Explanations for Some Results on Pages 11-13
Consider the model illustrated in Fig. 1.9 on p. 13.
Let's refer to the paths coming from U and V as u and v, respectively.
Using the structural equations style presented on p. 23, we have
A = tT + uU
and
B = tT + vV.
(Note: My use of boldface is not intended to denote matrices or vectors. Rather I'm just using it to better distinguish the random components.)
Recalling that all of the variables have mean 0 and variance 1, and noting that T, U, and V are uncorrelated, we have
ρAB = Cov(A,B) = E(AB) =
E([tT + uU][tT + vV]) = t2E(T2)
= t2Var(T)
= t2,
in agreement with the result given on p. 13.
(Note: My derivation isn't based on using Wright's rules or regression results. Rather, it just uses probability along with the model equations and the facts given about the variables.)
We also have that (noting that T and U are uncorrelated)
1 = Var(A) =
Var(tT + uU) =
t2Var(T)
+ u2Var(U)
= t2
+ u2,
and so u equals the square root of 1 - t2, in agreement with p. 13.
(Similarly, it can be shown that v equals the square root of 1 - t2.)
Now consider the model illustrated in Fig. 1.8 on p. 11.
Using the structural equations style presented on p. 23, we have
C = aA + bB
+ dX.
Recalling that all of the variables have mean 0 and variance 1 (and noting that A and X are uncorrelated), we have that
ρAC = Cov(A,C) = E(AC)
equals
E(A[aA + bB + dX]) = aE(A2)
+ bE(AB)
+ dE(AX)
= aVar(A)
+ bρAB
+ dρA,X = a + bc.
in agreement with the result given on p. 11.
(Note: As before, this derivation isn't based on using Wright's rules or regression results. Rather, it just uses probability along with the model equations and the facts given about the variables.)
Noting that X is uncorrelated with A and B, we also have that
Var(C), which equals 1, is just
Var(aA + bB + dX) =
a2Var(A)
+ b2Var(B)
+ d2Var(X)
+ 2abCov(A,B)
= a2
+ b2
+ d2 + 2abc,
and so d equals the square root of 1 - a2 - b2 - 2abc.
Plugging in the values given for a, b, and c on p. 11, we have that
d = ( 1 - (0.4)2 - (0.5)2 - 2(0.4)(0.5)(0.5) )1/2 = 0.391/2,
in agreement with p. 12.