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Examples; Asthma: State Transitions in a Clinical Trial

Asthma: State Transitions in a Clinical Trial

An example from the BUGS book lunn:2012:BUGS 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 qi 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

$$\begin{aligned} y_{ij} &\sim \text{Multinomial}(M_i, (q_{i1}, \ldots, q_{i5})) \quad\quad i=1, \ldots, 3 \\\ \bm{q}_i &\sim \text{Dirichlet}(1, \ldots, 1) \end{aligned}$$

where yij is the number of transitions from state i to j

Analysis Program

asthma.jl

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