On the definition of a confounder

“M E ] 2 A pr 2 01 3 The Annals of Statistics 2013, Vol. 41, No. 1, 196–220 DOI: 10.1214/12-AOS1058 c© Institute of Mathematical Statistics, 2013 ON THE DEFINITION OF A CONFOUNDER1 By Tyler J. VanderWeele and Ilya Shpitser”

“M E ] 2 A pr 2 01 3 The Annals of Statistics 2013, Vol. 41, No. 1, 196–220 DOI: 10.1214/12-AOS1058 c© Institute of Mathematical Statistics, 2013 ON THE DEFINITION OF A CONFOUNDER1 By Tyler J. VanderWeele and Ilya Shpitser”

“Such absence of confounding is alternatively referred to as “ignorability” or “ignorable treatment assignment” [Rubin (1978)], “exchangeability” [Greenland and Robins (1986)], “no unmeasured confounding” [Robins (1992)], “selection on observables” [Barnow, Cain and Goldberger (1980), Imbens (2004)]…”

“As noted in the penultimate section of the paper, the restriction to pre-exposure covariates could, in the context of causal diagrams [Pearl (1995, 2009)], be replaced to that of nondescendents of exposure A.”

“Kleinbaum et al. (1982), in a textbook on epidemiologic research, gave as a de nition of a "confounder" a variable that is "a member of a su¢ cient confounder group" where a su¢ cient confounder group is de ned as "a minimal set of one or more risk factors whose simultaneous control in the analysis will correct for jo…”

“The authors thank Sander Greenland, James Robins and Miguel Hernán for helpful comments on this paper.”

“Hernán (2008) accordingly speaks of a confounder as a variable that "can be used to block a backdoor path between exposure and outcome" (p.”

“Such absence of confounding is alternatively referred to as “ignorability” or “ignorable treatment assignment” [Rubin (1978)], “exchangeability” [Greenland and Robins (1986)], “no unmeasured confounding” [Robins (1992)], “selection on observables” [Barnow, Cain and Goldberger (1980), Imbens (2004)]…”

“This assumption is sometimes referred to as SUTVA, the stable unit treatment value assumption [Rubin (1990)] or as a no-interference assumption [Cox (1958)].”

“VanderWeele and Shpitser (2011) note that if for covariate set S we have that Ya ⊥⊥A|S, then if a backward selection procedure is applied to S such that variables are iteratively discarded that are independent of Y conditional on both exposure A and the members of S that have not yet been…”

“Note, however, that some authors also consider “confounding” to be scale-dependent [Greenland and Robins (1986, 2009), Greenland and Morgenstern (2001)] and use “ignorability” to refer to the notion of unconfoundedness in the distribution of counterfactuals as given above.”

“Statisticians and epidemiologists had traditionally conceived of a confounder as a pre-exposure variable that was associated with exposure and associated also with the outcome conditional on the exposure, possibly conditional also on other covariates [Miettinen (1974)].”

“Confounders have also sometimes been defined in terms of empirical collapsibility [Miettinen (1976), Breslow and Day (1980)], that is, if one obtains the same estimate with or without adjustment for a variable, then it is not a confounder.”

“Miettinen and Cook (1981) conceive of a confounder as any variable that is helpful in reducing bias.”

“M E ] 2 A pr 2 01 3 The Annals of Statistics 2013, Vol. 41, No. 1, 196–220 DOI: 10.1214/12-AOS1058 c© Institute of Mathematical Statistics, 2013 ON THE DEFINITION OF A CONFOUNDER1 By Tyler J. VanderWeele and Ilya Shpitser”

“Such absence of confounding is alternatively referred to as "ignorability" or "ignorable treatment assignment" (Rubin, 1978), "exchangeability" (Greenland and Robins, 1986), "no unmeasured confounding" (Robins, 1992), "selection on observables" (Barnow et al.”

“Note, however, that some authors also consider “confounding” to be scale-dependent [Greenland and Robins (1986, 2009), Greenland and Morgenstern (2001)] and use “ignorability” to refer to the notion of unconfoundedness in the distribution of counterfactuals as given above.”

“See Greenland and Pearl (2011) and Ogburn and VanderWeele (2012) for properties of such surrogate confounders.”

“See Greenland and Pearl (2011) and Ogburn and VanderWeele (2012) for properties of such surrogate confounders.”

“See Greenland and Pearl (2011) and Ogburn and VanderWeele (2012) for properties of such surrogate confounders.”

“and Robins (1999), Glymour and Greenland (2008), Pearl (2009)]. The literature has moved away from formal language about “confounders” and instead places the conceptual emphasis on “confounding.” See Morabia (2011) for historical discussion of this point. The causal inference literature has provided a formal definition…”

“Geng et al. (2002) for example, considered the analogous definition for the effect of the exposure on the exposed rather than the overall effect of the exposure on the population; one could likewise consider the analogue of definition in 5 for effects conditional on X rather than standardized over X or alternatively f…”

“See Greenland, Robins and Pearl (1999), Geng et al. (2001) and Geng and Li (2002) for further discussion of the relationship between, and general nonequivalence of, confounding and collapsibility.”

“The result is frequently termed confounding, the situation when the relationship between two variables is distorted because of the strength of the relationships between either one or both of them and a third variable included in the analysis (see, for example, Kish 1959; Morabia 2011; VanderWheele and Shpitser 2013).”

“Competing exposures are not always included in sufficient adjustment sets, unless they are caused by one or more unobserved confounders for which they can serve as a surrogate confounder.(22)”

“(29) and VanderWeele and Shpitser (30).”

“Type: apparent; source: (ii) confounding factors Confounding occurs when two ormore independent variables, X and C, are correlated such that their effects on dependent variable Y cannot be disentangled [40,41].”