Test of Independence
To carry out a chi-squared test of independence, we’ll examine the
association between income and educational attainment in the United
States. college is a categorical variable with values
degree and no degree, indicating whether or
not the respondent has a college degree (including community college),
and finrela gives the respondent’s self-identification of
family income—either far below average,
below average, average,
above average, far above average, or
DK (don’t know).
This is what the relationship looks like in the sample data:
If there were no relationship, we would expect to see the purple bars reaching to the same height, regardless of income class. Are the differences we see here, though, just due to random noise?
First, to calculate the observed statistic, we can use
specify() and calculate().
# calculate the observed statistic
observed_indep_statistic <- gss %>%
specify(college ~ finrela) %>%
hypothesize(null = "independence") %>%
calculate(stat = "Chisq")The observed \(\chi^2\) statistic is 30.6825. Now, we want to compare this statistic to a null distribution, generated under the assumption that these variables are not actually related, to get a sense of how likely it would be for us to see this observed statistic if there were actually no association between education and income.
We can generate the null distribution in one of two
ways—using randomization or theory-based methods. The randomization
approach approximates the null distribution by permuting the response
and explanatory variables, so that each person’s educational attainment
is matched up with a random income from the sample in order to break up
any association between the two.
# generate the null distribution using randomization
null_dist_sim <- gss %>%
specify(college ~ finrela) %>%
hypothesize(null = "independence") %>%
generate(reps = 1000, type = "permute") %>%
calculate(stat = "Chisq")Note that, in the line specify(college ~ finrela) above,
we could use the equivalent syntax
specify(response = college, explanatory = finrela). The
same goes in the code below, which generates the null distribution using
theory-based methods instead of randomization.
# generate the null distribution by theoretical approximation
null_dist_theory <- gss %>%
specify(college ~ finrela) %>%
assume(distribution = "Chisq")To get a sense for what these distributions look like, and where our
observed statistic falls, we can use visualize():
# visualize the null distribution and test statistic!
null_dist_sim %>%
visualize() +
shade_p_value(observed_indep_statistic,
direction = "greater")
We could also visualize the observed statistic against the
theoretical null distribution. To do so, use the assume()
verb to define a theoretical null distribution and then pass it to
visualize() like a null distribution outputted from
generate() and calculate().
# visualize the theoretical null distribution and test statistic!
gss %>%
specify(college ~ finrela) %>%
assume(distribution = "Chisq") %>%
visualize() +
shade_p_value(observed_indep_statistic,
direction = "greater")
To visualize both the randomization-based and theoretical null
distributions to get a sense of how the two relate, we can pipe the
randomization-based null distribution into visualize(), and
further provide method = "both".
# visualize both null distributions and the test statistic!
null_dist_sim %>%
visualize(method = "both") +
shade_p_value(observed_indep_statistic,
direction = "greater")
Either way, it looks like our observed test statistic would be quite
unlikely if there were actually no association between education and
income. More exactly, we can approximate the p-value with
get_p_value:
# calculate the p value from the observed statistic and null distribution
p_value_independence <- null_dist_sim %>%
get_p_value(obs_stat = observed_indep_statistic,
direction = "greater")
p_value_independence## # A tibble: 1 × 1
## p_value
## <dbl>
## 1 0Thus, if there were really no relationship between education and income, our approximation of the probability that we would see a statistic as or more extreme than 30.6825 is approximately 0.
To calculate the p-value using the true \(\chi^2\) distribution, we can use the
pchisq function from base R. This function allows us to
situate the test statistic we calculated previously in the \(\chi^2\) distribution with the appropriate
degrees of freedom.
pchisq(observed_indep_statistic$stat, 5, lower.tail = FALSE)## X-squared
## 1.082e-05Note that, equivalently to the theory-based approach shown above, the
package supplies a wrapper function, chisq_test, to carry
out Chi-Squared tests of independence on tidy data. The syntax goes like
this:
chisq_test(gss, college ~ finrela)## # A tibble: 1 × 3
## statistic chisq_df p_value
## <dbl> <int> <dbl>
## 1 30.7 5 0.0000108
