Assessing Mass Opinion Polarization in the US Using Relative Distribution Method

Abstract

Through an analysis of the cumulative data of the American National Election Studies between 1984 and 2008, this study presents evidence of growing mass polarization in terms of standard ANES measures of ideological orientation using the public policy issue dimensions. The empirical findings here suggest that the degree of polarization among US citizens increased as the distributional center of measures of political ideology have progressively declined, though the opinion distribution of the later periods do not dramatically exhibit a text-book style polarized distribution (e.g., bimodal distribution). According to the findings, attitudes toward government guarantees have shifted back and forth between more liberal and more conservative positions while public opinion on cultural issues has generally moved more liberal positions over years.

Appendices

Appendix 1: An Exclusion of “Don’t Knows (Haven’t Thought Much about It)” Category from Moderates

As mentioned above, it is conventional to include “don’t knows (haven’t thought much about it)” group to the middle of the roaders in the scholarly literature of mass opinion polarization (e.g., Abramowitz and Saunders 2008, 544; Campbell 2006, 157, Fiorina and Abrams 2011). In this appendix, these “don’t knows (DK)” answers are recoded as missing values, namely these are regarded separately from moderates.

Figure 4 provides two separate sets of kernel density plots comparing between when DKs are considered as the middle of the roaders (1st column) and when those answers are excluded from moderates (2nd column). Focusing on the aspect of growing distributional polarization, density reduction in the middle becomes much clearer on the government guarantee scale comparing ideological distributions between 1984 and 2008, as I remove the DKs from moderates. The density gap on the center areas (around scale 4) between the distribution of 1984 (a dotted line) and the distribution of 2008 (a line) is greater in the second column excluding DKs than in the first column including DKs for the government guarantee issues scale. That is, this increased gap of density differences could be attributed to an exclusion of those non-ideologicals from moderates. In contrast to the primary government guarantee scale, the cultural issues scale does not present dramatic differences in terms of distributional polarization between the two columns in the Fig. 4, as this scale does not show strong evidence of growing ideological polarization over time.

Fig. 4

Kernel density plots of the two ideological issue scales

Corresponding to the exploratory kernel density analysis, relative distributional analysis also presents much clearer evidence of distributional polarization on the government guarantee issues scale as I exclude DKs from moderates. As shown in the Fig. 5, the overall RD effect evidences that relative density in distributional center comprising 4th through 6th deciles is dramatically reduced in 2008 compared to the distribution of 1984 while both extremes of distribution of the government guarantee issue scale—especially the liberal end of the distribution—gain significant amount of density during the period. The disappearing center appears to be even deeper as well as more apparent compared to the distributional analysis including DKs into moderates. Both the location and shape effects contribute together to the distributional changes for the period given that entire distribution of the primary scale shift to the left (more liberal side) while polarization is occurring on the distributional shape. Unlike the government guarantee scale however, the cultural issues scale does not exhibit significant differences in the RD analysis as I remove the DKs from moderates. In sum, both kernel density analysis and corresponding RD method indicate that growing polarization of mass ideological distribution over time are more clearly identified when the non-ideologicals (respondents who answered “haven’t thought much about it”) are handled separately from middle of roaders.

Fig. 5

Relative density of the two ideological issue scales (1980–2008)

Appendix 2: American National Election Studies (ANES) Questionnaires Used in the Study

As described in the main text, a group of indicators is adopted from a cumulative data file (1948-2008) to construct ideological scales used in this study. In this appendix, I describe each issue item used here and identify the years from which it was available.

< GOVERNMENT GUARANTEE ISSUES >

• - [VCF0803] R Placement: Liberal-Conservative Scale (available from 1972)

• - [VCF0806] R Placement: Government Health Insurance Scale (available from 1970)

• - [VCF0809] R Placement: Guaranteed Jobs and Income Scale (available from 1972)

• - [VCF0830] R Placement: Aid to Blacks Scale (available from 1970)

• - [VCF0839] R Placement: Government Services/Spending Scale (available from 1982)

• - [VCF0843] R Placement: Defense Spending Scale (available from 1980)

< CULTURAL ISSUE >

• - [VCF0834] R Placement: Women Equal Role Scale (available from 1972)

• - [VCF0837] R Opinion: When Should Abortion Be Allowed (available in 1972, 1976, and 1980)

