Section 5.5 — Hierarchical models

Section 5.5 — Hierarchical models#

This notebook contains the code examples from Section 5.5 Hierarchical models from the No Bullshit Guide to Statistics.

Notebook setup#

# load Python modules
import os
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt

# Figures setup
plt.clf()  # needed otherwise `sns.set_theme` doesn"t work
from plot_helpers import RCPARAMS
RCPARAMS.update({"figure.figsize": (5, 3)})   # good for screen
# RCPARAMS.update({"figure.figsize": (5, 1.6)})  # good for print
sns.set_theme(
    context="paper",
    style="whitegrid",
    palette="colorblind",
    rc=RCPARAMS,
)

# High-resolution please
%config InlineBackend.figure_format = "retina"

<Figure size 640x480 with 0 Axes>

# set random seed for repeatability
np.random.seed(42)
#######################################################

# silence statsmodels kurtosistest warning when using n < 20
import warnings
warnings.filterwarnings("ignore", category=FutureWarning)
warnings.filterwarnings("ignore", category=UserWarning)

Definitions#

Hierarchical (multilevel) models#

Model formula#

A Bayesian hierarchical model is described by the following equations:

\(\def\calN{\mathcal{N}} \def\Tdist{\mathcal{T}} \def\Expon{\mathrm{Expon}}\)

\[\begin{align*} X_j &\sim \calN(M_{j}, \, \Sigma_X), \\ M_{j} &= B_0 + B_{0j}, \\ \Sigma_X &\sim \Tdist^+\!(\nu_{\Sigma_X}, \sigma_{\Sigma_X}), \\ B_0 &\sim \calN(\mu_{B_0},\sigma_{B_0}), \\ B_{0j} &\sim \calN(0,\Sigma_{B_{0j}}), \\ \Sigma_{B_{0j}} &\sim \Expon(\lambda). \end{align*}\]

Radon dataset#

https://bambinos.github.io/bambi/notebooks/radon_example.html

Description: Contains measurements of radon levels in homes across various counties.
Source: Featured in Andrew Gelman and Jennifer Hill’s book Data Analysis Using Regression and Multilevel/Hierarchical Models.
Application: Demonstrates partial pooling and varying intercepts/slopes in hierarchical modeling.

Loading the data#

radon = pd.read_csv("../datasets/radon.csv")
radon.shape

(919, 6)

radon.head()

	idnum	state	county	floor	log_radon	log_uranium
0	5081	MN	AITKIN	ground	0.788457	-0.689048
1	5082	MN	AITKIN	basement	0.788457	-0.689048
2	5083	MN	AITKIN	basement	1.064711	-0.689048
3	5084	MN	AITKIN	basement	0.000000	-0.689048
4	5085	MN	ANOKA	basement	1.131402	-0.847313

Descriptive statistics#

radon["log_radon"].describe()

count    919.000000
mean       1.224623
std        0.853327
min       -2.302585
25%        0.641854
50%        1.280934
75%        1.791759
max        3.875359
Name: log_radon, dtype: float64

len(radon["county"].unique())

radon["floor"].value_counts()

floor
basement    766
ground      153
Name: count, dtype: int64

from ministats.book.figures import plot_counties
sel_counties = [
  "LAC QUI PARLE", "AITKIN", "KOOCHICHING", "DOUGLAS",
  "HENNEPIN", "STEARNS", "RAMSEY", "ST LOUIS",
]
plot_counties(radon, counties=sel_counties);

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
Cell In[10], line 1
----> 1 from ministats.book.figures import plot_counties
      2 sel_counties = [
      3   "LAC QUI PARLE", "AITKIN", "KOOCHICHING", "DOUGLAS",
      4   "HENNEPIN", "STEARNS", "RAMSEY", "ST LOUIS",
      5 ]
      6 plot_counties(radon, counties=sel_counties);

ModuleNotFoundError: No module named 'ministats.book'

Example 1: complete pooling model#

= common linear regression model for all counties

Bayesian model#

We can pool all the data and estimate one big regression to asses the influence of the floor variable on radon levels across all counties.

\[\begin{align*} R &\sim \calN(M_R, \, \Sigma_R), \\ M_R &= B_0 + B_{\!f}\!\cdot\!f, \\ \Sigma_R &\sim \Tdist^+\!(4, 1), \\ B_0 &\sim \calN(1, 2), \\ B_f &\sim \calN(0, 5). \end{align*}\]

The variable \(f\) corresponds to the column floor in the radon data frame, which will be internally coded as binary with \(0\) representing basement, and \(1\) representing ground floor.

By ignoring the county feature, we do not differenciate on counties.

