.. index:: Examples; Inhalers: Ordered Categorical Data
An example from OpenBUGS :cite:`openbugs:2014:ex` and Ezzet and Whitehead :cite:`ezzet:1993:REM` concerning a two-treatment, two-period crossover trial comparing salbutamol inhalation devices in 286 asthma patients.
Treatment responses are modelled as
R_{i,t} &= j \quad \text{if}\; Y_{i,t} \in [a_{j-1}, a_j) \quad\quad i=1,\ldots,4; t=1,2; j=1,\ldots,3 \\
\operatorname{logit}(Q_{i,t,j}) &= -(a_j + \mu_{s_i,t} + b_i) \\
\mu_{1,1} &= \beta / 2 + \pi / 2 \\
\mu_{1,2} &= -\beta / 2 - \pi / 2 - \kappa \\
\mu_{2,1} &= -\beta / 2 + \pi / 2 \\
\mu_{2,2} &= \beta / 2 - \pi / 2 + \kappa \\
b_i &\sim \text{Normal}(0, \sigma) \\
a[1] &\sim \text{Flat}(-1000, a[2]) \\
a[2] &\sim \text{Flat}(-1000, a[3]) \\
a[3] &\sim \text{Flat}(-1000, 1000) \\
\beta &\sim \text{Normal}(0, 1000) \\
\pi &\sim \text{Normal}(0, 1000) \\
\kappa &\sim \text{Normal}(0, 1000) \\
\sigma^2 &\sim \text{InverseGamma}(0.001, 0.001),
where R_{i,t} is a 4-point ordinal rating of the device used by patient i, and Q_{i,t,j} is the cumulative probability of the rating in treatment period t being worse than category j.
.. literalinclude:: inhalers.jl
:language: julia