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Journal Articleβ€’DOIβ€’

Selective Ignorability Assumptions in Causal Inference

Marshall M. Joffe1, Wei Peter Yang, Harold I. Feldmanβ€’
University of Pennsylvania1
05 Mar 2010-The International Journal of Biostatistics (Int J Biostat)-Vol. 6, Iss: 2, pp 11
TL;DR: This paper outlines selective ignorability assumptions mathematically and sketches how they may be used along with otherwise standard G-estimation or likelihood-based methods to obtain inference on structural nested models to derive valid causal inferences.
Abstract: Most attempts at causal inference in observational studies are based on assumptions that treatment assignment is ignorable. Such assumptions are usually made casually, largely because they justify the use of available statistical methods and not because they are truly believed. It will often be the case that it is plausible that conditional independence holds at least approximately for a subset but not all of the experience giving rise to one's data. Such selective ignorability assumptions may be used to derive valid causal inferences in conjunction with structural nested models. In this paper, we outline selective ignorability assumptions mathematically and sketch how they may be used along with otherwise standard G-estimation or likelihood-based methods to obtain inference on structural nested models. We also consider use of these assumptions in the presence of selective measurement error or missing data when the missingness is not at random. We motivate and illustrate our development by considering an analysis of an observational database to estimate the effect of erythropoietin use on mortality among hemodialysis patients.

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Summary

  • Most attempts at causal inference in observational studies are based on assumptions that treatment assignment is ignorable.
  • Such assumptions are usually made casually, largely because they justify the use of available statistical methods and not because they are truly believed.
  • It will often be the case that it is plausible that conditional independence holds at least approximately for a subset but not all of the experience giving rise to one's data.
  • Such selective ignorability assumptions may be used to derive valid causal inferences in conjunction with structural nested models.
  • The authors outline selective ignorability assumptions mathematically and sketch how they may be used along with otherwise standard G-estimation or likelihood-based methods to obtain inference on structural nested models.
  • The authors also consider use of these assumptions in the presence of selective measurement error or missing data when the missingness is not at random.

