Hi Yuto,

it is not clear what you really want to know. When you have a relationship between x and y and s is a suppressor (hence it is uncorrelated with y) then

a) you will get a partial correlation between s and y when you control x

b) the partial effect of x (controlling for s) will be higher as the unconditional correlation/relationship

You can simulate this easily in R

S = rnorm(1000)

C = rnorm(1000)

X = .8*S + .6*C+ rnorm(1000)

Y = .7*C + rnorm(1000)

df = data.frame(X,S,Y)

summary(lm(Y~ X, data=df)) # Effect of X

summary(lm(Y~ S, data=df)) # No effect of S

summary(lm(Y ~ X + S, data=df)) #Suppression

Note that the most probable scenario underlying suppression (that I used for the simulation) is that the X-effect is confounded (in this case, by C) and controlling for X opens a spurious path from S to Y, see

Kim, Y. (2019). The causal structure of suppressor variables. Journal of Educational and Behavioral Statistics, 1076998619825679. https://doi.org/10.3102/1076998619825679

All the best,

Holger

The way I would put it is that the zero order correlation between X and Y is not statistically significant, but that relationship becomes significant after controlling for the suppressor variable.

This is a basic feature of partial regression coefficients, so you should not need a citation for it. But if you want one, you can do a Google search for: suppressor variable in regression. And if you limit that search to Books, you will get references to several statistical texts that cover suppression.

Hi David,

I might err, but isn't the classical definition of a suppressor that it is

a) is uncorrelated with Y but nonetheless

b) increase the prediction of X when added to the equation and

c) then suddenly gets an effect.

The scenerio that you describe would rather stem from controlling a collider (a common effect of both x and y).

The Kim paper is a nice piece as it presents two scenerios that cause aforementioned phenomenon (I simulated on of them above).

All the best!

Holger

The definition of suppression that I was taught it broader than that, so I will be interested in what others have to say about this.