Fabio Oliveira’s blog

Report - Gender Bias in Research Funding

Introduction

Motivation

This analysis is motivated by topics presented in the course of the Data Science Professional Certificate, a set of MOOCs presented by HarvardX and hosted on the edX distance learning platform.

This report is not focused on presenting any novel insight on the data or the subject, but rather on applying principles from this introductory set of courses. The original paper presenting this data contains additional data and levels of nuance and which are not addressed here.

In particular, this report hinges on the statistical analysis tools presented in course #4 (Inference and Modeling) and explores the R Markdown and Git features presented in course #5 (Productivity Tools). It also uses tools explored in course #2 (Data Visualization with ggplot2).

Dataset

The data under analysis in this report was taken from a 2015 PNAS paper analyzing success rates from research funding agencies in the Netherlands. The raw data is contained in the dslabs R package, a library with numerous datasets compiled for teaching purposes. Further information is available in this page.

Instructions

Given the simplicity of the analysis, the entire code is reproduced in the markdown file, without need for any additional script or saved data. The code is also entirely echoed in this report.

If you are interested in reproducing this analysis, the ResearchFundingGenderBias.R script available in the repository performs the same tasks.

Analysis

Housekeeping

The following code loads (and installs, if not yet installed) the libraries required for this analysis. It also loads the research_funding_rates data from the dslabs package.

if (! "tidyverse" %in% installed.packages()) {install.packages("tidyverse")}
if (! "ggplot2" %in% installed.packages()) {install.packages("ggplot2")}
if (! "dslabs" %in% installed.packages()) {install.packages("dslabs")}

library(tidyverse)
library(dslabs)
library(ggplot2)

data("research_funding_rates")

Raw data

First, we examine the raw data from the dslabs package:

# variables contained in the dataset
names(research_funding_rates)
##  [1] "discipline"          "applications_total"  "applications_men"   
##  [4] "applications_women"  "awards_total"        "awards_men"         
##  [7] "awards_women"        "success_rates_total" "success_rates_men"  
## [10] "success_rates_women"

# disciplines for which data is available
research_funding_rates$discipline
## [1] "Chemical sciences"   "Physical sciences"   "Physics"            
## [4] "Humanities"          "Technical sciences"  "Interdisciplinary"  
## [7] "Earth/life sciences" "Social sciences"     "Medical sciences"

# total number of applications
sum(research_funding_rates$applications_total)
## [1] 2823

# total number of grants
sum(research_funding_rates$awards_total)
## [1] 467

# total success rate
sum(research_funding_rates$awards_total) / sum(research_funding_rates$applications_total)
## [1] 0.165

The dataset includes, for each of the 9 disciplines, the number of applications, grants received and success rates. It also includes the same data discriminated by gender.

Analysis of the overall success rates

Based on the success rates and number of samples for both men and women, we can calculate a 95% confidence interval for the probability of receiving a grant. This analysis is based on the premise that the observed data is a sample distributed according to an underlying probability which is unknown.

# summary statistics on the data
combined_results <- 
  research_funding_rates %>%
  summarize(grants = sum(awards_total),
            applications = sum(applications_total),
            success_rate = grants/applications,
            standard_error_estimate = sqrt(success_rate*(1-success_rate)/applications),
            ci_lower = qnorm(0.025,success_rate,standard_error_estimate),
            ci_upper = qnorm(0.975,success_rate,standard_error_estimate))

combined_results
##   grants applications success_rate standard_error_estimate ci_lower ci_upper
## 1    467         2823        0.165                 0.00699    0.152    0.179

From this we can see that there is an overall 15.2-17.9% chance (with 95% confidence)of obtaining a grant, across different disciplines and genders.

