# What does a DW-NOMINATE score of zero mean?

In an earlier post, I talked about the asymmetrical polarization of US congress, and how DW-NOMINATE is used to quantify it. In this post, I’m going to discuss more technical details.  I’ll explain why I was initially skeptical, and why I came to accept the argument.

But before I get into the math, I should first emphasize that there are many arguments demonstrating that the US congress has become more polarized, and that the Republican party in particular is more extreme. I think those arguments stand on their own, with or without using any evidence from DW-NOMINATE. You can read some of those arguments here, or watch the Vox video I linked last time. It’s not just liberals who are saying this–one of the big proponents of asymmetrical polarization theory is Norman Ornstein, member of the conservative think tank American Enterprise Institute;

#### Translation symmetry

So here’s the issue I had with DW-NOMINATE scores. As I read in the primer, DW-NOMINATE is based on a maximum likelihood estimate of the coordinates of congress members, as well as the coordinates of “yay” and “nay” votes for each roll call. The problem is that this calculation has what physicists call “translation symmetry”. That means that the calculation remains the same if you shift all coordinates in a single direction by the same amount. A secondary problem is that the calculation has “scaling symmetry”, meaning that the calculation remains the same if you multiple all the coordinates by the same number.

If the DW-NOMINATE calculations truly have translation symmetry and scaling symmetry, then absolute positions and distances don’t mean anything. So we can use DW-NOMINATE to say:

Most Republicans are to the right of Democrats, and the overlap between the parties has dwindled to nothing.

Republicans are much farther away from Democrats than they are from each other.

But we can’t use DW-NOMINATE to say:

Republicans are far to the right on an absolute scale

Republicans are far away from Democrats.

Scaling symmetry problematizes the claim of polarization, because it seems you can just scale down the coordinates, and make polarization disappear. Translation symmetry problematizes the claim of asymmetrical polarization, because it seems you could just translate the coordinates until they are no longer asymmetrical.

The crux of the matter is this: What does a DW-NOMINATE score of zero mean? What does a score of one mean? If we establish a concrete meaning for a score of zero, then we can say “Republicans are moving away from zero, and doing so more quickly than the Democrats.”.

I had a hard time finding any source that would answer my question, because it’s a question that few people think to ask. But I eventually found a satisfactory answer that I will now share.

#### Moving towards historical extremes

There are some members of congress throughout history that have voted in such a consistently partisan way, that the maximum likelihood calculation causes their coordinates to blow up (i.e. run off to infinity). The NOMINATE calculation needs some way to handle these cases. Without getting into technical details (which are discussed in this paper), the algorithm sets the coordinates of those congress members to have a magnitude of 1.

Basically, a score of 1 corresponds to the most conservative members of congress, and a score of -1 corresponds to the most liberal members of congress. A score of zero is the position that bisects the two extremes.

There’s a little more to it than that. A score of 1 doesn’t necessarily mean the most conservative members of congress right now. It refers to the historical extremes. So if Republicans are moving closer to a score of 1, that means that the party as a whole is moving closer to the most extremely conservative members they’ve had throughout history.

We can compare current members of congress to members of congress past by looking at overlapping members. As explained by the voteview blog:

A sports analogy to the overlapping cohorts method is the “common opponents” statistic. If we want to compare two teams who have not played each other, we compare their performances against a common opponent(s).

To give an example, Barrack Obama and Elizabeth Warren never served on the senate at the same time. However, both of them overlapped with Bernie Sanders. We can tell Sanders is to the left of Obama, and Warren is to the left of Sanders, so Warren appears to the left of Obama. We can extend this method to compare any two senators throughout history as long as there is a continuous chain of senators.

In the NOMINATE calculation, members of congress are assumed to be in fixed positions, which makes it easy. In a DW-NOMINATE calculation, members of congress are allowed to shift over time. However, the shift over time is usually quite small, and the average velocity of individual members of congress is assumed to be zero. Asymmetrical polarization appears whether we use NOMINATE or DW-NOMINATE.1

#### Shifting by replacement

To be honest, the meaning of zero as “the bisector of historical extremes” is not that meaningful to me. What strikes me as more meaningful, is the concept of zero velocity. Zero velocity is “the average velocity of individual congress members”. Thus, if the Republicans have a positive average velocity, that means it is not driven by individual members of congress. Rather, Republican members of congress are being replaced, and the replacements are further to the right than the people they are replacing.

Every political science source I’ve read agrees why Democrats have moved further left in the past few decades. It’s because of the departure of Southern Democrats. What the NOMINATE scores show, is that Republicans have shown a similar trend of departing moderates and incoming extremists, albeit at a faster rate than the Democrats.

There is of course, one alternative interpretation. Perhaps individual members of the Republican party are all becoming more liberal as they grow older (relative to individual Democrats), and Republican voters have to keep on replacing their politicians in order to maintain the same political position. But this seems highly implausible.

Now that I better understand the math behind asymmetrical polarization, I find it hard to deny. Asymmetrical polarization means that Republicans are quickly being replaced by more conservative candidates over time. Asymmetrical polarization is real. So what will you do about it?

1. I also looked at a paper that discussed an alternative to NOMINATE, using a Bayesian approach. Surprisingly, in that model, the asymmetrical polarization appears to be in the opposite direction, with Democrats appearing further to the left than Republicans appear to the right. But the interpretation is very different in this model, and the authors say so. In this model, zero is the peak of the prior probability distribution. The fact that Democrats appear further from zero means Democrats’ votes make it easy to discriminate among Democrats, while Republicans’ votes make it hard to discriminate among Republicans. In other words, Republicans vote more cohesively, while Democrats have a more diverse range of preferences. (return)