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Volume 6, Issue 2 2010 Article 11
The International Journal of
Biostatistics
CAUSAL INFERENCE
Selective Ignorability Assumptions in Causal
Inference
Marshall M. Joffe, University of Pennsylvania School of
Medicine
Wei Peter Yang, University of Pennsylvania School of
Medicine
Harold I. Feldman, University of Pennsylvania School of
Medicine
Recommended Citation:
Joffe, Marshall M.; Yang, Wei Peter; and Feldman, Harold I. (2010) "Selective Ignorability
Assumptions in Causal Inference," The International Journal of Biostatistics: Vol. 6: Iss. 2,
Article 11.
DOI: 10.2202/1557-4679.1199
Selective Ignorability Assumptions in Causal
Inference
Marshall M. Joffe, Wei Peter Yang, and Harold I. Feldman
Abstract
Most attempts at causal inference in observational studies are based on assumptions that
treatment assignment is ignorable. Such assumptions are usually made casually, largely because
they justify the use of available statistical methods and not because they are truly believed. It will
often be the case that it is plausible that conditional independence holds at least approximately for
a subset but not all of the experience giving rise to one's data. Such selective ignorability
assumptions may be used to derive valid causal inferences in conjunction with structural nested
models. In this paper, we outline selective ignorability assumptions mathematically and sketch
how they may be used along with otherwise standard G-estimation or likelihood-based methods to
obtain inference on structural nested models. We also consider use of these assumptions in the
presence of selective measurement error or missing data when the missingness is not at random.
We motivate and illustrate our development by considering an analysis of an observational
database to estimate the effect of erythropoietin use on mortality among hemodialysis patients.
KEYWORDS: causal inference, ignorability, end-stage renal disease, anemia
Author Notes: This work was supported by an unrestricted grant from Amgen.
ξ˜ξ˜‚ξ˜ƒξ˜„ξ˜…ξ˜†ξ˜‡ξ˜ˆξ˜‰ξ˜„ξ˜Šξ˜†ξ˜ƒ
ξ˜ξ˜‚ξ˜ƒξ˜„ξ˜…ξ˜†ξ˜„ξ˜„ξ˜‡ξ˜ˆξ˜‰ξ˜„ξ˜ƒξ˜…ξ˜†ξ˜„ξ˜…ξ˜Šξ˜†ξ˜‹ξ˜ƒξ˜†ξ˜Œξ˜…ξ˜ξ˜Žξ˜ξ˜‡ξ˜ξ˜‡ξ˜Žξ˜Šξ˜‡ξ˜…ξ˜ξ˜Žξ˜…ξ˜‚ξ˜‘ξ˜ƒξ˜‡ξ˜ξ˜’ξ˜†ξ˜„ξ˜ξ˜‚ξ˜Žξ˜†ξ˜Œξ˜…ξ˜ƒξ˜„ξ˜‹ξ˜“ξ˜ξ˜‡ξ˜ƒξ˜…ξ˜†ξ˜ξ˜‡ξ˜…ξ˜‘ξ˜†ξ˜ƒξ˜‡ξ˜“ξ˜…ξ˜‚ξ˜Žξ˜…ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‰ξ˜„ξ˜ξ˜‚ξ˜Žξ˜ƒ
ξ˜„ξ˜”ξ˜†ξ˜„ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜…ξ˜†ξ˜ƒξ˜ƒξ˜ξ˜•ξ˜Žξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜…ξ˜ξ˜ƒξ˜…ξ˜ξ˜•ξ˜Žξ˜‚ξ˜ξ˜†ξ˜‘ξ˜Œξ˜‡ξ˜…ξ˜–ξ˜—ξ˜‚ξ˜ƒξ˜‡ξ˜Žξ˜‘ξ˜†ξ˜‹ξ˜ˆξ˜…ξ˜†ξ˜Žξ˜“ξ˜…ξ˜—ξ˜‹ξ˜‘ξ˜ξ˜Žξ˜…ξ˜˜ξ˜™ξ˜šξ˜›ξ˜œξ˜ξ˜…ξ˜ξ˜•ξ˜Žξ˜‚ξ˜ξ˜†ξ˜‘ξ˜ξ˜Œξ˜ξ˜„ξ˜ž
ξ˜ξ˜Žξ˜’ξ˜‚ξ˜Œξ˜’ξ˜‡ξ˜ƒξ˜…ξ˜Šξ˜‚ξ˜Žξ˜“ξ˜ξ˜„ξ˜ξ˜‚ξ˜Žξ˜†ξ˜Œξ˜…ξ˜ξ˜Žξ˜“ξ˜‡ξ˜‰ξ˜‡ξ˜Žξ˜“ξ˜‡ξ˜Žξ˜Šξ˜‡ξ˜…ξ˜‚ξ˜ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜…ξ˜†ξ˜Žξ˜“ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜‰ξ˜‚ξ˜„ξ˜‡ξ˜Žξ˜„ξ˜ξ˜†ξ˜Œξ˜…ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜ƒξ˜Ÿξ˜…ξ˜… ξ˜‹ξ˜Šξ˜”
ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‰ξ˜„ξ˜ξ˜‚ξ˜Žξ˜ƒξ˜… ξ˜†ξ˜ξ˜‡ξ˜… ξ˜‹ξ˜ƒξ˜‹ξ˜†ξ˜Œξ˜Œξ˜žξ˜… ξ˜ˆξ˜†ξ˜“ξ˜‡ξ˜… ξ˜Šξ˜†ξ˜ƒξ˜‹ξ˜†ξ˜Œξ˜Œξ˜ž!ξ˜… ξ˜Œξ˜†ξ˜ξ˜•ξ˜‡ξ˜Œξ˜žξ˜… ξ˜‘ξ˜‡ξ˜Šξ˜†ξ˜‹ξ˜ƒξ˜‡ξ˜… ξ˜„ξ˜”ξ˜‡ξ˜žξ˜… "ξ˜‹ξ˜ƒξ˜„ξ˜ξ˜ξ˜žξ˜… ξ˜„ξ˜”ξ˜‡ξ˜… ξ˜‹ξ˜ƒξ˜‡ξ˜… ξ˜‚ξ˜
ξ˜†ξ˜’ξ˜†ξ˜ξ˜Œξ˜†ξ˜‘ξ˜Œξ˜‡ξ˜…ξ˜ƒξ˜„ξ˜†ξ˜„ξ˜ξ˜ƒξ˜„ξ˜ξ˜Šξ˜†ξ˜Œξ˜…ξ˜ˆξ˜‡ξ˜„ξ˜”ξ˜‚ξ˜“ξ˜ƒξ˜…ξ˜†ξ˜Žξ˜“ξ˜…ξ˜Žξ˜‚ξ˜„ξ˜…ξ˜‘ξ˜‡ξ˜Šξ˜†ξ˜‹ξ˜ƒξ˜‡ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜žξ˜…ξ˜†ξ˜ξ˜‡ξ˜…ξ˜„ξ˜ξ˜‹ξ˜Œξ˜žξ˜…ξ˜‘ξ˜‡ξ˜Œξ˜ξ˜‡ξ˜’ξ˜‡ξ˜“ξ˜Ÿξ˜…ξ˜…#ξ˜„ξ˜…$ξ˜ξ˜Œξ˜Œξ˜…ξ˜‚ξ˜ξ˜„ξ˜‡ξ˜Žξ˜…ξ˜‘ξ˜‡
ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜Šξ˜†ξ˜ƒξ˜‡ξ˜…ξ˜„ξ˜”ξ˜†ξ˜„ξ˜…ξ˜ξ˜„ξ˜…ξ˜ξ˜ƒξ˜…ξ˜‰ξ˜Œξ˜†ξ˜‹ξ˜ƒξ˜ξ˜‘ξ˜Œξ˜‡ξ˜…ξ˜„ξ˜”ξ˜†ξ˜„ξ˜…ξ˜Šξ˜‚ξ˜Žξ˜“ξ˜ξ˜„ξ˜ξ˜‚ξ˜Žξ˜†ξ˜Œξ˜…ξ˜ξ˜Žξ˜“ξ˜‡ξ˜‰ξ˜‡ξ˜Žξ˜“ξ˜‡ξ˜Žξ˜Šξ˜‡ξ˜…ξ˜”ξ˜‚ξ˜Œξ˜“ξ˜ƒξ˜…ξ˜†ξ˜„ξ˜…ξ˜Œξ˜‡ξ˜†ξ˜ƒξ˜„ξ˜…ξ˜†ξ˜‰ξ˜‰ξ˜ξ˜‚%ξ˜ξ˜ˆξ˜†ξ˜„ξ˜‡ξ˜Œξ˜ž
ξ˜ξ˜‚ξ˜ξ˜…ξ˜†ξ˜…ξ˜ƒξ˜‹ξ˜‘ξ˜ƒξ˜‡ξ˜„ξ˜…ξ˜‘ξ˜‹ξ˜„ξ˜…ξ˜Žξ˜‚ξ˜„ξ˜…ξ˜†ξ˜Œξ˜Œξ˜…ξ˜‚ξ˜ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜‡%ξ˜‰ξ˜‡ξ˜ξ˜ξ˜‡ξ˜Žξ˜Šξ˜‡ξ˜…ξ˜•ξ˜ξ˜’ξ˜ξ˜Žξ˜•ξ˜…ξ˜ξ˜ξ˜ƒξ˜‡ξ˜…ξ˜„ξ˜‚ξ˜…ξ˜‚ξ˜Žξ˜‡&ξ˜ƒξ˜…ξ˜“ξ˜†ξ˜„ξ˜†ξ˜Ÿξ˜…ξ˜… ξ˜‹ξ˜Šξ˜”ξ˜…ξ˜ƒξ˜‡ξ˜Œξ˜‡ξ˜Šξ˜„ξ˜ξ˜’ξ˜‡
ξ˜ξ˜•ξ˜Žξ˜‚ξ˜ξ˜†ξ˜‘ξ˜ξ˜Œξ˜ξ˜„ξ˜žξ˜…ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‰ξ˜„ξ˜ξ˜‚ξ˜Žξ˜ƒξ˜…ξ˜ˆξ˜†ξ˜žξ˜…ξ˜‘ξ˜‡ξ˜…ξ˜‹ξ˜ƒξ˜‡ξ˜“ξ˜…ξ˜„ξ˜‚ξ˜…ξ˜“ξ˜‡ξ˜ξ˜ξ˜’ξ˜‡ξ˜…ξ˜’ξ˜†ξ˜Œξ˜ξ˜“ξ˜…ξ˜Šξ˜†ξ˜‹ξ˜ƒξ˜†ξ˜Œξ˜…ξ˜ξ˜Žξ˜ξ˜‡ξ˜ξ˜‡ξ˜Žξ˜Šξ˜‡ξ˜ƒξ˜…ξ˜ξ˜Žξ˜…ξ˜Šξ˜‚ξ˜Ž"ξ˜‹ξ˜Žξ˜Šξ˜„ξ˜ξ˜‚ξ˜Ž
