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(GLM-negative-binomial-regression)=
:::{post} September, 2023 :tags: negative binomial regression, generalized linear model, :category: beginner :author: Ian Ozsvald, Abhipsha Das, Benjamin Vincent :::
:::{include} ../extra_installs.md :::
import arviz as az
import numpy as np
import pandas as pd
import pymc as pm
import seaborn as sns
from scipy import stats
RANDOM_SEED = 8927
rng = np.random.default_rng(RANDOM_SEED)
%config InlineBackend.figure_format = "retina"
az.style.use("arviz-darkgrid")
This notebook closely follows the GLM Poisson regression example by Jonathan Sedar (which is in turn inspired by a project by Ian Osvald) except the data here is negative binomially distributed instead of Poisson distributed.
Negative binomial regression is used to model count data for which the variance is higher than the mean. The negative binomial distribution can be thought of as a Poisson distribution whose rate parameter is gamma distributed, so that rate parameter can be adjusted to account for the increased variance.
+++
As in the Poisson regression example, we assume that sneezing occurs at some baseline rate, and that consuming alcohol, not taking antihistamines, or doing both, increase its frequency.
First, let's look at some Poisson distributed data from the Poisson regression example.
# Mean Poisson values
theta_noalcohol_meds = 1 # no alcohol, took an antihist
theta_alcohol_meds = 3 # alcohol, took an antihist
theta_noalcohol_nomeds = 6 # no alcohol, no antihist
theta_alcohol_nomeds = 36 # alcohol, no antihist
# Create samples
q = 1000
df_pois = pd.DataFrame(
{
"nsneeze": np.concatenate(
(
rng.poisson(theta_noalcohol_meds, q),
rng.poisson(theta_alcohol_meds, q),
rng.poisson(theta_noalcohol_nomeds, q),
rng.poisson(theta_alcohol_nomeds, q),
)
),
"alcohol": np.concatenate(
(
np.repeat(False, q),
np.repeat(True, q),
np.repeat(False, q),
np.repeat(True, q),
)
),
"nomeds": np.concatenate(
(
np.repeat(False, q),
np.repeat(False, q),
np.repeat(True, q),
np.repeat(True, q),
)
),
}
)
df_pois.groupby(["nomeds", "alcohol"])["nsneeze"].agg(["mean", "var"])
Since the mean and variance of a Poisson distributed random variable are equal, the sample means and variances are very close.
Now, suppose every subject in the dataset had the flu, increasing the variance of their sneezing (and causing an unfortunate few to sneeze over 70 times a day). If the mean number of sneezes stays the same but variance increases, the data might follow a negative binomial distribution.
# Gamma shape parameter
alpha = 10
def get_nb_vals(mu, alpha, size):
"""Generate negative binomially distributed samples by
drawing a sample from a gamma distribution with mean `mu` and
shape parameter `alpha', then drawing from a Poisson
distribution whose rate parameter is given by the sampled
gamma variable.
"""
g = stats.gamma.rvs(alpha, scale=mu / alpha, size=size)
return stats.poisson.rvs(g)
# Create samples
n = 1000
df = pd.DataFrame(
{
"nsneeze": np.concatenate(
(
get_nb_vals(theta_noalcohol_meds, alpha, n),
get_nb_vals(theta_alcohol_meds, alpha, n),
get_nb_vals(theta_noalcohol_nomeds, alpha, n),
get_nb_vals(theta_alcohol_nomeds, alpha, n),
)
),
"alcohol": np.concatenate(
(
np.repeat(False, n),
np.repeat(True, n),
np.repeat(False, n),
np.repeat(True, n),
)
),
"nomeds": np.concatenate(
(
np.repeat(False, n),
np.repeat(False, n),
np.repeat(True, n),
np.repeat(True, n),
)
),
}
)
df
df.groupby(["nomeds", "alcohol"])["nsneeze"].agg(["mean", "var"])
As in the Poisson regression example, we see that drinking alcohol and/or not taking antihistamines increase the sneezing rate to varying degrees. Unlike in that example, for each combination of alcohol and nomeds, the variance of nsneeze is higher than the mean. This suggests that a Poisson distribution would be a poor fit for the data since the mean and variance of a Poisson distribution are equal.
+++
g = sns.catplot(x="nsneeze", row="nomeds", col="alcohol", data=df, kind="count", aspect=1.5)
# Make x-axis ticklabels less crowded
ax = g.axes[1, 0]
labels = range(len(ax.get_xticklabels(which="both")))
ax.set_xticks(labels[::5])
ax.set_xticklabels(labels[::5]);
+++
COORDS = {"regressor": ["nomeds", "alcohol", "nomeds:alcohol"], "obs_idx": df.index}
with pm.Model(coords=COORDS) as m_sneeze_inter:
a = pm.Normal("intercept", mu=0, sigma=5)
b = pm.Normal("slopes", mu=0, sigma=1, dims="regressor")
alpha = pm.Exponential("alpha", 0.5)
M = pm.ConstantData("nomeds", df.nomeds.to_numpy(), dims="obs_idx")
A = pm.ConstantData("alcohol", df.alcohol.to_numpy(), dims="obs_idx")
S = pm.ConstantData("nsneeze", df.nsneeze.to_numpy(), dims="obs_idx")
λ = pm.math.exp(a + b[0] * M + b[1] * A + b[2] * M * A)
y = pm.NegativeBinomial("y", mu=λ, alpha=alpha, observed=S, dims="obs_idx")
idata = pm.sample()
az.plot_trace(idata, compact=False);
# Transform coefficients to recover parameter values
az.summary(np.exp(idata.posterior), kind="stats", var_names=["intercept", "slopes"])
az.summary(idata.posterior, kind="stats", var_names="alpha")
The mean values are close to the values we specified when generating the data:
- The base rate is a constant 1.
- Drinking alcohol triples the base rate.
- Not taking antihistamines increases the base rate by 6 times.
- Drinking alcohol and not taking antihistamines doubles the rate that would be expected if their rates were independent. If they were independent, then doing both would increase the base rate by 3*6=18 times, but instead the base rate is increased by 3*6*2=36 times.
Finally, the mean of nsneeze_alpha is also quite close to its actual value of 10.
+++
See also, bambi's negative binomial example for further reference.
+++
- Created by Ian Ozsvald
- Updated by Abhipsha Das in August 2021
- Updated by Benjamin Vincent to PyMC v4 in June 2022
- Updated by Wesley Boelrijk to PyMC v5 in September 2023
%load_ext watermark
%watermark -n -u -v -iv -w -p pytensor,xarray
:::{include} ../page_footer.md :::