• - [VCF0838] R Opinion: By Law, When Should Abortion Be Allowed (available from 1980)

• - [VCF0876a] R Opinion Strength: Law Against Homosexual Discrimination (available from 1988)

• - [VCF0851] R Opinion: Newer Lifestyles Contributes to Society Breakdown (available from 1986)

• - [VCF0854] R Opinion: Tolerance of Different Moral Standards (Available from 1986)

Appendix 3: Relative Distribution Method (Handcock and Morris 1999)

Formally, let Y ₀ be a random outcome with its CDF (cumulative distribution function) F ₀(Y ₀) in the reference population. Suppose we also observe another outcome variable Y from a different (comparison) population of which CDF is given as F(Y). In general, Y is the distribution for a different group or the same group measured in different time periods. The RD (or “relative rank”) of Y to Y ₀ is defined as the distribution of a random variable R = F ₀(Y) and this implies “the proportion of the reference population whose values are at most Y” (Handcock and Morris 1999, 21; Hao and Naiman 2010, 65–66). R is obtained from Y by transforming the value by the CDF for the reference population F ₀. The transformation is also referred to as the “grade transformation” and the resulting data, which is continuous on the outcome space [0,1] is called relative data. As a random variable, the relative data R (and its realization r) has a CDF and a corresponding PDF (probability density function). The CDF of R, G(r) is expressed as follows;

$$G\left( r \right) = P\left[ {R \le r} \right] = P\left[ {F_{0} \left( Y \right) \le r} \right] = P\left[ {Y \le F_{0}^{ - 1} \left( r \right)} \right] = P\left[ {Y \le Q_{0} \left( r \right)} \right] = F(Q_{0} \left( r \right))$$

where $r \in [0,1]$ is the proportion of values, and $Q_{0} \left( r \right) = F_{0}^{ - 1} (r)$ is the quantile function of $F_{0}$ (Handcock and Morris 1999, 22; Hao and Naiman 2010, 66). The corresponding PDF (or relative density of R) is obtained as the derivative of CDF, G(r);³³

$$g\left( r \right) = \frac{{f(Q_{0} \left( r \right))}}{{f_{0} (Q_{0} \left( r \right))}} = \frac{{f(y_{r} )}}{{f_{0} (y_{r} )}}$$

where y _r denotes the rth quantile of R (${\text{i}} . {\text{e}} . ,Q_{0} (r)$) using the original measurement scale of the reference population.

According to Handcock and Morris (1999, 22–26; see also Hao and Naiman 2010, 68–69), the RD method is an intuitively appealing strategy for comparing two distributions because the relative data, and its density function have clear interpretation. The PDF $g(r)$ can be considered as a density ratio: the ratio of the fraction of values in the comparison cohort to the fraction in the reference cohort at a given level of outcome values. So if $F_{0} (Y_{0} )$ and $F(Y)$ are almost identical distributions, the display of the relative PDF will converge to the uniform distribution$g(r) = 1$. On the other hand, if a greater frequency is observed at a certain decile (percentile) of interest in the comparison distribution, the relative density is greater than $g\left( r \right) = 1.$ In contrast, if less frequency is observed in $F(Y)$ compared to the frequency observations in $F_{0} (Y)$, it is less than the horizontal line $g\left( r \right) = 1$.

Distributional differences can be decomposed into location and shape shifts (Handcock and Morris 1999, 41–47). Formally, if we assume a hypothetical distribution $Y_{0L}$ which is constructed to have the same location (median) with a comparison cohort ($Y$) while the distribution maintains same shape with a reference group ($Y_{0} )$. This location-adjusted distribution $Y_{0L}$ can be represented as $Y_{0} - \delta$ for an additive median shift, where $\delta$ is the difference between the medians of Y and $Y_{0} (i.e., median\left( Y \right) - median\left( {Y_{0} } \right)$). The corresponding CDF and PDF of $Y_{0L}$ can be denoted as $F_{0L} \left( Y \right) = F_{0} (Y + \delta )$ and $f_{0L} = f_{0} (Y + \delta )$ respectively.