Bambi model#

import bambi as bmb

priors1 = {
    "Intercept": bmb.Prior("Normal", mu=1, sigma=2),
    "floor": bmb.Prior("Normal", mu=0, sigma=5),
    "sigma": bmb.Prior("HalfStudentT", nu=4, sigma=1),
}

mod1 = bmb.Model(formula="log_radon ~ 1 + floor",
                 family="gaussian",
                 link="identity",
                 priors=priors1,
                 data=radon)
mod1

       Formula: log_radon ~ 1 + floor
        Family: gaussian
          Link: mu = identity
  Observations: 919
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 1.0, sigma: 2.0)
            floor ~ Normal(mu: 0.0, sigma: 5.0)
        
        Auxiliary parameters
            sigma ~ HalfStudentT(nu: 4.0, sigma: 1.0)

mod1.build()
mod1.graph()

../_images/87882ef7fbcd63ff3a912a4c74256f9699b4efa8be87198225ec17778257643e.svg

Model fitting and analysis#

idata1 = mod1.fit(random_seed=[42,43,44,45])

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [sigma, Intercept, floor]

Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 2 seconds.

import arviz as az

az.summary(idata1, kind="stats")

	mean	sd	hdi_3%	hdi_97%
Intercept	1.326	0.029	1.271	1.381
floor[ground]	-0.612	0.073	-0.753	-0.479
sigma	0.823	0.019	0.791	0.863

az.plot_posterior(idata1);

../_images/83cb5871d7102bd25cfb0e1c05605464e46eac207d3b9507bc203dbbfa9cf579.png

fig, axs = bmb.interpret.plot_predictions(mod1, idata1, conditional="floor")

means1 = az.summary(idata1, kind="stats")["mean"]
y0 = means1["Intercept"]
y1 = means1["Intercept"] + means1["floor[ground]"]
sns.lineplot(x=[0,1], y=[y0,y1], ax=axs[0]);

midpoint = [0.5, (y0+y1)/2 + 0.03]
slope = means1["floor[ground]"].round(2)
axs[0].annotate("$\\mu_{B_{f}}=%.2f$" % slope, midpoint);

Default computed for conditional variable: floor

../_images/10f6f772ec677f340a5c962c89ffeeec1bb8d8f2642960958f5d9d9672fd311d.png

Conclusion#

not using group membership, so we have lots of bias

Example 2: no pooling model#

= separate intercept for each county

Bayesian model#

If we treat different counties as independent, so each one gets an intercept term:

\[\begin{align*} R_j &\sim \calN(M_j, \, \Sigma_R), \\ M_j &= B_{0j} + B_{\!f}\!\cdot\!f, \\ \Sigma_R &\sim \Tdist^+\!(4, 1), \\ B_{0j} &\sim \calN(1, 2), \\ B_f &\sim \calN(0, 5). \end{align*}\]

Bambi model#

priors2 = {
    "county": bmb.Prior("Normal", mu=1, sigma=2),
    "floor": bmb.Prior("Normal", mu=0, sigma=5),
    "sigma": bmb.Prior("HalfStudentT", nu=4, sigma=1),
}

mod2 = bmb.Model("log_radon ~ 0 + county + floor",
                 family="gaussian",
                 link="identity",
                 priors=priors2,
                 data=radon)
mod2

       Formula: log_radon ~ 0 + county + floor
        Family: gaussian
          Link: mu = identity
  Observations: 919
        Priors: 
    target = mu
        Common-level effects
            county ~ Normal(mu: 1.0, sigma: 2.0)
            floor ~ Normal(mu: 0.0, sigma: 5.0)
        
        Auxiliary parameters
            sigma ~ HalfStudentT(nu: 4.0, sigma: 1.0)

mod2.build()
mod2.graph()

../_images/d94e90cacc230c99c28c570e3776e915d47a39f892e37b4717a893db35b2b7b7.svg

Model fitting and analysis#

idata2 = mod2.fit(random_seed=[42,43,44,45])

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [sigma, county, floor]

Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 4 seconds.

idata2sel = idata2.sel(county_dim=sel_counties)
az.summary(idata2sel, kind="stats")

	mean	sd	hdi_3%	hdi_97%
county[LAC QUI PARLE]	2.822	0.512	1.852	3.762
county[AITKIN]	0.840	0.377	0.094	1.526
county[KOOCHICHING]	0.814	0.285	0.260	1.341
county[DOUGLAS]	1.718	0.250	1.242	2.197
county[HENNEPIN]	1.359	0.074	1.221	1.496
county[STEARNS]	1.485	0.153	1.172	1.750
county[RAMSEY]	1.152	0.132	0.903	1.403
county[ST LOUIS]	0.866	0.071	0.725	0.989
floor[ground]	-0.706	0.073	-0.839	-0.565
sigma	0.756	0.019	0.722	0.792

axs = az.plot_forest(idata2sel, combined=True, figsize=(6,2.6))
axs[0].set_xticks(np.arange(-0.5,4.6,0.5))
axs[0].set_title(None);

../_images/a9aa0aefcf1a89ff9506f607db7f5b96ff32117a905b99406bb7eede36abc9ca.png

plot_counties(radon, idata_cp=idata1, idata_np=idata2);

