On The Spatial Model Of Public Opinion, Part One: Exploring Graphical Options
Last year I wrote an article about this popular graphic purporting to show “two dimensions” of public opinion:
Ever since then, I have been thinking a lot about these dimensional or spatial models of public opinion, and about different ways I might be able to visualize and understand instances of them. The spatial model keeps showing up, after all. The popularity of “political compass memes” is merely another example:
What Is The Spatial Model?:
The spatial or dimensional model of voting and public opinion is as follows: People can be modeled as points on a line or in an n-dimensional space, with different issues or bills or questions modeled as lines or planes that separate them.
For example, under the one-dimensional model, I should be able to line up all the respondents in a consistent way, so I can separate people by either their (binary) choice of favorite color:
Or by their (binary) choice of favorite ice cream:
Why A Graphical Approach?:
I have no mathematical or substantive issue with algorithms like “DW-Nominate” or “ideal”. It’s more about a general principle. In general, I would always prefer to have a data visualization, based on fairly simple computations, that I can check before the algorithm is run, not after. For comparison, before I run a linear regression, I might make a scatterplot of the independent and main dependent variables, to make sure linear regression is even the right approach.
In other words, I’d like to make sure that I can spot the gorilla — that I can inspect and explore the structure of the data and see what stands out — before doing any advanced math:
I think this is also important for persuasion. When DW-Nominate or ideal points are written about, people are often skeptical, and I think that’s because the algorithms seem like a black box, with the raw data driving the results invisible.
The One-Dimensional Case:
I will begin with the simplest case — a one-dimensional spatial model. Recall that the spatial model assigns people to points and divides them by questions or topics. (In general for public opinion and voting data, I will say that questions have “left-wing answers” or “right-wing answers”, although I’m not going to get into which one is chocolate and which one is vanilla.)
Consider the favorite-color-and-ice-cream example above. Note that, considering both lines, everyone is in one of three groups:
Everyone either likes chocolate and green (because they’re to the left of both lines), or they like vanilla and green (because they’re in between the two lines), or they like vanilla and yellow (because they’re to the right of both lines). There’s no way for people who like chocolate and yellow to fit into this model.
In other words: in a purely one-dimensional spatial model, for any Q1 and Q2, at most three out of the four possible combinations of (yes/no) on (Q1/Q2) will appear in the responses.
This mathematical fact has a clear visual interpretation if you plot the data in a particular way. I wrote a simple simulated “one-dimensional case”. For any question, I divided respondents into those who gave a “left-wing answer” and a “right-wing answer”, and compared what percentage of them gave the “left-wing answer” for every other question:
For example, 100% of the people who gave a left-wing answer to Q15 also gave a left-wing answer to each of Q13, Q9, Q12, Q2, and Q11, although they were split on the others, with only about 40% giving a left-wing answer to Q14. And 100% of the people who gave a right-wing answer to Q8 gave right-wing answers to each of Q14, Q16, Q7, Q10, etc, although they were split on the others, with about 15% giving a left-wing answer to Q1.
In other words, when I split by Q15,“easier” questions show up “on the y-axis”, since everyone who gave a left-wing answer to Q15 necessarily gave left-wing answers to those questions. And “harder” questions show up “on the x-axis”, since everyone who gave a right-wing answer to Q15 gave right-wing answers to those questions. Either way, in a one-dimensional model, whenever I split by a question, all the other questions show up “on the axes”, since only three of the four possible crosstabs are populated.
As a result, I can make the above plot for every question, and that same L-shaped pattern is evident:
Note that I could make this plot for a real-world data set, not just a simulated one. The closer a plot from a real-world data set would look to this simulated case, with every question ending up on the axes in the same order, the closer the real-world data set probably is to fitting a one-dimensional model, like the simulated case.
The (Simpler) Two-Dimensional Case:
The graphic applied to a pure one-dimensional case is fairly straightforward. What about a pure two-dimensional case?
The simplest two-dimensional case is when every question splits respondents entirely by one dimension or the other. In other words, each respondent is a point in two-dimensional space, but questions can only split them “horizontally” (call those questions “Topic 1”) or “vertically” (“Topic 2”).
Recall the earlier example, and say that favorite ice cream and favorite color are “Topic 1”, splitting respondents “horizontally”:
While some other question, maybe “window seat or aisle seat?”, divides the same respondents vertically:
(You can think of this as “two orthogonal copies of the one-dimensional case” instead of the “full two-dimensional case”, where there might also be questions corresponding to diagonal lines splitting the respondents, but that’s a little more complicated.)
I ran another basic simulation of the simpler two-dimensional case and made the above chart again for each question:
Note that, for questions in the same “topic” as the splitting question, it’s the same as in the one-dimensional case — they end up on the axes.
What about for questions in the other “topic” than the splitting question? They end up on the diagonal!
And note as well that, if I didn’t know it ahead of time, then this visualization would reveal which questions corresponded to which topics/dimensions.
These charts can therefore be interpreted by that rule of thumb: Compared to the splitting question, questions from the same dimension end up on the axes, questions from orthogonal dimensions end up on the diagonal, questions from dimensions that are somewhere in between end up somewhere in between.
Conclusion:
Can this graphic approach reveal interesting information about spatial structure in a “real-world” dataset? In the next part of this series, I’ll apply this graphic approach to a “real-world” dataset widely considered to be one of the best applications for spatial models: Congressional roll call voting.
