Estimation of voter transitions based on ecological inference: an empirical assessment of different approaches

Abstract

The analysis of voter transitions is an important area of electoral studies. A main strategy is to use aggregate data provided by the offices of statistics regarding districts, precincts, communities etc. and to rely on ecological inference. Ecological inference, however, is plagued by the well-known indeterminacy problem. In this article, we present the so far most extensive systematic empirical comparison of commonly used approaches for ecological inference of the analysis of voter transitions. Our evaluation is based on diverse simulations for multiple assumptions and scenarios. Based on recent election data for the German metropolitan city Munich, we are able to show that an application of the hierarchical multinomial-Dirichlet model, which is implemented in the R-library eiPack, exhibits the best overall estimation performance. Other prominent approaches frequently used by practitioners, e.g. the Thomsen logit approach, proved to be inconsistent. Furthermore, we demonstrate that appropriate data preprocessing is crucial for achieving reliable results.

Appendix

1.1 Model description Greiner and Quinn

As distributions at the district level for the Greiner and Quinn (2009) model several multinomial distributions are assumed:

$$\begin{aligned}&((\mathrm{CSU}_{1},\mathrm{CSU}_{2})_{i}, (\mathrm{CSU}_{1},\mathrm{SPD}_{2})_{i},\ldots , (\mathrm{CSU}_{1},\mathrm{NonVoters}_{2})_{i})\nonumber \\&\quad \sim \mathrm{Multinomial}(\beta _{\mathrm{CSU,CSU},i},\beta _{\mathrm{CSU,SPD},i}, \ldots , \beta _{\mathrm{CSU,NonVoters},i}, \mathrm{CSU}_{1,i}) \end{aligned}$$

(14)

Equation (14) presents the assumed multinomial distribution for one row. Instead of the margins like in the Rosen et al. (2001) proposal, the inner cells, e.g. $(\mathrm{CSU}_{1},\mathrm{CSU}_{2})_{i}$, of the table are considered and assumed to be dependent from the parameters

$$\begin{aligned} \beta _{\mathrm{CSU},i} \sim (\beta _{\mathrm{CSU,CSU},i},\beta _{\mathrm{CSU,SPD},i}, \ldots , \beta _{\mathrm{CSU,NonVoters},i}) \end{aligned}$$

(15)

and the corresponding row-margins $\mathrm{CSU}_{2,i}$. For each of the R-rows, one independent multinomial distribution is assumed. To hold the restrictions defined by the margins, indicator functions for the row and column sums are used.

For the second level of the models, the parameters are logit transformed using a reference column (party). If the non-voters would be used as reference party, the transformed $\beta$-parameters from (14) would have the following form.

$$\begin{aligned} \varOmega _{\mathrm{CSU},i}= & {} (\omega _{\mathrm{CSU,CSU},i},\omega _{\mathrm{CSU,SPD},i},\ldots ,\omega _{\mathrm{CSU,Others},i})\nonumber \\= & {} (log(\frac{\beta _{\mathrm{CSU,CSU},i}}{\beta _{\mathrm{CSU,NonVoters},i}}),log(\frac{\beta _{\mathrm{CSU,SPD},i}}{\beta _{\mathrm{CSU,NonVoters},i}}), \ldots ,log(\frac{\beta _{\mathrm{CSU,Others},i}}{\beta _{\mathrm{CSU,NonVoters},i}}))\nonumber \\ \end{aligned}$$

(16)

The parameters of each multinomial distribution (row) are transformed. Therefore, the $R\times C$ $\beta$-parameters are reduced to $Rx(C-1)$ $\omega$-parameters. As common distribution for the district $\omega$-parameters, a normal distribution is assumed at the second level.

$$\begin{aligned} (\varOmega _{\mathrm{CSU},i},\varOmega _{\mathrm{SPD},i},\ldots ,\varOmega _{\mathrm{NonVoters},i})^{T} \sim N_{R\times (C-1)}(\mu ,\varSigma ) \end{aligned}$$

(17)

As hyperpriors for the $\mu$ a normal distribution and for the $\varSigma$ an inverse wishart distribution are proposed by Greiner and Quinn (2009).

1.2 Description of the 2005 and 2009 Federal elections in Munich

See Fig. 4.

Fig. 4

Election results of the 2005 and 2009 Federal election in Munich. Results of the second vote (German: Zweitstimme) for the 314 time constant districts are presented. Voters by Mail are added to their corresponding ballot box districts. Each district is represented by one line for each party. For more details see Sect. 3.1

1.3 Additional model evaluation results

See Tables 9, 10, 11.

Table 9 Model evaluation Goodman, AD between model estimates is provided, unweighted [uw] or weighted [we] regression and two adjustment of estimates outside the admissible range, the truncation approach (3) [a] and our proposal (4) [o], are considered, for more details see Sect. 3.1

Table 10 Model evaluation hierarchical Multinomial-Dirichlet model (eiPack), AD between Model estimates are provided, different values for burning, thinning and chain length (B/T/C) were used, for details see Sect. 3.1

Table 11 Model evaluation Thomsen’s logit approach, AD between model estimates are provided, different reference parties analyzed (identical for both elections); all other parameters are standard, for details see Sect. 3.1