\section{Re-Analyzing Rainbow Coalition Potential and the Effects of Electoral
Structures on Black Representation}
\subsection{Our Methods of Analysis}
Our analysis re-examines Marschall et al.'s analyses in the following ways:
\begin{itemize}
\item We recalculate one of the key covariates used by Marschall et al. to measure
the potential power of rainbow coalitions to elect blacks to local legislatures.
Marschall et al. used the proportion of whites with BAs out of the white population
to estimate the potential for rainbow coalitions, including educated whites to vote
for black representatives. Yet, the proportion of whites in the voting age population
varies widely so their statistic is not interpretable in terms of voting power. We use
whites with BAs out of the entire voting age population to measure the potential
power of whites with BAs to form rainbow coalition voting blocks with blacks.
Lacking the location identifiers for the observations, census data was unavailable, so
we estimated the total population by assuming whites plus blacks plus hispanics
equal 100\% of the local population.
\item Marschall et al. found that single member district structures benefitted black
representation more than at large systems. However, they conducted no single
analysis that combined both SMD and at large data in order to measure the
treatment effect of these two electoral methods on black representation. Other than
separating the data and analysis, all the variables Marschall et al. used to analyze at
large and single member district systems use exactly the variables. We, therefore,
simply combine the data and include a dichotomous variable equaling one (1) when
the electoral method is at large, and zero (0) otherwise to directly estimate the
effect and significance of the electoral method on black representation.
\item We balance the data prior to conducting analyses in order to eliminate any
effects caused by confounding variables, in particular, we ensure that we have both
SMD and at-large electoral cases for the various high, medium, and low values of
hispanic, black, and white with BAs vote share, and also for various levels of
population and number of total legislative seats.
\item We properly analyze negative Logit coefficients as producing a positive, but
decreasing, rather than increasing, effect on the probability of electing blacks. In
other words, given an estimated negative Logit coefficient, the positive effect on
black representation may be large when the value of the covariate is small, and
small as the value of the covariate increases.
\item The above steps are a prelude to more fully answering the questions of how
electoral structures affect black representation and whether or not there is hope for
rainbow coalition groups to increase black representation. We derive and present a
algebraic model for how vote shares directly translate to legislative representation
that more accurately reflects the societal processes, including the real-life
interactions of legislative size, electoral structures, and share of voting age
population of potential rainbow coalition partners.
\item We use summary data to analyze the data, compare the original with the
matched data, and make model independent inferences.
\end{itemize}
We use the above methods to determine, using Marschall et al's own data and
models that there is potential, afterall, for rainbow coalition and to show that any
broad claim that SMD is always better for black representation than at-large
structures, or vice-versa, is incorrect but that it can usually be determined in any
individual jurisdiction which electoral structure is most beneficial for achieving
black represention.
In figure \ref{fig:pre-matchSchool} you can see the at-large observations are not
balanced by single member district observations, for instance, where percentage of
blacks out of voting age population is high and whites with BAs out of voting age
population is very low so that model dependence causes the coefficients to vary
depending on specification, including which other covariates are included in the
model. Thus, we cannot reasonably extract conclusions from data in this range.
Similar imbalances are evident for other covariates, in terms of trying to estimate
the treatment effects of electoral structures as well. We need to prune the data to
match the independent variables for at large versus single member districts and
then analyze. Because share of voting age population roughly translates to vote
shares translate to tallies determining legislative representation, we balance the
data with respect to electoral structures (SMD and at large) for share of black,
hispanic, and white with BA's voting age population before attempting to estimate
the effect of the electoral structure on black descriptive representation.
\begin{figure}[htb]
\centering
\includegraphics[width=0.8\textwidth]{prematch.png}
\caption{School Board data prior to Matching}
\label{fig:pre-matchSchool}
\end{figure}
\clearpage
\begin{figure}[htb]
\centering
\includegraphics[width=0.8\textwidth]{postmatch.png}
\caption{School Board data Post Matching}
\label{fig:post-matchSchool}
\end{figure}