We can also repeat this analysis, with data segregated between men and women:

# create distinct observations for men and women, group data by gender and calculate summary for each group
combined_results_by_gender <-
  research_funding_rates %>%
  mutate(Men = paste(awards_men,applications_men,success_rates_men),
         Women = paste(awards_women,applications_women,success_rates_women)) %>%
  select(discipline,Men,Women) %>%
  gather("gender","data",Men:Women) %>%
  mutate(grants = sapply(data,function(d){unlist(strsplit(d,split=" "))}[1]),
         applications = sapply(data,function(d){unlist(strsplit(d,split=" "))}[2]),
         success_rate = sapply(data,function(d){unlist(strsplit(d,split=" "))}[3])) %>%
  mutate(gender = as.factor(gender),
         grants = as.numeric(grants),
         applications = as.numeric(applications),
         success_rate = as.numeric(success_rate)/100) %>%
  select(-data) %>%
  group_by(gender) %>%
  summarize(grants = sum(grants),
            applications = sum(applications),
            success_rate = grants/applications,
            standard_error_estimate = sqrt(success_rate*(1-success_rate)/applications),
            ci_lower = qnorm(0.025,success_rate,standard_error_estimate),
            ci_upper = qnorm(0.975,success_rate,standard_error_estimate))

combined_results_by_gender
## # A tibble: 2 x 7
##   gender grants applications success_rate standard_error_esti~ ci_lower ci_upper
##   <fct>   <dbl>        <dbl>        <dbl>                <dbl>    <dbl>    <dbl>
## 1 Men       290         1635        0.177              0.00945    0.159    0.196
## 2 Women     177         1188        0.149              0.0103     0.129    0.169

The data shows that there is a difference of approximately 2.8% in the probability of obtaining a grant between men and women, which is similarly reflected in the confidence intervals. A visualization helps in evaluating the magnitudes:

# add observation with combined results and plot observed values and error bars
combined_results_by_gender %>%
  add_case(gender="Combined",
           grants=combined_results$grants,
           applications=combined_results$applications,
           success_rate=combined_results$success_rate,
           standard_error_estimate=combined_results$standard_error_estimate,
           ci_lower=combined_results$ci_lower,
           ci_upper=combined_results$ci_upper) %>%
  ggplot(aes(x=success_rate,y=gender)) +
  geom_vline(xintercept=combined_results$success_rate,
             color='red',size=2) +
  geom_vline(xintercept=combined_results$success_rate,
             color='red',size=2) +
  geom_point(size=5) +
  geom_errorbar(aes(xmin=ci_lower,xmax=ci_upper),
                size=2, width=.5) +
  ggtitle("Investigation of gender bias in research funding") +
  xlab("Success rate in obtaining a grant \n (95% confidence interval of the probability of receiving a grant according to gender)") +
  ylab("Gender") +
  scale_x_continuous(labels = scales::percent)

Despite confirming that there is a difference in the probability of receiving a grant between men and women, the graph shows that both confidence intervals have a significant overlap. Additionally, they both include the total observed success rate, depicted as the red vertical line in the figure.

Nevertheless, in order to evaluate the relevance of the observed discrepancy, it is useful to perform a chi-squared test on the data as well as calculate the odds ratio:

# create object with total yay/nay for each gender
totals <- research_funding_rates %>%
  select(-discipline) %>%
  summarize_all(sum) %>%
  summarize(yes_men = awards_men,
            no_men = applications_men - awards_men,
            yes_women = awards_women,
            no_women = applications_women - awards_women)

# format as two-by-two table used in chi-squared calculation
two_by_two <- 
  data.frame(gender = c("men","women"),
             yes = c(totals$yes_men,totals$yes_women), 
             no = c(totals$no_men,totals$no_women))

# perform chi-squared test
two_by_two %>%
  select(-gender) %>%
  chisq.test()
## 
##  Pearson's Chi-squared test with Yates' continuity correction
## 
## data:  .
## X-squared = 4, df = 1, p-value = 0.05

# calculate odds ratio
odds_ratio <-
  (totals$yes_men / totals$no_men) / 
  (totals$yes_women / totals$no_women)
odds_ratio
## [1] 1.23

The test reveals that the associated p-value is approximately 0.051, which indicates that, under the null hypothesis (no actual difference between men an women), the probability of observing such a discrepancy due to random variability is around 5.1%.

