Interrupted Time Series for fixed-period interventions#

Interrupted time series experiments could be broken down into two types.

Interventions that occur at a point in time
Interventions that have a fixed period.

This notebook deals with the latter case, where there is no intervention up to a point where an intervention is in effect for a fixed period of time. Examples might include marketing promotions, price discounts, or public policy interventions.

When we are dealing with fixed-period interventions, we can make our interrupted time series experiment aware of this by providing treatment_end_time as a keyword argument. This then creates 3 distinct time periods:

Pre-intervention period: Before any treatment has taken effect. This is what we fit our model to.
Intervention period: When treatment is active (from treatment_time to treatment_end_time)
Post-intervention period: After treatment ends

This enables analysis of immediate effects, effect persistence, and decay patterns.

Note

For standard two-period ITS analysis (permanent interventions), see Bayesian Interrupted Time Series (ITS) Analysis in Python.

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

import causalpy as cp

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

Example: Marketing Campaign#

We simulate a 12-week marketing campaign with an immediate effect (+25 units) and partial persistence after it ends (+8 units, ~30% persistence).

Treatment starts: 2023-09-17 00:00:00
Treatment ends: 2023-12-10 00:00:00
Intervention period: 12 weeks
Post-intervention period: 56 weeks

Let’s first visualize the raw time series data to get an intuitive sense of the intervention effect

/var/folders/f0/rbz8xs8s17n3k3f_ccp31bvh0000gn/T/ipykernel_54515/4071325791.py:22: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown

Run the Analysis#

Specify treatment_end_time to enable three-period analysis:

result = cp.InterruptedTimeSeries(
    df,
    treatment_time=treatment_time,
    treatment_end_time=treatment_end_time,
    formula="y ~ 1 + t + C(month)",
    model=cp.pymc_models.LinearRegression(
        sample_kwargs={
            "random_seed": seed,
            "progressbar": False,
        }
    ),
).fit()

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]

Visualization#

The three-period design visualization adds a vertical line to mark where the treatment ends:

Solid red line: treatment_time (intervention start)
Dashed orange line: treatment_end_time (intervention end)

The plot shows three panels:

Top panel: Time series with observations, counterfactual predictions, and causal impact shading
Middle panel: Pointwise causal impact over time
Bottom panel: Cumulative causal impact

The vertical line at treatment_end_time clearly separates the intervention period from the post-intervention period, allowing you to visually assess effect persistence and decay.

fig, ax = result.plot()
plt.show()

/Users/carlostrujillo/Documents/github/_worktrees/CausalPy/migration-821-its-sensitivity/causalpy/experiments/base.py:759: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown
/var/folders/f0/rbz8xs8s17n3k3f_ccp31bvh0000gn/T/ipykernel_54515/1616332287.py:2: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown

Period-Specific Summaries#

Get separate summaries for each period using the period parameter:

# Intervention period
intervention_summary = result.effect_summary(period="intervention")
print(intervention_summary.text)

During the During intervention (2023-09-17 00:00:00 to 2023-12-03 00:00:00), the response variable had an average value of approx. 140.87. By contrast, in the absence of an intervention, we would have expected an average response of 116.31. The 95% interval of this counterfactual prediction is [112.84, 119.93]. Subtracting this prediction from the observed response yields an estimate of the causal effect the intervention had on the response variable. This effect is 24.55 with a 95% interval of [20.94, 28.02].

Summing up the individual data points during the During intervention, the response variable had an overall value of 1690.40. By contrast, had the intervention not taken place, we would have expected a sum of 1395.76. The 95% interval of this prediction is [1354.10, 1439.13]. The cumulative effect is 294.64 with a 95% HDI [251.27, 336.30].

The posterior probability of an increase is 1.000. For the cumulative effect, The posterior probability of an increase is 1.000. Relative to the counterfactual, the effect represents a 21.14% change (95% HDI [17.46%, 24.84%]).

This analysis assumes that the relationship between the time-based predictors and the response observed during the pre-intervention period remains stable throughout the post-intervention period. If the formula includes external covariates, it further assumes they were not themselves affected by the intervention. We recommend inspecting model fit, examining pre-intervention trends, and conducting sensitivity analyses (e.g., placebo tests) to support any causal conclusions drawn from this analysis.

