Sharp regression discontinuity with `pymc` models#

import causalpy as cp

%load_ext autoreload
%autoreload 2
%config InlineBackend.figure_format = 'retina'
seed = 42

df = cp.load_data("rd")

Linear, main-effects, and interaction model#

Note

The random_seed keyword argument for the PyMC sampler is not necessary. We use it here so that the results are reproducible.

result = cp.RegressionDiscontinuity(
    df,
    formula="y ~ 1 + x + treated + x:treated",
    model=cp.pymc_models.LinearRegression(sample_kwargs={"random_seed": seed}),
    treatment_threshold=0.5,
).fit()

fig, ax = result.plot()
fig

Initializing NUTS using jitter+adapt_diag...

Multiprocess sampling (4 chains in 4 jobs)

NUTS: [beta, y_hat_sigma]

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

Sampling: [y_hat]

Sampling: [y_hat]

Sampling: [y_hat]

Sampling: [y_hat]

Sampling: [beta, y_hat, y_hat_sigma]

Sampling: [y_hat]

Sampling: [y_hat]

../_images/c5adf79d498651b531cd3e922baae2c7ed9d9a7b22ba4e3a8ee8d06ddc358365.png

Though we can see that this does not give a good fit of the data almost certainly overestimates the discontinuity at threshold.

Using a bandwidth#

One way how we could deal with this is to use the bandwidth kwarg. This will only fit the model to data within a certain bandwidth of the threshold. If \(x\) is the running variable, then the model will only be fitted to data where \(threshold - bandwidth \le x \le threshold + bandwidth\).

result = cp.RegressionDiscontinuity(
    df,
    formula="y ~ 1 + x + treated + x:treated",
    model=cp.pymc_models.LinearRegression(sample_kwargs={"random_seed": seed}),
    treatment_threshold=0.5,
    bandwidth=0.3,
).fit()

fig, ax = result.plot()

Initializing NUTS using jitter+adapt_diag...

Multiprocess sampling (4 chains in 4 jobs)

NUTS: [beta, y_hat_sigma]

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

Sampling: [y_hat]

Sampling: [y_hat]

Sampling: [y_hat]

Sampling: [y_hat]

Sampling: [beta, y_hat, y_hat_sigma]

Sampling: [y_hat]

Sampling: [y_hat]

We could even go crazy and just fit intercepts for the data close to the threshold. But clearly this will involve more estimation error as we are using less data.

result = cp.RegressionDiscontinuity(
    df,
    formula="y ~ 1 + treated",
    model=cp.pymc_models.LinearRegression(sample_kwargs={"random_seed": seed}),
    treatment_threshold=0.5,
    bandwidth=0.2,
).fit()

fig, ax = result.plot()

Initializing NUTS using jitter+adapt_diag...

Multiprocess sampling (4 chains in 4 jobs)

NUTS: [beta, y_hat_sigma]

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

Sampling: [y_hat]

Sampling: [y_hat]

Sampling: [y_hat]

Sampling: [y_hat]

Sampling: [beta, y_hat, y_hat_sigma]

Sampling: [y_hat]

Sampling: [y_hat]

Using basis splines#

Though it could arguably be better to fit with a more complex model, fit example a spline. This allows us to use all of the data, and (depending on the situation) maybe give a better fit.

result = cp.RegressionDiscontinuity(
    df,
    formula="y ~ 1 + bs(x, df=6) + treated",
    model=cp.pymc_models.LinearRegression(sample_kwargs={"random_seed": seed}),
    treatment_threshold=0.5,
).fit()

fig, ax = result.plot()

Initializing NUTS using jitter+adapt_diag...

Multiprocess sampling (4 chains in 4 jobs)

NUTS: [beta, y_hat_sigma]

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

Sampling: [y_hat]

Sampling: [y_hat]

Sampling: [y_hat]

Sampling: [y_hat]

Sampling: [beta, y_hat, y_hat_sigma]

Sampling: [y_hat]

Sampling: [y_hat]

As with all of the models in this notebook, we can ask for a summary of the model coefficients.

result.summary()

Regression Discontinuity experiment
Formula: y ~ 1 + bs(x, df=6) + treated
Running variable: x
Threshold on running variable: 0.5
Bandwidth: inf
Donut hole: 0.0
Observations used for fit: 100

Results:
Discontinuity at threshold = 0.50$CI_{94\%}$[0.35, 0.67]


Model coefficients:
  Intercept        -0.16, 94% HDI [-0.26, -0.067]
  treated[T.True]  0.5, 94% HDI [0.34, 0.66]
  bs(x, df=6)[0]   -0.4, 94% HDI [-0.59, -0.2]
  bs(x, df=6)[1]   -1.4, 94% HDI [-1.6, -1.3]
  bs(x, df=6)[2]   0.27, 94% HDI [0.12, 0.43]
  bs(x, df=6)[3]   1.5, 94% HDI [1.4, 1.7]
  bs(x, df=6)[4]   0.89, 94% HDI [0.54, 1.3]
  bs(x, df=6)[5]   0.33, 94% HDI [0.12, 0.56]

We can get nicely formatted tables from our integration with the maketables package.

from maketables import ETable

result.set_maketables_options(hdi_prob=0.95)
ETable(result, coef_fmt="b:.3f \n [ci95l:.3f, ci95u:.3f]")

	y
	(1)
coef
treated=True	0.504 [0.337, 0.669]
bs(x, df=6)=0	-0.402 [-0.598, -0.194]
bs(x, df=6)=1	-1.425 [-1.572, -1.261]
bs(x, df=6)=2	0.268 [0.095, 0.427]
bs(x, df=6)=3	1.506 [1.350, 1.663]
bs(x, df=6)=4	0.893 [0.534, 1.274]
bs(x, df=6)=5	0.334 [0.108, 0.572]
Intercept	-0.165 [-0.272, -0.066]
stats
N	100
Bayesian R2	0.984
Format of coefficient cell: Coefficient [95% CI Lower, 95% CI Upper]

Effect Summary Reporting#

For decision-making, you often need a concise summary of the causal effect. The effect_summary() method provides a decision-ready report with key statistics. Note that for Regression Discontinuity, the effect is a single scalar (the discontinuity at the threshold), similar to Difference-in-Differences.

# Generate effect summary
stats = result.effect_summary()
stats.table

	mean	median	hdi_lower	hdi_upper	p_gt_0
discontinuity	0.504481	0.503704	0.338363	0.667881	1.0

print(stats.text)

The discontinuity at threshold was 0.50 (95% HDI [0.34, 0.67]). The posterior probability of an increase is 1.000.

You can customize the summary with different directions and ROPE thresholds:

Direction: Test for increase, decrease, or two-sided effect
Alpha: Set the HDI confidence level (default 95%)
ROPE: Specify a minimal effect size threshold

# Example: Two-sided test with ROPE
stats = result.effect_summary(
    direction="two-sided",
    alpha=0.05,
    min_effect=0.2,  # Region of Practical Equivalence
)
stats.table

	mean	median	hdi_lower	hdi_upper	p_two_sided	prob_of_effect	p_rope
discontinuity	0.504481	0.503704	0.338363	0.667881	0.0	1.0	1.0

print("\n" + stats.text)

The discontinuity at threshold was 0.50 (95% HDI [0.34, 0.67]). The two-sided tail probability is 0.000. Using the closed ROPE [-0.2, 0.2], the 95% HDI is entirely outside the ROPE; the effect is practically significant. Posterior mass is 0.000 below, 0.000 inside, and 1.000 above the ROPE.

Sharp regression discontinuity with pymc models#