../_images/e3b0b385ddf2f4ad4b505873bdeb3f30caf13a2720621eef040ecf518256a973.png

# fig, axs = bmb.interpret.plot_predictions(mod2, idata2, ["floor", "county"]);
# axs[0].get_legend().remove()

# post2 = idata2["posterior"]
# unpooled_means = post2.mean(dim=("chain", "draw"))
# unpooled_hdi = az.hdi(idata2)

# unpooled_means_iter = unpooled_means.sortby("county")
# unpooled_hdi_iter = unpooled_hdi.sortby(unpooled_means_iter.county)

# _, ax = plt.subplots(figsize=(12, 5))
# unpooled_means_iter.plot.scatter(x="county_dim", y="county", ax=ax, alpha=0.9)
# ax.vlines(
#     np.arange(len(radon["county"].unique())),
#     unpooled_hdi_iter.county.sel(hdi="lower"),
#     unpooled_hdi_iter.county.sel(hdi="higher"),
#     color="orange",
#     alpha=0.6,
# )
# ax.set(ylabel="Radon estimate", ylim=(-2, 4.5))
# ax.tick_params(rotation=90);

Conclusion#

treating each group independently, so we have lots of variance

# TODO:
# calculate the std.dev. of the county-specific slopes -- this is like \Sigma_J but w/o a prior.

Example 3: hierarchical model#

= partial pooling model = varying intercepts model

Bayesian hierarchical model#

\[\begin{align*} R_j &\sim \calN(M_j, \, \Sigma_R), \\ M_j &= B_0 + B_{0j} + B_f\!\cdot\!f, \\ \Sigma_R &\sim \Tdist^+\!(4, 1), \\ B_0 &\sim \calN(1,2), \\ B_{0j} &\sim \calN(0,\Sigma_{B_{0j}}), \\ B_f &\sim \calN(0, 5), \\ \Sigma_{B_{0j}} &\sim \Expon(1). \end{align*}\]

The partial pooling formula estimates per-county intercepts which drawn from the same distribution which is estimated jointly with the rest of the model parameters. The 1 is the intercept co-efficient. The estimates across counties will all have the same slope.

log_radon ~ 1 + (1|county_id) + floor

There is a middle ground to both of these extremes. Specifically, we may assume that the intercepts are different for each county as in the unpooled case, but they are drawn from the same distribution. The different counties are effectively sharing information through the common prior.

NOTE: some counties have very few sample; the hierarchical model will provide “shrinkage” for these groups, and use global information learned from all counties

radon["log_radon"].describe()

count    919.000000
mean       1.224623
std        0.853327
min       -2.302585
25%        0.641854
50%        1.280934
75%        1.791759
max        3.875359
Name: log_radon, dtype: float64

radon.groupby("floor")["log_radon"].describe()

	count	mean	std	min	25%	50%	75%	max
floor
basement	766.0	1.326744	0.782709	-2.302585	0.788457	1.360977	1.883253	3.875359
ground	153.0	0.713349	0.999376	-2.302585	0.095310	0.741937	1.308333	3.234749

Bambi model#

priors3 = {
    "Intercept": bmb.Prior("Normal", mu=1, sigma=2),
    "floor": bmb.Prior("Normal", mu=0, sigma=5),
    "1|county": bmb.Prior("Normal", mu=0, sigma=bmb.Prior("Exponential", lam=1)),
    "sigma": bmb.Prior("HalfStudentT", nu=4, sigma=1),
}

formula3 = "log_radon ~ 1 + (1|county) + floor"
mod3 = bmb.Model(formula=formula3,
                 family="gaussian",
                 link="identity",
                 priors=priors3,
                 data=radon,
                 noncentered=False)

mod3

       Formula: log_radon ~ 1 + (1|county) + floor
        Family: gaussian
          Link: mu = identity
  Observations: 919
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 1.0, sigma: 2.0)
            floor ~ Normal(mu: 0.0, sigma: 5.0)
        
        Group-level effects
            1|county ~ Normal(mu: 0.0, sigma: Exponential(lam: 1.0))
        
        Auxiliary parameters
            sigma ~ HalfStudentT(nu: 4.0, sigma: 1.0)

mod3.build()
mod3.graph()

../_images/9d7a8ba9110201b4838ab2017b28b95c343232ff978c0f1cf2547c577ec6a4a5.svg

Model fitting and analysis#

idata3 = mod3.fit(random_seed=[42,43,44,45])

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [sigma, Intercept, floor, 1|county_sigma, 1|county]

Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 3 seconds.

