A Primer on Covariate Selection

Abstract

In this primer, we unpack the generic concept of a “covariate” by reviewing three crucial roles that a variable can play in relation to an effect of interest (X → Y), namely mediator, confounder, and collider. We also describe some common variations and extensions, e.g., scenarios in which a variable is a mediator of a confounder, or a descendant of a collider.

Full text

A Primer on Covariate Selection
[Modified from the supplementary material of: Del Giudice, M., & Gangestad, S. W. (2021).
A traveler’s guide to the multiverse: Promises, pitfalls, and a framework for the evaluation of
analytic decisions. Advances in Methods and Practices in Psychological Science, 4, 1-15.]
In this primer, we unpack the generic concept of a “covariate” by reviewing three crucial
roles that a variable can play in relation to an effect of interest (X
→
Y), namely mediator,
confounder, and collider. We also describe some common variations and extensions, e.g.,
scenarios in which a variable is a mediator of a confounder, or a descendant of a collider (Figure
1).
Figure 1. Simple causal models that illustrate the effects of covariate selection on the estimation
of the effect of interest (X
→
Y). In (a), (b), and (c), controlling for Z reduces or eliminates the
indirect (mediated) effect of X on Y. In (d), (e), and (f), controlling for Z removes estimation bias
by de-confounding the X
→
Y effect. In (g), (h), and (i), controlling for Z adds estimation bias to
the X
→
Y effect.
Mediators
A mediator is a variable that lies on a causal path leading from X to Y, and thus serves as
an intermediate step through which X affects Y. The effect of X may be fully mediated by other
variables, as in Figure 1a; alternatively, X may also have a direct effect on Y that does not flow
through any mediators (or at least not ones that have been measured), as in Figure 1b.

In the causal model of Figure 2, the effect of inflammation on depression is partly
mediated by pain. If pain is included as a covariate, the path inflammation
→
pain
→
depression
is blocked, and the statistical model estimates the direct effect of inflammation. If instead pain is
excluded, the model estimates the total effect of inflammation, i.e., the sum of the direct and
mediated effects. Both are potentially meaningful; which one should be the focus of the analysis
depends on the theoretical background and goals of the study. If the direct effect is the focus of
the analysis, failing to include mediators as covariates (or otherwise blocking the mediated paths)
will bias the estimate (see Pearl et al., 2016; Rohrer, 2018). But if the quantity of interest is the
total effect of X, mediators must be left out of the statistical model to avoid biasing the estimate.
Figure 1c illustrates a slightly more complex scenario, in which Z is not a mediator itself
but a descendant of a mediator M (see Cinelli et al., 2019; Pearl et al., 2016). Because Z shares
variance with M, including Z is equivalent to partially controlling for M. If the focus of the
analysis is the total effect of X on Y, both M and Z must be excluded from the statistical model to
prevent bias. Conversely, if the effect of interest is the direct effect of X on Y, including Z as a
covariate does not completely remove bias, and M should be included instead.
Figure 2. Causal model of a hypothetical study of the effect of inflammation on depression.
Rectangles indicate observed variables; ellipses indicate unobserved latent constructs.
Confounders
A confounder is a variable that affects both the predictor X and the response Y, as in
Figure 1d. Being a common cause of X and Y, a confounder may spuriously inflate, deflate, or
even reverse the X
→
Y effect. In the model of Figure 2, the effect of inflammation on depression
is confounded by age, through the path inflammation
←
age
→
depression. Unbiased estimates of
the effect of interest require control of potential confounders by including them as covariates. Of
course, if a confounder has been measured with error, including it as a covariate only partially
corrects estimation bias (see Westfall & Yarkoni, 2016).

The causal model in Figure 1d shows the basic case of a confounder Z that directly
affects X and Y. However, the effects of a confounder may also be mediated by additional
variables, as illustrated in Figure 1e. In this example, Z mediates the effect of confounder U on
the predictor X. Including either Z or U as a covariate in the statistical model blocks the
confounding path X
←
Z
←
U
→
Y and corrects the estimation bias (Cinelli et al., 2019; Pearl et
al., 2016). Figure 1f shows another variation on this theme. Here, Z is a common cause of the
predictor X and of a variable M that mediates the effect of X on Y. The confounding effect of Z in
this scenario is indirect but no less real, and Z must be controlled to avoid bias.
Colliders
A collider is the mirror image of a confounder—a common effect of both X and Y rather
than a common cause (or, equivalently, a descendant of both X and Y; Figure 1g). In the model of
Figure 2, both inflammation and depression affect fatigue, which plays the role of a collider.
Whereas confounders add bias to estimation of the X
→
Y effect unless they are actively
controlled for (or the confounding paths are otherwise blocked), colliders introduce bias if they
are included as covariates (“conditioning on a collider;” see Elwert & Winship, 2014; Pearl et
al., 2016; Rohrer, 2018). In Figure 2, including fatigue as a covariate would unblock the
inflammation
→
fatigue
←
depression path and bias the estimated effect of inflammation on
depression. Specifically, if both inflammation and depression increase fatigue, controlling for the
level of fatigue introduces a spurious negative association between the two variables. The reason
is that, at any fixed level of fatigue, a larger contribution from inflammation implies a smaller
contribution from depression (and vice versa), all else being equal. This counterintuitive effect is
also known as Berkson’s paradox (Berkson, 1946; Snoep et al., 2014).
If a variable is a collider, it should not be included as a covariate in the statistical model,
unless the biasing path is blocked again by the inclusion of other variables (e.g., a mediator of
the effect of X or Y on the collider). The same applies if a variable is not a collider itself but a
descendent of a collider, as illustrated in Figure 1h. Here, Z is a descendant of collider W;
including Z as a covariate partly controls for W. Finally, Figure 1i depicts a scenario in which Z
is a descendant of Y, but is not directly affected by X. Even in this seemingly neutral case, Z is a
common effect of X (indirectly through Y) and Y, and can be expected to introduce estimation
bias if included as a covariate (Cinelli et al., 2019).
Implications for precision
Even if a potential covariate is neutral with respect to estimation bias, it may still affect
the precision of the estimate (Cinelli et al., 2019; Pearl et al., 2016). Figure 3 depicts three
illustrative scenarios. In Figure 3a, variable Z has a causal influence on the predictor X, but no
direct effect on the response variable Y. Including Z as a covariate does not affect bias on the X
→
Y effect, but reduces the variation of the predictor, and thus may decrease the precision of the
estimated effect. In the model of Figure 2, this would correspond to including proinflammatory
genotype as a covariate. (Note that genotype is a neutral control only if age has also been
controlled for; if not, including genotype as a covariate amplifies the confounding effect of age.
See Pearl [2012].)

Figure 3. Simple causal models that illustrate the effects of covariates on the precision of the
estimate of the effect of interest (X
→
Y). In (a), controlling for Z reduces the precision of the
estimate. In (b) and (c), controlling for Z increases the precision of the estimate.
In Figure 3b, variable Z has a causal effect on the response variable Y. Controlling for Z
reduces the variation of the outcome that is not explained by X, and in doing so may increase the
precision of the estimate. Likewise, controlling for Z in Figure 3c reduces the variation of
mediator M that is not explained by X, with a positive effect on precision.
References
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