The Best Of Both Worlds: Hierarchical Linear Regression in PyMC

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Author

Thomas Wiecki & Danne Elbers

Published

March 17, 2014

The best of both worlds: Hierarchical Linear Regression in PyMC3

  1. Thomas Wiecki & Danne Elbers 2020

The power of Bayesian modelling really clicked for me when I was first introduced to hierarchical modelling. In this blog post we will highlight the advantage of using hierarchical Bayesian modelling as opposed to non-hierarchical Bayesian modelling. This hierachical modelling is especially advantageous when multi-level data is used, making the most of all information available by its ‘shrinkage-effect’, which will be explained below.

Having multiple sets of measurements comes up all the time, in Psychology for example you test multiple subjects on the same task. You then might want to estimate a model that describes the behavior as a set of parameters relating to mental functioning. Often we are interested in individual differences of these parameters but also assume that subjects share similarities (being human and all). Software from our lab, HDDM, allows hierarchical Bayesian estimation of a widely used decision making model but we will use a more classical example of hierarchical linear regression here to predict radon levels in houses.

This is the 3rd blog post on the topic of Bayesian modeling in PyMC3, see here for the previous two:

The data set

Gelman et al.’s (2007) radon dataset is a classic for hierarchical modeling. In this dataset the amount of the radioactive gas radon has been measured among different households in all county’s of several states. Radon gas is known to be the highest cause of lung cancer in non-smokers. It is believed to enter the house through the basement. Moreover, its concentration is thought to differ regionally due to different types of soil.

Here we’ll investigate this difference and try to make predictions of radon levels in different countys and where in the house radon was measured. In this example we’ll look at Minnesota, a state that contains 85 county’s in which different measurements are taken, ranging from 2 till 80 measurements per county.

radon

First, we’ll load the data:

%matplotlib inline

import matplotlib.pyplot as plt
import numpy as np
import pymc3 as pm 
import pandas as pd

data = pd.read_csv('radon.csv')

county_names = data.county.unique()
county_idx = data['county_code'].values

The relevant part of the data we will model looks as follows:

data[['county', 'log_radon', 'floor']].head()
county log_radon floor
0 AITKIN 0.832909 1.0
1 AITKIN 0.832909 0.0
2 AITKIN 1.098612 0.0
3 AITKIN 0.095310 0.0
4 ANOKA 1.163151 0.0

As you can see, we have multiple radon measurements (log-converted to be on the real line) in a county and whether the measurement has been taken in the basement (floor == 0) or on the first floor (floor == 1). Here we want to test the prediction that radon concentrations are higher in the basement.

The Models

Pooling of measurements

Now you might say: “That’s easy! I’ll just pool all my data and estimate one big regression to asses the influence of measurement across all counties”. In math-speak that model would be:

\[radon_{i, c} = \alpha + \beta*\text{floor}_{i, c} + \epsilon\]

Where \(i\) represents the measurement, \(c\) the county and floor contains which floor the measurement was made. If you need a refresher on Linear Regressions in PyMC3, check out my previous blog post. Critically, we are only estimating one intercept and one slope for all measurements over all counties.

Separate regressions

But what if we are interested whether different counties actually have different relationships (slope) and different base-rates of radon (intercept)? Then you might say “OK then, I’ll just estimate \(n\) (number of counties) different regresseions – one for each county”. In math-speak that model would be:

\[radon_{i, c} = \alpha_{c} + \beta_{c}*\text{floor}_{i, c} + \epsilon_c\]

Note that we added the subindex \(c\) so we are estimating \(n\) different \(\alpha\)s and \(\beta\)s – one for each county.

This is the extreme opposite model, where above we assumed all counties are exactly the same, here we are saying that they share no similarities whatsoever which ultimately is also unsatisifying.

Hierarchical Regression: The best of both worlds

Fortunately there is a middle ground to both of these extreme views. Specifically, we may assume that while \(\alpha\)s and \(\beta\)s are different for each county, the coefficients all come from a common group distribution:

\[\alpha_{c} \sim \mathcal{N}(\mu_{\alpha}, \sigma_{\alpha}^2)\] \[\beta_{c} \sim \mathcal{N}(\mu_{\beta}, \sigma_{\beta}^2)\]

We thus assume the intercepts \(\alpha\) and slopes \(\beta\) to come from a normal distribution centered around their respective group mean \(\mu\) with a certain standard deviation \(\sigma^2\), the values (or rather posteriors) of which we also estimate. That’s why this is called multilevel or hierarchical modeling.

How do we estimate such a complex model with all these parameters you might ask? Well, that’s the beauty of Probabilistic Programming – we just formulate the model we want and press our Inference Button(TM).

Note that the above is not a complete Bayesian model specification as we haven’t defined priors or hyperpriors (i.e. priors for the group distribution, \(\mu\) and \(\sigma\)). These will be used in the model implementation below but only distract here.

