Applied Bayesian Modelling - Free

50 HIERARCHICAL MIXTURE MODELSwith the total likelihood for subjects taking i cycles beingL i ˆ YC ikˆ1Program 2.3 includes this approach with the data arranged as a single string of lengthC i .Under the constant probability model (within the two groups), the posterior meansfor the conception probabilities are p 1 ˆ 0:23 and p 2 ˆ 0:33, with 95% credible intervals(0.19, 0.27) and (0.31, 0.36). The predicted cycle distribution under this model, denotednhat[,] in Program 2.3, under-predicts both short and long cycles to conception frequencies(1 cycle and over 12 cycles) ± see Table 2.4. This brings into doubt the constantprobability model.For the heterogeneous case, with a mixture of conception probabilities over couples,attention is confined to couples where the female partner smokes. The data are thendefined as a vector of 100 couple level observations of cycles required, with 29 observationsrequiring one cycle, 16 requiring two and so on, with the couples needing over 12cycles coded to 13. As well as the artificial data device, a reparameterisation as inExample 2.2 is employed for the beta mixture parameters.A three chain run taken to 10 000 iterations shows apparent convergence from 1000iterations (the beta mixture parameters are relatively slow to converge according toGelman±Rubin criteria). The average conception probability from the beta parameters,namelyP ˆ a=(a b)is 0.29 with 95% credible interval from 0.22 to 0.37. Short and long cycles to conceptionfrequencies are much more closely reproduced under this model (Table 2.5). Thecredible interval for these frequencies includes the actual frequencies of 29 and 7,respectively. The benefits from allowing conception chances to vary over couples areevident.L ikTable 2.4Cycles to conception, predictions under constant probability modelSmokersNon-SmokersCycle Mean 2.5% 97.5% Mean 2.5% 97.5%1 22.6 18.7 26.7 161.3 149.8 172.92 17.4 15.2 19.6 107.7 103.6 111.43 13.5 12.4 14.4 71.9 71.6 72.04 10.4 10.0 10.6 48.1 46.2 49.65 8.1 7.7 8.2 32.1 29.8 34.36 6.3 5.6 6.6 21.5 19.2 23.77 4.8 4.1 5.4 14.4 12.4 16.48 3.8 3.0 4.4 9.6 8.0 11.49 2.9 2.2 3.6 6.4 5.1 7.910 2.3 1.6 2.9 4.3 3.3 5.411 1.8 1.2 2.4 2.9 2.1 3.812 1.4 0.9 1.9 1.9 1.4 2.6Over 12 4.9 2.4 8.4 3.9 2.5 5.8