# Post-intervention period
post_summary = result.effect_summary(period="post")
print(post_summary.text)

During the Post-intervention (2023-12-10 00:00:00 to 2024-12-29 00:00:00), the response variable had an average value of approx. 122.63. By contrast, in the absence of an intervention, we would have expected an average response of 116.24. The 95% interval of this counterfactual prediction is [111.77, 120.48]. Subtracting this prediction from the observed response yields an estimate of the causal effect the intervention had on the response variable. This effect is 6.39 with a 95% interval of [2.15, 10.86].

Summing up the individual data points during the Post-intervention, the response variable had an overall value of 6867.34. By contrast, had the intervention not taken place, we would have expected a sum of 6509.58. The 95% interval of this prediction is [6259.17, 6746.98]. The cumulative effect is 357.76 with a 95% HDI [120.36, 608.17].

The posterior probability of an increase is 0.998. For the cumulative effect, The posterior probability of an increase is 0.998. Relative to the counterfactual, the effect represents a 5.53% change (95% HDI [1.43%, 9.32%]).

This analysis assumes that the relationship between the time-based predictors and the response observed during the pre-intervention period remains stable throughout the post-intervention period. If the formula includes external covariates, it further assumes they were not themselves affected by the intervention. We recommend inspecting model fit, examining pre-intervention trends, and conducting sensitivity analyses (e.g., placebo tests) to support any causal conclusions drawn from this analysis.

Comparison Summary#

Use period='comparison' to get a comparative summary showing persistence metrics:

comparison_summary = result.effect_summary(period="comparison")
print(comparison_summary.text)

Effect persistence: The post-intervention effect (6.4, 95% HDI [2.1, 10.9]) was 26.0% of the intervention effect (24.6, 95% HDI [20.9, 28.0]), with a posterior probability of 1.00 that some effect persisted beyond the intervention period.

The comparison summary provides:

Post-intervention effect as percentage of intervention effect
Posterior probability that some effect persisted
HDI interval comparison between periods

Detailed Persistence Analysis#

The analyze_persistence() method automatically prints and returns a detailed summary of effect persistence:

persistence = result.analyze_persistence()

============================================================
Effect Persistence Analysis
============================================================

During intervention period:
  Mean effect: 24.55
  94% HDI: [21.13, 27.93]
  Total effect: 294.64

Post-intervention period:
  Mean effect: 6.39
  94% HDI: [2.15, 10.45]
  Total effect: 357.76

Persistence ratio: 0.260
  (26.0% of intervention effect persisted)
============================================================

# Access the returned dictionary:
print("\nAccessing results programmatically:")
print(f"  Mean effect during: {persistence['mean_effect_during']:.2f}")
print(f"  Mean effect post: {persistence['mean_effect_post']:.2f}")
print(
    f"  Persistence ratio: {persistence['persistence_ratio']:.3f} ({persistence['persistence_ratio'] * 100:.1f}%)"
)
print(f"  Total effect during: {persistence['total_effect_during']:.2f}")
print(f"  Total effect post: {persistence['total_effect_post']:.2f}")

Accessing results programmatically:
  Mean effect during: 24.55
  Mean effect post: 6.39
  Persistence ratio: 0.260 (26.0%)
  Total effect during: 294.64
  Total effect post: 357.76

Sensitivity checks in the pipeline#

Up to this point, we have estimated an effect and summarized how much of it persists. Sensitivity checks ask two follow-up questions before we get too comfortable with that story:

Check	Question it answers	Reassuring pattern
`PlaceboInTime`	If we move the intervention window into the pre-treatment period, do we still recover an effect this large?	Fake intervention windows look much smaller than the observed effect.
`PersistenceCheck`	How much of the estimated effect remains after treatment ends?	The persistence ratio matches the carryover story you want to tell.

For fixed-period ITS, a placebo fold has to shift the whole intervention window, not just the start date. If the real campaign lasted 12 weeks, each fake campaign should also last 12 weeks so the placebo problem matches the real one. The code below handles that with a small experiment_factory built on SensitivityAnalysis.default_for(cp.InterruptedTimeSeries).