The group level parameters

idata3sel = idata3.sel(county__factor_dim=sel_counties)
az.summary(idata3sel, kind="stats")

	mean	sd	hdi_3%	hdi_97%
1\|county[LAC QUI PARLE]	0.413	0.294	-0.173	0.941
1\|county[AITKIN]	-0.267	0.251	-0.734	0.201
1\|county[KOOCHICHING]	-0.376	0.227	-0.835	0.022
1\|county[DOUGLAS]	0.170	0.206	-0.219	0.545
1\|county[HENNEPIN]	-0.099	0.085	-0.260	0.063
1\|county[STEARNS]	0.017	0.144	-0.237	0.301
1\|county[RAMSEY]	-0.261	0.133	-0.509	-0.013
1\|county[ST LOUIS]	-0.572	0.085	-0.730	-0.413
1\|county_sigma	0.333	0.046	0.250	0.421
Intercept	1.463	0.052	1.371	1.567
floor[ground]	-0.694	0.070	-0.829	-0.568
sigma	0.757	0.018	0.723	0.791

The intercept offsets for each county are:

# sum( idata3["posterior"]["1|county"].stack(sample=("chain","draw")).values.mean(axis=1) )

# az.plot_forest(idata3, combined=True, figsize=(7,2),
#                var_names=["Intercept", "floor", "1|county_sigma", "sigma"]);

axs = az.plot_forest(idata3sel,
                     var_names=["Intercept", "1|county", "floor", "1|county_sigma", "sigma"],
                     combined=True, figsize=(6,3))
axs[0].set_xticks(np.arange(-0.8,1.6,0.2))
axs[0].set_title(None);

../_images/cf5e06b8bf391edfaf660ec6c698049e58dcf752d81431c69075b58c3276e623.png

# az.plot_forest(idata3, var_names=["1|county"], combined=True);

Compare models#

Compare complete pooling, no pooling, and partial pooling models

plot_counties(radon, idata_cp=idata1, idata_np=idata2, idata_pp=idata3);

../_images/5c004e233fb5eff70f2814fb18dae9bc1357f60f8cacc93804b7586c3b238ce1.png

# # Forest plot comparing `mod2` and `mod3` estimates
# # (to illustrate reduced uncertainty in estimates + shrinkage)
# post3 = idata3sel["posterior"]
# post3["county"] = post3["Intercept"] + post3["1|county"]
# axs = az.plot_forest([idata2sel, idata3sel], model_names=["mod2", "mod3"],
#                      var_names=["county", "floor", "1|county_sigma", "sigma"], combined=True, figsize=(6,3))
# ax = axs[0]
# rbar = radon["log_radon"].mean()
# ax.axvline(rbar, ls="--");

Conclusions#

Explanations#

Prior selection for hierarchical models#

?

Varying intercepts and slopes model#

= Group-specific slopes We can also make beta_x group-specific

The varying-slope, varying intercept model adds floor to the group-level co-efficients. Now estimates across counties will all have varying slope.

log_radon ~ 1 + floor + (1 + floor | county)

#######################################################
formula4 = "log_radon ~ 1 + (1|county) + floor + (floor|county)"

SigmaB0j = bmb.Prior("Exponential", lam=1)
SigmaBfj = bmb.Prior("Exponential", lam=1)

priors4 = {
    "Intercept": bmb.Prior("Normal", mu=1, sigma=2),
    "1|county": bmb.Prior("Normal", mu=0, sigma=SigmaB0j),
    "floor": bmb.Prior("Normal", mu=-1, sigma=2),
    "floor|county": bmb.Prior("Normal", mu=0, sigma=SigmaBfj),
    "sigma": bmb.Prior("HalfStudentT", nu=4, sigma=1),
}

mod4 = bmb.Model(formula=formula4,
                 family="gaussian",
                 link="identity",
                 priors=priors4,
                 data=radon,
                 noncentered=False)
mod4

       Formula: log_radon ~ 1 + (1|county) + floor + (floor|county)
        Family: gaussian
          Link: mu = identity
  Observations: 919
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 1.0, sigma: 2.0)
            floor ~ Normal(mu: -1.0, sigma: 2.0)
        
        Group-level effects
            1|county ~ Normal(mu: 0.0, sigma: Exponential(lam: 1.0))
            floor|county ~ Normal(mu: 0.0, sigma: Exponential(lam: 1.0))
        
        Auxiliary parameters
            sigma ~ HalfStudentT(nu: 4.0, sigma: 1.0)

mod4.build()
mod4.graph()

../_images/2213e6e5f2315916199457337755caa1a49056c67b90bf2990bb30bf77bddadc.svg

idata4 = mod4.fit(draws=2000, tune=3000, random_seed=[42,43,44,45], target_accept=0.9)

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [sigma, Intercept, floor, 1|county_sigma, 1|county, floor|county_sigma, floor|county]

Sampling 4 chains for 3_000 tune and 2_000 draw iterations (12_000 + 8_000 draws total) took 62 seconds.
There were 1002 divergences after tuning. Increase `target_accept` or reparameterize.
The rhat statistic is larger than 1.01 for some parameters. This indicates problems during sampling. See https://arxiv.org/abs/1903.08008 for details
The effective sample size per chain is smaller than 100 for some parameters.  A higher number is needed for reliable rhat and ess computation. See https://arxiv.org/abs/1903.08008 for details