$  ξ˜„ ξ˜” ξ˜… ξ˜ƒ ξ˜„  ξ˜‹  ξ˜„ ξ˜‹  ξ˜†  ξ˜…  ξ˜‡ ξ˜ƒ ξ˜„ ξ˜‡ ξ˜“ ξ˜…  ξ˜‚ ξ˜“ ξ˜‡  ξ˜ƒ ! ξ˜… $ ξ˜” ξ˜‡  ξ˜‡ ξ˜† ξ˜ƒ ξ˜…  ξ˜‡ ξ˜„ ξ˜” ξ˜‚ ξ˜“ ξ˜ƒ ξ˜… ξ˜‘ ξ˜† ξ˜ƒ ξ˜‡ ξ˜“ ξ˜… ξ˜‚  ξ˜… ξ˜† ξ˜ƒ ξ˜ƒ ξ˜‹    ξ˜• ξ˜…  ξ˜‹   ξ˜…  ξ˜•  ξ˜‚  ξ˜† ξ˜‘    ξ˜„ 
ξ˜†ξ˜ξ˜‡ξ˜…ξ˜‘ξ˜ξ˜†ξ˜ƒξ˜‡ξ˜“ξ˜…ξ˜ξ˜Žξ˜…ξ˜„ξ˜”ξ˜ξ˜ƒξ˜…ξ˜ƒξ˜‡ξ˜„ξ˜„ξ˜ξ˜Žξ˜•ξ˜Ÿξ˜…ξ˜…#ξ˜Žξ˜…ξ˜„ξ˜”ξ˜ξ˜ƒξ˜…ξ˜‰ξ˜†ξ˜‰ξ˜‡ξ˜!ξ˜…$ξ˜‡ξ˜…ξ˜‚ξ˜‹ξ˜„ξ˜Œξ˜ξ˜Žξ˜‡ξ˜…ξ˜ƒξ˜‡ξ˜Œξ˜‡ξ˜Šξ˜„ξ˜ξ˜’ξ˜‡ξ˜…ξ˜ξ˜•ξ˜Žξ˜‚ξ˜ξ˜†ξ˜‘ξ˜ξ˜Œξ˜ξ˜„ξ˜žξ˜…ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‰ξ˜„ξ˜ξ˜‚ξ˜Žξ˜ƒ
ξ˜ˆξ˜†ξ˜„ξ˜”ξ˜‡ξ˜ˆξ˜†ξ˜„ξ˜ξ˜Šξ˜†ξ˜Œξ˜Œξ˜ž!ξ˜…ξ˜“ξ˜ξ˜ƒξ˜Šξ˜‹ξ˜ƒξ˜ƒξ˜…ξ˜’ξ˜†ξ˜ξ˜ξ˜‚ξ˜‹ξ˜ƒξ˜…ξ˜’ξ˜†ξ˜ξ˜ξ˜†ξ˜Žξ˜„ξ˜ƒξ˜…ξ˜‚ξ˜ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜ƒξ˜‡ξ˜…ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‰ξ˜„ξ˜ξ˜‚ξ˜Žξ˜ƒ!ξ˜…ξ˜†ξ˜Žξ˜“ξ˜…ξ˜ƒ'ξ˜‡ξ˜„ξ˜Šξ˜”ξ˜…ξ˜”ξ˜‚$ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜ƒξ˜‡
ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‰ξ˜„ξ˜ξ˜‚ξ˜Žξ˜ƒξ˜…ξ˜ˆξ˜†ξ˜žξ˜…ξ˜‘ξ˜‡ξ˜…ξ˜‹ξ˜ƒξ˜‡ξ˜“ξ˜…ξ˜†ξ˜Œξ˜‚ξ˜Žξ˜•ξ˜…$ξ˜ξ˜„ξ˜”ξ˜…ξ˜‚ξ˜„ξ˜”ξ˜‡ξ˜$ξ˜ξ˜ƒξ˜‡ξ˜…ξ˜ƒξ˜„ξ˜†ξ˜Žξ˜“ξ˜†ξ˜ξ˜“ξ˜…()ξ˜‡ξ˜ƒξ˜„ξ˜ξ˜ˆξ˜†ξ˜„ξ˜ξ˜‚ξ˜Žξ˜…ξ˜ˆξ˜‡ξ˜„ξ˜”ξ˜‚ξ˜“ξ˜ƒξ˜…ξ˜„ξ˜‚
ξ˜‚ξ˜‘ξ˜„ξ˜†ξ˜ξ˜Žξ˜… ξ˜ξ˜Žξ˜ξ˜‡ξ˜ξ˜‡ξ˜Žξ˜Šξ˜‡ξ˜… ξ˜‚ξ˜Žξ˜… ξ˜ƒξ˜„ξ˜ξ˜‹ξ˜Šξ˜„ξ˜‹ξ˜ξ˜†ξ˜Œξ˜… ξ˜Žξ˜‡ξ˜ƒξ˜„ξ˜‡ξ˜“ξ˜… ξ˜ˆξ˜‚ξ˜“ξ˜‡ξ˜Œξ˜ƒξ˜Ÿξ˜… ξ˜… * ξ˜‡ξ˜… ξ˜ˆξ˜‚ξ˜„ξ˜ξ˜’ξ˜†ξ˜„ξ˜‡ξ˜… ξ˜†ξ˜Žξ˜“ξ˜… ξ˜ξ˜Œξ˜Œξ˜‹ξ˜ƒξ˜„ξ˜ξ˜†ξ˜„ξ˜‡ξ˜… ξ˜‚ξ˜‹ξ˜
ξ˜“ξ˜‡ξ˜’ξ˜‡ξ˜Œξ˜‚ξ˜‰ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜…ξ˜‘ξ˜žξ˜…ξ˜Šξ˜‚ξ˜Žξ˜ƒξ˜ξ˜“ξ˜‡ξ˜ξ˜ξ˜Žξ˜•ξ˜…ξ˜†ξ˜Žξ˜…ξ˜†ξ˜Žξ˜†ξ˜Œξ˜žξ˜ƒξ˜ξ˜ƒξ˜…ξ˜‚ξ˜ξ˜…ξ˜†ξ˜Žξ˜…ξ˜‚ξ˜‘ξ˜ƒξ˜‡ξ˜ξ˜’ξ˜†ξ˜„ξ˜ξ˜‚ξ˜Žξ˜†ξ˜Œξ˜…ξ˜“ξ˜†ξ˜„ξ˜†ξ˜‘ξ˜†ξ˜ƒξ˜‡ξ˜…ξ˜„ξ˜‚ξ˜…ξ˜‡ξ˜ƒξ˜„ξ˜ξ˜ˆξ˜†ξ˜„ξ˜‡ξ˜…ξ˜„ξ˜”ξ˜‡
ξ˜‡ξ˜ξ˜ξ˜‡ξ˜Šξ˜„ξ˜…ξ˜‚ξ˜ξ˜…ξ˜‡ξ˜ξ˜žξ˜„ξ˜”ξ˜ξ˜‚ξ˜‰ξ˜‚ξ˜ξ˜‡ξ˜„ξ˜ξ˜Žξ˜…ξ˜‹ξ˜ƒξ˜‡ξ˜…ξ˜‚ξ˜Žξ˜…ξ˜ˆξ˜‚ξ˜ξ˜„ξ˜†ξ˜Œξ˜ξ˜„ξ˜žξ˜…ξ˜†ξ˜ˆξ˜‚ξ˜Žξ˜•ξ˜…ξ˜”ξ˜‡ξ˜ˆξ˜‚ξ˜“ξ˜ξ˜†ξ˜Œξ˜žξ˜ƒξ˜ξ˜ƒξ˜…ξ˜‰ξ˜†ξ˜„ξ˜ξ˜‡ξ˜Žξ˜„ξ˜ƒξ˜Ÿξ˜…
ξ˜‹ξ˜Œξ˜†ξ˜„ξ˜ξ˜ƒξ˜„ξ˜Šξ˜Žξ˜ξ˜ξ˜†ξ˜ˆξ˜„ξ˜‰ξ˜†ξ˜‘ξ˜ξ˜’ξ˜“ξ˜ξ˜‰ξ˜Žξ˜ˆξ˜’ξ˜Žξ˜ξ˜ξ˜ξ˜”ξ˜”ξ˜ξ˜‰ξ˜„ξ˜’ξ˜“ξ˜ξ˜Žξ˜ƒξ˜‡ξ˜ξ˜ˆξ˜’ξ˜ˆξ˜Žξ˜
ξ˜Šξ˜‡ξ˜ξ˜ƒξ˜„ξ˜Šξ˜”ξ˜•ξ˜Šξ˜ƒξ˜–ξ˜ξ˜Žξ˜’ξ˜’ξ˜ˆξ˜‘ ξ˜—ξ˜„ξ˜Šξ˜†ξ˜ƒξ˜’
#  ξ˜… ξ˜„ ξ˜”  ξ˜ƒ ξ˜… ξ˜ƒ ξ˜‡  ξ˜„  ξ˜‚  ! ξ˜… $ ξ˜‡ ξ˜… ξ˜‚ ξ˜‹ ξ˜„    ξ˜‡ ξ˜… ξ˜† ξ˜…  ξ˜‚   ξ˜‡ ξ˜‰ ξ˜„ ξ˜‹ ξ˜†  ξ˜…   ξ˜†  ξ˜‡ $ ξ˜‚  ' ξ˜…  ξ˜‚  ξ˜…  ξ˜† ξ˜‹ ξ˜ƒ ξ˜†  ξ˜… ξ˜‡   ξ˜‡  ξ˜„ ξ˜ƒ ξ˜… ξ˜‘ ξ˜† ξ˜ƒ ξ˜‡ ξ˜“ ξ˜… ξ˜‚ 
ξ˜‰ξ˜‚ξ˜„ξ˜‡ξ˜Žξ˜„ξ˜ξ˜†ξ˜Œξ˜…ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜ƒξ˜Ÿξ˜…ξ˜…* ξ˜‡ξ˜…ξ˜“ξ˜‚ξ˜…ξ˜„ξ˜”ξ˜ξ˜ƒξ˜…ξ˜ξ˜ξ˜ξ˜ƒξ˜„ξ˜…ξ˜ξ˜‚ξ˜ξ˜…ξ˜ƒξ˜Šξ˜†ξ˜Œξ˜†ξ˜ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜ƒξ˜…$ξ˜”ξ˜ξ˜Šξ˜”ξ˜…ξ˜“ξ˜‚ξ˜…ξ˜Žξ˜‚ξ˜„ξ˜…ξ˜’ξ˜†ξ˜ξ˜žξ˜…$ξ˜ξ˜„ξ˜”
ξ˜„ξ˜ξ˜ˆξ˜‡!ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜Žξ˜…ξ˜•ξ˜‡ξ˜Žξ˜‡ξ˜ξ˜†ξ˜Œξ˜+ξ˜‡ξ˜…ξ˜„ξ˜‚ξ˜…ξ˜„ξ˜ξ˜ˆξ˜‡)ξ˜’ξ˜†ξ˜ξ˜žξ˜ξ˜Žξ˜•ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜ƒξ˜Ÿξ˜…
ξ˜‹   ξ˜™  ξ˜‘    ξ˜ƒ ξ˜›  ξ˜…   ξ˜ƒ ξ˜„  ξ˜„ ξ˜…   ξ˜„ ξ˜‘  ξ˜ƒ ξ˜„ ξ˜’
, ξ˜” ξ˜‡ ξ˜… ξ˜‰ ξ˜‚ ξ˜„ ξ˜‡  ξ˜„  ξ˜†  ξ˜… ξ˜‚ ξ˜‹ ξ˜„  ξ˜‚  ξ˜‡ ξ˜ƒ ξ˜…  ξ˜‚ ξ˜“ ξ˜‡  ξ˜… ξ˜– - ξ˜‡   ξ˜†  ξ˜…  ξ˜™ ξ˜™ .  ξ˜— ξ˜‹ ξ˜‘   ξ˜…  ξ˜™ / 0  ξ˜… ξ˜‰  ξ˜‚ ξ˜’  ξ˜“ ξ˜‡ ξ˜ƒ ξ˜… ξ˜† ξ˜… ξ˜‹ ξ˜ƒ ξ˜‡  ξ˜‹ 
ξ˜ξ˜ξ˜†ξ˜ˆξ˜‡$ξ˜‚ξ˜'ξ˜…ξ˜ξ˜‚ξ˜ξ˜…ξ˜“ξ˜‡ξ˜ξ˜ξ˜Žξ˜ξ˜Žξ˜•ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜‡ξ˜ξ˜ξ˜‡ξ˜Šξ˜„ξ˜ƒξ˜…ξ˜‚ξ˜ξ˜…ξ˜†ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜…ξ˜‚ξ˜ξ˜…ξ˜‡%ξ˜‰ξ˜‚ξ˜ƒξ˜‹ξ˜ξ˜‡ξ˜Ÿξ˜…ξ˜…1ξ˜‚ξ˜ξ˜…ξ˜†ξ˜…ξ˜ƒξ˜Šξ˜†ξ˜Œξ˜†ξ˜ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„!
ξ˜Œξ˜‡ξ˜„ξ˜…ξ˜ξ˜…ξ˜“ξ˜‡ξ˜Žξ˜‚ξ˜„ξ˜‡ξ˜…ξ˜†ξ˜…ξ˜ƒξ˜‹ξ˜‘"ξ˜‡ξ˜Šξ˜„&ξ˜ƒξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„!ξ˜…ξ˜‚ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡!ξ˜…ξ˜†ξ˜Žξ˜“ξ˜…ξ˜ƒξ˜…ξ˜ˆξ˜‡ξ˜†ξ˜ƒξ˜‹ξ˜ξ˜‡ξ˜“ξ˜…ξ˜‰ξ˜ξ˜‡ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„
ξ˜Šξ˜‚ξ˜’ξ˜†ξ˜ξ˜ξ˜†ξ˜„ξ˜‡ξ˜ƒξ˜Ÿξ˜…ξ˜…2ξ˜‡ξ˜„ξ˜… ξ˜…ξ˜“ξ˜‡ξ˜Žξ˜‚ξ˜„ξ˜‡ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜…ξ˜„ξ˜”ξ˜†ξ˜„ξ˜…$ξ˜‚ξ˜‹ξ˜Œξ˜“ξ˜…ξ˜‘ξ˜‡ξ˜…ξ˜ƒξ˜‡ξ˜‡ξ˜Žξ˜…ξ˜ξ˜Žξ˜…ξ˜†ξ˜…ξ˜ƒξ˜‹ξ˜‘"ξ˜‡ξ˜Šξ˜„ξ˜…$ξ˜‡ξ˜ξ˜‡ξ˜…ξ˜„ξ˜”ξ˜†ξ˜„
ξ˜ƒξ˜‹ξ˜‘"ξ˜‡ξ˜Šξ˜„ξ˜… ξ˜„ξ˜‚ξ˜… ξ˜ξ˜‡ξ˜Šξ˜‡ξ˜ξ˜’ξ˜‡ξ˜… ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜… ξ˜Œξ˜‡ξ˜’ξ˜‡ξ˜Œξ˜… ξ˜„ξ˜Ÿξ˜… ξ˜… 3ξ˜„ξ˜… ξ˜„ξ˜”ξ˜‡ξ˜… ξ˜ξ˜Žξ˜“ξ˜ξ˜’ξ˜ξ˜“ξ˜‹ξ˜†ξ˜Œξ˜… ξ˜Œξ˜‡ξ˜’ξ˜‡ξ˜Œ!ξ˜… ξ˜Šξ˜†ξ˜‹ξ˜ƒξ˜†ξ˜Œξ˜… ξ˜‡ξ˜ξ˜ξ˜‡ξ˜Šξ˜„ξ˜ƒξ˜… ξ˜†ξ˜ξ˜‡
ξ˜Šξ˜‚ξ˜Žξ˜„ξ˜ξ˜†ξ˜ƒξ˜„ξ˜ƒξ˜…ξ˜‚ξ˜ξ˜…ξ˜“ξ˜ξ˜ξ˜ξ˜‡ξ˜ξ˜‡ξ˜Žξ˜„ξ˜…ξ˜‰ξ˜‚ξ˜„ξ˜‡ξ˜Žξ˜„ξ˜ξ˜†ξ˜Œξ˜…ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜ƒξ˜… ξ˜…ξ˜†ξ˜Žξ˜“ξ˜… !ξ˜… ξ˜Ÿξ˜…ξ˜…3ξ˜„ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜•ξ˜ξ˜‚ξ˜‹ξ˜‰ξ˜…ξ˜Œξ˜‡ξ˜’ξ˜‡ξ˜Œ!
