, 0.101404619251168,
> -0.650808634694079, 0.825505383799838, -0.201924224081044, -2.68900641120357,
> 0.99035517739857, -0.87284572591653, 0.852254281633406, -0.711369463267391,
> -0.0383747801012487, 0.485647005777027, 0.625092768956708,
> -0.283606183962768,
> -0.629655775957537, -0.422939936152345, -1.09910611522213, 0.494294801707532,
> -0.563337583057439, -1.36314895174426, -0.0883968731937448,
> 0.506910467343079,
> -0.421265706720007, 0.401037809314724, -0.245785031403988, 0.643333171381075,
> 0.439331726510418, -0.44574468183068, -0.438884527550954, 1.44052952689377,
> 0.341187100257255, 0.393861998965560, 0.841160286439434, -0.403785224956695,
> 0.378268632598794, 0.459112333533235, -0.0264085906969822, 0.850929713880627,
> 0.0693805019055929, 0.7445268479806, -0.741234673289254, 1.92163930399897,
> 0.41794101781863, -0.607906980038515, -0.0713919488472551,
> -0.212512408844271,
> -1.51057614114355, 0.390561312742683, -0.342115465405608,
> -0.0259278231450088,
> 0.558938833080513, 0.104875051264156, -0.410332941379152, 0.699092185883475,
> -0.0702551795807514, -0.52900873140823, 0.161455026953328,
> -0.892566084871266,
> 0.102540237722249, -0.377849721082014, 0.539144069995910, -0.8048997620704,
> -1.28565381563743, -1.15158522513273, 0.916797029468736, 0.46193417987513,
> 1.79337835976295, -0.209150185037746, 0.263540933647109, -0.901632948344635,
> -0.605513576208318, -0.138283499118962, 0.77530643160563, -0.183712201072288,
> 1.11705936013196, -0.667651442991455, -1.11879175374123, -0.61670685398736,
> 0.0250341939072417, 1.51475794394235, 0.50876493113721, -0.150199045978622,
> 0.745400538648151, -0.0900685102168363, -0.328020875611095,
> 0.487493151564222,
> 0.340147982405991, -0.734918396477195, -0.061351224753963,
> -0.704731484858476,
> -0.653050067550829, -1.17681487311360, 0.83224161173267, -0.159158099197796,
> 0.949677108154374, 0.530833300388768, -0.61936723366974, 0.681583352730048,
> -0.0314961824308881, 0.521859508532552, -0.520422371048633,
> 0.815622280532758,
> -0.0792672174818522, -0.110009438892181, 0.512039585384339,
> -1.35444147779143,
> 0.0298502117823569, -0.229440932055461, 0.349372859575004,
> -0.0108339086297197,
> -0.482833524874656, 0.819548031849175, 0.825742711958275, -0.670476489182971,
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> -0.070040708297565, -0.63009932690837, -0.91093837438548,
> 0.00754053139830496,
> -0.0328349710367747, 0.761364637204696, -1.42718275436943,
> -0.311706730066955,
> -0.260626388474995, 0.597412256722584, -0.253705464097676, 1.51729326343767,
> -0.881523092789801, 0.412239895865782, 0.269496631411631, 0.237848399073548,
> -1.23693269135033, 0.950682924493945, -0.36898365219508, -0.544230485458481,
> 0.158664115599743, -0.149917612384833, -0.107388119306422,
> 0.0416931568720789,
> -0.275203506831297, 0.362883358439414, 0.292085730666601, -0.327479623914981,
> -0.988187917484935, 0.573131232595114, -0.395142240347669,
> -0.0886587434135838,
> 0.336215288596743, -0.0328902386379526, -0.0688843083940802,
> 0.261555779761879, -0.499589799841197, -0.205435428476091,
> -0.404868045840446,
> -0.778773421746447, -0.955161012078008, -0.764596807369313,
> -0.104520490211723,
> -0.625961754845854, -0.966419758720353, -0.274752608020697, 1.60273083141038,
> 0.817689227995274, 0.39344702443475, 0.264668140895748, -0.405633295101381,
> 0.311436881438525, -0.193897905857856, -1.28109971241264, 1.79498837403237,
> -0.97776528891676, 1.34874765999540, -1.17147393222698, -0.406820364528985,
> -0.0810442975927692, 0.302232745852411, 0.229150841625598, 1.42516752443191,
> -0.524690855708216, 1.39638738287062, 0.244541584097404, 0.0658594058607683,
> 0.742036311856972, -0.348209766344051, -0.990618245949897,
> -0.569035281115429,
> -0.406441756940526, -0.569878424992982, 0.253254611043749, 0.256342237747865,
> -0.97218116095324, -0.403692648067440, -1.04402164938636, -0.349183510399362,
> 0.75380552868092, -0.211401067263695, 1.02620022256753, -0.990412593867658,
> 0.472403756297723, -0.663191318542109, 0.281532355764858, 0.051925633534339,
> -1.27589963228010, 1.04114865009781, -0.841234702385585, 0.362835921947747,
> 0.689239382939566, 0.539317648568145, 1.84073941863830, 1.01951251161427,
> -0.604073658965897, -1.06231079777414, -1.35449435995121, 0.46087060896684,
> 1.49530025536671, 0.134735638888674, -0.963936823234674, 0.281354205377584,
> -0.857755222752967, 0.328354325381843, 0.53418830192534, -1.43640287729360,
> -0.0480195857652559, 0.394763133841527, -0.489850380620958,
> 0.0600907703569357,
> -0.0374227535061644, -0.584595340799979, 1.63082309174468,
> 0.0966164057879005,
> 0.0739579750134796, -0.450491403243188, 1.14154699146042, -2.13269540788465,
> 0.938480211584413, -0.219312227853993, 0.388744567437591, -0.483585846164443,
