Confounding, Confounding Fact... Health Article

CONFOUNDING, CONFOUNDING FACTORS

The word confounding has been used to refer to at least three distinct concepts. In the oldest and most widespread usage, confounding is a source of bias in estimating causal effects. This bias is sometimes informally described as mixing of effects of extraneous factors (called confounders) with the effect of interest. This usage predominates in nonexperimental research, especially in epidemiology and sociology. In a second and more recent usage originating in statistics, confounding is a synonym for change in an effect measure upon stratification or adjustment for extraneous factors (a phenomenon called noncollapsibility or Simpson's paradox). In a third usage, originating in the experimental-design literature, confounding refers to inseparability to main effects and interactions under a particular design. The three concepts are closely related and are not always distinguished from one another. In particular, the concepts of confounding as a bias in effect estimation and as noncollapsibility are often treated as equivalent, even though they are not. Only the former concept will be described here.

CONFOUNDING AS A BIAS IN EFFECT ESTIMATION

A classic discussion of confounding in which explicit reference is made is to "confounded effects" is found in John Stuart Mill's A System of Logic, although Mill lays out the primary issues and acknowledges Francis Bacon as a forerunner in dealing with them. Mill lists a requirement for experiment intended to determine causal relations: " …none of the circumstances [of the experiment] that we do know shall have effects susceptible of confounded with those of the agents whose properties we with to study [emphasis added]."

In Mill's time, the world experiment referred to an observation in which some circumstances were under the control of the observer, as it still is used in ordinary English, rather than to the notion of a comparative trial. Nonetheless, Mill's requirement suggests that a comparison is to be made between the outcome of one's "experiment" (which is essentially, an uncontrolled trial) and what one would expect the outcome to be if the agents one wished to study had been absent. If the outcome is not as one would expect in the absence of the study agents, then Mill's requirement ensures that the unexpected outcome was not brought about by extraneous "circumstances" (factors). If, however, those circumstances do bring about the unexpected outcome, and that outcome is mistakenly attributed to effects of the study agents, then the mistake is one of confounding (or confusion) of the extraneous effects with the agent effects.

Much of the modern literature follows the same informal conceptualization give by Mill. Terminology is now more specific, with "treatment" used to refer to an agent administered by the investigator and "exposure" often used to denote an unmanipulated agent. The chief development beyond Mill is that the expectation for the outcome in the absence of the study exposure is now almost always explicitly derived from observation of a control group that is untreated or unexposed. For example, D. Clayton and M. Hills (1993) state of observational studies:

there is always the possibility that an important influence on the outcome … differs systematically between the comparison [exposed and unexposed] groups. It is then possible [that] part of the apparent effect of exposure is due to these differences, [in which case] the comparison of the exposure groups is said to be confounded [emphasis in the original].

In fact, confounding is also possible in randomized experiments owing to systematic improprieties in treatment allocation, administration, and compliance. A further and somewhat controversial point that confounding (as per Mill's original definition) can also occur perfect randomized trials due to random differences between comparison groups.

THE COUNTERFACTUAL APPROACH

Various mathematical formalizations of confounding have been proposed for use in statistical analyses. Perhaps the one closest to Mill's concept is based on the counterfactual model for casual effects. Suppose one wishes to consider how a health-status (outcome) measure of a population would change in response to an intervention (population treatment). More precisely, suppose one's objective is to determine the effect that applying a treatment x₁ had or would have an outcome measure µ relative to applying treatment x₀ to a specific target population A. For example, A could be a cohort of breast-cancer patients, treatment x₁ could be a new hormone therapy, x₀ could be a placebo therapy, and the measure µ could be a five-year survival probability. The treatment x₁ is sometimes called the index treatment; and x₀ is sometimes called the control or reference treatment (which if often a standard or placebo treatment).

The counterfactual model posits that, in population A, µ will equal µ_A1 if x₁ is applied, µ_A0 is applied; the casual effect of x₁ relative to x₀ is defined as the change from µ_A0 to µ_A1, which might be measured as µ_A1 − µ_A0 or µ_A1/µ_A0. If A is given treatment x₁ then µ will equal µ_A1 and µ_A1 will be observable, but µ_A0 will be unobserved. Suppose, however, we expect µ_A0 to equal µ_B0, where µ_B0 is the value of the outcome µ observed or estimated for a population B that was administered treatment x₀. The latter population is sometimes called the control or reference population. Confounding is said to be present if in fact µ_A0 [.notequal] µ_B0, for then there must be some difference between populations A and B (other than treatment) that is affecting µ.

If confounding is present, a naïve (crude) association measure obtained by substituting µ_B0 for µ_A0 is an effect measure will not equal the effect measure, and the association measure is said to be confounded. For example, if µ_B0 [.notequal] µ_A0 then µ_A1 − µ_A1, which measure the association of treatments with outcomes across the populations, is confounded for µ_A1 − µ_A0, which measures the effect of treatment x₁ on population A. Thus, saying an association measure such as µ_A1 − µB0 is confounded for an effect measure such as µ_A1 − µ_A0 is synonymous with saying the two measures are not equal.

The preceding counterfactual approach to confounding gradually emerged through attempts to separate effect measures into a component due to the effect of interest and a component due to the effect of interest and a component due to extraneous effects. One noteworthy aspect of this approach is that confounding depends on the outcome measure. For example, suppose populations A and B have a different five-year survival probability µ under placebo treatment x₀; that is, suppose µ_B0 [.notequal] µ_A0, so that µ_A1 − µ_B0 is confounded for the actual effect µ_A1 − µ_B0 of treatment on five-year survival. It is then still possible that ten-year survival, µ, under the placebo would be identical in both populations; that is, µ_A0 could equal µ_B0, so that µ_A1 − µ_B0 is not confounded for the actual effect of treatment on ten-year survival. (We should generally expect no confounding for 200-year survival, since no treatment is likely to raise the 200-year survival probability of human patients above zero.)

A second noteworthy point is that confounding depends on the target population of inference. The preceding example, with A as the target, had different five-year survivals µ_A0 and µ_A0 for A and B under placebo therapy, and hence µ_A1 − µ_B0 was confounded for the effect µ_A1 − µ_A0 of treatment on population A. A lawyer or ethicist may also be interested in what effect the hormone treatment would have had on population B. Writing µ_B1 for the (unobserved) outcome of B under treatment, this effect on B may measured by µ_B1 − µ_B0. Substituting µ_A1 for the unobserved µ_B1 yields µ_A1− µ_B0. This measure of association is confounded for µ_B1 − µ_B0 (the effect of treatment x₁ on five-year survival in population B) if and only if µ_A1 [.notequal] µ_B1. Thus, the same measure of association, µ_A1, may be confounded for the effect of treatment on neither, one, or both of populations A and B, and may or may not be confounded for the effect of treatment on other targets.