ξ˜Šξ˜†ξ˜‹ξ˜ƒξ˜†ξ˜Œξ˜… ξ˜‡ξ˜ξ˜ξ˜‡ξ˜Šξ˜„ξ˜ƒξ˜… ξ˜†ξ˜ξ˜‡ξ˜… ξ˜Šξ˜‚ξ˜Žξ˜„ξ˜ξ˜†ξ˜ƒξ˜„ξ˜ƒξ˜… ξ˜‚ξ˜ξ˜… ξ˜“ξ˜ξ˜ƒξ˜„ξ˜ξ˜ξ˜‘ξ˜‹ξ˜„ξ˜ξ˜‚ξ˜Žξ˜ƒξ˜… ξ˜‚ξ˜ξ˜… ξ˜‰ξ˜‚ξ˜„ξ˜‡ξ˜Žξ˜„ξ˜ξ˜†ξ˜Œξ˜… ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜ƒξ˜… ξ˜‹ξ˜Žξ˜“ξ˜‡ξ˜ξ˜… ξ˜“ξ˜ξ˜ξ˜ξ˜‡ξ˜ξ˜‡ξ˜Žξ˜„
ξ˜Šξ˜‚ξ˜Žξ˜“ξ˜ξ˜„ξ˜ξ˜‚ξ˜Žξ˜ƒξ˜…ξ˜ξ˜‚ξ˜ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜ƒξ˜†ξ˜ˆξ˜‡ξ˜…ξ˜•ξ˜ξ˜‚ξ˜‹ξ˜‰ξ˜ξ˜…ξ˜ξ˜Ÿξ˜‡ξ˜Ÿ!ξ˜…ξ˜‚ξ˜ξ˜… ξ˜…ξ˜†ξ˜Žξ˜“ξ˜… ξ˜Ÿξ˜…ξ˜…,ξ˜”ξ˜‡ξ˜ƒξ˜‡ξ˜…ξ˜Šξ˜†ξ˜‹ξ˜ƒξ˜†ξ˜Œξ˜…ξ˜‡ξ˜ξ˜ξ˜‡ξ˜Šξ˜„ξ˜ƒξ˜…ξ˜†ξ˜ξ˜‡
ξ˜Žξ˜‚ξ˜„ξ˜… ξ˜‚ξ˜‘ξ˜ƒξ˜‡ξ˜ξ˜’ξ˜†ξ˜‘ξ˜Œξ˜‡!ξ˜… ξ˜ƒξ˜ξ˜Žξ˜Šξ˜‡ξ˜… ξ˜… ξ˜†ξ˜Žξ˜“ξ˜… ξ˜… ξ˜†ξ˜ξ˜‡ξ˜… ξ˜Žξ˜‚ξ˜„ξ˜… ξ˜ƒξ˜ξ˜ˆξ˜‹ξ˜Œξ˜„ξ˜†ξ˜Žξ˜‡ξ˜‚ξ˜‹ξ˜ƒξ˜Œξ˜žξ˜… ξ˜‚ξ˜‘ξ˜ƒξ˜‡ξ˜ξ˜’ξ˜†ξ˜‘ξ˜Œξ˜‡ξ˜Ÿξ˜… ξ˜… 2ξ˜‡ξ˜„
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Joffe et al.: Selective Ignorability Assumptions
ξ˜…ξ˜“ξ˜‡ξ˜Žξ˜‚ξ˜„ξ˜‡ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜ƒξ˜‡ξ˜„ξ˜…ξ˜‚ξ˜ξ˜…ξ˜†ξ˜Œξ˜Œξ˜…ξ˜‰ξ˜‚ξ˜„ξ˜‡ξ˜Žξ˜„ξ˜ξ˜†ξ˜Œξ˜…ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜ƒξ˜…ξ˜†ξ˜Žξ˜“ξ˜…ξ˜ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜ƒξ˜‡ξ˜„ξ˜…ξ˜‚ξ˜ξ˜…ξ˜†ξ˜Œξ˜Œξ˜…ξ˜‰ξ˜‚ξ˜ƒξ˜ƒξ˜ξ˜‘ξ˜Œξ˜‡
ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜ƒξ˜Ÿ
, ξ˜” ξ˜‡ ξ˜… ξ˜† ξ˜ƒ ξ˜ƒ ξ˜‹  ξ˜‰ ξ˜„  ξ˜‚  ξ˜… ξ˜‚  ξ˜… ξ˜ƒ ξ˜„  ξ˜‚  ξ˜•   ξ˜…  ξ˜•  ξ˜‚  ξ˜† ξ˜‘  ξ˜‡ ξ˜… ξ˜„  ξ˜‡ ξ˜† ξ˜„  ξ˜‡  ξ˜„ ξ˜… ξ˜† ξ˜ƒ ξ˜ƒ  ξ˜•   ξ˜‡  ξ˜„ ξ˜… ξ˜– ξ˜— ξ˜‚ ξ˜ƒ ξ˜‡  ξ˜‘ ξ˜† ξ˜‹  ξ˜… ξ˜†  ξ˜“
ξ˜— ξ˜‹ ξ˜‘   ξ˜…  ξ˜™  ξ˜›  ξ˜… ξ˜… ξ˜†   ξ˜‚ $ ξ˜ƒ ξ˜…  ξ˜“ ξ˜‡  ξ˜„     ξ˜† ξ˜„  ξ˜‚  ξ˜… ξ˜‚  ξ˜…  ξ˜† ξ˜‹ ξ˜ƒ ξ˜†  ξ˜… ξ˜‡   ξ˜‡  ξ˜„ ξ˜ƒ ξ˜…   ξ˜‚  ξ˜… ξ˜‚ ξ˜‘ ξ˜ƒ ξ˜‡  ξ˜’ ξ˜‡ ξ˜“ ξ˜… ξ˜“ ξ˜† ξ˜„ ξ˜† ξ˜… ξ˜‚  ξ˜… ξ˜„  ξ˜‡ ξ˜† ξ˜„  ξ˜‡  ξ˜„ !
ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡!ξ˜…ξ˜†ξ˜Žξ˜“ξ˜…ξ˜Šξ˜‚ξ˜’ξ˜†ξ˜ξ˜ξ˜†ξ˜„ξ˜‡ξ˜ƒξ˜Ÿξ˜…ξ˜…,ξ˜”ξ˜‡ξ˜…ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‰ξ˜„ξ˜ξ˜‚ξ˜Žξ˜…ξ˜ˆξ˜†ξ˜žξ˜…ξ˜‘ξ˜‡ξ˜…$ξ˜ξ˜ξ˜„ξ˜„ξ˜‡ξ˜Žξ˜…ξ˜†ξ˜ƒ
4 ξ˜†   ξ˜†  ξ˜„ ξ˜ƒ ξ˜… ξ˜‚  ξ˜… ξ˜„ ξ˜” ξ˜‡ ξ˜… ξ˜† ξ˜ƒ ξ˜ƒ ξ˜‹  ξ˜‰ ξ˜„  ξ˜‚  ξ˜… ξ˜” ξ˜† ξ˜’ ξ˜‡ ξ˜… ξ˜†  ξ˜ƒ ξ˜‚ ξ˜… ξ˜‘ ξ˜‡ ξ˜‡  ξ˜… ξ˜‰  ξ˜‚ ξ˜‰ ξ˜‚ ξ˜ƒ ξ˜‡ ξ˜“ ξ˜… $ ξ˜”   ξ˜” ξ˜… ξ˜ƒ ξ˜„    ξ˜… ξ˜†   ξ˜‚ $ ξ˜…  ξ˜“ ξ˜‡  ξ˜„     ξ˜† ξ˜„  ξ˜‚ 
ξ˜‚ξ˜ξ˜… ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜… ξ˜‡ξ˜ξ˜ξ˜‡ξ˜Šξ˜„ξ˜ƒξ˜… ξ˜–(ξ˜ξ˜‡ξ˜‡ξ˜Žξ˜Œξ˜†ξ˜Žξ˜“ξ˜… ξ˜†ξ˜Žξ˜“ξ˜…ξ˜—ξ˜‚ξ˜‘ξ˜ξ˜Žξ˜ƒξ˜… ξ˜˜ξ˜™ξ˜š56ξ˜‡ξ˜ξ˜Žξ˜†ξ˜Žξ˜…ξ˜†ξ˜Žξ˜“ξ˜… ξ˜—ξ˜‚ξ˜‘ξ˜ξ˜Žξ˜ƒξ˜… 7..ξ˜™ξ˜œξ˜ξ˜… ξ˜ξ˜‚ξ˜
ξ˜‡%ξ˜†ξ˜ˆξ˜‰ξ˜Œξ˜‡!ξ˜…ξ˜‚ξ˜Žξ˜‡ξ˜…ξ˜ˆξ˜ξ˜•ξ˜”ξ˜„ξ˜…ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‡ξ˜…ξ˜„ξ˜”ξ˜†ξ˜„ξ˜…
8 ξ˜‚  ξ˜“  ξ˜„  ξ˜‚  ξ˜… ξ˜ξ˜‚ξ˜ƒ!ξ˜… ξ˜„ξ˜‚ξ˜•ξ˜‡ξ˜„ξ˜”ξ˜‡ξ˜ξ˜… $ξ˜ξ˜„ξ˜”ξ˜… ξ˜„ξ˜”ξ˜‡ξ˜… ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‰ξ˜„ξ˜ξ˜‚ξ˜Žξ˜… ξ˜„ξ˜”ξ˜†ξ˜„ξ˜… !ξ˜… ξ˜ξ˜ƒξ˜… ξ˜ƒξ˜‹ξ˜ξ˜ξ˜ξ˜Šξ˜ξ˜‡ξ˜Žξ˜„ξ˜… ξ˜„ξ˜‚
ξ˜ξ˜“ξ˜‡ξ˜Žξ˜„ξ˜ξ˜ξ˜žξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜‡ξ˜ξ˜ξ˜‡ξ˜Šξ˜„ξ˜…ξ˜‚ξ˜ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜… ξ˜‚ξ˜Žξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜‡ξ˜“ξ˜…ξ˜ξ˜‚ξ˜ξ˜…ξ˜†ξ˜…ξ˜‘ξ˜ξ˜Žξ˜†ξ˜ξ˜žξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„!ξ˜…ξ˜†ξ˜Žξ˜“!ξ˜…ξ˜ˆξ˜‚ξ˜ξ˜‡
ξ˜• ξ˜‡  ξ˜‡  ξ˜†    ! ξ˜… ξ˜‰  ξ˜‚ ξ˜’  ξ˜“ ξ˜‡ ξ˜ƒ ξ˜… ξ˜† ξ˜… ξ˜‘ ξ˜† ξ˜ƒ  ξ˜ƒ ξ˜…  ξ˜‚  ξ˜… ξ˜‡ ξ˜ƒ ξ˜„   ξ˜† ξ˜„  ξ˜‚  ξ˜…   ξ˜… ξ˜ƒ ξ˜„  ξ˜‹  ξ˜„ ξ˜‹  ξ˜†  ξ˜…  ξ˜‡ ξ˜ƒ ξ˜„ ξ˜‡ ξ˜“ ξ˜…  ξ˜‚ ξ˜“ ξ˜‡  ξ˜ƒ  ξ˜… ξ˜… 3  ξ˜„ ξ˜‡   ξ˜† ξ˜„  ξ˜’ ξ˜‡   !
ξ˜„ ξ˜”ξ˜‡ ξ˜…  ξ˜‚  ξ˜“  ξ˜„  ξ˜‚  ξ˜†  ξ˜…   ξ˜“ ξ˜‡ ξ˜‰ ξ˜‡  ξ˜“ ξ˜‡   ξ˜‡ ξ˜… ξ˜† ξ˜ƒ ξ˜ƒ ξ˜‹  ξ˜‰ ξ˜„  ξ˜‚  ξ˜ƒ ξ˜… ξ˜‡  ξ˜„ ξ˜†   ξ˜‡ ξ˜“ ξ˜… ξ˜‘ ξ˜žξ˜…  ξ˜„ξ˜ƒ ξ˜… ξ˜†  ξ˜“ ξ˜…  ξ˜‚ ξ˜ƒ ξ˜…  ξ˜†  ξ˜… ξ˜‘ ξ˜‡ ξ˜… $   ξ˜„ ξ˜„ ξ˜‡  ξ˜… ξ˜† ξ˜ƒ ξ˜…
ξ˜‚ξ˜ξ˜… !ξ˜…ξ˜ξ˜‡ξ˜ƒξ˜‰ξ˜‡ξ˜Šξ˜„ξ˜ξ˜’ξ˜‡ξ˜Œξ˜žξ˜Ÿ
ξ˜‹  ξ˜‹ ξ˜™  ξ˜‘   ξ˜›  ξ˜… ξ˜•  ξ˜ƒ ξ˜–  ξ˜„ ξ˜…   ξ˜„ ξ˜‘  ξ˜ƒ ξ˜„ ξ˜’
1 ξ˜‚  ξ˜… ξ˜„   ξ˜‡ ) ξ˜’ ξ˜†     ξ˜• ξ˜… ξ˜„  ξ˜‡ ξ˜† ξ˜„  ξ˜‡  ξ˜„ ξ˜ƒ ! ξ˜… ξ˜ƒ ξ˜‡ 9 ξ˜‹ ξ˜‡  ξ˜„  ξ˜†  ξ˜… ξ˜’ ξ˜‡  ξ˜ƒ  ξ˜‚  ξ˜ƒ ξ˜… ξ˜‚  ξ˜…  ξ˜•  ξ˜‚  ξ˜† ξ˜‘    ξ˜„  ξ˜… ξ˜” ξ˜† ξ˜’ ξ˜‡ ξ˜… ξ˜‘ ξ˜‡ ξ˜‡  ξ˜… ξ˜‰  ξ˜‚ ξ˜‰ ξ˜‚ ξ˜ƒ ξ˜‡ ξ˜“ 
2 ξ˜‡ ξ˜„ ξ˜… ξ˜… ξ˜“ ξ˜‡  ξ˜‚ ξ˜„ ξ˜‡ ξ˜… ξ˜„ ξ˜” ξ˜‡ ξ˜… ξ˜’ ξ˜†  ξ˜‹ ξ˜‡ ξ˜ƒ ξ˜… ξ˜‚  ξ˜… ξ˜„  ξ˜‡ ξ˜† ξ˜„  ξ˜‡  ξ˜„ ξ˜… ξ˜‰  ξ˜‚ ξ˜’  ξ˜“ ξ˜‡ ξ˜“ ξ˜… ξ˜† ξ˜„ ξ˜… ξ˜„   ξ˜‡ ξ˜… ξ˜…!ξ˜… ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜’ξ˜†ξ˜Œξ˜‹ξ˜‡ξ˜…ξ˜‚ξ˜ξ˜…ξ˜Šξ˜‚ξ˜’ξ˜†ξ˜ξ˜ξ˜†ξ˜„ξ˜‡ξ˜ƒ
ξ˜ˆξ˜‡ξ˜†ξ˜ƒξ˜‹ξ˜ξ˜‡ξ˜“ξ˜…ξ˜†ξ˜„ξ˜…ξ˜…!ξ˜… ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜’ξ˜†ξ˜Œξ˜‹ξ˜‡ξ˜…ξ˜‚ξ˜ξ˜…ξ˜†ξ˜…ξ˜ξ˜‡ξ˜‰ξ˜‡ξ˜†ξ˜„ξ˜‡ξ˜“ξ˜…ξ˜ˆξ˜‡ξ˜†ξ˜ƒξ˜‹ξ˜ξ˜‡ξ˜ƒξ˜…ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜…ξ˜†ξ˜„ξ˜…ξ˜…!ξ˜…ξ˜†ξ˜Žξ˜“ξ˜… ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜’ξ˜†ξ˜Œξ˜‹ξ˜‡
