Confidence interval for the ratio of two related proportions (glass ceiling index)

Question

I would like to compute the confidence interval of the glass ceiling index (GCI), which aims to measure gender inequality in Academia. According to "She Figures", the GCI is defined as follow:

The GCI measures the relative chance for women, as compared with men, of reaching a top position. The GCI compares the proportion of women in grade A positions (equivalent to Full Professors in most countries) to the proportion of women in academia (grade A, B, and C), indicating the opportunity, or lack of it, for women to move up the hierarchical ladder in their profession.

Hence it is defined $GCI=\frac{p_{W, A}}{p_W}$ where $p_{W,A}=n_{W, A}/n_A$ is the proportion of women in grade A, and $p_{W}=n_W/n=\frac{n_{W, A}+n_{W, B}+n_{W, C}}{n}$ the proportion of women in academia.

I have seen here a solution for computing the confidence interval for two independant proportion: Confidence interval around the ratio of two proportions

But in this case, the proportions are not independent. Do you know any method to approximate an interval of such kind of ratio? Thanks for your answer!

Answer 1

A quick and dirty solution using parametric bootstrap that will give you a confidence interval would look like the following in R:

# Made up numbers, substitute with your own
n_wa <- 50
n_wb <- 50
n_wc <- 50
n_a <- 100
n_b <- 100
n_c <- 100

# Parametric bootstrap
no_boot_samp <- 10000
n_wa_boot <- rbinom(no_boot_samp, size = n_a, prob = n_wa / n_a)
n_wb_boot <- rbinom(no_boot_samp, size = n_b, prob = n_wb / n_b)
n_wc_boot <- rbinom(no_boot_samp, size = n_c, prob = n_wc / n_c)
p_wa_boot <- n_wa_boot / n_a
p_w_boot <- (n_wa_boot + n_wb_boot + n_wc_boot) / (n_a + n_b + n_c)
gci_boot <- p_wa_boot / p_w_boot

# Calculating the 95% quantile bootstrap confidence intervall
quantile(gci_boot, c(0.025, 0.975))

Using these made up data the output is here:

  2.5%  97.5% 
0.8382 1.1600