The odds ratio of 1.23, on the other hand, does not show a significant difference among genders.

Monte Carlo simulation

A Monte Carlo simulation is a convenient way of repeating an experiment multiple times, under a certain set of assumptions, to determine if the results fit our conclusions and the observed data.

Here, we perform 10000 iteractions under the following assumptions:

At each iteraction, a total of 1635 men and 1188 women are drawn, corresponding to the total amount of applications for each gender in the original data;
Each researcher will be randomly assigned a response of 1 (in case the application is granted) or 0 (in case the application is rejected), with a probability corresponding to the combined success rate observed in the data (with no gender bias);
After the simulation, the spread (difference in success rate between men and women) is calculated for each iteraction;
The 2.5% and 97.5% quantiles for the spread are calculated, indicating the range where 95% of the observations are contained;
The percentage of cases where the spread obtained was equal to or greater than the actual data in the dataset is calculated, indicating the likelihood that such a discrepancy is observed under the simulation assumptions.

# set number of simulation iteractions
N <- 10000

# run simulation
simulation <- replicate(N,{
  results_men <- 
    sample(c(1,0),
           combined_results_by_gender$applications[combined_results_by_gender$gender=="Men"],
           prob = c(combined_results$success_rate,1-combined_results$success_rate),
           replace = TRUE) %>%
    sum()
  results_women <- 
    sample(c(1,0),
           combined_results_by_gender$applications[combined_results_by_gender$gender=="Women"],
           prob = c(combined_results$success_rate,1-combined_results$success_rate),
           replace = TRUE) %>%
    sum()
  c(results_men,results_women)
})

# gather results into a data frame
monte_carlo_results <-
  data.frame(grants_men = simulation[1,],
             grants_women = simulation[2,],
             applications_men = combined_results_by_gender$applications
             [combined_results_by_gender$gender=="Men"],
             applications_women = combined_results_by_gender$applications
             [combined_results_by_gender$gender=="Women"]) %>%
  mutate(success_rate_men = grants_men / applications_men,
         success_rate_women = grants_women / applications_women,
         spread = success_rate_men - success_rate_women)

# calculate quantiles corresponding to the 95% range
monte_carlo_results$spread %>% quantile(c(0.025,0.975))
##    2.5%   97.5% 
## -0.0283  0.0278

The 2.5% and 97.5% quantiles are -0.028 and 0.028. The actual difference observed in the data is slightly above the 97.5% quantile, at 2.8%.

In fact, we can see the percentage of iteractions which showed a spread as big or bigger than the actual data:

# calculate the spread from the dataset
actual_spread <-
  combined_results_by_gender$success_rate[combined_results_by_gender$gender == "Men"] - 
  combined_results_by_gender$success_rate[combined_results_by_gender$gender == "Women"]

# estimate the standard error based on the number of observations
standard_error_estimate <-
  sqrt(
  combined_results_by_gender$standard_error_estimate[combined_results_by_gender$gender == "Men"]^2 +
  combined_results_by_gender$standard_error_estimate[combined_results_by_gender$gender == "Women"]^2)
  
# evaluate the simulation results for the percentage of cases with discrepancy matching the data
spread_extreme <- mean(monte_carlo_results$spread >= actual_spread | monte_carlo_results$spread <= -actual_spread)
spread_extreme
## [1] 0.0465

When comparing the results from the Monte Carlo simulation with the actual spread observed in the dataset, we see that 4.6% of the iteractions generated a discrepancy with a magnitude at least as large as the one observed in the data, with no bias towards any of the genders programmed into the simulation.

