Chapter 17 interactive notes

Author

R Saidi

Load the libraries and data

library(tidyverse)
library(openintro)
library(tidymodels)
data("gss")

A question in two variables

Do women and men join political parties at different rates?

Let p be the proportion that are democrats.

\(H_o: p_f = p_m\) \(H_a: p_f \neq p_m\)

We’re curious to know: do men and women join political parties at different rates?

Let’s let p be the true proportion that join democrats.

We can then phrase this question as the null hypothesis that the difference in the proportion of men and the proportion of women that join democrats is zero.

The alternative hypothesis would then be that that difference is non-zero.

Do women and men have different political affiliations?

ggplot(gss, aes(x = sex, fill = partyid)) +
  geom_bar()

Let’s take a look at how these proportions compare in the gss data set. The data live in two columns, partyid and sex, so we can map sex to the x-axis and their political party to the color fill of the bars. If we add a geom_bar layer, we get a stacked bar chart that shows us that we have more females in our data set than males and that opinions are split.

ggplot(gss, aes(x = sex, fill = partyid)) +
  geom_bar(position = "fill")

We can convert these to proportions by adding the position equals “fill” argument. It looks like the proportion for men is a bit lower than the proportion for women.

Create p-hat values

Sample proportions for males and females to identify with democrats - then calculate the difference of the proportions using d_hat

p_hats <- gss |>
  group_by(sex) |>
  summarize(mean(partyid == "dem", na.rm = TRUE)) |>
  pull()

d_hat <- diff(p_hats)
d_hat

[1] 0.06499174

We can calculate the difference in these proportions by using our normal summarize method of calculating a proportion, but add in a group_by line to indicate we want to calculate that proportion for men and women separately. The result is a vector of two proportions. We take their difference with the diff function and save it as d-hats, which we learn is 0.064.

Generating data from Ho

\(H_o: p_f = p_m\)

There is no association between political party affiliation sex of a subject.

The variable partyid is independent from the variable sex.

⇒ Generate data by permutation

Mutate partyid to be a binary variable

gss1 <- gss |>
  mutate(two_party = ifelse(partyid == "dem", "democrat", "not-democrat"))

Do women and men have different political party affiliations?

gss1 |>
  specify(
    two_party ~ sex,  # this line is new
    success = "democrat"
  ) |>
  hypothesize(null = "independence") |>
  generate(reps = 1, type = "permute")

Response: two_party (factor)
Explanatory: sex (factor)
Null Hypothesis: independence
# A tibble: 500 × 3
# Groups:   replicate [1]
   two_party    sex    replicate
   <fct>        <fct>      <int>
 1 not-democrat male           1
 2 not-democrat female         1
 3 not-democrat male           1
 4 democrat     male           1
 5 not-democrat male           1
 6 not-democrat female         1
 7 not-democrat female         1
 8 not-democrat female         1
 9 not-democrat female         1
10 not-democrat female         1
# ℹ 490 more rows

Build up a full null distribution

null <- gss1 |>
  specify(two_party ~ sex, success = "democrat") |>
  hypothesize(null = "independence") |>
  generate(reps = 500, type = "permute") |>
  calculate(stat = "diff in props", order = c("female", "male"))
null

Response: two_party (factor)
Explanatory: sex (factor)
Null Hypothesis: independence
# A tibble: 500 × 2
   replicate     stat
       <int>    <dbl>
 1         1 -0.0553 
 2         2 -0.0393 
 3         3  0.0329 
 4         4 -0.0553 
 5         5 -0.0152 
 6         6  0.00884
 7         7  0.0409 
 8         8 -0.0553 
 9         9  0.0249 
10        10  0.0409 
# ℹ 490 more rows

Plot a density curve

ggplot(null, aes(x = stat)) +
  geom_density() +
  geom_vline(xintercept = d_hat, color = "red")

This distribution of data suggests that there may be a difference between sexes in proportion of affiliation to democrats with females at higher proportions than males..

We need a p-value to determine the true effect.

# Compute two-tailed p-value
null |>
  summarize(
    one_tailed_pval = mean(stat >= d_hat),
    two_tailed_pval = 2 * one_tailed_pval
  ) |>
  pull(two_tailed_pval)

[1] 0.136

The two-tailed p-value is 0.172. We fail to reject the null. There is no compelling evidence that there is a difference in proportions of females and males affiliated to the democratic party.

Create a 95% Bootstrap CI

# Create the bootstrap distribution
boot <- gss1 |>
  # Specify the variables and success
  specify(two_party ~ sex, success = "democrat") |>
  # Generate 500 bootstrap reps
  generate(reps = 500, type = "bootstrap") |>
  # Calculate the statistics
  calculate(stat = "diff in props", order = c("female", "male"))

# Compute the standard error
SE <- boot |>
  summarize(se = sd(stat)) |>
  pull()
  
# Form the CI (lower, upper)
c(d_hat - 2 * SE, d_hat + 2 * SE)

[1] -0.02261768  0.15260116

We are 95% confident that the true difference in proportions of male and female affiliation to the democratic party is between -1.79% and 14.79%. Because zero is included there is no compelling evidence that there is a difference in sex with respect to affiliation to the democratic party.