Chi-Square tests of Independence

Cleaning up the data

library(tidyverse)
library(infer)
load("../Data/gss.RData")
gss2016 <- gss |>
  filter(year == 2016) |> 
  # I am filtering to get a similar set used in DataCamp
  filter(!is.na(consci)) |> 
  filter(!is.na(cappun)) |> 
  filter(!is.na(sex)) |> 
  filter(!is.na(partyid)) |> 
  filter(!is.na(postlife)) |> 
  filter(!is.na(natspac)) |> 
  filter(!is.na(natarms)) |> 
  filter(!is.na(happy)) |> 
  filter(!is.na(region))
# Recoding levels of consci---similar to data camp???
gss2016$consci <- ifelse(gss2016$consci == "A GREAT DEAL", "High", "Low")
set.seed(321)  # Make same size as data camp...randomly
ind <- sample(1:nrow(gss2016), 150, replace = FALSE)
gss2016 <- gss2016[ind, ]
# Create new variable party
gss2016$party <- ifelse(gss2016$partyid %in% c("STRONG DEMOCRAT",
                                               "NOT STR DEMOCRAT"),
                        "Dem", ifelse(gss2016$partyid %in% 
                          c("IND,NEAR DEM", "IND,NEAR REP",
                            "INDEPENDENT"), "Ind",
                          ifelse(gss2016$partyid %in% 
                                   c("STRONG REPUBLICAN", 
                                     "NOT STR REPUBLICAN"), 
                                 "Rep", "Oth")))
###
# Subset data
gss_party <- gss2016 |>
  # Filter out the "Oth"
  filter(party != "Oth") 
tab_politics <- gss_party |>
  # Select columns of interest
  select(natarms, party) |>
  # Create table
  table()
# View the result
tab_politics

             party
natarms       Dem Ind Rep
  TOO LITTLE   17  16  26
  ABOUT RIGHT  19  21  10
  TOO MUCH     16  18   1

Test of Independence Hypotheses

We would like to test the null hypothesis that opinions on how much money is spend on defense (natarms) is independent of party (party) affiliation. The alternative hypothesis is that there is an association between opinions on how much money is spend on defense and party affiliation.

Graphing the Counts

gss_party |> 
  ggplot(aes(x = party, fill = natarms)) +
  geom_bar() + 
  theme_bw()

If we want to look at the proportions we would code the graph as follows:

gss_party |> 
  ggplot(aes(x = party, fill = natarms)) +
  geom_bar(position = "fill") + 
  theme_bw()

Computing the expected counts

tab_politics -> obs
obs

             party
natarms       Dem Ind Rep
  TOO LITTLE   17  16  26
  ABOUT RIGHT  19  21  10
  TOO MUCH     16  18   1

chisq.test(obs)$expected -> exp
exp

             party
natarms            Dem      Ind       Rep
  TOO LITTLE  21.30556 22.53472 15.159722
  ABOUT RIGHT 18.05556 19.09722 12.847222
  TOO MUCH    12.63889 13.36806  8.993056

Computing the test statistic

chi_sq_stat <- sum((obs - exp)^2/exp)
chi_sq_stat

[1] 20.98965

Computing the p-value

pchisq(chi_sq_stat, 4, lower = FALSE)

[1] 0.0003181692

# Making assumptions and using chisq.test()
chisq.test(obs, correct = FALSE)

    Pearson's Chi-squared test

data:  obs
X-squared = 20.99, df = 4, p-value = 0.0003182

Verifying the result without assumptions

null_natarms <- gss_party |> 
  specify(natarms ~ party) |> 
  hypothesize(null = "independence") |> 
  generate(reps = 10000, type = "permute") |> 
  calculate(stat = "Chisq")
visualize(null_natarms) + 
  shade_p_value(chi_sq_stat, direction = "right")

get_p_value(null_natarms, obs_stat = chi_sq_stat, direction = "right")

# A tibble: 1 × 1
  p_value
    <dbl>
1  0.0003

Doing the same resampling process with a for loop

set.seed(31)
(T1 <- table(gss_party$natarms, gss_party$party))

             
              Dem Ind Rep
  TOO LITTLE   17  16  26
  ABOUT RIGHT  19  21  10
  TOO MUCH     16  18   1

(obs_stat <- chisq.test(T1, correct = FALSE)$stat)

X-squared 
 20.98965 

P <- 10^4
chi_stat <- numeric(P)
for(i in 1:P){
  chi_stat[i] <- chisq.test(table(gss_party$natarms, sample(gss_party$party)), correct = FALSE)$stat
}
hist(chi_stat)

(p_value <- mean(chi_stat >= obs_stat))

[1] 4e-04

Conclusion

Given the p-value (\(4\times 10^{-4}\)) is significantly less than \(\alpha = 0.05\) we reject the null hypothesis and conclude there is an association between a persons opinion on the how much the country is spending on national defense and their party affiliation.