This is a skeletal post to work up an answer to a twitter question using Wright’s rules of path models. Using this figure

from Panel A of a figure from Hernan and Cole. The scribbled red path coefficients are added

from Panel A of a figure from Hernan and Cole. The scribbled red path coefficients are added

the question is I want to know about A->Y but I measure A* and Y*. So in figure A, is the bias the backdoor path from A* to Y* through A and Y?

Short answer: the bias is the path as stated divided by the path from A to Y.

Medium answer: We want to estimate β, the true effect of A on Y. We have measured the proxies, A⋆ and Y⋆. The true effect of A on A⋆ is α1 and the true effect of Y on Y⋆ is α2.

So what we want is β but what we estimate is α1βα2 (the path from A⋆ to Y⋆) so the bias is α1α2.

But, this is really only true for standardized effects. If the variables are not variance-standardized (and why should they be?), the bias is a bit more complicated.

TL;DR answer: In terms of β the estimated effect is

α1α2σA2σA⋆2β

so the bias is

k=α1α2σA2σA⋆2

The derivation is scratched out here:

Old fashion doodle deriving the bias of the estimate of \beta when proxies of A and Y are measured. The bias is k.

Old fashion doodle deriving the bias of the estimate of β when proxies of A and Y are measured. The bias is k.

Part 2 of the derivation. Substituting the non-standardized coefficients in for the standardized coefficients.

Part 2 of the derivation. Substituting the non-standardized coefficients in for the standardized coefficients.

If the data are standardized (all variables have unit variance)

This is easy and just uses Wright’s rules of adding up effects along a path

n <- 10^6
beta <- 0.6 # true effect
alpha_1 <- 0.9 # standardized effect of A on A* -- this is the correlation of A with proxy
alpha_2 <- 0.8 # standardized effect of Y o Y* -- this is the correlation of Y with proxy
A <- rnorm(n)
Y <- beta*A + sqrt(1 - beta^2)*rnorm(n)
astar <- alpha_1*A + sqrt(1 - alpha_1^2)*rnorm(n) # proxy for A
ystar <- alpha_2*Y + sqrt(1 - alpha_2^2)*rnorm(n) # proxy for Y

β is the true effect and the expected estimated effect is α1βα2 (using Wright rules) so α1α2 is the bias. Note this isn’t added to the true effect as in omitted variable bias (confounding). We can check this with the fake data.

alpha_1*beta*alpha_2 # expected measured effect
## [1] 0.432
coef(lm(ystar ~ astar)) # measured effect
##  (Intercept)        astar 
## 0.0007277386 0.4313778378

check some other measures

var(A) # should be 1
## [1] 1.002161
var(Y) # should be 1
## [1] 1.001362
var(astar) # should be 1
## [1] 1.00204
var(ystar) # should be 1
## [1] 0.998893
cor(ystar, astar) # should be equal to expected measured effect
## [1] 0.4320569

if the data are not standardized

n <- 10^5
rho_alpha_1 <- 0.9 # correlation of A and A*
rho_alpha_2 <- 0.8 # correlation of Y and Y*
rho_b <- 0.6 # standardized true effect of A on Y
sigma_A <- 2 # total variation in A
sigma_Y <- 10 # total variation in Y
sigma_astar <- 2.2 # total variation in A*
sigma_ystar <- 20 # total variation in Y* 
alpha_1 <- rho_alpha_1*sigma_astar/sigma_A # effect of A on astar
alpha_2 <- rho_alpha_2*sigma_ystar/sigma_Y # effect of Y on ystar
beta <- rho_b*sigma_Y/sigma_A # effect of A on Y (the thing we want)
A <- rnorm(n, sd=sigma_A)
R2_Y <- (beta*sigma_A)^2/sigma_Y^2 # R^2 for E(Y|A)
Y <- beta*A + sqrt(1-R2_Y)*rnorm(n, sd=sigma_Y)
R2_astar <- (alpha_1*sigma_A)^2/sigma_astar^2 # R^2 for E(astar|A)
astar <- alpha_1*A + sqrt(1-R2_astar)*rnorm(n, sd=sigma_astar)
R2_ystar <- (alpha_2*sigma_Y)^2/sigma_ystar^2 # R^2 for E(ystar|Y)
ystar <- alpha_2*Y + sqrt(1-R2_ystar)*rnorm(n, sd=sigma_ystar)

Now let’s check our math in the figure above. Here is the estimated effect

coef(lm(ystar ~ astar))
## (Intercept)       astar 
## 0.004357835 3.950256435

And the expected estimated effect using just the standardized coefficients

rho_alpha_1*rho_alpha_2*rho_b*sigma_ystar/sigma_astar
## [1] 3.927273

And the expected estimated effect using the equation kβ, where k is the bias (this is in the top image of the derivation)

k <- rho_alpha_1*sigma_A/sigma_Y*rho_alpha_2*sigma_ystar/sigma_astar
k*beta
## [1] 3.927273

And finally, the expected estimated effect using the bias as a function of the unstandardized variables (this is in the bottom – part 2– image of the derivation)

k <- alpha_1*alpha_2*sigma_A^2/sigma_astar^2
k*beta
## [1] 3.927273

And the true effect?

beta
## [1] 3

Some other checks

coef(lm(ystar ~ Y))
## (Intercept)           Y 
## -0.03612649  1.60271514
alpha_2
## [1] 1.6
coef(lm(ystar ~ A))
## (Intercept)           A 
## -0.00254846  4.80956599
alpha_2*beta
## [1] 4.8
coef(lm(astar ~ A))
##  (Intercept)            A 
## -0.002796505  0.991465329
alpha_1
## [1] 0.99
sd(A)
## [1] 2.014547
sd(Y)
## [1] 10.04232
sd(astar)
## [1] 2.215202
sd(ystar)
## [1] 20.09823
cor(A, astar)
## [1] 0.9016573
cor(Y, ystar)
## [1] 0.8008157

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