Probabilistic Programming

Individual/non-hierarchical model

To really highlight the effect of the hierarchical linear regression we’ll first estimate the non-hierarchical Bayesian model from above (separate regressions). For each county a new estimate of the parameters is initiated. As we have no prior information on what the intercept or regressions could be we are placing a Normal distribution centered around 0 with a wide standard-deviation. We’ll assume the measurements are normally distributed with noise \(\epsilon\) on which we place a Half-Cauchy distribution.

# takes about 45 minutes
indiv_traces = {}
for county_name in county_names:
    # Select subset of data belonging to county
    c_data = data.loc[data.county == county_name]
    c_data = c_data.reset_index(drop=True)
    
    c_log_radon = c_data.log_radon
    c_floor_measure = c_data.floor.values
    
    with pm.Model() as individual_model:
        # Intercept prior
        a = pm.Normal('alpha', mu=0, sigma=1)
        # Slope prior
        b = pm.Normal('beta', mu=0, sigma=1)
    
        # Model error prior
        eps = pm.HalfCauchy('eps', beta=1)
    
        # Linear model
        radon_est = a + b * c_floor_measure
    
        # Data likelihood
        y_like = pm.Normal('y_like', mu=radon_est, sigma=eps, observed=c_log_radon)

        # Inference button (TM)!
        trace = pm.sample(progressbar=False)
        
    indiv_traces[county_name] = trace

Hierarchical Model

Instead of initiating the parameters separatly, the hierarchical model initiates group parameters that consider the county’s not as completely different but as having an underlying similarity. These distributions are subsequently used to influence the distribution of each county’s \(\alpha\) and \(\beta\).

with pm.Model() as hierarchical_model:
    # Hyperpriors
    mu_a = pm.Normal('mu_alpha', mu=0., sigma=1)
    sigma_a = pm.HalfCauchy('sigma_alpha', beta=1)
    mu_b = pm.Normal('mu_beta', mu=0., sigma=1)
    sigma_b = pm.HalfCauchy('sigma_beta', beta=1)
    
    # Intercept for each county, distributed around group mean mu_a
    a = pm.Normal('alpha', mu=mu_a, sigma=sigma_a, shape=len(data.county.unique()))
    # Intercept for each county, distributed around group mean mu_a
    b = pm.Normal('beta', mu=mu_b, sigma=sigma_b, shape=len(data.county.unique()))
    
    # Model error
    eps = pm.HalfCauchy('eps', beta=1)
    
    # Expected value
    radon_est = a[county_idx] + b[county_idx] * data.floor.values
    
    # Data likelihood
    y_like = pm.Normal('y_like', mu=radon_est, sigma=eps, observed=data.log_radon)
with hierarchical_model:
    hierarchical_trace = pm.sample()
Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (2 chains in 2 jobs)
NUTS: [eps, beta, alpha, sigma_beta, mu_beta, sigma_alpha, mu_alpha]
Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 29 seconds.
There were 3 divergences after tuning. Increase `target_accept` or reparameterize.
There were 47 divergences after tuning. Increase `target_accept` or reparameterize.
The acceptance probability does not match the target. It is 0.7013813577935659, but should be close to 0.8. Try to increase the number of tuning steps.
The rhat statistic is larger than 1.05 for some parameters. This indicates slight problems during sampling.
The estimated number of effective samples is smaller than 200 for some parameters.
100.00% [4000/4000 00:18<00:00 Sampling 2 chains, 50 divergences] 
pm.traceplot(hierarchical_trace);

The marginal posteriors in the left column are highly informative. mu_a tells us the group mean (log) radon levels. mu_b tells us that the slope is significantly negative (no mass above zero), meaning that radon concentrations are higher in the basement than first floor. We can also see by looking at the marginals for a that there is quite some differences in radon levels between counties; the different widths are related to how much measurements we have per county, the more, the higher our confidence in that parameter estimate.

After writing this blog post I found out that the chains here (which look worse after I just re-ran them) are not properly converged, you can see that best for sigma_beta but also the warnings about “diverging samples” (which are also new in PyMC3). If you want to learn more about the problem and its solution, see my more recent blog post “Why hierarchical models are awesome, tricky, and Bayesian”.

Posterior Predictive Check

The Root Mean Square Deviation

To find out which of the models works better we can calculate the Root Mean Square Deviaton (RMSD). This posterior predictive check revolves around recreating the data based on the parameters found at different moments in the chain. The recreated or predicted values are subsequently compared to the real data points, the model that predicts data points closer to the original data is considered the better one. Thus, the lower the RMSD the better.

