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 Conservatives.

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 American Journal of Political Science (AJPS) is the flagship journal of the Midwest Political Science Association and is published by Wiley.

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