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arxiv: 1906.08726 · v1 · pith:5WEJ5RHGnew · submitted 2019-06-20 · 📊 stat.AP · econ.EM· stat.ME· stat.OT

On the probability of a causal inference is robust for internal validity

Tenglong Li , Kenneth A. Frank This is my paper

Pith reviewed 2026-05-25 19:02 UTC · model grok-4.3

classification 📊 stat.AP econ.EMstat.MEstat.OT
keywords causal inferenceinternal validityrobustnesscounterfactualsnull hypothesis testingobservational studiessensitivity analysisPIV
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The pith

The PIV is the probability that a null hypothesis rejection on observed data persists after adding counterfactual outcomes, serving as a robustness index for causal inferences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper defines the probability of a causal inference being robust for internal validity (PIV) to quantify how secure a causal claim remains when the unconfoundedness assumption is questionable. Counterfactuals are treated as an unobserved additional sample, and PIV is the conditional probability that the null hypothesis is rejected again once those outcomes are included, given that it was already rejected on the observed sample alone. Bounds on the PIV follow from bounded beliefs about the counterfactuals, available under either frequentist or Bayesian reasoning. The index equals statistical power when the test is imagined to have already used the full data including counterfactuals. An eight-step procedure is given for applying the index, demonstrated on an education example.

Core claim

The paper establishes that the PIV, defined as the probability of rejecting the null hypothesis again based on both the observed sample and the counterfactuals given that the same null was already rejected on the observed sample alone, functions as a robustness index. Under either frequentist or Bayesian framework, the PIV of an inference can be bounded from bounded beliefs about the counterfactuals, which is useful when the unconfoundedness assumption is dubious. The PIV is equivalent to statistical power when the NHST is considered to be based on both the observed sample and the counterfactuals.

What carries the argument

The PIV itself, the conditional probability that a null hypothesis rejection on the observed sample alone continues after the counterfactual outcomes are folded in as an additional sample.

If this is right

  • A researcher can place numerical bounds on the robustness of a causal claim without observing the counterfactuals, by stating bounds on beliefs about them.
  • When the test is viewed as already incorporating counterfactuals, the PIV reduces exactly to ordinary statistical power.
  • The eight-step procedure supplies a concrete workflow for evaluating internal validity of any observational causal inference.
  • The same bounding logic applies under both frequentist and Bayesian interpretations of the test.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The PIV framing could be applied to missing-data problems outside causal inference by treating the missing cases as the counterfactual sample.
  • One could test the bounding procedure by generating data with known counterfactual distributions and checking whether the derived bounds contain the realized rejection probability.
  • The approach supplies a probabilistic alternative to deterministic sensitivity analyses that vary one parameter at a time.
  • Integration with existing software for power analysis might allow direct computation of PIV bounds once belief intervals on counterfactual means or variances are supplied.

Load-bearing premise

Counterfactual outcomes can be treated as an additional sample whose influence on the test statistic permits probabilistic bounding from subjective beliefs about those outcomes.

What would settle it

A simulation in which the true distribution of counterfactual outcomes is fully known, the actual frequency of continued null rejection with the full data is computed, and this frequency is checked against the interval obtained from the paper's bounding procedure applied to deliberately limited beliefs.

Figures

Figures reproduced from arXiv: 1906.08726 by Kenneth A. Frank, Tenglong Li.

Figure 1
Figure 1. Figure 1: illustrates the conceptualization of the unobserved sample in Hong & Raudenbush (2005) for the simple estimator. The observed outcome , ob Yri symbolizes the reading score of any retained student whose counterfactual outcome is , un Ypi . Likewise, the observed outcome , ob Ypj represents the reading score of any promoted student whose counterfactual outcome is , un Yrj . The unobserved sample consists of … view at source ↗
read the original abstract

The internal validity of observational study is often subject to debate. In this study, we define the counterfactuals as the unobserved sample and intend to quantify its relationship with the null hypothesis statistical testing (NHST). We propose the probability of a causal inference is robust for internal validity, i.e., the PIV, as a robustness index of causal inference. Formally, the PIV is the probability of rejecting the null hypothesis again based on both the observed sample and the counterfactuals, provided the same null hypothesis has already been rejected based on the observed sample. Under either frequentist or Bayesian framework, one can bound the PIV of an inference based on his bounded belief about the counterfactuals, which is often needed when the unconfoundedness assumption is dubious. The PIV is equivalent to statistical power when the NHST is thought to be based on both the observed sample and the counterfactuals. We summarize the process of evaluating internal validity with the PIV into an eight-step procedure and illustrate it with an empirical example (i.e., Hong and Raudenbush (2005)).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes the probability of a causal inference is robust for internal validity (PIV) as a robustness index for causal inferences from observational data. Formally, PIV is defined as the conditional probability of rejecting the null hypothesis H0 again when the test is based on both the observed sample and the counterfactual outcomes, given that H0 was already rejected based on the observed sample alone. The authors claim that PIV can be bounded using subjective beliefs about the counterfactuals under either frequentist or Bayesian frameworks, that it is equivalent to statistical power when the test incorporates both samples, and that an eight-step procedure can be used to evaluate internal validity, illustrated with the Hong and Raudenbush (2005) example.

