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Appendix

Consider causal relationship between Z and Y , where Z is the cause and Y is the effect.

Let X be the third variable.

Definition. X is not a confounder in the relationship of Z → Y if and only if

(i) the relationship of Z → Y does not depend on X = x, and

(ii) X and Y are conditionally independent conditioned on Z, or X and Z are

conditionally independent conditioned on Y .

Proposition 1. Suppose that (X, Y, Z) follows three-dimensional normal distribution. Then ρzx = 0, or ρxy = 0 if X is not a confounder in the Z → Y relationships,

where ρzx and ρxy are Pearson correlation coefficients between X and Z, and X and Y ,

respectively.

Proof. Suppose that X is not a confounder in the Z → Y relationship, then by the

definition the conditional Pearson correlation coefficient between Y and Z conditioned on

X = x, denoted by ρ(y, z|x), does not depend on X = x and X and Y are conditionally

10

Y. Yoshida and T. Yanagawa

independent conditioned on Z, or ρ(y, z|x), does not depend on X = x and X and

Z are conditionally independent conditioned on Y . Since X and Y are conditionally

independent conditioned on Z if and only if ρxy = ρyz ρzx , it follows that

1 − ρ2zx

ρyz − ρxy ρzx

ρ(y, z|x) = √

= ρyz

1 − ρ2yz ρ2zx

(1 − ρ2 )(1 − ρ2 )

xy

zx

Thus, ρ(y, z|x) does not depend on X = x if and only if ρzx = 0. Next, suppose

that ρ(y, z|x), does not depend on X = x and X and Z are conditionally independent

conditioned on Y . Then, similarly as above we may show ρxy = 0. (QED)

The proposition shows that if ρzx ̸= 0 and ρxy ̸= 0, then X is the confounder in the

relationship of Z → Y . Furthermore, we have the following proposition in the context

of regression of Y on Z and X.

Proposition 2. Suppose that (X, Y, Z) follows three-dimensional normal distribution. Then, X and Y are conditionally independent conditioned on Z if and only if

E(Y |z, x) = E(Y |z), where E(Y |z, x) is the conditional expectation of Y given Z = z

and X = x, and E(Y |z) is the conditional expectation of Y given Z = z.

Proof. Let µx , µy and µz be expectations of X, Y and Z, respectively, and σx , σy

and σz be standard deviations of X, Y and Z, Then

σy (ρyz − ρxy ρzx )

σy (ρxy − ρyz ρzx )

(x − µx ) + ( )

(z − µz ),

σx

(1 − ρ2zx )

σz

(1 − ρ2zx )

E(Y |z, x)

µy + (

E(Y |z)

σy

ρyz (z − µz )

σz

Thus, if X and Y are conditionally independent conditioned on Z, that is ρ(x, y|z) =

ρxy − ρzx ρyz = 0, then E(Y |z, x) = E(Y |z). The reverse is trivial. (QED).

Received: February 21, 2023

Revised: March 2, 2023

Accept: March 4, 2023

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