Analysis of the discrepancies between men and women for each discipline

Another relevant point of analysis is whether there is any pattern in the different success rates across different disciplines. Here we look at the data for each particular discipline, arranged according to the observed spread in success rates:

# calculate spread, estimated standard error and favored gender and arrange according to spread
totals_per_discipline <-
  research_funding_rates %>%
  mutate(success_rates_men = success_rates_men/100,
         success_rates_women = success_rates_women/100) %>%
  mutate(se_men = sqrt(success_rates_men*(1-success_rates_men)/applications_men),
         se_women = sqrt(success_rates_women*(1-success_rates_women)/applications_women),
         spread = success_rates_men - success_rates_women,
         se_spread = sqrt(se_men^2 + se_women^2)) %>%
  select(discipline,success_rates_men,success_rates_women,se_men,se_women,spread,se_spread) %>%
  add_case(discipline = "Combined",
           spread = actual_spread,
           se_spread = standard_error_estimate) %>%
  mutate(p_null = pnorm(0,spread,se_spread),
         ci_lower = qnorm(0.025,spread,se_spread),
         ci_upper = qnorm(0.975,spread,se_spread),
         include_zero = 0>qnorm(0.025,spread,se_spread) & 0<qnorm(0.975,spread,se_spread),
         favorite = ifelse(spread>0,"Men","Women")) %>%
  arrange(spread)

# print relevant variables
totals_per_discipline %>%
  filter(discipline != "Combined") %>%
  select(discipline,spread,ci_lower,ci_upper,favorite)
##            discipline spread ci_lower ci_upper favorite
## 1   Interdisciplinary -0.104 -0.21396  0.00596    Women
## 2  Technical sciences -0.051 -0.16500  0.06300    Women
## 3          Humanities -0.050 -0.12517  0.02517    Women
## 4   Physical sciences -0.038 -0.18609  0.11009    Women
## 5   Chemical sciences  0.009 -0.15766  0.17566      Men
## 6     Social sciences  0.038 -0.00812  0.08412      Men
## 7             Physics  0.047 -0.24454  0.33854      Men
## 8    Medical sciences  0.076  0.01385  0.13815      Men
## 9 Earth/life sciences  0.101  0.01001  0.19199      Men

We see that, even though the total success rate favors male researchers, this rate is actually more positive towards women in 4 disciplines.

The spread data per discipline, along with the calculated confidence intervals, are represented in this figure:

# create and populate plot
totals_per_discipline %>%
  ggplot(aes(x=spread,y=discipline)) +
  geom_point(size=5) +
  labs(title = "Gender bias in research funding",
       subtitle = "based on data from the research_funding_rates database \r
       contained in the dslabs library") +
  xlab("Gender bias \n (95% confidence interval of the difference \r 
       in probability of receiving a grant according to gender)") +
  ylab("Discipline") +
  scale_y_discrete(limits = c("Technical sciences",
                              "Social sciences",
                              "Physics",
                              "Physical sciences",
                              "Medical sciences",
                              "Interdisciplinary",
                              "Humanities",
                              "Earth/life sciences",
                              "Chemical sciences",
                              "Combined")) +
  scale_x_continuous(breaks = seq(-.40,.40,.1), 
                     minor_breaks=seq(-.40,.40,.01),
                     labels = scales::percent) +
  coord_cartesian(xlim=c(-.35,.35),
                  ylim=c(1.1,9.7)) +
  geom_vline(xintercept = actual_spread, color = 'green', size = 15, alpha = 0.2) +
  geom_vline(xintercept = .2, color = 'blue', size = 96, alpha = 0.2) +
  geom_vline(xintercept = -.2, color = 'red', size = 96, alpha = 0.2) +
  geom_vline(xintercept=0,color='red', size = 2) +
  geom_errorbar(aes(xmin = spread-1.96*se_spread, xmax = spread+1.96*se_spread),
              size=2,width=.5) +
  geom_errorbar(aes(y = "Combined",
                    x = actual_spread,
                    xmin = qnorm(0.025,actual_spread,standard_error_estimate),
                    xmax = qnorm(0.975,actual_spread,standard_error_estimate)),
                size=2,width=.5,color='blue') +
  geom_point(aes(y = "Combined",
                 x = actual_spread),
             size=5, color='blue')

<img 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

Fabio Oliveira's blog

an Electrical Engineer with background in the Automotive and Aerospace industries, now delving into Data Science & Machine Learning and documenting his exploits