> 1.37883473420675, 0.0397128552449173, 0.237777623050931, 0.635094392799893,
> -0.304473072222319, -0.617244349534207, -0.200536559907023,
> 0.0663412805499233,
> 0.193327263445334, 0.872648235085903, 0.414117494993621, 0.746146987755067,
> 0.39998744624149, 0.516163972942518, -1.12977247337538, -1.65873833164563,
> -0.288976737018402, 0.336607679589844, -0.805130005230272,
> -0.161927654964284,
> -0.0733530165122028, -1.10610315112596, -0.0292695293369888,
> 0.272110377669371, 1.20149603175768, 0.164324635606736, -0.180374827529047,
> -0.337178089117294, -0.417701447165887, -0.524349267297216,
> -0.478780936814164,
> 0.870211928501177, -0.80892766060025, 0.554727232356038, -0.734099773099188,
> -0.622771711189445, 0.552867441906036, 0.462670317019126, 0.658403944661937,
> 1.31156763902789, -0.507450833011692, 1.07489464697887, 0.37024754287166,
> 0.508679652766496, 0.978013944933652, -0.590053126139065, -1.67088691957439,
> 0.917201629575048, -0.0300033921582482, -0.375324670208786,
> -1.10041243900434,
> 0.117186607668980, -0.219726850121263, -0.133903931883344,
> -0.0903486546895271,
> 0.413685778181764, 1.15733315970167, 0.0247986146635587, 0.643327005619584,
> -0.473147776668208, 0.191741165582149, -1.41169313375439, -0.498355189067551,
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> 0.961715826859577,
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> -0.528078129085878,
> -0.698358898262289, 1.07920271591172, -0.880558835381087, 1.64258455397624,
> 0.00371727830975332, -0.385540297057388, -1.19231334907844,
> 0.716028074656116,
> -0.0910344276139589, 1.24885973485630, -1.46832608885572, 1.19163120656298,
> -1.35329445417639, -0.680412488383345, -0.326213533793055,
> -0.900144656852019,
> -1.09423395178476, -0.598726949275079, -0.504710937186593, -0.67355826707978,
> -0.665019103743323, -0.96427474914904, -0.461447371504625, 0.84911364886565,
> 0.199144932989593, -0.635453756438417, 0.772347983390234, 0.556890757763254,
> 0.816073908123003, -0.745507810095904, 1.05361437540234, -1.04294237070362,
> -0.139644975953926, -0.605781103668443, -0.601441464666243, 2.00272834917963,
> -0.661204669848463, -0.222977281776556, -0.0522990223040301,
> 0.951540526818452, 0.589081260327396, 1.14054439435737, -0.695887290174514,
> -0.270708914210209, -0.711380268098953, -0.110722774618135, 1.04628130728235,
> -0.511462995356823, -0.167334743891058, -0.276346895423433,
> -1.23586964091865,
> 0.707901423833829, 0.610436190650904, -1.57331062822986, 0.179985533535713,
> 0.554155461403677, 0.29132915762825, -1.16946182818641, 1.11131317861817,
> 0.202678735944249, 0.350389496212382, -0.0188997098773617, 0.220723398659665,
> -1.08964837647398, 0.34511793140342, -1.68226294199260, 0.42372307489144,
> -0.0707894361041884, -0.299314150859737, 0.19778694080626, 1.04686501579577,
> -0.0135981310993740, 1.09097945900444, 0.0346151682731844,
> -0.197802459374791,
> -1.47514497226055, 0.407207984279559, -0.309004451018228, -0.36725925131416,
> 1.30945036479418, 0.765328207381141, 1.30184900895539, 1.62183617798371,
> 0.564485540193846, -0.892685774787704, -0.41708531390867, -1.82656464269386,
> 0.254683325636118, -0.304363230935357, -0.349802115911927, -0.91133574351325,
> -1.23231058438745, -0.308931641851249, 0.298146719690391, 0.430237494557323,
> 1.42949014693376, 0.93109091891645, 0.56110944691244, -0.268290089653620,
> -0.651203391833492, -0.299477176380969, 1.00123461378704, -0.132486843148446,
> 1.37848452519519, -0.431433999132537, -1.55023463925404, -1.36465421332819,
> 0.0339502755094938, 0.397492874858985, 0.383095985651148,
> 0.00568678112217202,
> 0.617176121879459, 0.150103498407917, -0.351107961069875, 0.215606643314364,
> -0.852552887011086, -0.892052149513662, -0.195285146867997,
> -0.46025895465704,
> -0.375836754636786, 0.153241148807489, 0.754634996062549, -0.322830422102971,
> -1.12570002174003, -0.371595866755367, -0.675509053412837, 0.97407630412936,
> 0.199279992105377, 0.145756186739396, -1.45985395704062, 0.634084076326643,
> -0.839175345416791, -0.811343569105302, -1.53394510038748, -1.06507227404272,
> 1.08274843159052, 0.307840840323339, -0.135123087179747, -0.315506811176714,
> -0.292291579345138, 0.601793929269907, 1.68835971398503, -0.418231348707417,
> -0.527953059264142, -1.34171700632720, 0.877855944027478, 2.07116931866773,
> 0.125286188715722, 0.821505571498239, -0.620181753764083), .Dim = c(100,
> 8)), r = c(1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1,
> 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1,
> 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1,
> 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0,
> 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0))
>
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