Dynamic Pie: A Strategy for Modeling Trade-offs in Compositional Variables Over Time

The forthcoming article “Dynamic Pie: A Strategy for Modeling Trade-offs in Compositional Variables Over Time” by Andrew Q. Philips, Amanda Rutherford and Guy D. Whitten is summarized here:
Much of the substance of politics and policy making involves trade-offs. These trade-offs reflect
the competitive nature of the processes that political scientists study. For instance, when public
support for one party increases, this usually comes from a corresponding decline in the support for
other parties or in the proportion of the public that was undecided. Similarly, when policymakers
increase spending in one area of their budget, this will typically be offset by spending cuts in other
areas. In both situations, we can think about the dependent variable of interest as analogous to a
pie that is repeatedly divided into portions. Although researchers have developed a wide range of
theories about the processes that shape these types of zero-sum trade-off relationships over time,
most have limited their analyses of this type of variable over time to models of the size of a single
piece of the pie.

We propose a research strategy for testing theories about trade-off relationships that shape
compositional variables over time. This approach improves on current methods used by political
scientists to analyze compositional dependent variables by addressing two limitations in the current
literature. First, although political scientists have considered compositional dependent variables
before, they have done so either in contexts that are not dynamic or in contexts in which they
ignore the dynamic nature of their data. Second, current approaches to graphical presentations
become unwieldy when the compositional dependent variable has more than three categories. Our
strategy combines a set of existing tools to overcome both of these problems, resulting in expanded
theories and a richer set of inferences that more closely resemble real world trade-offs.

Generally, we need to move from theorizing about and modeling the impact of independent
variables on the level or change in the level of one category of the dependent variable to theorizing
about and modeling the impact of independent variables on the relative levels or relative changes
in levels of dependent variable categories. For example, in the case of a dependent variable with
four categories (A, B, C, and D), each independent variable is linked to six pairwise trade-off
relationships (A/B, A/C, A/D, B/C, B/D, and C/D). When evaluated over time, these trade-offs (or
ratios), may vary depending on environmental shocks to the overall composition.

We start by following the suggestion of Aitchison (1986) by expressing the component parts of
the compositional dependent variable through a log-ratio transformation. Next, we argue that the
best approach for modeling compositional time series is through error correction models (ECMs).
ECMs estimate the rate at which the dependent variable returns to an equilibrium point after
changes in independent variable values. Finally, for compositions with more than two component
parts, we follow the recommendation of previous scholars in advocating the use of a seemingly
unrelated regression (SUR) estimation approach.

The task of interpreting the estimates from models specified according to our proposed approach may at first seem daunting. While the voluminous raw results can be useful for making assessments about whether or not these individual pairwise marginal effects are statistically significant, we follow the growing convention in political science of making substantively-meaningful inferences about the effects of variables through graphic displays of model-based simulations. To conduct these simulations, we take the results from our ECM-SUR to produce 1000 parameter estimates using the Clarify program developed by Tomz et al.(2003).

Figure 3

Figure 3: Dynamic simulation of an increase in the average evaluation of the Liberal Democratic leader

To illustrate the utility of our modeling strategy, we provide two applied examples in which we
are able to estimate changes in compositions over time–monthly UK party support (2004-2010)
and annual US budget categories (1947-2009). In Figure 3 (figure numbers consistent with the
paper), for example, we present a dynamic simulation of UK party support for a scenario in which
there is a one standard deviation increase in the average evaluation of the Liberal Democratic
leader. The immediate impact of this change is an increase in support for the Liberal Democrats
to around 25 percent and a decline in support for both Labour and the Conservatives. All of these
effects are statistically significant. After an initial fall back toward its starting value, support for
the Liberal Democrats plateaus at a level that is statistically significantly higher than it was before
month nine. Interestingly, support for Labour quickly returns to its starting value, while support
for the Conservatives levels off at a lower value that is statistically significant from the beginning
of the scenario. In summary, we find that an increase in evaluations of the leader of the Liberal
Democratic Party has an initial short-run impact on support for both of the other major parties but
ultimately does the most damage to the electoral prospects of the other major opposition party, the

In Figure 5, we display the simulated effects of a one standard deviation decrease in policy
mood liberalism on US budget allocations. This shift in policy mood leads to significant or
borderline-significant changes in almost every spending category. We see slight decreases in relative
spending on welfare and “other” categories with increases in relative spending on social
security and interest payment. The biggest surprise in this scenario is the relative drop in spending on defense. Most budgeting models expect that all else equal, conservatives favor increased
spending on defense.

Figure 5

Figure 5: Dynamic simulation of a decrease in policy mood liberalism

While we have demonstrated the utility of our approach using two prominent applied examples,
there are many additional areas in which this approach should improve our understanding
of important political phenomena. These include studies of taxation policy, campaign emphasis,
and media attention to particular issues. We argue that once researchers start looking for dynamic
compositional variables, they will find them everywhere.

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The Editor of the AJPS is at Michigan State University and the Editorial Office is supported by
the Michigan State University Department of Political Science and the School of Social Sciences.

  Michigan State University 
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