oxford.rst

Examples; Oxford: Smooth Fit to Log-Odds Ratios

Oxford: Smooth Fit to Log-Odds Ratios

An example from OpenBUGS openbugs:2014:ex and Breslow and Clayton breslow:1993:AIG concerning the association between death from childhood cancer and maternal exposure to X-rays, for subjects partitioned into 120 age and birth-year strata.

Model

Deaths are modelled as

$$\begin{aligned} r^0_i &\sim \text{Binomial}(n^0_i, p^0_i) \quad\quad i=1,\ldots,120 \\\ r^1_i &\sim \text{Binomial}(n^1_i, p^1_i) \\\ \operatorname{logit}(p^0_i) &= \mu_i \\\ \operatorname{logit}(p^1_i) &= \mu_i + \log(\psi_i) \\\ \log(\psi) &= \alpha + \beta_1 \text{year}_i + \beta_2 (\text{year}^2_i - 22) + b_i \\\ \mu_i &\sim \text{Normal}(0, 1000) \\\ b_i &\sim \text{Normal}(0, \sigma) \\\ \alpha, \beta_1, \beta_2 &\sim \text{Normal}(0, 1000) \\\ \sigma^2 &\sim \text{InverseGamma}(0.001, 0.001), \end{aligned}$$

where r_i⁰ is the number of deaths among unexposed subjects in stratum i, r_i¹ is the number among exposed subjects, and year_i is the stratum-specific birth year (relative to 1954).

Analysis Program

oxford.jl

Results

Iterations = 2502:12500
Thinning interval = 2
Chains = 1,2
Samples per chain = 5000

Empirical Posterior Estimates:
          Mean          SD         Naive SE        MCSE         ESS
beta2  0.005477119 0.0035675748 0.00003567575 0.00033192987 115.519023
beta1 -0.043336269 0.0161754258 0.00016175426 0.00133361554 147.112695
alpha  0.565784774 0.0630050896 0.00063005090 0.00468384860 180.944576
   s2  0.026238992 0.0307989154 0.00030798915 0.00302056007 103.967091

Quantiles:
           2.5%         25.0%         50.0%         75.0%         97.5%
beta2 -0.0010499046  0.0028489198  0.0056500394  0.0077473623  0.013630865
beta1 -0.0745152363 -0.0543180318 -0.0434425931 -0.0321216097 -0.009920787
alpha  0.4438257884  0.5238801187  0.5675039159  0.6051427125  0.695968063
   s2  0.0007134423  0.0033352655  0.0146737037  0.0397132522  0.118202258