This long thread about odds ratios Should one derive risk difference from the odds ratio?contained a brief exchange about the idea of “random confounding.” After seeing experts debate this term online, I wondered whether the controversy might be fuelling confusion among students of epidemiology, statistics, and medicine.

Reading around this topic led me to the following article:

This publication seems important. Is the idea of confounding “in measure” widely recognized or accepted by statisticians, who seem to speak exclusively (?) about confounding “in expectation”? Does the epidemiologic notion of confounding “in measure” underlie historically common MD critical appraisal practices (e.g., scrutinizing Table 1 in RCTs for between-arm covariate imbalances)? At least some MDs were taught to do this by MD instructors who had an epidemiology background, only later to hear statisticians tell them, in the immortal words of Bob Newhart, to “Just Stop It !!”

Could some of the heated exchanges in the Odds Ratio thread stem from disagreement about the “in-distribution” versus “in-measure” conceptualization of confounding and effect modification?

Some notable excerpts from the article (with bolded and underlined text inserted by me for emphasis):

*“This general notion of confounding or exchangeability can be defined both with respect to the* **distribution***of potential outcomes and with respect to a* * specific measure*.

*The distinction has been drawn before (Greenland et al., 1999)…*

**…A further distinction can be drawn between confounding “in expectation” and “realized” confounding***(Fisher, 1935; Rothman, 1977; Greenland, 1990; Greenland et al., 1999). In a randomized trial the groups receiving the placebo and the treatment will be comparable in their potential outcomes* * on average over repeated experiments*. However, for any

**given***experiment, the*

*particular*

*randomization may result in chance imbalances due to the*

*particular*

*allocation. Such a scenario would be one in which there is no confounding “*

*in expectation*” but there is*realized*

*confounding*

*for the particular experiment*

*(conditional on the allocation).*

**Some authors (Greenland et al., 1999; Greenland and Robins, 2009) prefer to restrict the use of “no confounding” to that that is realized; a number of authors (e.g. Rubin, 1991; Robins, 1992; Stone, 1993) use terms like “no confounding” to refer to that in expectation; here we will adopt the latter practice…****…We have seen that a distinction can be drawn between confounding in distribution and confounding in measure. A similar distinction can in fact also be drawn with regard to effect modification…**

*…More recently, the expression "effect-* **measure***modification" (Rothman, 2002; Brumback and Berg, 2008)* *has been used in place of the expression, “ effect modification.” This has arguably occurred for two reasons. First, as has often been pointed out (Miettinen, 1974; Rothman, 2002; Brumback and Berg, 2008; Rothman et al., 2008), there may be effect modification for*

*one*

*measure (e.g. the risk*

*) but not for*

**difference***another*

*(e.g. the risk*

*).*

**ratio***Effect modification in*

**measure***is thus*

*scale-dependent*

*and the expression "effect-*

**measure***modification" makes this more explicit. Second, with*

*observational*

*data, control for confounding is often inadequate; the quantities we estimate from data may not reflect true causal effects. The expression "effect-*

**measure***modification" suggests only that our*

**measures***(which may not reflect*

**causal**effects) vary across strata of Q, rather than the

**effects themselves***(which we may not be able to consistently estimate).*

*…Confounding and effect modification, as conceived in this paper, and in much of modern epidemiology are* **causal***concepts: they relate to the distribution of counterfactual variables. In* * practice*, however,

**statistical***models are often used to reason about the presence or absence of confounding and effect modification.*

*…Third,* **the distinction between confounding in****distribution****versus****measure****becomes important when considering “collapsibility” approaches to confounding assessment i.e. in settings in which an investigator evaluates confounding by comparing an****adjusted****and****un****adjusted estimate.***Greenland et al. (1999) showed that for the risk difference and the risk ratio scales, collapsibility follows from no-confounding and vice versa. However, this implication holds for confounding in* * measure*,

*not*

*confounding in*

*. One may have collapsibility on the risk*

**distribution**

**difference***scale and therefore conclude that a particular variable is not a confounder of the risk*

**difference**(conditional on the other covariates); however, this does not imply that the variable is not a confounder for the risk**ratio**; it might be necessary to make control for that variable in evaluating the risk*.*

**ratio***Collapsibility of the risk*

**difference***implies no confounding in*

**measure***for the risk*

*;*

**difference***collapsibility of the risk*

**ratio***implies no confounding in*

**measure***for the risk*

*;*

**ratio***however*,

**neither***implies no confounding in*

*.*

**distribution***One must be careful when changing scales - not only in assessing effect modification - but also when thinking about confounding.*

**Question**: Do statisticians subscribe to the idea of confounding “in measure”? If not, why not? Do these differences of opinion (if they exist) relate to statisticians’ focus on RCTs (as compared with the observational study focus of epidemiologists)? And where do these disagreements leave students, who will struggle to reconcile the views of their epidemiology and statistics instructors?