ξ˜‚ξ˜ξ˜… ξ˜†ξ˜… ξ˜ξ˜†ξ˜ξ˜Œξ˜‹ξ˜ξ˜‡)ξ˜„ξ˜ξ˜ˆξ˜‡ξ˜… ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜Ÿξ˜… ξ˜… * ξ˜‡ξ˜… ξ˜ξ˜Žξ˜Šξ˜Œξ˜‹ξ˜“ξ˜‡ξ˜… ξ˜ξ˜Žξ˜… ξ˜… ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜ƒξ˜… ξ˜†ξ˜„ξ˜… ξ˜…ξ˜…ξ˜–ξ˜ξ˜Ÿξ˜‡ξ˜Ÿ!ξ˜…ξ˜…ξ˜†ξ˜Žξ˜“
ξ˜…ξ˜–ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜„ξ˜ξ˜ˆξ˜‡ξ˜…ξ˜‚ξ˜ξ˜…ξ˜ξ˜†ξ˜ξ˜Œξ˜‹ξ˜ξ˜‡ξ˜ƒξ˜…ξ˜‚ξ˜Šξ˜Šξ˜‹ξ˜ξ˜ξ˜Žξ˜•ξ˜…ξ˜‘ξ˜‡ξ˜„$ξ˜‡ξ˜‡ξ˜Žξ˜…ξ˜…ξ˜ξ˜˜ ξ˜… ξ˜†  ξ˜“ ξ˜… ξ˜…ξ˜œ  ξ˜… ξ˜… * ξ˜‡ ξ˜… ξ˜‹ ξ˜ƒ ξ˜‡
ξ˜‚ξ˜’ξ˜‡ξ˜ξ˜‘ξ˜†ξ˜ξ˜ƒξ˜… ξ˜„ξ˜‚ξ˜… ξ˜“ξ˜‡ξ˜Žξ˜‚ξ˜„ξ˜‡ξ˜… ξ˜”ξ˜ξ˜ƒξ˜„ξ˜‚ξ˜ξ˜žξ˜… ξ˜‚ξ˜ξ˜… ξ˜†ξ˜… ξ˜’ξ˜†ξ˜ξ˜ξ˜†ξ˜‘ξ˜Œξ˜‡ξ˜ξ˜… ξ˜„ξ˜”ξ˜‹ξ˜ƒ!ξ˜… ξ˜…ξ˜ξ˜ƒξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜”ξ˜ξ˜ƒξ˜„ξ˜‚ξ˜ξ˜žξ˜…ξ˜‚ξ˜
ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜…ξ˜„ξ˜”ξ˜ξ˜‚ξ˜‹ξ˜•ξ˜”ξ˜…ξ˜…!ξ˜…ξ˜†ξ˜Žξ˜“ξ˜… ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜”ξ˜ξ˜ƒξ˜„ξ˜‚ξ˜ξ˜žξ˜…ξ˜‚ξ˜ξ˜…ξ˜Šξ˜‚ξ˜’ξ˜†ξ˜ξ˜ξ˜†ξ˜„ξ˜‡ξ˜ƒξ˜…ξ˜–ξ˜†ξ˜Žξ˜“ξ˜…ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜œξ˜…ξ˜„ξ˜”ξ˜ξ˜‚ξ˜‹ξ˜•ξ˜”ξ˜…ξ˜…ξ˜Ÿξ˜…ξ˜…* ξ˜‡
ξ˜‹ξ˜ƒξ˜‡ξ˜… ξ˜„ξ˜”ξ˜‡ξ˜… ξ˜„ξ˜ξ˜Œξ˜“ξ˜‡ξ˜… ξ˜ƒξ˜žξ˜ˆξ˜‘ξ˜‚ξ˜Œξ˜… ξ˜„ξ˜‚ξ˜… ξ˜“ξ˜‡ξ˜Žξ˜‚ξ˜„ξ˜‡ξ˜… ξ˜ξ˜‹ξ˜„ξ˜‹ξ˜ξ˜‡ξ˜… ξ˜’ξ˜†ξ˜Œξ˜‹ξ˜‡ξ˜ƒξ˜… ξ˜‚ξ˜ξ˜… ξ˜†ξ˜… ξ˜’ξ˜†ξ˜ξ˜ξ˜†ξ˜‘ξ˜Œξ˜‡ξ˜ξ˜… ξ˜„ξ˜”ξ˜‹ξ˜ƒ!
ξ˜Ÿξ˜…ξ˜…
: ξ˜‚ ξ˜„ ξ˜‡  ξ˜„  ξ˜†  ξ˜… ξ˜‚ ξ˜‹ ξ˜„  ξ˜‚  ξ˜‡ ξ˜ƒ ξ˜… ξ˜†  ξ˜“ ξ˜…  ξ˜† ξ˜‹ ξ˜ƒ ξ˜†  ξ˜… ξ˜‡   ξ˜‡  ξ˜„ ξ˜ƒ ξ˜…  ξ˜‹ ξ˜ƒ ξ˜„ ξ˜…  ξ˜‚ $ ξ˜… ξ˜‘ ξ˜‡ ξ˜…   ξ˜“ ξ˜‡ % ξ˜‡ ξ˜“ ξ˜… ξ˜‘  ξ˜… ξ˜„  ξ˜‡ ξ˜† ξ˜„  ξ˜‡  ξ˜„
ξ˜”  ξ˜ƒ ξ˜„ ξ˜‚   ξ˜‡ ξ˜ƒ  ξ˜… ξ˜… , ξ˜” ξ˜‹ ξ˜ƒ ! ξ˜… $ ξ˜‡ ξ˜… $   ξ˜„ ξ˜‡ ξ˜… ξ˜„ ξ˜” ξ˜‡ ξ˜… ξ˜‚ ξ˜‹ ξ˜„  ξ˜‚  ξ˜‡ ξ˜… ξ˜„ ξ˜” ξ˜† ξ˜„ ξ˜… $ ξ˜‚ ξ˜‹  ξ˜“ ξ˜… ξ˜‘ ξ˜‡ ξ˜… ξ˜ƒ ξ˜‡ ξ˜‡  ξ˜… ξ˜† ξ˜„ ξ˜… ξ˜…ξ˜…ξ˜‹ξ˜Žξ˜“ξ˜‡ξ˜ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„
ξ˜”  ξ˜ƒ ξ˜„ ξ˜‚   ξ˜… ξ˜… ξ˜† ξ˜ƒ ξ˜… ξ˜… ξ˜†  ξ˜“ ξ˜… ξ˜„ ξ˜” ξ˜‡ ξ˜… ξ˜’ ξ˜‡  ξ˜„ ξ˜‚  ξ˜… ξ˜‚  ξ˜…  ξ˜‹ ξ˜„ ξ˜‹  ξ˜‡ ξ˜… ξ˜‚ ξ˜‹ ξ˜„  ξ˜‚  ξ˜‡ ξ˜ƒ ξ˜… ξ˜† ξ˜„ ξ˜… ξ˜…ξ˜…ξ˜†ξ˜ƒξ˜… ξ˜Ÿξ˜…ξ˜…ξ˜…;ξ˜‡ξ˜Žξ˜‚ξ˜„ξ˜‡
ξ˜ξ˜„ξ˜ƒ
ξ˜ξ˜‚ξ˜ƒ
2
The International Journal of Biostatistics, Vol. 6 [2010], Iss. 2, Art. 11
DOI: 10.2202/1557-4679.1199
ξ˜‘ξ˜žξ˜… ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜…ξ˜”ξ˜ξ˜ƒξ˜„ξ˜‚ξ˜ξ˜žξ˜… ξ˜…ξ˜„ξ˜”ξ˜ξ˜‚ξ˜‹ξ˜•ξ˜”ξ˜…ξ˜…ξ˜…ξ˜ξ˜‚ξ˜Œξ˜Œξ˜‚$ξ˜‡ξ˜“ξ˜…ξ˜‘ξ˜žξ˜…ξ˜ξ˜‡ξ˜Šξ˜‡ξ˜ξ˜’ξ˜ξ˜Žξ˜•ξ˜…ξ˜Žξ˜‚ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„
ξ˜‘ξ˜‡ξ˜žξ˜‚ξ˜Žξ˜“ξ˜…ξ˜„ξ˜”ξ˜†ξ˜„ξ˜…ξ˜‰ξ˜‚ξ˜ξ˜Žξ˜„ξ˜…ξ˜ξ˜Žξ˜…ξ˜„ξ˜ξ˜ˆξ˜‡ξ˜Ÿξ˜…ξ˜…,ξ˜”ξ˜‹ξ˜ƒ!ξ˜… ξ˜…ξ˜“ξ˜‡ξ˜Žξ˜‚ξ˜„ξ˜‡ξ˜ƒξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜…ξ˜„ξ˜”ξ˜†ξ˜„ξ˜…$ξ˜‚ξ˜‹ξ˜Œξ˜“ξ˜…ξ˜‘ξ˜‡ξ˜…ξ˜ƒξ˜‡ξ˜‡ξ˜Ž
ξ˜†ξ˜„ξ˜…ξ˜†ξ˜…   ξ˜… ξ˜ƒ ξ˜‚  ξ˜‡ ξ˜‚  ξ˜‡ ξ˜…  ξ˜‚   ξ˜‚ $ ξ˜‡ ξ˜“ ξ˜… ξ˜”ξ˜ ξ˜ƒ ξ˜… ξ˜‚ ξ˜‘ ξ˜ƒ ξ˜‡  ξ˜’ ξ˜‡ ξ˜“ ξ˜… ξ˜„  ξ˜‡ ξ˜† ξ˜„  ξ˜‡  ξ˜„ ξ˜… ξ˜”ξ˜ ξ˜ƒ ξ˜„ ξ˜‚  ξ˜žξ˜… ξ˜„ ξ˜”ξ˜ ξ˜‚ ξ˜‹ ξ˜•ξ˜”ξ˜… ξ˜…ξ˜… ξ˜„ ξ˜”ξ˜‡  ξ˜…  ξ˜‡  ξ˜‡  ξ˜’ ξ˜‡ ξ˜“ ξ˜…  ξ˜‚
ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜…ξ˜ƒξ˜‹ξ˜‘ξ˜ƒξ˜‡9ξ˜‹ξ˜‡ξ˜Žξ˜„ξ˜Œξ˜žξ˜Ÿξ˜…ξ˜… ξ˜ξ˜ˆξ˜ξ˜Œξ˜†ξ˜ξ˜Œξ˜ž! ξ˜…ξ˜ξ˜Žξ˜“ξ˜ξ˜Šξ˜†ξ˜„ξ˜‡ξ˜ƒξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜’ξ˜‡ξ˜Šξ˜„ξ˜‚ξ˜ξ˜…ξ˜‚ξ˜ξ˜…ξ˜‰ξ˜‚ξ˜„ξ˜‡ξ˜Žξ˜„ξ˜ξ˜†ξ˜Œξ˜…ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜ƒ
ξ˜‘ξ˜‡ξ˜žξ˜‚ξ˜Žξ˜“ξ˜…ξ˜…ξ˜…ξ˜„ξ˜”ξ˜†ξ˜„ξ˜…$ξ˜‚ξ˜‹ξ˜Œξ˜“ξ˜…ξ˜”ξ˜†ξ˜’ξ˜‡ξ˜…ξ˜‘ξ˜‡ξ˜‡ξ˜Žξ˜…ξ˜ƒξ˜‡ξ˜‡ξ˜Žξ˜…ξ˜”ξ˜†ξ˜“ξ˜…ξ˜†ξ˜…ξ˜ƒξ˜‹ξ˜‘"ξ˜‡ξ˜Šξ˜„ξ˜…ξ˜ξ˜‡ξ˜Šξ˜‡ξ˜ξ˜’ξ˜‡ξ˜“ξ˜…ξ˜”ξ˜ξ˜ƒξ˜…ξ˜‚ξ˜ξ˜…ξ˜”ξ˜‡ξ˜ξ˜…ξ˜‚ξ˜‘ξ˜ƒξ˜‡ξ˜ξ˜’ξ˜‡ξ˜“
ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜…ξ˜„ξ˜”ξ˜ξ˜‚ξ˜‹ξ˜•ξ˜”ξ˜…ξ˜…ξ˜…ξ˜ξ˜‚ξ˜Œξ˜Œξ˜‚$ξ˜‡ξ˜“ξ˜…ξ˜‘ξ˜žξ˜…ξ˜Žξ˜‚ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜…ξ˜ƒξ˜‹ξ˜‘ξ˜ƒξ˜‡9ξ˜‹ξ˜‡ξ˜Žξ˜„ξ˜Œξ˜žξ˜Ÿξ˜…ξ˜…
<  ξ˜‡ ξ˜… ξ˜• ξ˜‡  ξ˜‡  ξ˜†   + ξ˜† ξ˜„  ξ˜‚  ξ˜… ξ˜‚  ξ˜… $ ξ˜‡ ξ˜† ' ξ˜… ξ˜’ ξ˜‡  ξ˜ƒ  ξ˜‚  ξ˜… ξ˜‚  ξ˜…  ξ˜•  ξ˜‚  ξ˜† ξ˜‘    ξ˜„  ξ˜… ξ˜ξ˜‚ξ˜ƒξ˜…ξ˜„ξ˜‚ξ˜…ξ˜„ξ˜”ξ˜ξ˜ƒξ˜…ξ˜ƒξ˜‡ξ˜„ξ˜„ξ˜ξ˜Žξ˜•ξ˜…ξ˜ξ˜ƒξ˜…
ξ˜‚ξ˜!ξ˜…ξ˜‡9ξ˜‹ξ˜ξ˜’ξ˜†ξ˜Œξ˜‡ξ˜Žξ˜„ξ˜Œξ˜ž!ξ˜…
ξ˜— ξ˜‚ ξ˜‘   ξ˜ƒ ξ˜… ξ˜‡ ξ˜„ ξ˜… ξ˜†   ξ˜… ξ˜– ξ˜— ξ˜‚ ξ˜‘   ξ˜ƒ ! ξ˜… =  ξ˜‡ ξ˜’   ξ˜ƒ ! ξ˜… ξ˜—  ξ˜„ ξ˜„ ξ˜‡  ξ˜… ξ˜†  ξ˜“ ξ˜… * ξ˜‹   ξ˜ƒ ξ˜‚ ξ˜”  ξ˜…  ξ˜™ ξ˜™ 7  ξ˜… ξ˜‰  ξ˜‚ ξ˜’  ξ˜“ ξ˜‡ ξ˜ƒ ξ˜… ξ˜’ ξ˜†   ξ˜†  ξ˜„
ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‰ξ˜„ξ˜ξ˜‚ξ˜Žξ˜ƒ!ξ˜…ξ˜ƒξ˜‚ξ˜ˆξ˜‡ξ˜…ξ˜‚ξ˜ξ˜…$ξ˜”ξ˜ξ˜Šξ˜”ξ˜…ξ˜†ξ˜ξ˜‡ξ˜…ξ˜ˆξ˜‚ξ˜ξ˜‡ξ˜…ξ˜†'ξ˜ξ˜Žξ˜…ξ˜„ξ˜‚ξ˜…ξ˜ƒξ˜„ξ˜ξ˜‚ξ˜Žξ˜•ξ˜…ξ˜ξ˜•ξ˜Žξ˜‚ξ˜ξ˜†ξ˜‘ξ˜ξ˜Œξ˜ξ˜„ξ˜žξ˜Ÿξ˜…ξ˜…,ξ˜”ξ˜‡ξ˜ƒξ˜‡ξ˜…ξ˜ƒξ˜‡9ξ˜‹ξ˜‡ξ˜Žξ˜„ξ˜ξ˜†ξ˜Œ
