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. 1
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
. Our analysis resembles that of Lawrence, Maltzman and Smith (1999), who plot each member’s win rate (the
proportion of all votes on which a member is on the winning side) against his or her left-right scale position. We,
however, look not at all votes but rather only at final-passage votes; and we examine roll rates rather than win rates.
Roll rates are not simply the inverse of win rates, as they identify one particular class of defeat—voting against a bill
that nonetheless passes—and leave another sort of defeat—voting for a bill that fails—explicitly out of the picture.
1
(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.
Basics
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. 2
2
. The size of the slope (and the extent to which the relationship is linear) both depend on the distribution of status
quo points.
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.
The right
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.3 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 www.mmccubbins.com/revisited.htm.
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
3
. In this analysis, we include final passage votes on bills in the H.R. series only.
proceeds upward.4 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.5
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
4
. The other figures from Democratic Congresses posted at www.mmccubbins.com/revisited.htm also look like Nike
swooshes, although the 84th is a bit flat on the left.
5
. One can also run these analyses with common-space Nominate scores, in which case it is possible to locate the
filibuster pivots in the Senate and translate them into the House distribution of ideal points. When one does this, one
finds that the right pivot has a roll rate of 15-35% in the Democratic Congresses; while the left pivot has a roll rate
of 15-40% in the Republican Congresses. Similar findings arise if one defines the left and right pivots simply as the
40th and 60th percentiles of the House distribution.
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 incompleteinformation 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.6 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-ofstatus-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.
Roll rates
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),
6
. Recall that members’ errors are independent.
changes with member i’s ideal point, xi. More formally, we consider the Ri(xi) curve’s slope,
Ri
.
xi
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
Ri
>
xi
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,
Ri
depends on the proportion of bills that propose leftward moves,
xi
PLeft, as follows (see the appendix for a formal derivation):
Ri
= PLeft rbq  1  PLeft rqb .
xi
(1)
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 majorityparty (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
In this section, we examine a somewhat different proposition—viz., that majority status
lowers members’ roll rates, holding constant ideological location. Under the cartel agenda
model, members of the majority party are protected from certain votes--those which would roll at
least half of them--ever taking place. Under the floor agenda model, once one controls for each
member’s ideological location, majority status should not matter.7
To test this proposition, we regress each member’s roll rate on a constant, a variable
measuring the distance between the member’s ideal point and the floor median, and a variable
indicating that the member’s party is in the minority. As our data cover over a century of
congressional history, during which time there have been substantial changes in the majority
party’s advantage in setting the agenda, we allow the effect of our three variables to adjust when:
(1) Reed’s rules were adopted (1890); (2) the revolt against “czar rule” occurred (1910); (3) the
Rules Committee was packed (1961); and (4) the House was reformed (1974). If our previous
claims asserting the primacy of Reed’s rules are correct (see chapter 4), then the floor agenda
model may fit the pre-Reed data reasonably well—but it will not fit any of the post-Reed
periods.
Our results, presented Table 10.2, can be characterized as follows.8 Members whose
ideal points are further from the floor median are more likely to be rolled in all periods of
congressional history we examine. Controlling for each member’s distance from the floor
median, however, there are (1) significant differences between majority-party and minority-party
members in every post-Reed period; and (2) significant changes in members’ roll rates
associated with each major watershed in House procedure.
. From the appendix, the probability that member i votes for the left-of-cutpoint alternative is pij = (dj(cj-xi)/si).
Fixing each member’s ideal point (and standard error), the set of p ij’s for a given roll call, j, is a function solely of
the roll call parameters, dj and cj, or equivalently, bj (the bill location) and qj (the status quo location). Under the
floor agenda model, bj = Fj, the floor median on the jth dimension, while q j is drawn by Nature from a fixed
distribution. Thus, each member’s roll rate is purely a function of the distribution of status quo points. As this
distribution does not depend on which party is currently in the majority, majority status should be irrelevant.
8
. As in previous chapters, we use an extended beta-binomial procedure to estimate the model.
7
Table 10.2 about here.
Consider the difference between majority- and minority-party members first. Prior to the
adoption of Reed’s rules in 1890, there was no difference. Each member’s roll rate was a
function of how extreme that member was, relative to the floor median—but whether they were
in the majority or minority party did not matter. After passage of Reed’s rules, however, a large
difference opens up between the parties: specifically, a minority-party member is substantially
more likely to be rolled than a majority-party member equally distant from the floor median.
While this minority-party disadvantage shifts after the revolt against czar rule in 1910, again
after the packing of the Rules Committee in 1961, and yet again after the reform of the House in
1974, it remains large and statistically significant in all post-Reed periods.
If minority-party members have higher roll rates on final-passage votes because of the
agenda power exercised by the majority party, as envisioned by the cartel model, then their
disadvantage should grow when congressional reforms bolster the power of the majority on the
margin. Three events since Reconstruction stand out as obvious points at which the majority’s
grip on the agenda may have strengthened: the adoption of Reed’s rules, the packing of the
Rules Committee, and the adoption of broader reforms in the 94th Congress. In addition, the
revolt against czar rule in 1910 is conventionally depicted as diminishing the power of the
majority party’s central leadership. Thus, an additional test of the cartel model is provided by
the interaction terms in Table 10.2, which allow the effect of minority status to shift after each of
the four events just noted. The cartel model predicts that the discrepancy in roll rates between
members of the minority and majority party should widen after the three majority-strengthening
rule changes but shrink after the majority-weakening change in 1910.