# az.autocorr(idata4["posterior"]["sigma"].values.flatten())[0:10]

idata4sel = idata4.sel(county__factor_dim=sel_counties)
az.summary(idata4sel, kind="stats")

	mean	sd	hdi_3%	hdi_97%
1\|county[LAC QUI PARLE]	0.392	0.296	-0.174	0.936
1\|county[AITKIN]	-0.282	0.250	-0.753	0.187
1\|county[KOOCHICHING]	-0.389	0.234	-0.835	0.040
1\|county[DOUGLAS]	0.167	0.198	-0.211	0.531
1\|county[HENNEPIN]	-0.091	0.088	-0.252	0.077
1\|county[STEARNS]	0.030	0.144	-0.253	0.293
1\|county[RAMSEY]	-0.271	0.130	-0.494	-0.002
1\|county[ST LOUIS]	-0.576	0.087	-0.747	-0.423
1\|county_sigma	0.333	0.046	0.250	0.422
Intercept	1.459	0.053	1.364	1.563
floor[ground]	-0.677	0.082	-0.829	-0.518
floor\|county[ground, LAC QUI PARLE]	0.125	0.266	-0.290	0.729
floor\|county[ground, AITKIN]	0.025	0.243	-0.462	0.513
floor\|county[ground, KOOCHICHING]	0.030	0.221	-0.394	0.485
floor\|county[ground, DOUGLAS]	0.036	0.249	-0.461	0.533
floor\|county[ground, HENNEPIN]	-0.074	0.168	-0.432	0.221
floor\|county[ground, STEARNS]	-0.080	0.211	-0.540	0.287
floor\|county[ground, RAMSEY]	0.178	0.257	-0.208	0.739
floor\|county[ground, ST LOUIS]	0.037	0.150	-0.263	0.334
floor\|county_sigma[ground]	0.230	0.140	0.021	0.454
sigma	0.752	0.019	0.714	0.785

az.summary(idata4, var_names="sigma", kind="stats")

	mean	sd	hdi_3%	hdi_97%
sigma	0.752	0.019	0.714	0.785

The estimated mean standard deviation is the lowest of all the models we considered so far.

plot_counties(radon, idata_cp=idata1, idata_np=idata2, idata_pp=idata3, idata_pp2=idata4);

../_images/607f2769d6166c1ba4c28937c9ff008e8eb2961b58f13aeb1392c681b257dc23.png

idata4sel = idata4.sel(county__factor_dim=sel_counties)
var_names = ["Intercept",
             "1|county",
             "floor",
             "floor|county",
             "1|county_sigma",
             "floor|county_sigma",
             "sigma"]
axs = az.plot_forest(idata4sel, combined=True, var_names=var_names, figsize=(6,4))
axs[0].set_xticks(np.arange(-1.6,1.6,0.2))
axs[0].set_title(None);

../_images/5e35d7429acef128b40cf129b0df4ad53097dbc2c4c64996cf537fd2a4878895.png

Comparing models#

from ministats.utils import loglevel
with loglevel("ERROR", module="pymc"):
    idata1ll = mod1.fit(idata_kwargs={"log_likelihood": True}, random_seed=[42,43,44,45], progressbar=False)
    idata2ll = mod2.fit(idata_kwargs={"log_likelihood": True}, random_seed=[42,43,44,45], progressbar=False)
    idata3ll = mod3.fit(idata_kwargs={"log_likelihood": True}, random_seed=[42,43,44,45], progressbar=False)
    idata4ll = mod4.fit(idata_kwargs={"log_likelihood": True}, random_seed=[42,43,44,45], progressbar=False,
                        draws=2000, tune=3000, target_accept=0.9)

There were 1002 divergences after tuning. Increase `target_accept` or reparameterize.
The effective sample size per chain is smaller than 100 for some parameters.  A higher number is needed for reliable rhat and ess computation. See https://arxiv.org/abs/1903.08008 for details

Compare models based on their expected log pointwise predictive density (ELPD).

The ELPD is estimated either by Pareto smoothed importance sampling leave-one-out cross-validation (LOO) or using the widely applicable information criterion (WAIC). We recommend loo. Read more theory here - in a paper by some of the leading authorities on model comparison dx.doi.org/10.1111/1467-9868.00353

radon_models = {
    "mod1 (complete pooling)": idata1ll,
    "mod2 (no pooling)": idata2ll,
    "mod3 (varying intercepts)": idata3ll,
    "mod4 (varying int. and slopes)": idata4ll,
}
compare_results = az.compare(radon_models)
compare_results

	rank	elpd_loo	p_loo	elpd_diff	weight	se	dse	warning	scale
mod3 (varying intercepts)	0	-1074.193864	49.481974	0.000000	6.425957e-01	28.159665	0.000000	False	log
mod4 (varying int. and slopes)	1	-1075.716479	62.934270	1.522615	2.812520e-01	28.776081	1.920789	True	log
mod2 (no pooling)	2	-1096.112086	83.676430	21.918221	1.705775e-14	28.273724	6.038902	True	log
mod1 (complete pooling)	3	-1126.868605	3.724237	52.674741	7.615230e-02	25.547441	10.576320	False	log

az.plot_compare(compare_results);

