Concentration index

Author

Gibran Hemani

Published

March 21, 2025

Background

How do concentration index and Lorenz curves respond to different simulation scenarios?

Simulations

Random

GWAS attention is independent of DALY

library(dplyr)

Attaching package: 'dplyr'
The following objects are masked from 'package:stats':

    filter, lag
The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union
library(rineq)

ndisease <- 450
diseases <- tibble(
    disease = 1:ndisease, 
    daly = rbeta(ndisease, 1, 0.7), 
    gwas_attention = rbeta(ndisease, 1, 0.7))
ci1 <- ci(ineqvar=diseases$gwas_attention, outcome=diseases$daly, method="direct")
summary(ci1)
Call:
[1] "ci(ineqvar = diseases$gwas_attention, outcome = diseases$daly, "
[2] "    method = \"direct\")"                                       

Type of Concentration Index:
 CI 

Health Concentration Index:
 0.002435111 

Variance:
 0.0001686933 

95% Confidence Interval:
 -0.02302129 0.02789151 
plot(ci1)

Try again with uniform distribtuion

diseases <- tibble(
    disease = 1:ndisease, 
    daly = rbeta(ndisease, 1, 1), 
    gwas_attention = rbeta(ndisease, 1, 1))
ci1 <- ci(ineqvar=diseases$gwas_attention, outcome=diseases$daly, method="direct")
summary(ci1)
Call:
[1] "ci(ineqvar = diseases$gwas_attention, outcome = diseases$daly, "
[2] "    method = \"direct\")"                                       

Type of Concentration Index:
 CI 

Health Concentration Index:
 -0.01067268 

Variance:
 0.0002296126 

95% Confidence Interval:
 -0.04037195 0.0190266 
plot(ci1)

Proportional

GWAS attention is the same as DALY

ndisease <- 450
diseases <- tibble(
    disease = 1:ndisease, 
    daly = rbeta(ndisease, 1, 0.7), 
    gwas_attention = daly + rnorm(ndisease, 0, sd=0.001))
ci1 <- ci(ineqvar=diseases$gwas_attention, outcome=diseases$daly, method="direct")
summary(ci1)
Call:
[1] "ci(ineqvar = diseases$gwas_attention, outcome = diseases$daly, "
[2] "    method = \"direct\")"                                       

Type of Concentration Index:
 CI 

Health Concentration Index:
 0.277959 

Variance:
 0.0001193012 

95% Confidence Interval:
 0.2565513 0.2993668 
plot(ci1)

Again with uniform distribution

diseases <- tibble(
    disease = 1:ndisease, 
    daly = rbeta(ndisease, 1, 1), 
    gwas_attention = daly + rnorm(ndisease, 0, sd=0.001))
ci1 <- ci(ineqvar=diseases$gwas_attention, outcome=diseases$daly, method="direct")
summary(ci1)
Call:
[1] "ci(ineqvar = diseases$gwas_attention, outcome = diseases$daly, "
[2] "    method = \"direct\")"                                       

Type of Concentration Index:
 CI 

Health Concentration Index:
 0.3434778 

Variance:
 0.0001280117 

95% Confidence Interval:
 0.3213023 0.3656532 
plot(ci1)

Unequal

GWAS attention grows faster than DALY

diseases <- tibble(
    disease = 1:ndisease, 
    daly = 1:ndisease, 
    gwas_attention = daly^2)
ci1 <- ci(ineqvar=diseases$gwas_attention, outcome=diseases$daly, method="direct")
summary(ci1)
Call:
[1] "ci(ineqvar = diseases$gwas_attention, outcome = diseases$daly, "
[2] "    method = \"direct\")"                                       

Type of Concentration Index:
 CI 

Health Concentration Index:
 0.3325926 

Variance:
 0.0001307385 

95% Confidence Interval:
 0.3101822 0.355003 
plot(ci1)

sessionInfo()
R version 4.4.3 (2025-02-28)
Platform: aarch64-apple-darwin20
Running under: macOS Sonoma 14.6.1

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.0

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

time zone: Europe/London
tzcode source: internal

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

other attached packages:
[1] rineq_0.2.3 dplyr_1.1.4

loaded via a namespace (and not attached):
 [1] digest_0.6.37     utf8_1.2.4        R6_2.5.1          fastmap_1.2.0    
 [5] tidyselect_1.2.1  xfun_0.48         magrittr_2.0.3    glue_1.8.0       
 [9] tibble_3.2.1      knitr_1.48        pkgconfig_2.0.3   htmltools_0.5.8.1
[13] rmarkdown_2.27    generics_0.1.3    lifecycle_1.0.4   cli_3.6.3        
[17] fansi_1.0.6       vctrs_0.6.5       compiler_4.4.3    tools_4.4.3      
[21] pillar_1.9.0      evaluate_1.0.1    yaml_2.3.10       rlang_1.1.4      
[25] jsonlite_1.8.9    htmlwidgets_1.6.4