Example: Generalized Linear Mixed Models

The UCBadmit data is sourced from the study [1] of gender biased in graduate admissions at UC Berkeley in Fall 1973:

UCBadmit dataset

dept

male

applications

admit

0

1

825

512

0

0

108

89

1

1

560

353

1

0

25

17

2

1

325

120

2

0

593

202

3

1

417

138

3

0

375

131

4

1

191

53

4

0

393

94

5

1

373

22

5

0

341

24

This example replicates the multilevel model m_glmm5 at [3], which is used to evaluate whether the data contain evidence of gender biased in admissions across departments. This is a form of Generalized Linear Mixed Models for binomial regression problem, which models

  • varying intercepts across departments,

  • varying slopes (or the effects of being male) across departments,

  • correlation between intercepts and slopes,

and uses non-centered parameterization (or whitening).

A more comprehensive explanation for binomial regression and non-centered parameterization can be found in Chapter 10 (Counting and Classification) and Chapter 13 (Adventures in Covariance) of [2].

References:

  1. Bickel, P. J., Hammel, E. A., and O’Connell, J. W. (1975), “Sex Bias in Graduate Admissions: Data from Berkeley”, Science, 187(4175), 398-404.

  2. McElreath, R. (2018), “Statistical Rethinking: A Bayesian Course with Examples in R and Stan”, Chapman and Hall/CRC.

  3. https://github.com/rmcelreath/rethinking/tree/Experimental#multilevel-model-formulas

../_images/ucbadmit.png
import argparse
import os

import matplotlib.pyplot as plt
import numpy as np

from jax import random
import jax.numpy as jnp
from jax.scipy.special import expit

import numpyro
import numpyro.distributions as dist
from numpyro.examples.datasets import UCBADMIT, load_dataset
from numpyro.infer import MCMC, NUTS, Predictive


def glmm(dept, male, applications, admit=None):
    v_mu = numpyro.sample("v_mu", dist.Normal(0, jnp.array([4.0, 1.0])))

    sigma = numpyro.sample("sigma", dist.HalfNormal(jnp.ones(2)))
    L_Rho = numpyro.sample("L_Rho", dist.LKJCholesky(2, concentration=2))
    scale_tril = sigma[..., jnp.newaxis] * L_Rho
    # non-centered parameterization
    num_dept = len(np.unique(dept))
    z = numpyro.sample("z", dist.Normal(jnp.zeros((num_dept, 2)), 1))
    v = jnp.dot(scale_tril, z.T).T

    logits = v_mu[0] + v[dept, 0] + (v_mu[1] + v[dept, 1]) * male
    if admit is None:
        # we use a Delta site to record probs for predictive distribution
        probs = expit(logits)
        numpyro.sample("probs", dist.Delta(probs), obs=probs)
    numpyro.sample("admit", dist.Binomial(applications, logits=logits), obs=admit)


def run_inference(dept, male, applications, admit, rng_key, args):
    kernel = NUTS(glmm)
    mcmc = MCMC(
        kernel,
        num_warmup=args.num_warmup,
        num_samples=args.num_samples,
        num_chains=args.num_chains,
        progress_bar=False if "NUMPYRO_SPHINXBUILD" in os.environ else True,
    )
    mcmc.run(rng_key, dept, male, applications, admit)
    return mcmc.get_samples()


def print_results(header, preds, dept, male, probs):
    columns = ["Dept", "Male", "ActualProb", "Pred(p25)", "Pred(p50)", "Pred(p75)"]
    header_format = "{:>10} {:>10} {:>10} {:>10} {:>10} {:>10}"
    row_format = "{:>10.0f} {:>10.0f} {:>10.2f} {:>10.2f} {:>10.2f} {:>10.2f}"
    quantiles = jnp.quantile(preds, jnp.array([0.25, 0.5, 0.75]), axis=0)
    print("\n", header, "\n")
    print(header_format.format(*columns))
    for i in range(len(dept)):
        print(row_format.format(dept[i], male[i], probs[i], *quantiles[:, i]), "\n")


def main(args):
    _, fetch_train = load_dataset(UCBADMIT, split="train", shuffle=False)
    dept, male, applications, admit = fetch_train()
    rng_key, rng_key_predict = random.split(random.PRNGKey(1))
    zs = run_inference(dept, male, applications, admit, rng_key, args)
    pred_probs = Predictive(glmm, zs)(rng_key_predict, dept, male, applications)[
        "probs"
    ]
    header = "=" * 30 + "glmm - TRAIN" + "=" * 30
    print_results(header, pred_probs, dept, male, admit / applications)

    # make plots
    fig, ax = plt.subplots(figsize=(8, 6), constrained_layout=True)

    ax.plot(range(1, 13), admit / applications, "o", ms=7, label="actual rate")
    ax.errorbar(
        range(1, 13),
        jnp.mean(pred_probs, 0),
        jnp.std(pred_probs, 0),
        fmt="o",
        c="k",
        mfc="none",
        ms=7,
        elinewidth=1,
        label=r"mean $\pm$ std",
    )
    ax.plot(range(1, 13), jnp.percentile(pred_probs, 5, 0), "k+")
    ax.plot(range(1, 13), jnp.percentile(pred_probs, 95, 0), "k+")
    ax.set(
        xlabel="cases",
        ylabel="admit rate",
        title="Posterior Predictive Check with 90% CI",
    )
    ax.legend()

    plt.savefig("ucbadmit_plot.pdf")


if __name__ == "__main__":
    assert numpyro.__version__.startswith("0.8.0")
    parser = argparse.ArgumentParser(
        description="UCBadmit gender discrimination using HMC"
    )
    parser.add_argument("-n", "--num-samples", nargs="?", default=2000, type=int)
    parser.add_argument("--num-warmup", nargs="?", default=500, type=int)
    parser.add_argument("--num-chains", nargs="?", default=1, type=int)
    parser.add_argument("--device", default="cpu", type=str, help='use "cpu" or "gpu".')
    args = parser.parse_args()

    numpyro.set_platform(args.device)
    numpyro.set_host_device_count(args.num_chains)

    main(args)

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