../_images/32b497664543ce3ad72493e939f1d426a5d44d3cf87e37ae727a0fb87c6b6c06.png

ELPD and elpd_loo#

The ELPD is the theoretical expected log pointwise predictive density for a new dataset (Eq 1 in VGG2017), which can be estimated, e.g., using cross-validation. elpd_loo is the Bayesian LOO estimate of the expected log pointwise predictive density (Eq 4 in VGG2017) and is a sum of N individual pointwise log predictive densities. Probability densities can be smaller or larger than 1, and thus log predictive densities can be negative or positive. For simplicity the ELPD acronym is used also for expected log pointwise predictive probabilities for discrete models. Probabilities are always equal or less than 1, and thus log predictive probabilities are 0 or negative.

via https://mc-stan.org/loo/reference/loo-glossary.html

Frequentist multilevel models#

We can use statsmodels to fit multilevel models too.

Random intercepts model using `statsmodels`#

import statsmodels.formula.api as smf
sm3 = smf.mixedlm("log_radon ~ 1 + floor",   # Fixed effects (no intercept and floor as a fixed effect)
                  groups="county",           # Grouping variable for random effects
                  re_formula="1",            # Random effects = intercept
                  data=radon)
res3 = sm3.fit()
res3.summary().tables[1]

	Coef.	Std.Err.	z	P>\|z\|	[0.025	0.975]
Intercept	1.462	0.052	28.164	0.000	1.360	1.563
floor[T.ground]	-0.693	0.071	-9.818	0.000	-0.831	-0.555
county Var	0.108	0.041

# slope
#######################################################
res3.params["floor[T.ground]"]

-0.6929937406558049

# sigma-hat
np.sqrt(res3.scale)

0.7555891484188184

# standard deviation of the variability among county-specific Intercepts
np.sqrt(res3.cov_re)

	county
county	0.328222

# intercept deviation for first country in res3
res3.random_effects["LAC QUI PARLE"].values

array([0.40649212])

# compare with corresponding Bayesian point estimate
lqp = idata3.sel(county__factor_dim="LAC QUI PARLE")
post3_lqp_means = lqp["posterior"].mean()
post3_lqp_means["1|county"].values

array(0.41299475)

This is very close to the mean of the random effect coefficient for AITKIN in the Bayesian model mod3 which was \(-0.268\).

Random intercepts and slopes model using `statsmodels` (BONUS TOPIC)#

sm4 = smf.mixedlm("log_radon ~ 1 + floor",   # Fixed effects (no intercept and floor as a fixed effect)
                  groups="county",           # Grouping variable for random effects
                  re_formula="1 + floor",    # Random effects: 1 for intercept, floor for slope
                  data=radon)
res4 = sm4.fit()
res4.summary().tables[1]

	Coef.	Std.Err.	z	P>\|z\|	[0.025	0.975]
Intercept	1.463	0.054	26.977	0.000	1.356	1.569
floor[T.ground]	-0.681	0.089	-7.624	0.000	-0.856	-0.506
county Var	0.122	0.049
county x floor[T.ground] Cov	-0.040	0.057
floor[T.ground] Var	0.118	0.120

# slope estimate
res4.params["floor[T.ground]"]

-0.6810977572101944

# sigma-hat
np.sqrt(res4.scale)

0.7461559982563549

# standard deviation of the variability among county-specific Intercepts
county_var_int = res4.cov_re.loc["county", "county"]
np.sqrt(county_var_int)

0.34867221107257784

# standard deviation of the variability among county-specific slopes
county_var_slopes = res4.cov_re.loc["floor[T.ground]", "floor[T.ground]"]
np.sqrt(county_var_slopes)

0.3435539748320311

# correlation between Intercept and slope group-level coefficients
county_floor_cov = res4.cov_re.loc["county", "floor[T.ground]"]
county_floor_cov / (np.sqrt(county_var_int)*np.sqrt(county_var_slopes))

-0.337230360777921

Discussion#

Alternative notations for hierarchical models#

IMPORT FROM Gelman & Hill Section 12.5 printout
watch the subscripts!

