Results - Noninformative Priors Sample Clauses

Results - Noninformative Priors. The relative influence of the prior and data on the posterior belief depends on how much weight is given to the prior and the strength of the data. In general, if the sample was small with an informative prior, then the prior distribution would have a relatively greater influence on the posterior belief about the parameter of interest. However, a large data sample would tend to have a predominant impact on the posterior belief on the parameter of interest unless the prior was informative [Xxxxxxx, 2006a]. Since the sample size is small (n = 10) in this study, noninformative priors are used to avoid the possible negative influence on the posterior belief of the parameter of number of stages by wrongly defined informative priors. Likelihoods are computed based on different data assumptions such as Poisson, Binomial and Weibull distribution. Results from different noninformative priors are displayed in Table 3.4. Figure 3.3 shows the mean and 95% confidence intervals of cancer rates estimated by Bayesian Armitage-Doll model and compares with the observed colon cancer mortality rate. The model fits the data well except at older age. Prior m Posterior m∗ p value Likelihood x x xxxxxxxxxx x x x x X (0, 000) N (5, 106) NA -31.57 5.80 < .001 < .001 N (0, 106) Gamma(.1, 100) NA -31.60 5.80 .51 .52 Poisson N (0, σ2) N (0, 106) N (5, σ2) Gamma(α, β) σ2 ∼ Gamma(.1, 100) α ∼ Exp(5); -31.61 -31.57 5.78 5.80 .26 .85 .28 .88 β ∼ Gamma(.1, 100) N (0, 106) N (0, 106) NA -31.58 5.80 .67 .67 N (0, 106) Gamma(.1, 100) NA -31.58 5.80 .20 .19 Xxxxxxxx X (0, x0) X (0, 000) X (0, x0) Xxxxx(x, x) σ2 ∼ Gamma(.1, 100) α ∼ Exp(5); -31.58 -31.58 5.80 5.80 .83 .83 .77 .82 β ∼ Gamma(.1, 100) Table 3.4: Different likelihood and noninformative priors in estimating the posterior for colon cancer Figure 3.3: Fitting of Bayesian Armitage-Doll model