ξ˜ξ˜•ξ˜Žξ˜‚ξ˜ξ˜†ξ˜‘ξ˜ξ˜Œξ˜ξ˜„ξ˜žξ˜…ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‰ξ˜„ξ˜ξ˜‚ξ˜Žξ˜ƒξ˜…$ξ˜‚ξ˜‹ξ˜Œξ˜“ξ˜…ξ˜‘ξ˜‡ξ˜…"ξ˜‹ξ˜ƒξ˜„ξ˜ξ˜ξ˜ξ˜‡ξ˜“ξ˜…ξ˜‘ξ˜žξ˜…ξ˜†ξ˜…ξ˜ƒξ˜‡9ξ˜‹ξ˜‡ξ˜Žξ˜„ξ˜ξ˜†ξ˜Œξ˜Œξ˜žξ˜…ξ˜ξ˜†ξ˜Žξ˜“ξ˜‚ξ˜ˆξ˜+ξ˜‡ξ˜“ξ˜…ξ˜„ξ˜ξ˜ξ˜†ξ˜Œξ˜Ÿξ˜…ξ˜…#
ξ˜ƒξ˜‹ξ˜Šξ˜”ξ˜…ξ˜†ξ˜…ξ˜„ξ˜ξ˜ξ˜†ξ˜Œ!ξ˜…ξ˜†ξ˜„ξ˜…ξ˜‡ξ˜†ξ˜Šξ˜”ξ˜…ξ˜„ξ˜ξ˜ˆξ˜‡ξ˜…ξ˜…!ξ˜…ξ˜“ξ˜‡ξ˜Šξ˜ξ˜ƒξ˜ξ˜‚ξ˜Ž)ξ˜ˆξ˜†'ξ˜‡ξ˜ξ˜ƒξ˜…ξ˜‚ξ˜‘ξ˜ƒξ˜‡ξ˜ξ˜’ξ˜‡ξ˜…ξ˜†ξ˜…ξ˜ƒξ˜‹ξ˜‘"ξ˜‡ξ˜Šξ˜„&ξ˜ƒξ˜…ξ˜Šξ˜‚ξ˜’ξ˜†ξ˜ξ˜ξ˜†ξ˜„ξ˜‡ξ˜…ξ˜Œξ˜‡ξ˜’ξ˜‡ξ˜Œξ˜ƒξ˜…ξ˜†ξ˜Žξ˜“
ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜ƒξ˜…ξ˜„ξ˜”ξ˜ξ˜‚ξ˜‹ξ˜•ξ˜”ξ˜…ξ˜„ξ˜”ξ˜†ξ˜„ξ˜…ξ˜‰ξ˜‚ξ˜ξ˜Žξ˜„ξ˜…ξ˜ξ˜Žξ˜…ξ˜„ξ˜ξ˜ˆξ˜‡!ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜Žξ˜…ξ˜†ξ˜ƒξ˜ƒξ˜ξ˜•ξ˜Žξ˜…ξ˜†ξ˜…ξ˜ξ˜†ξ˜Žξ˜“ξ˜‚ξ˜ˆξ˜+ξ˜†ξ˜„ξ˜ξ˜‚ξ˜Žξ˜…ξ˜‰ξ˜ξ˜‚ξ˜‘ξ˜†ξ˜‘ξ˜ξ˜Œξ˜ξ˜„ξ˜žξ˜…ξ˜„ξ˜‚ξ˜…ξ˜„ξ˜”ξ˜‡
ξ˜ƒξ˜‹ξ˜‘"ξ˜‡ξ˜Šξ˜„ξ˜…ξ˜‘ξ˜†ξ˜ƒξ˜‡ξ˜“ξ˜…ξ˜‚ξ˜Žξ˜…ξ˜„ξ˜”ξ˜†ξ˜„ξ˜…ξ˜”ξ˜ξ˜ƒξ˜„ξ˜‚ξ˜ξ˜žξ˜Ÿξ˜…,ξ˜”ξ˜‡ξ˜…ξ˜‰ξ˜ξ˜‚ξ˜‘ξ˜†ξ˜‘ξ˜ξ˜Œξ˜ξ˜„ξ˜žξ˜…ξ˜‚ξ˜ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜…ξ˜ˆξ˜†ξ˜žξ˜…ξ˜’ξ˜†ξ˜ξ˜žξ˜…$ξ˜ξ˜„ξ˜”ξ˜…ξ˜Šξ˜‹ξ˜ξ˜ξ˜‡ξ˜Žξ˜„ξ˜…ξ˜‚ξ˜
ξ˜‰ξ˜ξ˜‡ξ˜’ξ˜ξ˜‚ξ˜‹ξ˜ƒξ˜…ξ˜Šξ˜‚ξ˜’ξ˜†ξ˜ξ˜ξ˜†ξ˜„ξ˜‡ξ˜ƒξ˜…ξ˜†ξ˜Žξ˜“ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜Ÿξ˜…ξ˜…,ξ˜”ξ˜‡ξ˜…ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‰ξ˜„ξ˜ξ˜‚ξ˜Žξ˜…ξ˜”ξ˜†ξ˜ƒξ˜…ξ˜‘ξ˜‡ξ˜‡ξ˜Žξ˜…ξ˜„ξ˜‡ξ˜ξ˜ˆξ˜‡ξ˜“ξ˜…>ξ˜Žξ˜‚ξ˜…ξ˜‹ξ˜Žξ˜ˆξ˜‡ξ˜†ξ˜ƒξ˜‹ξ˜ξ˜‡ξ˜“
ξ˜Šξ˜‚ξ˜Žξ˜ξ˜‚ξ˜‹ξ˜Žξ˜“ξ˜‡ξ˜ξ˜ƒ?ξ˜… ξ˜… ξ˜–ξ˜—ξ˜‚ξ˜‘ξ˜ξ˜Žξ˜ƒξ˜… ξ˜˜ξ˜™ξ˜™7ξ˜ξ˜—ξ˜‚ξ˜‘ξ˜ξ˜Žξ˜ƒξ˜… ξ˜‡ξ˜„ξ˜… ξ˜†ξ˜Œξ˜Ÿξ˜… ξ˜˜ξ˜™ξ˜™7!ξ˜… ξ˜ƒξ˜ξ˜Žξ˜Šξ˜‡ξ˜… ξ˜„ξ˜”ξ˜‡ξ˜ξ˜‡ξ˜… $ξ˜‚ξ˜‹ξ˜Œξ˜“ξ˜… ξ˜‘ξ˜‡ξ˜… ξ˜Žξ˜‚
ξ˜Šξ˜‚ξ˜Žξ˜ξ˜‚ξ˜‹ξ˜Žξ˜“ξ˜ξ˜Žξ˜•ξ˜…ξ˜ξ˜ξ˜…ξ˜‡ξ˜Žξ˜‚ξ˜‹ξ˜•ξ˜”ξ˜…ξ˜Šξ˜‚ξ˜Žξ˜ξ˜‚ξ˜‹ξ˜Žξ˜“ξ˜ξ˜Žξ˜•ξ˜…ξ˜’ξ˜†ξ˜ξ˜ξ˜†ξ˜‘ξ˜Œξ˜‡ξ˜ƒξ˜…ξ˜”ξ˜†ξ˜“ξ˜…ξ˜‘ξ˜‡ξ˜‡ξ˜Žξ˜…ξ˜ˆξ˜‡ξ˜†ξ˜ƒξ˜‹ξ˜ξ˜‡ξ˜“ξ˜…ξ˜„ξ˜‚ξ˜…ξ˜ξ˜‡ξ˜Žξ˜“ξ˜‡ξ˜ξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„
ξ˜†ξ˜„ξ˜…ξ˜…ξ˜…ξ˜†ξ˜Žξ˜“ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜…ξ˜‰ξ˜‚ξ˜„ξ˜‡ξ˜Žξ˜„ξ˜ξ˜†ξ˜Œξ˜…ξ˜‚ξ˜‹ξ˜„ξ˜Šξ˜‚ξ˜ˆξ˜‡ξ˜ƒξ˜…ξ˜Šξ˜‚ξ˜Žξ˜“ξ˜ξ˜„ξ˜ξ˜‚ξ˜Žξ˜†ξ˜Œξ˜Œξ˜žξ˜…ξ˜ξ˜Žξ˜“ξ˜‡ξ˜‰ξ˜‡ξ˜Žξ˜“ξ˜‡ξ˜Žξ˜„ξ˜Ÿξ˜…
     ξ˜‰ ξ˜„  ξ˜›    ξ˜– ξ˜ƒ ξ˜† ξ˜…      ξ˜„ ξ˜•   ξ˜‘ ξ˜† ξ˜„  ξ˜›  ξ˜„  ξ˜† ξ˜ƒ   ξ˜ƒ ξ˜‡  ξ˜” ξ˜† ξ˜… ξ˜‘    ξ˜„  ξ˜† ξ˜ƒ
; ξ˜‚ ξ˜‹ ξ˜‘ ξ˜„ ξ˜ƒ ξ˜… ξ˜† ξ˜‘ ξ˜‚ ξ˜‹ ξ˜„ ξ˜… ξ˜„ ξ˜” ξ˜‡ ξ˜… ξ˜† ξ˜‰ ξ˜‰  ξ˜‚ ξ˜‰   ξ˜† ξ˜„ ξ˜‡  ξ˜‡ ξ˜ƒ ξ˜ƒ ξ˜… ξ˜‚  ξ˜…  ξ˜•  ξ˜‚  ξ˜† ξ˜‘    ξ˜„  ξ˜… ξ˜† ξ˜ƒ ξ˜ƒ ξ˜‹  ξ˜‰ ξ˜„  ξ˜‚  ξ˜ƒ ξ˜… ξ˜– ξ˜‘ ξ˜‚ ξ˜„ ξ˜” ξ˜… ξ˜ƒ   ξ˜‰  ξ˜‡ ξ˜… ξ˜†  ξ˜“
ξ˜ƒξ˜‡9ξ˜‹ξ˜‡ξ˜Žξ˜„ξ˜ξ˜†ξ˜Œξ˜œξ˜… ξ˜ξ˜Žξ˜… ξ˜‰ξ˜ξ˜†ξ˜Šξ˜„ξ˜ξ˜Šξ˜‡ξ˜… ξ˜”ξ˜†ξ˜ƒξ˜… ξ˜Œξ˜‡ξ˜“ξ˜… ξ˜„ξ˜‚ξ˜… ξ˜†ξ˜Œξ˜„ξ˜‡ξ˜ξ˜Žξ˜†ξ˜„ξ˜ξ˜’ξ˜‡ξ˜… ξ˜†ξ˜‰ξ˜‰ξ˜ξ˜‚ξ˜†ξ˜Šξ˜”ξ˜‡ξ˜ƒξ˜… ξ˜„ξ˜‚ξ˜… ξ˜ξ˜Žξ˜ξ˜‡ξ˜ξ˜‡ξ˜Žξ˜Šξ˜‡ξ˜Ÿξ˜… ξ˜… ,ξ˜”ξ˜‡ξ˜ƒξ˜‡
ξ˜†ξ˜Œξ˜„ξ˜‡ξ˜ξ˜Žξ˜†ξ˜„ξ˜ξ˜’ξ˜‡ξ˜ƒξ˜… ξ˜ξ˜Žξ˜Šξ˜Œξ˜‹ξ˜“ξ˜‡ξ˜… ξ˜‡%ξ˜†ξ˜ˆξ˜ξ˜Žξ˜ξ˜Žξ˜•ξ˜… ξ˜ƒξ˜‡ξ˜Žξ˜ƒξ˜ξ˜„ξ˜ξ˜’ξ˜ξ˜„ξ˜žξ˜… ξ˜‚ξ˜ξ˜… ξ˜ξ˜Žξ˜ξ˜‡ξ˜ξ˜‡ξ˜Žξ˜Šξ˜‡ξ˜… ξ˜„ξ˜‚ξ˜… ξ˜“ξ˜‡ξ˜‰ξ˜†ξ˜ξ˜„ξ˜‹ξ˜ξ˜‡ξ˜ƒξ˜… ξ˜ξ˜ξ˜‚ξ˜ˆ
ξ˜ξ˜•ξ˜Žξ˜‚ξ˜ξ˜†ξ˜‘ξ˜ξ˜Œξ˜ξ˜„ξ˜žξ˜ξ˜… ξ˜ˆξ˜‡ξ˜„ξ˜”ξ˜‚ξ˜“ξ˜ƒξ˜… ξ˜ξ˜‚ξ˜ξ˜… ξ˜„ξ˜”ξ˜ξ˜ƒξ˜… ξ˜”ξ˜†ξ˜’ξ˜‡ξ˜… ξ˜‘ξ˜‡ξ˜‡ξ˜Žξ˜… ξ˜‰ξ˜ξ˜‚ξ˜‰ξ˜‚ξ˜ƒξ˜‡ξ˜“ξ˜… ξ˜ξ˜‚ξ˜ξ˜… ξ˜‘ξ˜‚ξ˜„ξ˜”ξ˜… ξ˜ƒξ˜„ξ˜†ξ˜„ξ˜ξ˜Šξ˜… ξ˜–(ξ˜ξ˜‡ξ˜‡ξ˜Žξ˜Œξ˜†ξ˜Žξ˜“
 ξ˜™ ξ˜™ 5  ξ˜— ξ˜‚ ξ˜ƒ ξ˜‡  ξ˜‘ ξ˜† ξ˜‹  ξ˜… 7 . . 7  ξ˜… ξ˜†  ξ˜“ ξ˜… ξ˜„   ξ˜‡ ) ξ˜’ ξ˜†     ξ˜• ξ˜… ξ˜– =  ξ˜‹  ξ˜‘ ξ˜†  ' ! ξ˜… 6 ξ˜‡   ξ˜†  ! ξ˜… 6 ξ˜†  ξ˜‡ ξ˜‹ ξ˜ƒ ξ˜‡ ξ˜… ξ˜†  ξ˜“ ξ˜… ξ˜— ξ˜‚ ξ˜‘   ξ˜ƒ
7 . . 0  ξ˜— ξ˜‚ ξ˜‘   ξ˜ƒ ! ξ˜… ξ˜— ξ˜‚ ξ˜„   ξ˜„ + '  ξ˜… ξ˜†  ξ˜“ ξ˜…  ξ˜” ξ˜†   ξ˜ƒ ξ˜„ ξ˜‡   ξ˜… 7 . . .  ξ˜… ξ˜„  ξ˜‡ ξ˜† ξ˜„  ξ˜‡  ξ˜„ ξ˜ƒ  ξ˜… ξ˜… < ξ˜„ ξ˜” ξ˜‡  ξ˜… ξ˜† ξ˜‰ ξ˜‰  ξ˜‚ ξ˜†  ξ˜” ξ˜‡ ξ˜ƒ ξ˜…     ξ˜‹ ξ˜“ ξ˜‡
ξ˜‡%ξ˜†ξ˜ˆξ˜ξ˜Žξ˜ξ˜Žξ˜•ξ˜… ξ˜‘ξ˜‚ξ˜‹ξ˜Žξ˜“ξ˜ƒξ˜… ξ˜ξ˜‚ξ˜ξ˜… ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜… ξ˜‡ξ˜ξ˜ξ˜‡ξ˜Šξ˜„ξ˜ƒξ˜… ξ˜–ξ˜ξ˜†ξ˜Žξ˜ƒ'ξ˜ξ˜… ξ˜˜ξ˜™ξ˜™.ξ˜ξ˜ξ˜†ξ˜Žξ˜ƒ'ξ˜ξ˜… ξ˜˜ξ˜™ξ˜™5ξ˜œξ˜… ξ˜ξ˜ˆξ˜‰ξ˜‚ξ˜ƒξ˜‡ξ˜“
ξ˜Žξ˜‚ξ˜Žξ˜‰ξ˜†ξ˜ξ˜†ξ˜ˆξ˜‡ξ˜„ξ˜ξ˜ξ˜Šξ˜†ξ˜Œξ˜Œξ˜žξ˜…ξ˜‚ξ˜ξ˜… ξ˜‘ξ˜†ξ˜ƒξ˜‡ξ˜“ξ˜…ξ˜‚ξ˜Žξ˜…ξ˜†ξ˜“ξ˜“ξ˜ξ˜„ξ˜ξ˜‚ξ˜Žξ˜†ξ˜Œξ˜…ξ˜†ξ˜ƒξ˜ƒξ˜‹ξ˜ˆξ˜‰ξ˜„ξ˜ξ˜‚ξ˜Žξ˜ƒξ˜ξ˜… ξ˜„ξ˜”ξ˜‡ξ˜ƒξ˜‡ξ˜…ξ˜ˆξ˜‡ξ˜„ξ˜”ξ˜‚ξ˜“ξ˜ƒξ˜…ξ˜”ξ˜†ξ˜’ξ˜‡ξ˜…ξ˜‘ξ˜‡ξ˜‡ξ˜Ž
ξ˜“ξ˜‡ξ˜’ξ˜‡ξ˜Œξ˜‚ξ˜‰ξ˜‡ξ˜“ξ˜…ξ˜Œξ˜†ξ˜ξ˜•ξ˜‡ξ˜Œξ˜žξ˜…ξ˜ξ˜‚ξ˜ξ˜…ξ˜ƒξ˜„ξ˜†ξ˜„ξ˜ξ˜Šξ˜…ξ˜„ξ˜ξ˜‡ξ˜†ξ˜„ξ˜ˆξ˜‡ξ˜Žξ˜„ξ˜ƒξ˜Ÿξ˜…ξ˜…@ξ˜ƒξ˜‹ξ˜†ξ˜Œξ˜Œξ˜ž!ξ˜…ξ˜„ξ˜”ξ˜‡ξ˜ƒξ˜‡ξ˜…ξ˜†ξ˜‰ξ˜‰ξ˜ξ˜‚ξ˜†ξ˜Šξ˜”ξ˜‡ξ˜ƒξ˜…ξ˜“ξ˜‚ξ˜…ξ˜Žξ˜‚ξ˜„ξ˜…ξ˜Œξ˜‡ξ˜†ξ˜“ξ˜…ξ˜„ξ˜‚
ξ˜ξ˜…ξ˜ƒ
ξ˜ξ˜†ξ˜ƒ
3
Joffe et al.: Selective Ignorability Assumptions
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Book Chapterβ€’DOIβ€’
Eight myths about causality and structural equation models