We use four placebo folds here because one or two can give a noisy impression of ordinary pre-treatment variation, while many more folds slow the notebook and leave less pre-period data for each fake intervention.

The pipeline in the next cell runs once. The cells after that only interpret the stored results.

Tip

The sampler settings below are intentionally smaller than the earlier model fit so the workflow stays quick to rerun in a notebook. For substantive analyses, increase draws, tune steps, and placebo folds.

Note

Under the lazy experiment lifecycle, an experiment_factory must return a fitted experiment, so make_placebo_its calls .fit() on each placebo fold. The placebo formula pins the month levels (C(month, levels=[1, ..., 12])) because a shifted fold only spans part of the year; without pinned levels, patsy infers the categories from the first slice and then rejects folds that contain months missing from it.

fit_sample_kwargs = {
    "draws": 300,
    "tune": 300,
    "chains": 2,
    "cores": 1,
    "progressbar": False,
    "random_seed": seed,
    "compute_convergence_checks": False,
}
placebo_sample_kwargs = {
    "draws": 100,
    "tune": 100,
    "chains": 2,
    "cores": 1,
    "progressbar": False,
    "random_seed": seed,
    "compute_convergence_checks": False,
}
n_placebo_folds = 4
intervention_length = treatment_end_time - treatment_time
time_step = df.index[1] - df.index[0]
formula = "y ~ 1 + t + C(month, levels=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12])"


def make_placebo_its(data, pseudo_treatment_time):
    # Shift the full intervention window for fixed-period ITS placebo folds.
    # The lazy lifecycle requires the factory to return a fitted experiment.
    return cp.InterruptedTimeSeries(
        data,
        treatment_time=pseudo_treatment_time,
        treatment_end_time=pseudo_treatment_time + intervention_length - time_step,
        formula=formula,
        model=cp.pymc_models.LinearRegression(sample_kwargs=fit_sample_kwargs),
    ).fit()


default_sensitivity = cp.SensitivityAnalysis.default_for(cp.InterruptedTimeSeries)

for check in default_sensitivity.checks:
    if isinstance(check, cp.checks.PlaceboInTime):
        check.n_folds = n_placebo_folds
        check.sample_kwargs.update(placebo_sample_kwargs)
        check.experiment_factory = make_placebo_its

its_sensitivity = cp.SensitivityAnalysis(
    checks=[*default_sensitivity.checks, cp.checks.PersistenceCheck()]
)

pipeline_result = cp.Pipeline(
    data=df,
    steps=[
        cp.EstimateEffect(
            method=cp.InterruptedTimeSeries,
            treatment_time=treatment_time,
            treatment_end_time=treatment_end_time,
            formula=formula,
            model=cp.pymc_models.LinearRegression(sample_kwargs=fit_sample_kwargs),
        ),
        its_sensitivity,
    ],
).run()

sensitivity_results = {
    check_result.check_name: check_result
    for check_result in pipeline_result.sensitivity_results
}
placebo_result = sensitivity_results["PlaceboInTime"]
persistence_result = sensitivity_results["PersistenceCheck"]

Initializing NUTS using jitter+adapt_diag...

Sequential sampling (2 chains in 1 job)

NUTS: [beta, y_hat_sigma]

Sampling 2 chains for 300 tune and 300 draw iterations (600 + 600 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]

/Users/carlostrujillo/Documents/github/_worktrees/CausalPy/migration-821-its-sensitivity/causalpy/steps/sensitivity.py:209: UserWarning: PlaceboInTime placebo windows span 12 observation(s) but the actual effect is summarised over 68 post-intervention observation(s). The actual cumulative impact therefore accumulates over a longer span than the null distribution it is compared against, which inflates P(actual outside null). Lengthen intervention_length, or interpret the verdict as an upper bound.

Initializing NUTS using jitter+adapt_diag...

Sequential sampling (2 chains in 1 job)

NUTS: [beta, y_hat_sigma]

Sampling 2 chains for 300 tune and 300 draw iterations (600 + 600 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]

Initializing NUTS using jitter+adapt_diag...

Sequential sampling (2 chains in 1 job)

NUTS: [beta, y_hat_sigma]

Sampling 2 chains for 300 tune and 300 draw iterations (600 + 600 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]

Initializing NUTS using jitter+adapt_diag...

