# Should only confounders (but not covariates) be added to a multiple regression?

Up until now, it was my understanding that multiple regression can be used to estimate the influence of an independent variable (IV) on an outcome (DV) while controlling for the influence of other possibly interfering covariates (by entering them as additional predictors; e.g., age, gender, income, etc.). Then, by looking at the coefficient of the IV of interest, I will get an impression of this variable's influence without the additional impact of the covariates (which are kept constant).

In a way, this implies a "partialling out" interpretation of multiple regression, as stated in this textbook on p. 83.

However, today I stumbled upon a recent 2020 paper (published with relatively high impact!), which claims that this is fundamentally wrong. The author claims that you should only control for so-called "confounding variables" in your regression analyses, i.e., variables that affect both the IV and the outcome. The author further explains that previous research on life satisfaction is all wrong, because it made the mistake of using "income" as a covariate in a regression between age (IV) and life satisfaction (outcome).

Could someone help me wrap my head around this? In my psychological understanding, the criticized procedure is absolutely correct: If I want to estimate the sheer influence of age on life satisfaction without the impact of another interfering variable such as money, I should keep the latter constant by adding it as a covariate, no?

Thank you very much for your help.

• You need to decide on the specific effect you want to estimate. If you want the total effect of age on life satisfaction, you should not control for any variables potentially caused by age. If you want the direct effect of age on life satisfaction that does not pass through income, then you need to include income in the model. – Noah Jan 25 at 4:27
• Thank you, @Noah, very comprehensible answer, and it helped me to once again understand the relationship between regression and mediation better. :) – JPS Jan 25 at 13:21

The statement : "Regression models should control only ‘confounding’ variables; that is, variables that are causally prior to the dependent variable and the core independent variable of interest." (Bartram, 2020 - abstract) is wrong.

Only controlling for control variables that are correlated prior to the independent variable and the outcome is wrong. Of course, age can not be "caused", but there could be a suppressor or mediating effect, i.e., an effect intervening between IVs and the outcome. Confounding, suppressor and mediating effects are statistically the same, but conceptually different.

   Here an example build in R:

> #hypothetical example
> n=1000000
> age=rnorm(n)
> anxiety=.5*age+sqrt(1-.5^2)*rnorm(n)
> satisfaction = .4*age-.4*anxiety+sqrt(1-(.4^2+.4^2+2*.4*(-.4)*.5))*rnorm(n)
>
> #correlation
> cov(cbind(age,anxiety,satisfaction))
age    anxiety satisfaction
age          0.9989992  0.5001790    0.2000404
anxiety      0.5001790  0.9991424   -0.1977109
satisfaction 0.2000404 -0.1977109    0.9992774
>
> #regression with both IV
> lm(satisfaction~age+anxiety)

Call:
lm(formula = satisfaction ~ age + anxiety)

Coefficients:
(Intercept)          age      anxiety
0.0001752    0.3994308   -0.3978389

>
> #regression with only the one correlated to
> lm(satisfaction~age)

Call:
lm(formula = satisfaction ~ age)

Coefficients:
(Intercept)          age
3.786e-05    2.002e-01


Here, if you don't control for anxiety, you have a biased effect of age $$=.20$$ instead of the real $$.40$$, this is because, anxiety is actually a suppressor variable (or mediating effect, depending on how you conceptualized your model).

You can read more on this topic here.

I am not familiar on the topic of age and satisfaction. What strikes me, is the lack of concern for age$$^2$$, which would point at the "U-shaped" relation (if there is one), and that there is a preference for splitting data in two, which is a questionnable practice at best.

• Thank you so much for your quick reply. That's exactly what I thought: It's a common (and important) practice to also control for suppressor or mediator effects in regression So, my intuition was correct and the cited paper is just...statistical bollocks? How incredible that such confident claims are published in high-level journals then... – JPS Jan 21 at 21:54
• Thank you @JPS. You would be surprised how often it happens actually... – POC Jan 21 at 22:03