Differential susceptibility explained by interactors and mediators with different distributions

interactions
multi-ancestry
Author

Gibran Hemani

Published

June 9, 2023

Background

What leads to different effect estimates in different populations? If the DAG is the same, then different distributions of mediators and interactions could explain different effects

n <- 100000

g1 <- rbinom(n, 2, 0.3)
g2 <- rbinom(n, 2, 0.3)

# The liability for the mediator 
# Influenced by g (same effect in both populations)
# Influenced by some other variable that gives it a different mean in the two populations
# Now liability of mediator has different means
ml1 <- plogis(scale(g1) + rnorm(n, 2, 1))
ml2 <- plogis(scale(g2) + rnorm(n, 0, 1))

hist(ml1)

hist(ml2)

# Generate mediators - they have different means
m1 <- rbinom(n, 1, ml1)
m2 <- rbinom(n, 1, ml2)
mean(m1)
[1] 0.81663
mean(m2)
[1] 0.49808

Disease influenced by mediator and noise (same effects)

d1 <- rbinom(n, 1, plogis(m1 + rnorm(n)))
d2 <- rbinom(n, 1, plogis(m2 + rnorm(n)))

mean(d1)
[1] 0.66026
mean(d2)
[1] 0.59796

Slightly different prevalence in different populations.

The genotypic effect is now different in the two populations due to different

glm(d1 ~ g1, family="binomial")

Call:  glm(formula = d1 ~ g1, family = "binomial")

Coefficients:
(Intercept)           g1  
     0.5736       0.1538  

Degrees of Freedom: 99999 Total (i.e. Null);  99998 Residual
Null Deviance:      128200 
Residual Deviance: 128000   AIC: 128000
glm(d2 ~ g2, family="binomial")

Call:  glm(formula = d2 ~ g2, family = "binomial")

Coefficients:
(Intercept)           g2  
     0.2551       0.2405  

Degrees of Freedom: 99999 Total (i.e. Null);  99998 Residual
Null Deviance:      134800 
Residual Deviance: 134200   AIC: 134200
sessionInfo()
R version 4.3.0 (2023-04-21)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Monterey 12.6.2

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_1.6.2 compiler_4.3.0    fastmap_1.1.1     cli_3.6.1        
 [5] tools_4.3.0       htmltools_0.5.5   rstudioapi_0.14   yaml_2.3.7       
 [9] rmarkdown_2.22    knitr_1.43        xfun_0.39         digest_0.6.31    
[13] jsonlite_1.8.5    rlang_1.1.1       evaluate_0.21