Sequential sampling (2 chains in 1 job)

NUTS: [beta, y_hat_sigma]

Sampling 2 chains for 300 tune and 300 draw iterations (600 + 600 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]

Initializing NUTS using jitter+adapt_diag...

Sequential sampling (2 chains in 1 job)

NUTS: [beta, y_hat_sigma]

Sampling 2 chains for 300 tune and 300 draw iterations (600 + 600 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]

Initializing NUTS using jitter+adapt_diag...

Sequential sampling (2 chains in 1 job)

NUTS: [mu_status_quo, tau_status_quo, fold_z]

Sampling 2 chains for 100 tune and 100 draw iterations (200 + 200 draws total) took 1 seconds.

Sampling: [theta_new]

============================================================
Effect Persistence Analysis
============================================================

During intervention period:
  Mean effect: 24.53
  94% HDI: [21.12, 27.50]
  Total effect: 294.32

Post-intervention period:
  Mean effect: 6.37
  94% HDI: [2.60, 10.67]
  Total effect: 356.67

Persistence ratio: 0.260
  (26.0% of intervention effect persisted)
============================================================

from IPython.display import Markdown, display

display(
    Markdown(
        f"""
The pipeline above ran once. The cells below reuse `pipeline_result.sensitivity_results`, so the remaining sections interpret stored results rather than rerunning the checks.

- `PlaceboInTime` asks whether fake pre-treatment windows can mimic the observed effect.
- `PersistenceCheck` asks how much of the estimated effect remains after treatment ends.
- Default ITS checks: `{[type(check).__name__ for check in default_sensitivity.checks]}`
- Checks run in this example: `{list(sensitivity_results)}`
- Configured placebo folds in this notebook: `{n_placebo_folds}`
- Why four? More folds give a better sense of ordinary pre-treatment variation, but each extra fold requires another full refit.
"""
    )
)

The pipeline above ran once. The cells below reuse pipeline_result.sensitivity_results, so the remaining sections interpret stored results rather than rerunning the checks.

PlaceboInTime asks whether fake pre-treatment windows can mimic the observed effect.
PersistenceCheck asks how much of the estimated effect remains after treatment ends.
Default ITS checks: ['PlaceboInTime']
Checks run in this example: ['PlaceboInTime', 'PersistenceCheck']
Configured placebo folds in this notebook: 4
Why four? More folds give a better sense of ordinary pre-treatment variation, but each extra fold requires another full refit.

PlaceboInTime#

Start with the intuition: pretend the campaign happened earlier, entirely inside the pre-treatment period, and fit the same ITS model again. If those fake campaigns regularly produce effects close to the observed one, then the original effect is less distinctive.

Because this is a fixed-period ITS, each placebo fold shifts the whole 12-week intervention window backward. That keeps the placebo problem comparable to the real one: same duration, same model, different timing.

In this notebook we use four folds so you can see several fake intervention windows rather than only one or two. Each new fold starts one intervention-length earlier than the next, and each row in the table below corresponds to one fake intervention window.

placebo_fold_summary = pd.DataFrame(
    [
        {
            "fold": fold.fold,
            "pseudo_treatment_time": fold.pseudo_treatment_time,
            "placebo_mean": fold.fold_mean,
            "placebo_sd": fold.fold_sd,
        }
        for fold in placebo_result.metadata["fold_results"]
    ]
)
p_outside = placebo_result.metadata["p_effect_outside_null"]
actual_cumulative_impact = placebo_result.metadata["actual_cumulative_mean"]
placebo_fold_summary["pct_of_observed_effect"] = (
    100 * placebo_fold_summary["placebo_mean"].abs() / abs(actual_cumulative_impact)
)
max_placebo_pct = placebo_fold_summary["pct_of_observed_effect"].max()
placebo_interpretation = (
    "supports the original ITS effect"
    if placebo_result.passed
    else "does not clearly support the original ITS effect"
)

display(
    Markdown(
        f"""
`PlaceboInTime` asks whether we would recover effects of a similar size if we pretended the intervention started earlier in the pre-period. Here, the observed cumulative impact is `{actual_cumulative_impact:.2f}`.