Computational challenges associated with hierarchical models#

centred vs. noncentred representations

Benefits of multilevel models#

TODO LIST
Better than repeated measures ANOVA because:
- tells you the direction and magnitude of effect
- can handle more multiple grouping scenarios (e.g. by-item, and by-student)
- works for categorical predictors

Applications#

Need for hierarchical models often occurs in social sciences (better than ANOVA)
Hierarchical models are often used for Bayesian meta-analysis

Exercises#

Exercise: mod1u#

Same model as Example 3 but also include the predictor log_uranium.

import bambi as bmb

covariate_priors = {
    "floor": bmb.Prior("Normal", mu=0, sigma=10),
    "log_uranium": bmb.Prior("Normal", mu=0, sigma=10),
    "1|county": bmb.Prior("Normal", mu=0, sigma=bmb.Prior("Exponential", lam=1)),
    "sigma": bmb.Prior("Exponential", lam=1),
}

mod1u = bmb.Model(formula="log_radon ~ 1 + floor + (1|county) + log_uranium",
                  priors=covariate_priors,
                  data=radon)

mod1u

       Formula: log_radon ~ 1 + floor + (1|county) + log_uranium
        Family: gaussian
          Link: mu = identity
  Observations: 919
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 1.2246, sigma: 2.1322)
            floor ~ Normal(mu: 0.0, sigma: 10.0)
            log_uranium ~ Normal(mu: 0.0, sigma: 10.0)
        
        Group-level effects
            1|county ~ Normal(mu: 0.0, sigma: Exponential(lam: 1.0))
        
        Auxiliary parameters
            sigma ~ Exponential(lam: 1.0)

Exercise: pigs dataset#

https://bambinos.github.io/bambi/notebooks/multi-level_regression.html

import statsmodels.api as sm
dietox = sm.datasets.get_rdataset("dietox", "geepack").data
dietox

	Pig	Evit	Cu	Litter	Start	Weight	Feed	Time
0	4601	Evit000	Cu000	1	26.5	26.50000	NaN	1
1	4601	Evit000	Cu000	1	26.5	27.59999	5.200005	2
2	4601	Evit000	Cu000	1	26.5	36.50000	17.600000	3
3	4601	Evit000	Cu000	1	26.5	40.29999	28.500000	4
4	4601	Evit000	Cu000	1	26.5	49.09998	45.200001	5
...	...	...	...	...	...	...	...	...
856	8442	Evit000	Cu175	24	25.7	73.19995	83.800003	8
857	8442	Evit000	Cu175	24	25.7	81.69995	99.800003	9
858	8442	Evit000	Cu175	24	25.7	90.29999	115.200001	10
859	8442	Evit000	Cu175	24	25.7	96.00000	133.200001	11
860	8442	Evit000	Cu175	24	25.7	103.50000	151.400002	12

861 rows × 8 columns

pigsmodel = bmb.Model("Weight ~ Time + (Time|Pig)", dietox)
pigsidata = pigsmodel.fit()

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [sigma, Intercept, Time, 1|Pig_sigma, 1|Pig_offset, Time|Pig_sigma, Time|Pig_offset]

Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 10 seconds.
The rhat statistic is larger than 1.01 for some parameters. This indicates problems during sampling. See https://arxiv.org/abs/1903.08008 for details

az.summary(pigsidata, var_names=["Intercept", "Time", "1|Pig_sigma", "Time|Pig_sigma", "sigma"])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
Intercept	15.755	0.575	14.625	16.777	0.021	0.015	748.0	1402.0	1.01
Time	6.939	0.079	6.787	7.080	0.004	0.002	497.0	1417.0	1.01
1\|Pig_sigma	4.534	0.438	3.797	5.403	0.013	0.009	1229.0	1824.0	1.00
Time\|Pig_sigma	0.664	0.061	0.556	0.779	0.002	0.001	1176.0	1919.0	1.00
sigma	2.460	0.065	2.337	2.579	0.001	0.001	5497.0	3107.0	1.00

Exercise: educational data#

cf. https://mc-stan.org/users/documentation/case-studies/tutorial_rstanarm.html

1.1 Data example We will be analyzing the Gcsemv dataset (Rasbash et al. 2000) from the mlmRev package in R. The data include the General Certificate of Secondary Education (GCSE) exam scores of 1,905 students from 73 schools in England on a science subject. The Gcsemv dataset consists of the following 5 variables:

school: school identifier
student: student identifier
gender: gender of a student (M: Male, F: Female)
written: total score on written paper
course: total score on coursework paper

import pyreadr

# Gcsemv_r = pyreadr.read_r('/Users/ivan/Downloads/mlmRev/data/Gcsemv.rda')
# Gcsemv_r["Gcsemv"].dropna().to_csv("../datasets/gcsemv.csv", index=False)

gcsemv = pd.read_csv("../datasets/gcsemv.csv")
gcsemv.head()

	school	student	gender	written	course
0	20920	27	F	39.0	76.8
1	20920	31	F	36.0	87.9
2	20920	42	M	16.0	44.4
3	20920	101	F	49.0	89.8
4	20920	113	M	25.0	17.5

import bambi as bmb
m1 = bmb.Model(formula="course ~ 1 + (1 | school)", data=gcsemv)
m1