Significance. If the central definition were rigorously grounded, the PIV could provide a quantitative index for sensitivity to unconfoundedness violations. The manuscript offers no machine-checked proofs, reproducible code, parameter-free derivations, or falsifiable predictions; the contribution rests entirely on the conceptual proposal and the eight-step procedure.

major comments (3)
  1. [Abstract] Abstract (formal definition of PIV): The quantity P(reject H0 on observed+counterfactuals | reject H0 on observed) is not formally defined in the frequentist NHST framework employed for the original test. Counterfactual outcomes are fixed but unobserved; without an explicit joint distribution or worst-case measure over their values, the conditional probability and its bounds cannot be derived.
  2. [Abstract] Abstract (equivalence claim): The stated equivalence of PIV to statistical power when the NHST is 'thought to be based on both' inherits the same definitional ambiguity, because power is defined with respect to a sampling distribution under the null or alternative, not over fixed potential outcomes.
  3. [Abstract] Abstract and eight-step procedure: All numerical illustrations and the robustness index rest on the load-bearing assumption that counterfactuals can be treated as an additional sample whose distribution is subjectively bounded to compute a conditional rejection probability; this assumption is not justified within standard frequentist potential-outcomes theory.
minor comments (2)
  1. [Abstract] The abstract refers to 'bounded belief about the counterfactuals' without specifying how this belief is translated into numerical bounds on the rejection probability.
  2. Notation for counterfactual outcomes should be introduced explicitly and distinguished from random variables to avoid conflating fixed potential outcomes with a probability space.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript proposing the PIV as a robustness measure for causal inferences. We address each of the major comments point by point below. While we maintain that the conceptual contribution is valuable for sensitivity analysis in observational studies, we acknowledge the need for greater formal rigor in some aspects and will revise accordingly where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract (formal definition of PIV): The quantity P(reject H0 on observed+counterfactuals | reject H0 on observed) is not formally defined in the frequentist NHST framework employed for the original test. Counterfactual outcomes are fixed but unobserved; without an explicit joint distribution or worst-case measure over their values, the conditional probability and its bounds cannot be derived.

    Authors: We agree that a more precise formalization is needed. In the revised manuscript, we will define PIV more rigorously by specifying that the bounds on counterfactual outcomes are incorporated via a set of possible distributions or values consistent with the analyst's beliefs, and the conditional probability is computed as the infimum or range over these possibilities. This treats the counterfactuals as fixed but unknown, with the probability arising from the sampling distribution of the observed data conditional on the bounds. This is an extension of standard NHST to include sensitivity parameters. revision: yes

  2. Referee: [Abstract] Abstract (equivalence claim): The stated equivalence of PIV to statistical power when the NHST is 'thought to be based on both' inherits the same definitional ambiguity, because power is defined with respect to a sampling distribution under the null or alternative, not over fixed potential outcomes.

    Authors: The equivalence is conceptual: PIV represents the power of a test that would be conducted if the counterfactual outcomes were observed, but since they are not, we bound it using beliefs about them. We will revise the manuscript to clarify that it is not a direct equivalence but an analogy to power under the extended sample, and remove any implication of strict mathematical equivalence without the additional bounding framework. revision: yes

  3. Referee: [Abstract] Abstract and eight-step procedure: All numerical illustrations and the robustness index rest on the load-bearing assumption that counterfactuals can be treated as an additional sample whose distribution is subjectively bounded to compute a conditional rejection probability; this assumption is not justified within standard frequentist potential-outcomes theory.

    Authors: This is the core of our proposal, which intentionally extends beyond standard theory to provide a practical tool for assessing internal validity when unconfoundedness may be violated. Similar to other sensitivity analyses (e.g., those using Rosenbaum's bounds or partial identification), we allow subjective input on counterfactual distributions. The eight-step procedure makes these assumptions explicit and falsifiable by the reader. We do not claim justification within unmodified standard theory but as a new robustness index; thus, no change to this aspect is planned. revision: no

Circularity Check

0 steps flagged

No circularity in PIV definition or bounding claim.

full rationale

The paper introduces PIV via an explicit formal definition as the conditional probability P(reject H0 on observed+counterfactuals | reject H0 on observed). It states that bounds follow from subjective beliefs about counterfactuals under frequentist or Bayesian views and notes an equivalence to power under a joint-sample interpretation. No quoted equations, procedures, or self-citations reduce this definition or the bounding claim to a fitted parameter, prior self-result, or input by construction. The eight-step procedure and empirical illustration rest on the definitional proposal itself rather than any hidden reduction, so the derivation chain is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Limited information from abstract only; no free parameters or additional axioms explicitly stated.

axioms (1)
  • domain assumption Counterfactuals can be treated as an unobserved sample for the purposes of null hypothesis statistical testing.
    This is foundational to the definition of PIV as per the abstract.
invented entities (1)
  • PIV no independent evidence
    purpose: Robustness index for internal validity of causal inference
    Newly defined probability measure without mentioned external evidence or validation in abstract.

pith-pipeline@v0.9.0 · 5724 in / 1298 out tokens · 39217 ms · 2026-05-25T19:02:00.108116+00:00 · methodology

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