This is indeed what we find (see Table 10.2). By far the largest increase in the minorityparty disadvantage occurs in 1890, with the adoption of Reed’s rules; smaller but still
statistically significant increases occur in 1961, with the packing of the Rules Committee, and
1974, with the adoption of a series of reforms. Between these majority-strengthening moves lies
the important revolt of 1910, sparking a substantial shrinkage of the minority-party disadvantage
(although the disadvantage remains large and positive).
These results appear hard to explain from a perspective that takes majority-party agenda
power as nil, weak or tenuous. While one might seek to explain the finding that minority-party
members have higher roll rates in a given period by appealing to an unusual distribution of status
quo points in that Congress, there is no reason to believe that the distribution of status quo points
was unfavorable to the minority in every period, nor that the distribution would become less
favorable after each majority-strengthening reform and more favorable after the main majorityweakening reform.
Conclusion
In this chapter, we have investigated how individual members’ roll rates vary with their
ideological location. The basic empirical pattern that we find, in all Congresses, is the following.
First, the members with the lowest roll rates are slightly more extreme than the majority-party
median. That is, in Democratic Houses, members slightly to the left of the Democratic median
are the least often rolled; in Republican Houses, members slightly to the right of the Republican
median are the least often rolled. Second, to either side of this minimum point, members’ roll
rates increase monotonically.
Theoretically, this pattern is inconsistent with a complete-information cartel model.
When one recalls that members’ ideological locations are only estimates, and that voting
behavior exhibits some variation around the central tendencies reflected in estimated ideal
points, the data become intelligible from a cartel viewpoint. As we show above, with stochastic
voting, members’ roll rates are highly sensitive to the proportion of bills that propose to move
policy leftward (or rightward). As we have already seen in Chapter 10, the proportion of leftward
moves is in turn a direct function of which party has the gavel. Putting these two observations
together, one can provide the following explanation for the empirical patterns in a typical
Democratic House. Most of the bills that reach final passage propose to move policy leftward
(typically about 80%). On these bills, more conservative members should have higher roll rates.
On the 20% of bills proposing to move policy rightward, in contrast, more liberal members
should have higher roll rates. Averaging across a large set of leftward moves and a small set of
rightward moves produces the patterns shown in Figure 10.4.
Table 10.1:
The relationship between member ideology and roll rate
Post-Reconstruction roll call data set (45th to 105th Congress)
Region name
Cartel
Estimated slope
Std error
prediction
Far Left
Negative
-.212
.008
Moderate Left
Zero
.225
.017
Right
Positive
.550
.005
Notes: The dependent variable is each member’s roll rate. Estimation by *. All coefficients
significantly differ from zero at the .001 level.
Table 10.2
Pre-Reed
Constant
Distance of
member
from floor
median
Member is
in minority
party
-1.88
1.73
Interactions
Post-Reed
-0.35
-0.73
Interactions
Post-Revolt
-0.07 ns
0.48
Interactions
Post-packing
-0.27
-0.04 ns
Interactions
Post-reform
-0.33
0.95
-0.03 ns
1.73
-0.88
0.17
0.09
Notes: The dependent variable is each member’s roll rate in a given Congress. There were
25,142 observations.
Figure 10.1
Average Democratic Roll Rate, 83rd through 105th Congresses
.5
R
R
Average
Democratic
Roll Rate
R
D
D
D
D
D
D D
D
D
D
D
D
D
D
D
D
D
D
D D
0
83
85
87
89
91
93
95
97
99
101 103 105
Congress
Note: The symbol "R" indicates a Congress in which the Republicans held a majority of House
seats, while the symbol "D" indicates a Congress in which the Democrats held a majority.
Figure 10.2
Average Republican Roll Rate, 83rd through 105th Congresses
.5
D
D
D
D
D
D
D
D
D
D
D
Average
Republican
Roll Rate
D
D
D
D
D
D D
D D
R
R R
0
83
85
87
89
91
93
95
97
99
101 103 105
Congress
Note: The symbol "R" indicates a Congress in which the Republicans held a majority of House
seats, while the symbol "D" indicates a Congress in which the Democrats held a majority.
Figure 10.3: Expected relationship between ideological location and roll rate under the
cartel model
Predicted
Roll
Rate
M
F
Figure 10.4.a. How members’ roll rates vary with their left-right ideological position. Congress
86.
Figure 10.4.b. How members’ roll rates vary with their left-right ideological position. Congress
92.
Figure 10.4.c. How members’ roll rates vary with their left-right ideological position. Congress
96.
Figure 10.4.d. How members’ roll rates vary with their left-right
ideological position. Congress 83.
Figure 10.4.e. How members’ roll rates vary with their left-right ideological position. Congress
104.
Figure 10.4.f. How members’ roll rates vary with their left-right ideological position. Congress
105.