Lister*

import pymc as pm
import numpy as np
import arviz as az
from pymc.math import invlogit

1. Lister*#

Adapted from Unit 10: lister.odc.

Problem Statement#

Sir Joseph Lister (1827–1912), Professor of Surgery at Glasgow University, was influenced by Pasteur’s ideas. He discovered that wounds wrapped in bandages treated with Carbolic acid (phenol) often did not become infected.

Here are Lister’s data on treating open fractures and amputations:

Period

Carbolic Acid Used

Results

1864–1866

No

Treated 34 patients: 15 died, 19 recovered

1867–1870

Yes

Treated 40 patients: 6 died, 34 recovered

Estimate and interpret the risk difference, risk ratio, and odds ratio.

Notes#

The results are a bit different than BUGS. I’m not sure if that’s from the slightly different prior (WishartBartlett vs Wishart), or something else I’ve missed.

n_untreated = 34
n_untreated_fail = 15
n_treated = 40
n_treated_fail = 6
mu0 = np.array([-0.5, -1.5])
S = np.array([[0.1, 0], [0, 0.1]])
S_init = np.array([[1, 0], [0, 1]])

with pm.Model() as m:
    tau = pm.WishartBartlett("tau", S=S, nu=4, initval=S_init)
    sigma = pm.Deterministic("sigma", pm.math.matrix_inverse(tau))
    mu = pm.MvNormal("mu", mu=mu0, tau=tau, shape=2)

    p1 = invlogit(mu[0])
    p2 = invlogit(mu[1])

    pm.Binomial("y_untreated", p=p1, n=n_untreated, observed=n_untreated_fail)
    pm.Binomial("y_treated", p=p2, n=n_treated, observed=n_treated_fail)

    pm.Deterministic("rd", p1 - p2)
    pm.Deterministic("rr", p1 / p2)
    pm.Deterministic("or", (p1 / (1 - p1)) / (p2 / (1 - p2)))

    trace = pm.sample(3000, target_accept=0.95)

Hide code cell output

 
Added new variable tau_c to model diagonal of Wishart.
Added new variable tau_z to model off-diagonals of Wishart.
Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [tau_c, tau_z, mu]
100.00% [16000/16000 00:07<00:00 Sampling 4 chains, 0 divergences] 
Sampling 4 chains for 1_000 tune and 3_000 draw iterations (4_000 + 12_000 draws total) took 8 seconds.

az.summary(trace, var_names=["~tau"], filter_vars="like")
mean sd hdi_3% hdi_97% mcse_mean mcse_sd ess_bulk ess_tail r_hat
mu[0] -0.253 0.349 -0.896 0.408 0.003 0.003 14065.0 9108.0 1.0
mu[1] -1.788 0.443 -2.639 -0.983 0.004 0.003 11314.0 7297.0 1.0
sigma[0, 0] 43.124 837.373 0.576 86.710 8.108 5.733 8565.0 6194.0 1.0
sigma[0, 1] 6.524 489.723 -47.465 51.318 5.054 3.574 11237.0 7516.0 1.0
sigma[1, 0] 6.524 489.723 -47.465 51.318 5.054 3.574 11237.0 7516.0 1.0
sigma[1, 1] 40.100 491.767 0.521 81.160 6.329 4.475 9202.0 4896.0 1.0
rd 0.287 0.100 0.101 0.470 0.001 0.001 13492.0 8897.0 1.0
rr 3.348 1.643 1.106 6.191 0.018 0.014 11531.0 7110.0 1.0
or 5.469 3.534 1.105 11.696 0.037 0.028 12107.0 8051.0 1.0 
%load_ext watermark
%watermark -n -u -v -iv -p pytensor

Last updated: Mon Nov 13 2023

Python implementation: CPython
Python version       : 3.11.5
IPython version      : 8.15.0

pytensor: 2.17.1

numpy: 1.25.2
pymc : 5.9.0
arviz: 0.16.1