[...]

Kenneth A. Bollen1, Judea Pearl2β€’
University of North Carolina at Chapel Hill1, University of California2
01 Jan 2013
TL;DR: In this paper, structural equation models (SEMs) and their role in causal analysis have been discussed and a variety of misunderstandings and myths about the nature of SEMs have emerged, and their repetition has led some to believe they are true.
Abstract: Causality was at the center of the early history of structural equation models (SEMs) which continue to serve as the most popular approach to causal analysis in the social sciences. Through decades of development, critics and defenses of the capability of SEMs to support causal inference have accumulated. A variety of misunderstandings and myths about the nature of SEMs and their role in causal analysis have emerged, and their repetition has led some to believe they are true. Our chapter is organized by presenting eight myths about causality and SEMs in the hope that this will lead to a more accurate understanding. More specifically, the eight myths are the following: (1) SEMs aim to establish causal relations from associations alone, (2) SEMs and regression are essentially equivalent, (3) no causation without manipulation, (4) SEMs are not equipped to handle nonlinear causal relationships, (5) a potential outcome framework is more principled than SEMs, (6) SEMs are not applicable to experiments with randomized treatments, (7) mediation analysis in SEMs is inherently noncausal, and (8) SEMs do not test any major part of the theory against the data. We present the facts that dispel these myths, describe what SEMs can and cannot do, and briefly present our critique of current practice using SEMs. We conclude that the current capabilities of SEMs to formalize and implement causal inference tasks are indispensible; its potential to do more is even greater.