When computing the RMSD (code not shown) we get the following result:

  • individual/non-hierarchical model: 0.13
  • hierarchical model: 0.08

As can be seen above the hierarchical model performs a lot better than the non-hierarchical model in predicting the radon values. Following this, we’ll plot some examples of county’s showing the true radon values, the hierarchial predictions and the non-hierarchical predictions.

selection = ['CASS', 'CROW WING', 'FREEBORN']
fig, axis = plt.subplots(1, 3, figsize=(12, 6), sharey=True, sharex=True)
axis = axis.ravel()
for i, c in enumerate(selection):
    c_data = data.loc[data.county == c]
    c_data = c_data.reset_index(drop = True)
    z = list(c_data['county_code'])[0]

    xvals = np.linspace(-0.2, 1.2)
    for a_val, b_val in zip(indiv_traces[c]['alpha'][::10], indiv_traces[c]['beta'][::10]):
        axis[i].plot(xvals, a_val + b_val * xvals, 'b', alpha=.05)
    axis[i].plot(xvals, indiv_traces[c]['alpha'][::10].mean() + indiv_traces[c]['beta'][::10].mean() * xvals, 
                 'b', alpha=1, lw=2., label='individual')
    for a_val, b_val in zip(hierarchical_trace['alpha'][::10][z], hierarchical_trace['beta'][::10][z]):
        axis[i].plot(xvals, a_val + b_val * xvals, 'g', alpha=.05)
    axis[i].plot(xvals, hierarchical_trace['alpha'][::10][z].mean() + hierarchical_trace['beta'][::10][z].mean() * xvals, 
                 'g', alpha=1, lw=2., label='hierarchical')
    axis[i].scatter(c_data.floor + np.random.randn(len(c_data))*0.01, c_data.log_radon, 
                    alpha=1, color='k', marker='.', s=80, label='original data')
    axis[i].set_xticks([0,1])
    axis[i].set_xticklabels(['basement', 'first floor'])
    axis[i].set_ylim(-1, 4)
    axis[i].set_title(c)
    if not i%3:
        axis[i].legend()
        axis[i].set_ylabel('log radon level')

In the above plot we have the data points in black of three selected counties. The thick lines represent the mean estimate of the regression line of the individual (blue) and hierarchical model (in green). The thinner lines are regression lines of individual samples from the posterior and give us a sense of how variable the estimates are.

When looking at the county ‘CASS’ we see that the non-hierarchical estimation has huge uncertainty about the radon levels of first floor measurements – that’s because we don’t have any measurements in this county. The hierarchical model, however, is able to apply what it learned about the relationship between floor and radon-levels from other counties to CASS and make sensible predictions even in the absence of measurements.

We can also see how the hierarchical model produces more robust estimates in ‘CROW WING’ and ‘FREEBORN’. In this regime of few data points the non-hierarchical model reacts more strongly to individual data points because that’s all it has to go on.

Having the group-distribution constrain the coefficients we get meaningful estimates in all cases as we apply what we learn from the group to the individuals and vice-versa.

Shrinkage

Shrinkage describes the process by which our estimates are “pulled” towards the group-mean as a result of the common group distribution – county-coefficients very far away from the group mean have very low probability under the normality assumption. In the non-hierachical model every county is allowed to differ completely from the others by just using each county’s data, resulting in a model more prone to outliers (as shown above).

hier_a = hierarchical_trace['alpha'].mean(axis=0)
hier_b = hierarchical_trace['beta'].mean(axis=0)
indv_a = [indiv_traces[c]['alpha'].mean() for c in county_names]
indv_b = [indiv_traces[c]['beta'].mean() for c in county_names]
fig = plt.figure(figsize=(10, 10))
ax = fig.add_subplot(111, xlabel='Intercept', ylabel='Floor Measure', 
                     title='Hierarchical vs. Non-hierarchical Bayes', 
                     xlim=(0.25, 2), ylim=(-2, 1.5))

ax.scatter(indv_a,indv_b, s=26, alpha=0.4, label = 'non-hierarchical')
ax.scatter(hier_a,hier_b, c='red', s=26, alpha=0.4, label = 'hierarchical')
for i in range(len(indv_b)):  
    ax.arrow(indv_a[i], indv_b[i], hier_a[i] - indv_a[i], hier_b[i] - indv_b[i], 
             fc="k", ec="k", length_includes_head=True, alpha=0.4, head_width=.02)
ax.legend();

In the shrinkage plot above we show the coefficients of each county’s non-hierarchical posterior mean (blue) and the hierarchical posterior mean (red). To show the effect of shrinkage on a single coefficient-pair (alpha and beta) we connect the blue and red points belonging to the same county by an arrow. Some non-hierarchical posteriors are so far out that we couldn’t display them in this plot (it makes the axes to wide). Interestingly, all hierarchical posteriors of the floor-measure seem to be around -0.6 confirming out prediction that radon levels are higher in the basement than in the first floor. The differences in intercepts (which we take for type of soil) differs among countys indicating that meaningful regional differences exist in radon concentration. This information would have been difficult to find when just the non-hierarchial model had been used and estimates for individual counties would have been much more noisy.

Summary

In this post, co-authored by Danne Elbers, we showed how a multi-level hierarchical Bayesian model gives the best of both worlds when we have multiple sets of measurements we expect to have similarity. The naive approach either pools all data together and ignores the individual differences, or treats each set as completely separate leading to noisy estimates as shown above. By placing a group distribution on the individual sets we can learn about each set and the group simultaneously. Probabilistic Programming in PyMC then makes Bayesian estimation of this model trivial.

References