# What assumptions are necessary to establish that a variable is confounding?

Suppose that an article claims that there is a cause-effect relationship between a certain explanatory variable A and a response variable B.

If I were to test whether a confounding variable C exists in the study, what are the statements that I would have to prove in order to establish C as a confounding variable?

I know that I must show that the following statements are true:

• Variable C must have an association with variable A.

• At the same time, variable C must have an association with variable B.

• Also, variable C must not be an intermediate in the cause-effect relationship (i.e. it cannot be A $\rightarrow$ C $\rightarrow$ B).

Are there any more statements I must prove in order to establish variable C as a confounding variable?

Can someone provide an example of a well-established confounding variable?

Also, just a side question, is it possible to have a confounding variable in an experiment? Or, do confounding variables only appear in observational studies?

• Confounding variables "should" not be in experiments, but they can and will appear when the experiment is not well-controlled. – Kevin Jul 21 '17 at 23:58
• @Kevin So confounding variables can appear in both experiments and observational studies? – user8081591 Jul 22 '17 at 0:10
• Confounding variables can occur in stratified randomized trials as well. Let's say men are assigned treatment .25 of the time, but women are assigned treatment .5 of the time by design of the researcher. If gender is related to the outcome, then gender is a confounder in this experiment. – Noah Jul 27 '17 at 22:51

Those statements are not enough because they are too weak. Association is not enough to determine confounding; you need causation, which has within it a temporal assumption.

• C is a cause of A.
• C is a cause of B beyond its effect on A.

An example in my area of research (higher education) is the following: for the causal effect of a highly selective academic enrichment program on college GPA, high school achievement is a confounder. High school achievement causes selection into the program. High school achievement causes variability in college GPA, beyond its effect through the program. Without adjusting for high school achievement (e.g., by regressing college GPA on program participation and high school GPA), a possible observed difference in college GPA between program participants and non-participants could be attributable not just to program participation but also to high school achievement, thereby underestimating the unique effect of participation.

I know that I must show that the following statements are true:

• Variable C must have an association with variable A.

• At the same time, variable C must have an association with variable B.

• Also, variable C must not be an intermediate in the cause-effect relationship (i.e. it cannot be A $\rightarrow$ C $\rightarrow$ B).

This is not correct, you should check some of the things here, here, and here. For example, your statement would fail to recognize a collider, where all these statements would be true, yet $C$ would not be a confounder. So you might want to adjust your claim by saying, for instance, that $C$ is a common cause of $A$ and $B$ and its confounding effect was not blocked by the study's design.

Can someone provide an example of a well-established confounding variable?

Yes, just think of ice-cream sales and diseases that are more common during warm seasons. These things will be correlated due to the common cause (warm season).

Also, just a side question, is it possible to have a confounding variable in an experiment? Or, do confounding variables only appear in observational studies?

Yes, it's possible to have confounding in experiments. The difference between observational studies and experiments is of degree. In an experiment you hopefully (physically/by design) control/block some of the confounding factors and randomize to account for the ones you can't control. But you can only do this up to a certain point. After randomization patients might not follow the protocol, you might have patients that drop the study, etc. Several things might happen. You might also want to check this answer here.

No, there are not more statements you need to prove.

In clinical studies, I simply view all the variables that I don't want to study as nuisance factors, or junk variables. In regression modeling, a confounder first has to be a significant predictor of the outcome, and then also has to be associated with the exposure or experimental treatment groups. I commonly work backwards and look for junk variables that are significant across treatment groups first, and if significant, then find out if the effect size of treatment changes when the junk variable is added(removed) from the model. Any junk variables that are not significantly different across treatments fail the second criterion, so you don't have to consider them in modeling the crude and adjusted effect size. So it's a way to save time, that is, don't bother thinking about whether a confounder is first associated (predicts) outcome, but rather find out if it's associated (significantly different) across treatment groups (exposure) first. If not, they don't need to be even considered in modeling -- which is far more time consuming. You want a fast screen for confounders, so just test the equality of means(proportions) across exposure (treatment) groups so you know if you have to include them in modeling assessments.

This latter part can be confirmed by just finding out if the group-specific means of a junk variable are significantly different across treatment groups, i.e., by using ANOVA(Kruskal-Wallis), or t-test(Mann-Whitney). Significant differences of the proportions of counts (frequencies) of category levels across the treatment groups can be identified by contingency table analysis (chi-squared tests).

• Can you please provide an example of proving those statements in a scenario? How would you formulate a statement to propose that a confounding variable exists? – user8081591 Jul 22 '17 at 1:53
• OP also needs that C is not a consequent of B. – Noah Jul 27 '17 at 23:03