Individual Roll Rates
Thus far, we have tested hypotheses derived from the cartel agenda model regarding the aggregate roll rates of the majority and minority parties. In this chapter, we will derive and test hypotheses regarding individual members’ roll rates--that is, how often each member of the House votes against a bill that passes.
We begin by presenting some simple averages of individual roll rates by party and Congress, to see how majority status affects how often members suffer rolls. These analyses are similar in some ways to our previous investigations of party roll rates, although as will be seen the average roll rate within a party is not the same as the aggregate roll rate for that party.
We then take a closer look at variations of members’ roll rates within each Congress. In particular, we ask how roll rates vary with two factors: ideological location and majority status. This level of analysis gives us better leverage than we enjoyed when examining aggregate party roll rates, as we can explicitly control for members’ ideological locations when examining the influence of majority status.
To set the stage for our more detailed within-Congress analyses, we first derive predictions from our cartel model regarding how a member’s probability of being rolled should vary with that member’s ideological location; and then test those predictions against the empirical record of 60 post-Reconstruction Congresses. Our examination shows that the complete-information version of the cartel model does not fully fit the observed data. Nonetheless, when we introduce an element of uncertainty to our idealized cartel agenda model (specifically, by incorporating stochastic voting along the lines laid out in the previous chapter) the predictions of the cartel model are consistent with the evidence on individual roll rates.
We conclude by conducting a multivariate analysis in which a member’s roll rate is predicted as a function of (1) the member’s ideological location relative to the floor median and (2) whether the member’s party is in the majority. We find that the probability of a member suffering the passage of a bill against his or her wishes is significantly increased as that member’s ideological location becomes more extreme, relative to the floor median; and also when that member’s party is in the minority. That is, minority-party members are more likely to be rolled than are equally extreme majority-party members.
Some preliminary comparisons
To get an initial feel for how often members of the House are rolled, consider the following summary statistics from the 83rd-105th Congresses. On average, Democrats were rolled 11.4% of the time, when they held a majority, and 33.1% of the time, when they lacked a majority. Similarly, Republicans were rolled 7.7% of the time, when they held a majority, and 29.0% of the time, when they lacked a majority.
We present two graphs to show the data at a finer-grained, Congress-by-Congress, level. Recall that the only Houses in this period with Republican majorities were the very first in the series (the 83rd) and the last two (the 104th and 105th). With this in mind, look at Figure 11.1, which displays average roll rates by Congress for the Democrats. As can be seen, the three highest averages are achieved in the three Republican Houses. The drop from the 83rd (Republican-controlled) to the 84th (Democrat-controlled) House is relatively modest (10 percentage points); but the surge from the 103rd (Democrat-controlled) to the 104th (Republican-controlled) is more substantial (over 35 percentage points). In-between the initial and ending peaks, the Democratic averages show relatively little variation, with perhaps a mild decline from the 84th to 103rd Houses.
Figure 10.1 about here
Turning to the Republican averages displayed in Figure 10.2, one finds a similar but more nuanced picture. The three lowest average Republican roll rates occur in the three Houses that they controlled. In-between, there is an initial peak in the 86th-88th Congresses, at slightly below 40%; followed by a valley reaching bottom in the 91st-92nd Congresses, at slightly below 15%; followed by another peak near 40% in the 98th-103rd Congresses.
Figure 10.2 about here
These simple statistics seem consistent with the idealized cartel model, in that average roll rates are lower in the majority party. In the next few sections, we specify in greater detail what one should expect from the cartel model, when one looks in more detail at within-Congress variation in roll rates. We then turn to various empirical analyses of the within-Congress data that control for member preferences.
Individual rolls: Non-stochastic theory
In this section, we consider what the idealized, complete-information cartel model predicts about individual rolls. Recall that an individual member is rolled on a particular final passage vote if and only if s/he votes against the bill in question, yet it passes. We consider another sort of defeat--voting for a bill that fails--later in the book.
In the idealized cartel model, an individual’s roll rate is a simple function of his or her ideal point’s location, relative to the majority-party median (M) and floor median (F). To explain this point, we shall henceforth focus on the case of a Democratic majority (for which M < F).
The cartel model predicts that a member’s roll rate should be zero, if his or her ideal point lies between the Democratic median (M) and the floor median (F). Such members could be rolled if a bill, b, were voted against a status quo, q, such that the cutpoint c = (b+q)/2 lay between M and F. But the idealized cartel model specifically denies that such (bill, status quo) pairs are allowed on the floor, leading to the prediction that “moderate leftists” (those with ideal points between M and F) are never rolled.
