# Kidney: Weibull Regression with Random Effects¶

An example from OpenBUGS [44] and McGilchrist and Aisbett [62] concerning time to first and second infection recurrence in 38 kidney patients on dialysis.

## Model¶

Time to recurrences are modelled as

where is the time of infection in patient , is a vector of covariates, and are patient-specific random effects.

## Analysis Program¶

```
using Mamba
## Data
kidney = Dict{Symbol, Any}(
:t => reshape(
[8, 16, 23, NaN, 22, 28, 447, 318, 30, 12, 24, 245, 7, 9, 511, 30, 53, 196,
15, 154, 7, 333, 141, NaN, 96, 38, NaN, NaN, 536, NaN, 17, NaN, 185, 177,
292, 114, NaN, NaN, 15, NaN, 152, 562, 402, NaN, 13, 66, 39, NaN, 12, 40,
NaN, 201, 132, 156, 34, 30, 2, 25, 130, 26, 27, 58, NaN, 43, 152, 30, 190,
NaN, 119, 8, NaN, NaN, NaN, 78, 63, NaN],
2, 38)',
:tcensor => reshape(
[0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0,
0, 149, 70, 0, 25, 0, 4, 0, 0, 0, 0, 22, 159, 0, 108, 0, 0, 0, 24, 0, 0, 0,
46, 0, 0, 113, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 5, 0, 0, 54,
16, 6, 0, 0, 8],
2, 38)',
:age => reshape(
[28, 28, 48, 48, 32, 32, 31, 32, 10, 10, 16, 17, 51, 51, 55, 56, 69, 69, 51,
52, 44, 44, 34, 34, 35, 35, 42, 42, 17, 17, 60, 60, 60, 60, 43, 44, 53, 53,
44, 44, 46, 47, 30, 30, 62, 63, 42, 43, 43, 43, 57, 58, 10, 10, 52, 52, 53,
53, 54, 54, 56, 56, 50, 51, 57, 57, 44, 45, 22, 22, 42, 42, 52, 52, 60, 60],
2, 38)',
:sex => [0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1,
1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0],
:disease => [1, 2, 1, 1, 1, 1, 2, 2, 3, 2, 3, 1, 3, 3, 1, 3, 1, 1, 2, 1, 4,
1, 3, 3, 3, 3, 2, 3, 2, 2, 3, 3, 4, 2, 1, 1, 4, 4],
:N => 38,
:M => 2
)
kidney[:Dx] = Int[
kidney[:disease][i] == j ? 1 : 0
for i in 1:38, j in 2:4
]
## Model Specification
model = Model(
t = Stochastic(2,
(alpha, beta_age, age, beta_sex, sex, Dx, beta_Dx, b, r,
tcensor, N, M) ->
begin
beta_dis = Dx * beta_Dx
UnivariateDistribution[
begin
mu = alpha + beta_age * age[i, j] + beta_sex * sex[i] + beta_dis[i] +
b[i]
lambda = exp(-mu / r)
0 < lambda < Inf ?
Truncated(Weibull(r, lambda), tcensor[i, j], Inf) :
Uniform(0, Inf)
end
for i in 1:N, j in 1:M
]
end,
false
),
b = Stochastic(1,
s2 -> Normal(0, sqrt(s2)),
false
),
s2 = Stochastic(
() -> InverseGamma(0.001, 0.001)
),
alpha = Stochastic(
() -> Normal(0, 100)
),
beta_age = Stochastic(
() -> Normal(0, 100)
),
beta_sex = Stochastic(
() -> Normal(0, 100)
),
beta_Dx = Stochastic(1,
() -> Normal(0, 100)
),
r = Stochastic(
() -> Gamma(1, 1000)
)
)
## Initial Values
inits = [
Dict(:t => kidney[:t], :alpha => 0, :beta_age => 0, :beta_sex => 0,
:beta_Dx => zeros(3), :s2 => 3, :r => 1.0, :b => zeros(kidney[:N])),
Dict(:t => kidney[:t], :alpha => 1, :beta_age => -1, :beta_sex => 1,
:beta_Dx => ones(3), :s2 => 1, :r => 1.5, :b => zeros(kidney[:N]))
]
## Sampling Scheme
scheme = [MISS(:t),
Slice([:alpha, :beta_age, :beta_sex, :beta_Dx], 0.1),
Slice(:b, 0.01),
Slice(:s2, 0.1),
Slice(:r, 0.001)]
setsamplers!(model, scheme)
## MCMC Simulations
sim = mcmc(model, kidney, inits, 20000, burnin=2500, thin=2, chains=2)
describe(sim)
```