Using these three distributions—$f_{0} \left( {Y_{0} } \right), f\left( Y \right), and f_{0L} (Y)$—we can derive two relative densities (RDs) that account for the location and shape effects respectively. We define the RD of $Y to Y_{0}$ as $R_{0} \equiv F_{0} \left( Y \right),$ where $F_{0}$ is a CDF of the reference group $Y_{0}$. Hence, the first RD of $f_{0L} (Y)$ to $f_{0} (Y_{0} )$, or $R_{0}^{0L} = F_{0} \left( {Y_{0L} } \right) = F_{0} (Y_{0} - \delta )$ represents the location shift between the comparison and the reference group. This location effect will display the uniform distribution defined in [0,1] if the reference and the comparison distributions have the same median. In contrast, if they have different medians, the location shift will display monotonically increasing (or decreasing) graphs in r (relative data), if the comparison median is greater (or smaller) than the reference median. Second, the RD of $f(Y)$ to $f_{0L} (Y)$, or $R_{0L} = F_{0L} \left( Y \right) = F_{0} (Y + \delta )$ represents the shape shift isolated from the location effect between the two distributions. Thus, the shape effect will converge to the uniform distribution defined in [0,1] if the comparison distribution and the location-adjusted reference distribution have almost identical distributional shape. Moreover, this shape effect allows researchers to determine if distributional upgrading or downgrading—the shift of distributional mass to the upper or lower tails—occurred, or otherwise, if convergence toward the median happened in the distribution. Finally, these two RDs compose together the overall relative density $R_{0} = F_{0} (Y)$ between the comparison $f(y)$ and the reference $f_{0} (Y)$ distributions.³⁴

While graphical analysis is a key component of the RD method, readers might still want to have measures to summarize distributional differences or changes. To meet the needs, the RD method also provides summary measures which are robust to both outliers and to violations of parametric assumptions (Handcock and Morris 1999, Chapter 5).³⁵ The median relative polarization (MRP) index evaluates to what extent a comparison distribution is more polarized than a reference distribution (Hao and Naiman 2010, 86). The measure compares a reference and a comparison distribution at their tails to see if the former stretches wider (narrower) or has heavier (lighter) tails than the latter. As we have discussed above, assessing polarization is often relevant to a shape-effect, so locations are adjusted to isolate differences due to the distributional shape. Thus, MRP is defined in terms of the RD of the comparison distribution to the location-matched reference distribution, where the location is adjusted to equalize the median (or mean) between the two distributions (Handcock and Morris 1999, 70; Hao and Naiman 2010, 86).³⁶ Intuitively, the polarization statistic measures the deviations of the relative density from the uniform distribution, where the uniform relative density indicates that the reference and comparison distributions coincide. We defined above $R_{0L}$ as the RD between the comparison distribution ($Y$) and the location-matched reference distribution ($Y_{0L}$). That is, $R_{0L} = F_{0L} (Y) = F_{0} \left( {Y + \delta } \right)$ where $\delta = Q\left( \frac{1}{2} \right) - Q_{0} \left( \frac{1}{2} \right)$ is the difference between the medians of Y and $Y_{0}$. We then measure the mean absolute deviation from ½ and the relative polarization index is constructed using a linear transformation (four times the mean absolute deviation from ½ minus 1).³⁷ Namely, the median relative polarization index of Y relative $Y_{0}$ is defined as (Handcock and Morris 1999, 70–71);

$$MRP\left( {F;F_{0} } \right) = 4E\left[ {\left| {R_{0L} - \frac{1}{2}} \right|} \right] - 1 = 4\mathop \smallint \limits_{0}^{1} \left| {r - \frac{1}{2}} \right|g_{0L} \left( r \right)dr - 1$$

where $g_{0L}$ is the density of $R_{0L}$.

After a linear transformation, the index ranges from −1 to 1. The zero score means there are no differences in distributional shapes, the positive values of MRP (closer to 1) index implies that the comparison distribution is more polarized than the reference distribution, and the negative values (closer to −1) indicates the comparison is less polarized than the reference distribution.³⁸ Intuitively, the value of the MRP can be interpreted as a proportional shift of the population in the distribution from more central to less central locations. For example, a MRP value of 0.382 implies 38.2 % of the population shift from the middle of the distribution to the upper and lower percentiles. In contrast, a MRP of −0.287 means that 28.7 % of the population converges from the two tails toward the center of the distribution.