Outside of this “no roll zone,” the cartel model predicts that a member’s roll rate will increase monotonically, the further he or she is from this zone. That is, the idealized cartel model predicts that the further left (or the further right) of the interval [M,F] is a member’s ideal point, the higher that member’s roll rate will be.
All told, the idealized cartel model’s predictions can be diagrammed as in Figure 10.3. As shown in the bottom portion of the figure, the left-right spectrum can be divided into three regions: the “far left” (to the left of M); the “moderate left” (between M and F); and the “right” (to the right of F). We will derive predictions from the idealized cartel model for each of these segments. 
Figure 10.3 about here.
The far left
Within this region, the idealized cartel model predicts that for members of either party, the further left a member’s ideal point lies, the larger his or her roll rate will be. In terms of a regression of an individual’s roll rate on his or her ideal point’s location, the theory predicts a negative slope in this region.
The moderate left
Members whose ideal points lie between M and F are protected from rolls in the idealized cartel model because no bills that would roll them are ever scheduled. Since all members in this region should have zero roll rates, there should be no relationship between ideal point location and roll rate for a member whose ideal point is located in this region. In terms of a regression of an individual’s roll rate on his or her ideal point’s location, the idealized cartel model predicts a zero slope in this region.
Roll rates should increase monotonically with ideal points, for members whose ideal points lie to the right of the floor median. In terms of a regression of an individual’s roll rate on his or her ideal point’s location, the idealized cartel model predicts a positive slope in this region.
Individual rolls: Evidence
In order to test the predictions just stated, we use our post-Reconstruction dataset that covers every final passage vote on House bills occurring in the 45th-105th Congresses. For each such vote, we code whether each voting member was rolled (that is, whether each voted against a bill that passed). We then compute each member’s roll rate in each Congress (denoted Ri for member i)--that is, the number of times the member was rolled, divided by the number of final passage votes in which he or she participated.
The main independent variable in the analysis is each member’s ideological location, xi. Operationally, we use the first dimension of each member’s DW-Nominate score (Poole 1998) to measure xi.
In Figure 10.4, we plot each member’s roll rate (on the vertical axis) against each member’s ideological location (on the horizontal axis), for three Republican Congresses (the 83rd, 104th and 105th) and three Democratic Congresses (the 86th, 92nd, and 96th). These Congresses were selected to illustrate the range of results we have obtained. A complete set of figures, for all 61 Congresses, is posted at
Figure 10.4 about here
Along with the raw data, each graph also displays a lowess regression line (a kind of running average) and the location of the majority-party median (M) and the floor median (F). The lowess lines in all Democratic Congresses (see Figures 10.4a-10.4c) look roughly like the Nike swoosh: starting at the far left, the curve first declines briefly, then reverses direction and proceeds upward. The members with the lowest roll rates are consistently slightly to the left of the Democratic median. In stark contrast, the lowess lines in all Republican Congresses (Figures 10.4d-10.4f and those posted on the web) look roughly like mirror-image Nike swooshes: starting at the far right, the curve first declines briefly, then reverses direction and proceeds upward. The members with the lowest roll rates are consistently slightly to the right of the Republican median.
Interestingly, the floor median has a roll rate above 12% in the Congresses pictured, and this is typical of all the Congresses studied. The members with the lowest roll rate are, as already noted, slightly more extreme than the majority-party median. When it comes to avoiding the passage of bills that propose unwanted policies, those near the floor median do not do the best; rather, those near (and more extreme than) the majority-party median do the best.
In order to explore the relationship between ideological location and roll rate more systematically, we run separate regressions of Ri on xi for each of the three regions noted above: the far left, the moderate left, and the right. We include all Democratic Congresses in each regression, allowing the constant term to shift (via the inclusion of dummy variables representing each Congress, excluding the 84th) but pooling the slope coefficient on xi. The results are presented in Table 10.1.
Table 10.1 about here
Our findings can be summarized as follows. In the far left and in the right, more extreme members should be rolled more often, and that is what we find. In the moderate left, the idealized cartel model predicts a nil slope but in fact the data show a significant positive slope.
Because it fails in the moderate left region, we reject the complete-information version of our model. In the remainder of the chapter, however, we use the more realistic incomplete-information version of the cartel model, developed in Chapter 10, and show how it fits the data displayed in Figure 10.4. In particular, we offer a partial explanation for why the lowest roll rate is observed for members somewhat to the left of the Democratic median, with roll rates increasing monotonically on either side of that minimum.
