Gibran Hemani’s lab book - Rsq in simulations

Background

What determines $R^{2}$ between X and Y in a linear model under confounding

\begin{aligned} Y & = a + b X + E \\ b & = c o v (X, Y) / v a r (X) \\ R & = c o v (X, Y) / [s d (X) s d (Y)] \\ = b * s d (x) / s d (y) \end{aligned}

In OLS the b will be different to the IV b if there is unmeasured confounding

\begin{aligned} X & = a + β_{1} G + β_{2} U + ϵ \\ Y & = a + b_{1} X + b_{2} U + E \\ b_{O L S} & = c o v (X, Y) / v a r (X) \\ = c o v (β_{1} G + β_{2} U, b_{1} (β_{1} G + β_{2} U) + b_{2} U) / v a r (X) \\ = c o v (β_{1} G + β_{2} U, b_{1} β_{1} G + b_{1} β_{2} U + b_{2} U) / v a r (X) \\ = [b_{1} β_{1}^{2} v a r (G) + (b_{1} β_{2}^{2} + b_{2} β_{2}) v a r (U)] / v a r (X) \end{aligned}

and

$R_{O L S} = b_{O L S} s d (x) / s d (y)$

therefore to get the OLS $R^{2}$ of X,Y

$R^{2} = {[\frac{b_{1} β_{1}^{2} v a r (G) + (b_{1} β_{2}^{2} + b_{2} β_{2}) v a r (U)}{s d (x) s d (y)}]}^{2}$

Note that

$\begin{aligned} v a r (x) & = s d (x)^{2} = β_{1}^{2} v a r (G) + β_{2}^{2} v a r (U) + v a r (ϵ) \\ v a r (y) & = s d (y)^{2} = b_{1}^{2} v a r (X) + b_{2}^{2} v a r (U) + v a r (E) \end{aligned}$

So ultimately if you want to fix $R^{2}$ for different parameters of effects you should be able to scale $v a r (ϵ)$ and $v a r (E)$ , the residual variances, according to these formulae.

By contrast the variance explained by the causal effect of X is

$R_{I V, x, y}^{2} = b_{1}^{2} v a r (X) / v a r (Y)$

Check

set.seed(1)
b1 <- 0.2
b2 <- 3
beta1 <- 4
beta2 <- 5
n <- 10000
u <- rnorm(n)
g <- rnorm(n)
x <- u * beta2 + g * beta1 + rnorm(n)
y <- u * b2 + x * b1 + rnorm(n, sd=0)

Beta

cov(x, y)/var(x)

[1] 0.562716

summary(lm(y ~ x))$coef[2,1]

[1] 0.562716

( b1*beta1^2*var(g) + (b1*beta2^2 + b2*beta2) * var(u) ) / var(x)

[1] 0.5557103

Correlation

((( b1*beta1^2*var(g) + (b1*beta2^2 + b2*beta2) * var(u) ) / var(x)) * sd(x) / sd(y))^2

[1] 0.768337

cor(x, y)^2

[1] 0.7878316

sessionInfo()

R version 4.3.2 (2023-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Ventura 13.6

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0

locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8

time zone: Europe/London
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] htmlwidgets_1.6.3 compiler_4.3.2    fastmap_1.1.1     cli_3.6.1        
 [5] tools_4.3.2       htmltools_0.5.7   yaml_2.3.7        rmarkdown_2.25   
 [9] knitr_1.45        jsonlite_1.8.7    xfun_0.41         digest_0.6.33    
[13] rlang_1.1.2       evaluate_0.23