How to read the table:
- `pseudo_treatment_time`: when the fake intervention starts
- `placebo_mean`: the estimated cumulative effect under that fake intervention
- `placebo_sd`: the uncertainty for that fake intervention
- `pct_of_observed_effect`: how large the placebo estimate is **in absolute value** relative to the observed effect

Some placebo folds are negative in this example. That is not a problem by itself. The key question is whether placebo effects, in either direction, are comparable in magnitude to the observed effect.

`p_effect_outside_null` is a compact summary of how separated the observed effect is from the placebo-based null distribution. In plain language, it is the estimated probability that the real cumulative effect is more extreme than what these fake intervention windows usually generate. It is not a frequentist p-value; it is a Bayesian separation score.

In this example, the largest placebo estimate in absolute value is only about `{max_placebo_pct:.1f}%` of the observed effect. The separation score is `{p_outside:.3f}`, and `placebo_result.passed` is `{placebo_result.passed}`, so this check **{placebo_interpretation}**.
"""
    )
)
display(
    placebo_fold_summary.round(
        {"placebo_mean": 2, "placebo_sd": 2, "pct_of_observed_effect": 1}
    )
)

PlaceboInTime asks whether we would recover effects of a similar size if we pretended the intervention started earlier in the pre-period. Here, the observed cumulative impact is 650.99.

How to read the table:

pseudo_treatment_time: when the fake intervention starts
placebo_mean: the estimated cumulative effect under that fake intervention
placebo_sd: the uncertainty for that fake intervention
pct_of_observed_effect: how large the placebo estimate is in absolute value relative to the observed effect

Some placebo folds are negative in this example. That is not a problem by itself. The key question is whether placebo effects, in either direction, are comparable in magnitude to the observed effect.

p_effect_outside_null is a compact summary of how separated the observed effect is from the placebo-based null distribution. In plain language, it is the estimated probability that the real cumulative effect is more extreme than what these fake intervention windows usually generate. It is not a frequentist p-value; it is a Bayesian separation score.

In this example, the largest placebo estimate in absolute value is only about 32.7% of the observed effect. The separation score is 0.980, and placebo_result.passed is True, so this check supports the original ITS effect.

	fold	pseudo_treatment_time	placebo_mean	placebo_sd	pct_of_observed_effect
0	1	2022-10-16	-201.65	354.96	31.0
1	2	2023-01-08	-212.92	287.71	32.7
2	3	2023-04-02	104.92	331.32	16.1
3	4	2023-06-25	78.05	37.95	12.0

PersistenceCheck#

PersistenceCheck asks a different question from PlaceboInTime: not “could this be a spurious break?” but “how much of the effect remains after treatment ends?”

This is the same substantive persistence question we already explored with period="comparison" and analyze_persistence(). The advantage here is that persistence now appears inside the same pipeline as the placebo check, so both diagnostics can be reported together.

The most important quantity is the persistence ratio, which compares the average post-intervention effect with the average during-intervention effect. A value near 0 means the effect fades quickly. A value near 1 means the effect is roughly as strong after treatment ends as it was during treatment.

persistence = persistence_result.metadata["persistence"]
persistence_ratio = persistence["persistence_ratio"]
mean_effect_during = persistence["mean_effect_during"]
mean_effect_post = persistence["mean_effect_post"]
intervention_weeks = int(
    ((df.index >= treatment_time) & (df.index < treatment_end_time)).sum()
)
post_weeks = int((df.index >= treatment_end_time).sum())
persistence_summary = pd.DataFrame(
    [
        {
            "period": "intervention",
            "weeks_in_period": intervention_weeks,
            "mean_effect": mean_effect_during,
            "total_effect": persistence["total_effect_during"],
        },
        {
            "period": "post_intervention",
            "weeks_in_period": post_weeks,
            "mean_effect": mean_effect_post,
            "total_effect": persistence["total_effect_post"],
        },
    ]
)

if persistence_ratio < 0.1:
    persistence_interpretation = (
        "little evidence that the effect persists once treatment ends"
    )
elif persistence_ratio < 0.5:
    persistence_interpretation = (
        "partial persistence with substantial decay after treatment ends"
    )
elif persistence_ratio < 1.0:
    persistence_interpretation = (
        "meaningful persistence, although the effect is smaller after treatment ends"
    )
else:
    persistence_interpretation = "a post-treatment effect that is at least as large as the during-treatment effect"

