As we wrote yesterday evening, the ever-increasing number of challenged ballots in Minnesota is making it more and more difficult to determine the extent to which Al Franken is in fact gaining ground in the state’s recount process. An analysis of precinct-by-precinct returns available on the Secretary of State’s website, however, suggests that Franken’s position is somewhat stronger than it appears, and that he may in fact be the favorite to prevail in the recount process.

Consider the following. In precincts where no challenges have been issued (these are the only precincts in which, in some sense, the results of the recount can be considered to be final and “official”) Franken has gained a total of 34 votes, and Coleman a total of 6 votes, for a net gain by Franken of 28 votes. Moreover, in precincts where just 1 challenge has been issued, Franken has gained a net of 31 votes on Coleman, and in precincts where exactly 2 challenges have been issued, Franken has gained a net of 32 votes on Coleman.

By contrast, in precincts where 5 or more ballots have been challenged between the two campaigns, Coleman has gained a net of 57 votes on Franken.

In other words, the fewer the number of challenged ballots, the better Franken is doing, and the higher the number of challenged ballots, the worse he is doing; the relationship is in fact quite strong.

Precinct-Level Returns Analysis

# Challenges n Franken Coleman Net

0 2233 +34 +6 Franken +28

1 419 -94 -125 Franken +31

2 154 -90 -122 Franken +32

3-4 133 -157 -171 Franken +14

5-9 59 -158 -116 Coleman -42

10+ 26 -156 -141 Coleman -15

It is not an accident, then, that as the number of challenges has increased with each day of the recount, Franken’s momentum appears to have stalled out. Very probably, a majority of the challenges are coming from Franken’s pile. This is somewhat irrespective of which campaign actually instigates the challenge, since as we suggested yesterday, a potential Franken undervote could be the subject of a challenge from either campaign depending on the initial ruling of the local elections judge.

We can address this phenomenon more systematically by means of a regression analysis. In the regression, we are attempting to predict a variable I’ve defined as franken_net, which is the net gain by Franken per 10,000 ballots cast in that precinct. The independent variables considered in the regression are as follows:

t: the proportion of the two-way vote received by Franken in the initial count (e.g. excluding votes for third parties)

c_f: the number of challenges initiated by the Franken campaign per 10,000 ballots counted in that precinct

c_c: the number of challenges initiated by the Coleman campaign per 10,000 ballots counted in that precinct

In addition, the regression analysis contains interaction terms between each combination of two variables, as well as an interaction term for all three variables, all of which are statistically significant. The regression is weighted by the square root of the number of ballots cast in that precinct.

The results of the regression are as follows:

franken_net Coef. t P>|t|

t 8.922 2.89 0.004

c_f -0.280 -3.99 0.000

c_c -0.926 -9.82 0.000

t * c_f -0.703 -8.59 0.000

t * c_c +0.565 2.89 0.004

c_f * c_c -0.013 -4.29 0.000

t * c_f * c_c +0.012 2.81 0.005

_constant -3.622 -2.36 0.019

This regression is a bit difficult to interpret, particularly with the presence of all the interaction terms, but the key intuition is as follows. Suppose that the number of challenges is zero — as will happen once the state canvassing board finishes considering all such challenges in December. In this case, all terms in the regression equation reduce to zero, except for the constant term and t, which is Franken’s share of the two-way vote in that precinct. We are thus left with the following:

franken_net = t * 8.922 – 3.622

Now, we can attempt to solve this equation at the statewide level. When we plug in a t of .499956 — Franken was picked on just slightly very less than half of the ballots during the initial count — we get a value for franken_net of .837. That is, Franken will gain a net of .837 votes for every 10,000 cast. With a total of 2,885,555 ballots having been recorded in the initial count, this works out to a projected gain of 242 votes for Franken statewide. Since Norm Coleman led by 215 votes in the initial count, this suggests that Franken will win by 27 votes once the recount process is complete (including specifically the adjudication of all challenged ballots).

The error bars on this regression analysis are fairly high, and so even if you buy my analysis, you should not regard Franken as more than a very slight favorite. Nevertheless, there is good reason to believe that the high rate of ballot challenges is in fact hurting Franken disproportionately, and that once such challenges are resolved, Franken stands to gain ground, perhaps enough to let him overtake Coleman.

(Note: it is also possible to build a multivariate regression model that attempts to solve for both Franken and Coleman’s totals in an absolute sense, rather than Franken’s gain relative to Coleman. This multivariate model produces a slightly more optimistic result for Franken, suggesting that he will gain 254 votes statewide and Coleman will lose 12, producing a net swing of 268 votes toward Franken.)