495Β citations

Cites background from "Selective Ignorability Assumptions ..."

  • ...Such conditions are extremely difficult to interpret (unaided by graphical tools) and β€œare usually made casually, largely because they justify the use of available statistical methods and not because they are truly believed” (Joffe et al. 2010)....

    [...]

Posted Contentβ€’
Interpretation and Identification of Causal Mediation

[...]

Judea Pearl1β€’
University of California, Los Angeles1
18 Sep 2013-Social Science Research Network
TL;DR: It is shown that natural effects can be identified by methods that go beyond standard adjustment for confounders, applicable to observational studies in which treatment assignment remains confounded with the mediator or with the outcome.
Abstract: This paper reviews the foundations of causal mediation analysis and offers a general and transparent account of the conditions necessary for the identification of natural direct and indirect effects, thus facilitating a more informed judgment of the plausibility of these conditions in specific applications. We show that the conditions usually cited in the literature are overly restricted, and can be relaxed substantially, without compromising identification. In particular, we show that natural effects can be identified by methods that go beyond standard adjustment for confounders, applicable to observational studies in which treatment assignment remains confounded with the mediator or with the outcome. These identification conditions can be validated algorithmically from the diagramatic description of one's model, and are guaranteed to produce unbiased results whenever the description is correct. The identification conditions can be further relaxed in parametric models, possibly including interactions, and permit us to compare the relative importance of several pathways, mediated by interdependent variables.

298Β citations

Cites background from "Selective Ignorability Assumptions ..."

  • ...The verification of such independencies, often called β€œstrong ignorability,” β€œconditional ignorability,” or β€œsequential ignorability,” presents a formidable judgmental task to most researchers if unaided by structural models (Joffe et al., 2010)....

    [...]

Journal Articleβ€’DOIβ€’
Interpretation and identification of causal mediation.

[...]

Judea Pearl1β€’
University of California, Los Angeles1
02 Jun 2014-Psychological Methods
TL;DR: In this article, the authors provide a general and transparent account of the conditions necessary for the identification of natural direct and indirect effects, thus facilitating a more informed judgment of the plausibility of these conditions in specific applications.
Abstract: This article reviews the foundations of causal mediation analysis and offers a general and transparent account of the conditions necessary for the identification of natural direct and indirect effects, thus facilitating a more informed judgment of the plausibility of these conditions in specific applications. I show that the conditions usually cited in the literature are overly restrictive and can be relaxed substantially without compromising identification. In particular, I show that natural effects can be identified by methods that go beyond standard adjustment for confounders, applicable to observational studies in which treatment assignment remains confounded with the mediator or with the outcome. These identification conditions can be validated algorithmically from the diagrammatic description of one's model and are guaranteed to produce unbiased results whenever the description is correct. The identi- fication conditions can be further relaxed in parametric models, possibly including interactions, and permit one to compare the relative importance of several pathways, mediated by interdependent variables. Mediation analysis aims to uncover causal pathways along which changes are transmitted from causes to effects. Interest in mediation analysis stems from both scientific and practical con- siderations. Scientifically, mediation tells us how nature works, and practically, it enables us to predict behavior under a rich variety of conditions and policy interventions. For example, in coping with the age-old problem of gender discrimination (Bickel, Hammel, & O'Connell, 1975; Goldberger, 1984), a policymaker may be interested in assessing the extent to which gender disparity in hiring can be reduced by making hiring decisions gender-blind, compared with eliminating gender inequality in education or job qualifications. The former concerns the direct effect of gender on hiring, while the latter concerns the indirect effect or the effect mediated via job qualification. The example illustrates two essential ingredients of modern mediation analysis. First, the indirect effect is not merely a mod- eling artifact formed by suggestive combinations of parameters but an intrinsic property of reality that has tangible policy implica- tions. In this example, reducing employers' prejudices and launch- ing educational reforms are two contending policy options that involve costly investments and different implementation efforts. Knowing in advance which of the two, if successful, has a greater impact on reducing hiring disparity is essential for planning and depends critically on mediation analysis for resolution. Second, the policy decisions in this example concern the enabling and dis- abling of processes (hiring vs. education) rather than lowering or raising values of specific variables. These two considerations lead to the analysis of natural direct and indirect effects. Mediation analysis has its roots in the literature of structural equation models (SEMs), going back to Wright's (1923, 1934) method of path analysis and continuing in the social sciences from the 1960s to 1980s through the works of Baron and Kenny (1986), Bollen (1989), Duncan (1975), and Fox (1980). The bulk of this work was carried out in the context of linear models, in which effect sizes are represented as sums and products of structural coefficients. The definition, identification, and estimation of these coefficients required a commitment to a particular parametric and distributional model and fell short of providing a general, causally defensible measure of mediation (Glynn, 2012; Hayes, 2009; Kraemer, Kiernan, Essex, & Kupfer, 2008; MacKinnon, 2008).

295Β citations

Journal Articleβ€’DOIβ€’
Structural Nested Models and G-estimation: The Partially Realized Promise

[...]

Stijn Vansteelandt, Marshall M. Joffe
05 Mar 2015-arXiv: Methodology
TL;DR: An overview of the models and estimation methods as developed, primarily by James Robins, over the years are provided, and insight into their advantages over other methods is provided.
Abstract: Structural nested models (SNMs) and the associated method of G-estimation were first proposed by James Robins over two decades ago as approaches to modeling and estimating the joint effects of a sequence of treatments or exposures. The models and estimation methods have since been extended to dealing with a broader series of problems, and have considerable advantages over the other methods developed for estimating such joint effects. Despite these advantages, the application of these methods in applied research has been relatively infrequent; we view this as unfortunate. To remedy this, we provide an overview of the models and estimation methods as developed, primarily by Robins, over the years. We provide insight into their advantages over other methods, and consider some possible reasons for failure of the methods to be more broadly adopted, as well as possible remedies. Finally, we consider several extensions of the standard models and estimation methods.

113Β citations

Journal Articleβ€’DOIβ€’
Adjustments and their Consequencesβ€”Collapsibility Analysis using Graphical Models

[...]

Sander Greenland1, Judea Pearl1β€’
University of California, Los Angeles1
01 Dec 2011-International Statistical Review
TL;DR: Probabilistic and graphical rules for detecting situations in which a dependence of one variable on another is altered by adjusting for a third variable are considered, whether that dependence is causal or purely predictive.
Abstract: We consider probabilistic and graphical rules for detecting situations in which a dependence of one variable on another is altered by adjusting for a third variable (i.e., noncollapsibility), whether that dependence is causal or purely predictive. We focus on distinguishing situations in which adjustment will reduce, increase, or leave unchanged the degree of bias in an association of two variables when that association is taken to represent a causal effect of one variable on the other. We then consider situations in which adjustment may partially remove or introduce a potential source of bias in estimating causal effects, and some additional special cases useful for casecontrol studies, cohort studies with loss, and trials with noncompliance (nonadherence).

87Β citations

Related Papers (5)
Marginal Structural Models and Causal Inference in Epidemiology

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01 Sep 2000-Epidemiology
James M. Robins, Miguel A. HernΓ‘n, Babette Brumback
Causal diagrams for empirical research

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01 Dec 1995-Biometrika
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G-estimation of the effect of prophylaxis therapy for Pneumocystis carinii pneumonia on the survival of AIDS patients.

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01 Jul 1992-Epidemiology
James M. Robins, Donald Blevins, Grant Ritter, Michael WulfsohnΒ 
Causal diagrams for epidemiologic research.

[...]

01 Jan 1999-Epidemiology
Sander Greenland, Judea Pearl, James M. Robins
A new approach to causal inference in mortality studies with a sustained exposure periodβ€”application to control of the healthy worker survivor effect

[...]

01 Jan 1986-Mathematical Modelling
James M. Robins
Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "Selective ignorability assumptions in causal inference" ?

In this paper, the authors outline selective ignorability assumptions mathematically and sketch how they may be used along with otherwise standard G-estimation or likelihood-based methods to obtain inference on structural nested models.Β The authors also consider use of these assumptions in the presence of selective measurement error or missing data when the missingness is not at random.Β The authors motivate and illustrate their development by considering an analysis of an observational database to estimate the effect of erythropoietin use on mortality among hemodialysis patients.Β