display(
    Markdown(
        f"""
`PersistenceCheck` compares the intervention-period effect with the post-intervention effect. The mean effect falls from `{mean_effect_during:.2f}` during treatment to `{mean_effect_post:.2f}` afterward. The persistence ratio is `{persistence_ratio:.3f}`, which means the average post-intervention effect is about `{100 * persistence_ratio:.1f}%` of the average during-intervention effect. In this example, that suggests **{persistence_interpretation}**.

How to read the table:
- `weeks_in_period`: how long that period lasts
- `mean_effect`: the average effect per week in that period
- `total_effect`: the cumulative effect summed over the whole period

The post-intervention total can be larger than the intervention total even when the post-intervention mean is smaller, simply because the post period lasts much longer. For interpretation, the ratio and the period means are usually more informative than comparing totals alone.

Read this alongside `PlaceboInTime`: the placebo result asks whether the estimated effect looks distinctive, while the persistence ratio asks how much of that effect remains once treatment ends.
"""
    )
)
display(persistence_summary.round(3))

PersistenceCheck compares the intervention-period effect with the post-intervention effect. The mean effect falls from 24.53 during treatment to 6.37 afterward. The persistence ratio is 0.260, which means the average post-intervention effect is about 26.0% of the average during-intervention effect. In this example, that suggests partial persistence with substantial decay after treatment ends.

How to read the table:

weeks_in_period: how long that period lasts
mean_effect: the average effect per week in that period
total_effect: the cumulative effect summed over the whole period

The post-intervention total can be larger than the intervention total even when the post-intervention mean is smaller, simply because the post period lasts much longer. For interpretation, the ratio and the period means are usually more informative than comparing totals alone.

Read this alongside PlaceboInTime: the placebo result asks whether the estimated effect looks distinctive, while the persistence ratio asks how much of that effect remains once treatment ends.

	period	weeks_in_period	mean_effect	total_effect
0	intervention	12	24.526	294.317
1	post_intervention	56	6.369	356.672

What to do when a check looks weak#

Treat a weak check as a cue to slow down and narrow your claims, not as something to hide.

If PlaceboInTime reports large placebo effects, or most folds are skipped, the intervention effect may be hard to distinguish from ordinary pre-treatment fluctuation. Extend the pre-period if you can, revisit seasonality or trend terms, and report the estimate as fragile rather than definitive.
If PersistenceCheck shows a ratio near zero, the effect likely fades quickly after treatment ends. That can still be a useful result, but it argues against assuming long carryover.
If the two diagnostics disagree, treat that as information. Report the disagreement and explain which assumptions or domain mechanisms make the estimate plausible.

For broader interpretation guidance across methods, see Sensitivity Checks in Pipeline Workflows.

Summary#

The three-period design enables analysis of temporary interventions:

Immediate effects: effect_summary(period="intervention") analyzes effects during the active intervention
Persistence: effect_summary(period="post") measures how effects persist after the intervention ends
Comparison: effect_summary(period="comparison") provides a comparative summary with persistence metrics
Detailed analysis: analyze_persistence() automatically prints and returns a detailed summary with mean effects, persistence ratio (as decimal), and total effects

The persistence ratio (e.g., 0.30 = 30%) indicates how much of the intervention effect “carried over” into the post-intervention period. Note that the ratio can exceed 1.0 if the post-intervention effect is larger than the intervention effect.

In practice, persistence effects could be caused by various mechanisms. For example, in marketing contexts, persistence might reflect brand awareness effects that continue to influence consumer behavior even after the promotional campaign ends.

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

from maketables import ETable

ETable(result, coef_fmt="b:.3f")

	y
	(1)
coef
month=2	-4.764
month=3	-2.628
month=4	-0.149
month=5	4.771
month=6	6.121
month=7	10.142
month=8	16.225
month=9	16.588
month=10	11.874
month=11	8.790
month=12	5.859
t	0.146
Intercept	94.572
stats
N	135
Bayesian R2	0.713
Format of coefficient cell: Coefficient