Treatment Effects on Ordinal Outcomes: Causal Estimands and Sharp Bounds

Lu, Jiannan; Ding, Peng; Dasgupta, Tirthankar

doi:10.3102/1076998618776435

Cited by 28 publications

(54 citation statements)

References 84 publications

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“…measure the probability that the treatment is not worse than the control and the probability that the treatment is better than the control, respectively. Lu et al (2015) emphasize that for ordinal outcomes, δ 1 and δ 2 are well defined even though Inferring such parameters is arguably more challenging than most standard missing data problems. It is also an example where some theoretical development in causal inference lends to research on missing data.…”

Section: Partially Identified Parameters and Boundsmentioning

confidence: 99%

Causal Inference: A Missing Data Perspective

Ding¹,

Li²

2018

Statist. Sci.

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Inferring causal effects of treatments is a central goal in many disciplines. The potential outcomes framework is a main statistical approach to causal inference, in which a causal effect is defined as a comparison of the potential outcomes of the same units under different treatment conditions. Because for each unit at most one of the potential outcomes is observed and the rest are missing, causal inference is inherently a missing data problem. Indeed, there is a close analogy in the terminology and the inferential framework between causal inference and missing data. Despite the intrinsic connection between the two subjects, statistical analyses of causal inference and missing data also have marked differences in aims, settings and methods. This article provides a systematic review of causal inference from the missing data perspective.Focusing on ignorable treatment assignment mechanisms, we discuss a wide range of causal inference methods that have analogues in missing data analysis, such as imputation, inverse probability weighting and doubly-robust methods. Under each of the three modes of inference-Frequentist, Bayesian, and Fisherian randomization-we present the general structure of inference for both finite-sample and superpopulation estimands, and illustrate via specific examples. We identify open questions to motivate more research to bridge the two fields.

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Section: Partially Identified Parameters and Boundsmentioning

confidence: 99%

Causal Inference: A Missing Data Perspective

Ding¹,

Li²

2018

Statist. Sci.

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“…Related Literature. Worst case bounds on the distributions of potential outcomes and treatment effects and their quantiles have been analyzed by Heckman, Smith, and Clements (1997), Manski (1997), Firpo and Ridder (2008), Fan and Park (2010), Fan and Park (2012), Fan and Wu (2010), and Lu, Ding, and Dasgupta (2018). This literature uses theoretical results on dependency bounds for functions of several random variables which were developed among others by Cambanis, Simmons, and Stout (1976), Makarov (1982), Frank, Nelsen, and Schweizer (1987), and Williamson and Downs (1990).…”

Section: An Analytical Variance Estimator Definêmentioning

confidence: 99%

A Causal Bootstrap

Imbens¹,

Menzel²

2018

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The bootstrap, introduced by Efron (1982), has become a very popular method for estimating variances and constructing confidence intervals. A key insight is that one can approximate the properties of estimators by using the empirical distribution function of the sample as an approximation for the true distribution function. This approach views the uncertainty in the estimator as coming exclusively from sampling uncertainty. We argue that for causal estimands the uncertainty arises entirely, or partially, from a different source, corresponding to the stochastic nature of the treatment received. We develop a bootstrap procedure that accounts for this uncertainty, and compare its properties to that of the classical bootstrap.JEL Classification: C1, C14, C18

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“…Recently, Lu et al proposed the use of the following 2 causal parameters:

θ_{x} = Pr ({}, Y ((), x) \geq Y ((), 1 - x)),

φ_{x} = Pr ({}, Y ((), x) > Y ((), 1 - x)),

for x = 0, 1. For a binary outcome, neither θ x nor φ x is identical to Pr{ Y ( x ) = 1}:

\begin{matrix} {right left} \\ θ_{x} = Pr \{Y (x) = 1, Y (1 - x) = 0\} + Pr \{Y (x) = Y (1 - x) = 1\} + Pr \{Y (x) = Y (1 - x) = 0\} \\ = Pr \{Y (x) = 1\} + Pr \{Y (x) = Y (1 - x) = 0\}, \end{matrix}

…”

Section: Treatment Effect On An Ordinal Outcomementioning

confidence: 99%

“…Unfortunately, this is still not well defined for ordinal outcomes. Although several authors have discussed the definition, their definitions are limited in that they are not identical to the well‐defined causal risk for a binary outcome, which is the simplest ordinal outcome, although the relative treatment effects are identical when a difference scale is used.…”

Section: Introductionmentioning

confidence: 99%

Sharp nonparametric bounds and randomization inference for treatment effects on an ordinal outcome

Chiba

2017

Statistics in Medicine

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In clinical research, investigators are interested in inferring the average causal effect of a treatment. However, the causal parameter that can be used to derive the average causal effect is not well defined for ordinal outcomes. Although some definitions have been proposed, they are limited in that they are not identical to the well-defined causal risk for a binary outcome, which is the simplest ordinal outcome. In this paper, we propose the use of a causal parameter for an ordinal outcome, defined as the proportion that a potential outcome under one treatment condition would not be smaller than that under the other condition. For a binary outcome, this proportion is identical to the causal risk. Unfortunately, the proposed causal parameter cannot be identified, even under randomization. Therefore, we present a numerical method to calculate the sharp nonparametric bounds within a sample, reflecting the impact of confounding. When the assumption of independent potential outcomes is included, the causal parameter can be identified when randomization is in play. Then, we present exact tests and the associated confidence intervals for the relative treatment effect using the randomization-based approach, which are an extension of the existing methods for a binary outcome. Our methodologies are illustrated using data from an emetic prevention clinical trial.

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Treatment Effects on Ordinal Outcomes: Causal Estimands and Sharp Bounds

Cited by 28 publications

References 84 publications

Causal Inference: A Missing Data Perspective

Causal Inference: A Missing Data Perspective

A Causal Bootstrap

Sharp nonparametric bounds and randomization inference for treatment effects on an ordinal outcome

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