1 Undervoting for President, by Race: Difference in Two Binomial Proportions
Does undervoting for the US president differ by race? Intentional undervoting is when a voter chooses not to cast vote for an item on a ballot.
Tomz and Houweling (2003) analyze this phenomenon using two surveys:
- Voter News Service (VNS) exit poll for the 1992 election
- American National Election Studies (ANES) for the 1964–2000 elections
Each of these surveys asked voters whether they voted for president, as well as the race of the respondents.
The results of these surveys is contained in the undervote data frame.
The column undervote is the number of respondents who reported voting but not voting for president.
undervote <- tribble(
~survey, ~race, ~n, ~undervote,
"VNS", "black", 6537, 26,
"VNS", "white", 44531, 91,
"ANES", "black", 1101, 10,
"ANES", "white", 9827, 57
)| survey | race | n | undervote | Survey | Race | No. Voted | Didn’t vote for president |
|---|---|---|---|---|---|---|---|
| VNS | black | 6537 | 26 | VNS | black | 6537 | 26 |
| VNS | white | 44531 | 91 | VNS | white | 44531 | 91 |
| ANES | black | 1101 | 10 | ANES | black | 1101 | 10 |
| ANES | white | 9827 | 57 | ANES | white | 9827 | 57 |
We are interested in analyzing the difference in proportions for each of these surveys independently. We will model the proportions of each race and survey, \[ \begin{aligned}[t] y_i &\sim \mathsf{Binomial}(n_i, \pi_i) , \end{aligned} \] where \[ i \in \{ (\text{VNS},\text{black}), (\text{VNS},\text{white}), (\text{ANES},\text{black}), (\text{ANES},\text{white}) \} . \]
We will model the proportions independently by assigning them identical independent uninformative priors,
\[ \begin{aligned}[t] \pi_i &\sim \mathsf{Beta}(1, 1) . \end{aligned} \] The racial differences in undervoting in each survey are auxiliary quantities, \[ \begin{aligned}[t] \delta_{\text{VNS}} &= \pi_{\text{VNS},\text{black}} - \pi_{\text{VNS},\text{white}} ,\\ \delta_{\text{ANES}} &= \pi_{\text{ANES},\text{black}} - \pi_{\text{ANES},\text{white}} . \\ \end{aligned} \] We are also interested in the probability that black undervoting is greater than white undervoting, \(\Pr(\delta_j) > 0\), in each survey.
data {
int n[4];
int y[4];
vector[4] pi_a;
vector[4] pi_b;
}
parameters {
vector[4] pi;
}
model {
y ~ binomial(n, pi);
pi ~ beta(pi_a, pi_b);
}
generated quantities {
vector[2] delta;
int good[2];
delta[1] = pi[2] - pi[1];
delta[2] = pi[4] - pi[3];
good[1] = int_step(delta[1]);
good[2] = int_step(delta[2]);
}
# this analysis depends on the order of the data frame
undervote_data <-
list(y = undervote$undervote,
n = undervote$n,
N = nrow(undervote),
pi_a = rep(1, 4),
pi_b = rep(1, 4))undervote_fit
#> Inference for Stan model: undervote.
#> 4 chains, each with iter=2000; warmup=1000; thin=1;
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#>
#> mean se_mean sd 2.5% 25% 50% 75%
#> pi[1] 0.00 0.00 0.00 0.00 0.00 0.00 0.00
#> pi[2] 0.00 0.00 0.00 0.00 0.00 0.00 0.00
#> pi[3] 0.01 0.00 0.00 0.01 0.01 0.01 0.01
#> pi[4] 0.01 0.00 0.00 0.00 0.01 0.01 0.01
#> delta[1] 0.00 0.00 0.00 0.00 0.00 0.00 0.00
#> delta[2] 0.00 0.00 0.00 -0.01 -0.01 0.00 0.00
#> good[1] 0.00 0.00 0.03 0.00 0.00 0.00 0.00
#> good[2] 0.08 0.00 0.26 0.00 0.00 0.00 0.00
#> lp__ -1254.94 0.03 1.38 -1258.39 -1255.60 -1254.63 -1253.94
#> 97.5% n_eff Rhat
#> pi[1] 0.01 4000 1
#> pi[2] 0.00 4000 1
#> pi[3] 0.02 4000 1
#> pi[4] 0.01 4000 1
#> delta[1] 0.00 4000 1
#> delta[2] 0.00 4000 1
#> good[1] 0.00 4000 1
#> good[2] 1.00 2850 1
#> lp__ -1253.23 2020 1
#>
#> Samples were drawn using NUTS(diag_e) at Fri Apr 20 00:53:03 2018.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at
#> convergence, Rhat=1).References
Simon Jackman, “Undervoting for President, by Race: difference in two binomial proportions”, BUGS Examples 2007-07-24.
References
Tomz, Michael, and Robert P. Van Houweling. 2003. “How Does Voting Equipment Affect the Racial Gap in Voided Ballots?” American Journal of Political Science 47 (1). Wiley-Blackwell: 46–60. https://doi.org/10.1111/1540-5907.00004.