Individual roll rates
In this section, we consider how the roll rates of individuals change as their ideal points change, given that all members vote according to the stochastic model outlined in Chapter 10 (which itself is a version of the standard model proposed in Poole and Rosenthal 1997, among others). We begin by considering roll probabilities on a single vote, then aggregate across all the final passage votes held in a session.
Vote-specific roll probabilities
In order to derive conclusions about the relationship between a legislator’s ideal point and his/her roll rate, we assume for convenience that si = s for all i. If si = 0, then there are no additional considerations and we have the complete-information model with which we began in chapter 4. Here, however, we assume si > 0 throughout: members do not vote perfectly in accord with their ideal points; there are some “errors” or “additional considerations.”
Consider a particular policy dimension, j. Suppose that the status quo is qj and that a bill, bj, is being voted against the status quo in a final passage vote. What is the probability that member i will be rolled on this vote, denoted Rij(bj,qj) (sometimes shortened to Rij)?
In the appendix weshow that, when the bill is to the left of the status quo (bj < qj), the probability of a member being rolled on final passage increases as the member becomes more conservative (as his or her ideal point shifts rightward). The basic reason is as follows. The probability Rij(bj,qj) that a member i will be rolled on a particular final passage vote (pitting bj against qj) is the probability that member i votes against the bill, times the conditional probability that the bill passes, given that member i votes against it. This latter probability is unaffected by changes in the location of the members’ ideal point, xi. Thus, whether increasing xi increases i’s probability of being rolled depends on how increasing xi affects the probability that i will vote against the bill. But more conservative members are more likely to vote against the (left-of-status-quo) bill, hence they are more likely to be rolled.
On the other hand, when the bill is to the right of the status quo (qj < bj), the probability of a member being rolled on final passage decreases as the member becomes more conservative. The more conservative members are less likely to vote against the (right-of-status-quo) bill, hence they are less likely to be rolled.
Thus far we have imagined a single pair of alternatives (bj,qj) being voted on. In this section, we consider how member i’s roll rate across the roll calls held in a given session, denoted Ri(xi), changes with member i’s ideal point, xi. More formally, we consider the Ri(xi) curve’s slope, .
Our main point is that, in the stochastic version of the model, how Ri changes with xi depends heavily on the proportion of bills that are to the left of their respective status quo points. If all (bj,qj) pairs were such that bj < qj, then we know, from the results stated above, that > 0. In other words, if every voted-on bill proposes a leftward movement in policy, then roll rates will increase monotonically with conservatism. Alternatively, if every voted-on bill proposes a rightward movement in policy, then roll rates will decrease monotonically with conservatism.
More generally, depends on the proportion of bills that propose leftward moves, PLeft, as follows (see the appendix for a formal derivation):
Here, rbq represents how Ri changes with xi, on average, across the bills that are to the left of the status quo; while rqb represents how Ri changes with xi, on average, across the bills that are to the right of the status quo. Holding constant rbq and rqb, increasing PLeft increases the slope of the Ri(xi) curve. That is, the larger the proportion of bills that propose leftward policy moves, the more likely it is that roll rates will increase with conservatism.
Roll rates redux
In this section, we return to the relationship between members’ roll rates and their ideological locations, as displayed in Figure 10.4. Having developed the stochastic version of our model, we can now partly explain why the lowess lines deviate in the way they do from the predictions of the idealized complete-information cartel model. If members voted purely according to the distance between the proposals facing them and their respective ideal points, without the non-spatial errors or “additional considerations” introduced above, then the cartel model would indeed make the prediction given in Figure 10.3.
When members’ preferences are not purely stochastic, several things happen to predictions of our idealized model. First, the members with ideal points between the majority-party (M) and floor (F) medians are no longer expected to have zero roll rates: they have a positive probability of being rolled on any final-passage vote, although that outcome is unlikely when spatial errors are small. Second, as the proportion of final-passage bills that move policy leftward increases, the minimum point of the R(x) curve shifts to the left (see the appendix). Third, one expects the R(x) curve to increase monotonically on either side of its minimum.
Thus, given (1) moderate variances in the errors or “additional considerations” (i.e., moderate values of (s1,…sT)); and (2) a heavy preponderance of moves toward the majority party; one expects something similar to what all the lowess graphs in fact show. In particular, the minimum of the R(x) curve is more extreme than the majority-party median (because of the heavy preponderance of bills proposing to move policy toward the majority party) and the curve bends upward on either side of its minimum. In particular, you see a positive slope for the R(x) curve to the right of M, in between M and F. The addition of stochastic voting thus accounts for the prediction errors in our idealized model.
A multivariate test