Asthma: State Transitions in a Clinical Trial¶
An example from the BUGS book [56] concerning transitions between five clinical states in a randomized trial of treatments (seretide and fluticasone) for asthma.
Model¶
A discrete-time Markov model (equivalent to independent multinomial models) is fit with probability vector governing the state in the following week conditionally on the current state. Possible states are successfully treated, unsuccessfully treated, hospital-managed exacerbation, primary care-managed exacerbation, and treatment failure. The fifth state, treatment failure, is absorbing (patients cannot move out of it). The model is given by
where is the number of transitions from state to
Analysis Program¶
using Mamba
## Data
asthma = Dict{Symbol, Any}(
:y =>
[210 60 0 1 1
88 641 0 4 13
1 0 0 0 1],
:M =>
[272, 746, 2]
)
## Model Specification
model = Model(
y = Stochastic(2,
(M, q) ->
MultivariateDistribution[
Multinomial(M[i], vec(q[i, :]))
for i in 1:length(M)
],
false
),
q = Stochastic(2,
M ->
MultivariateDistribution[
Dirichlet(ones(5))
for i in 1:length(M)
],
true
)
)
## Initial Values
inits = [
Dict{Symbol, Any}(
:y => asthma[:y],
:q => vcat([rand(Dirichlet(ones(5)))' for i in 1:3]...)
)
for i in 1:3
]
## Sampling Scheme
scheme = [SliceSimplex(:q)]
setsamplers!(model, scheme)
## MCMC Simulations
sim = mcmc(model, asthma, inits, 10000, burnin=2500, thin=2, chains=3)
describe(sim)
Results¶
Iterations = 2502:10000
Thinning interval = 2
Chains = 1,2,3
Samples per chain = 3750
Empirical Posterior Estimates:
Mean SD Naive SE MCSE ESS
q[1,1] 0.7615754849 0.0272201055 0.000256633616 0.001595328676 291.12484
q[1,2] 0.2204851131 0.0265594084 0.000250404504 0.001578283076 283.18288
q[1,3] 0.0034735444 0.0037556875 0.000035408962 0.000069146577 2950.10543
q[1,4] 0.0072778962 0.0053705520 0.000050634050 0.000107382077 2501.34876
q[1,5] 0.0071879614 0.0053672180 0.000050602617 0.000112487762 2276.60625
q[2,1] 0.1191655126 0.0121038180 0.000114115890 0.000530342898 520.87229
q[2,2] 0.8543825941 0.0130973639 0.000123483131 0.000564705907 537.92673
q[2,3] 0.0012103544 0.0013675802 0.000012893670 0.000023654051 3342.67820
q[2,4] 0.0066582050 0.0030978699 0.000029206998 0.000059253193 2733.39774
q[2,5] 0.0185833339 0.0051526335 0.000048579495 0.000120299727 1834.54864
q[3,1] 0.2936564126 0.1740764923 0.001641208908 0.005020530045 1202.21215
q[3,2] 0.1394405572 0.1262073820 0.001189894609 0.002598964624 2358.13563
q[3,3] 0.1424463856 0.1308387051 0.001233559141 0.002834635296 2130.48310
q[3,4] 0.1417886606 0.1328770997 0.001252777310 0.003532010313 1415.32558
q[3,5] 0.2826679840 0.1709210331 0.001611458954 0.004864906134 1234.36071
Quantiles:
2.5% 25.0% 50.0% 75.0% 97.5%
q[1,1] 0.706775811059 0.74365894715 0.7617692483 0.7804747794 0.8125428018
q[1,2] 0.169898760444 0.20150542266 0.2199140987 0.2386805795 0.2733246918
q[1,3] 0.000059528767 0.00083424954 0.0022254753 0.0048346451 0.0138405267
q[1,4] 0.000716682148 0.00327126033 0.0060441127 0.0099507177 0.0205431907
q[1,5] 0.000740554613 0.00319616653 0.0058781978 0.0098161771 0.0209389022
q[2,1] 0.096435484007 0.11060974070 0.1187015139 0.1271183450 0.1443574095
q[2,2] 0.828193845237 0.84541382625 0.8550642143 0.8634806977 0.8792757738
q[2,3] 0.000018088918 0.00028620290 0.0007503763 0.0016511135 0.0049571810
q[2,4] 0.002033379513 0.00434074296 0.0061948558 0.0084704327 0.0138428435
q[2,5] 0.010047515668 0.01486876182 0.0180566543 0.0217580995 0.0303924987
q[3,1] 0.035105559792 0.15629556479 0.2700039867 0.4070428885 0.6826474093
q[3,2] 0.002527526439 0.03939967503 0.1024828775 0.2077906692 0.4616490169
q[3,3] 0.002612503802 0.03901018402 0.1041953555 0.2089759155 0.4719934399
q[3,4] 0.002859691566 0.03827679857 0.1016367373 0.2072463302 0.4865686565
q[3,5] 0.033209396793 0.14610521565 0.2577690743 0.3965462343 0.6524841408