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.

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]);
}

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.