       Formula: course ~ 1 + (1 | school)
        Family: gaussian
          Link: mu = identity
  Observations: 1523
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 73.3814, sigma: 41.0781)
        
        Group-level effects
            1|school ~ Normal(mu: 0.0, sigma: HalfNormal(sigma: 41.0781))
        
        Auxiliary parameters
            sigma ~ HalfStudentT(nu: 4.0, sigma: 16.4312)

idata_m1 = m1.fit()

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [sigma, Intercept, 1|school_sigma, 1|school_offset]

Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 4 seconds.

az.summary(idata_m1)

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
1\|school[20920]	-6.951	5.170	-16.906	2.466	0.065	0.061	6392.0	2921.0	1.00
1\|school[22520]	-15.649	2.086	-19.413	-11.615	0.043	0.031	2323.0	2661.0	1.00
1\|school[22710]	6.309	3.717	-0.480	13.257	0.052	0.041	5155.0	3018.0	1.00
1\|school[22738]	-0.771	4.358	-8.600	7.725	0.054	0.072	6569.0	3082.0	1.00
1\|school[22908]	-0.789	5.966	-11.454	10.829	0.072	0.095	6852.0	3257.0	1.00
...	...	...	...	...	...	...	...	...	...
1\|school[84707]	4.024	7.309	-9.999	17.522	0.095	0.107	5986.0	2988.0	1.00
1\|school[84772]	8.665	3.565	1.413	14.947	0.047	0.036	5784.0	2940.0	1.00
1\|school_sigma	8.817	0.887	7.170	10.477	0.028	0.020	1024.0	1709.0	1.01
Intercept	73.798	1.101	71.743	75.898	0.032	0.023	1151.0	1724.0	1.00
sigma	13.978	0.258	13.507	14.477	0.003	0.002	6922.0	2869.0	1.00

76 rows × 9 columns

m3 = bmb.Model(formula="course ~ gender + (1 + gender|school)", data=gcsemv)
m3

       Formula: course ~ gender + (1 + gender|school)
        Family: gaussian
          Link: mu = identity
  Observations: 1523
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 73.3814, sigma: 53.4663)
            gender ~ Normal(mu: 0.0, sigma: 83.5292)
        
        Group-level effects
            1|school ~ Normal(mu: 0.0, sigma: HalfNormal(sigma: 53.4663))
            gender|school ~ Normal(mu: 0.0, sigma: HalfNormal(sigma: 83.5292))
        
        Auxiliary parameters
            sigma ~ HalfStudentT(nu: 4.0, sigma: 16.4312)

idata_m3 = m3.fit()

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [sigma, Intercept, gender, 1|school_sigma, 1|school_offset, gender|school_sigma, gender|school_offset]

Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 9 seconds.

Exercise: sleepstudy dataset#

Description: Contains reaction times of subjects under sleep deprivation conditions.
Source: Featured in the R package lme4.
Application: Demonstrates linear mixed-effects modeling with random slopes and intercepts.

Links:

sleepstudy = bmb.load_data("sleepstudy")
sleepstudy

	Reaction	Days	Subject
0	249.5600	0	308
1	258.7047	1	308
2	250.8006	2	308
3	321.4398	3	308
4	356.8519	4	308
...	...	...	...
175	329.6076	5	372
176	334.4818	6	372
177	343.2199	7	372
178	369.1417	8	372
179	364.1236	9	372

180 rows × 3 columns

Exercise: tadpoles (BONUS)#

https://www.youtube.com/watch?v=iwVqiiXYeC4

logistic regression model

Section 5.5 — Hierarchical models

Contents

Section 5.5 — Hierarchical models#

Notebook setup#

Definitions#

Hierarchical (multilevel) models#

Model formula#

Radon dataset#

Loading the data#

Descriptive statistics#

Example 1: complete pooling model#

Bayesian model#

Bambi model#

Model fitting and analysis#

Conclusion#

Example 2: no pooling model#

Bayesian model#

Bambi model#

Model fitting and analysis#

Conclusion#

Example 3: hierarchical model#

Bayesian hierarchical model#

Bambi model#

Model fitting and analysis#

Compare models#

Conclusions#

Explanations#

Prior selection for hierarchical models#

Varying intercepts and slopes model#

Comparing models#

ELPD and elpd_loo#

Frequentist multilevel models#

Random intercepts model using statsmodels#

Random intercepts and slopes model using statsmodels (BONUS TOPIC)#

Discussion#

Alternative notations for hierarchical models#

Computational challenges associated with hierarchical models#

Benefits of multilevel models#

Applications#

Exercises#

Exercise: mod1u#

Exercise: pigs dataset#

Exercise: educational data#

Exercise: sleepstudy dataset#

Exercise: tadpoles (BONUS)#

Links#

EXTRA MATERIAL#

Random intercepts model using `statsmodels`#

Random intercepts and slopes model using `statsmodels` (BONUS TOPIC)#