As you know, my forthcoming thesis is on whether a minority party in a legislative district ought to use the single-shot strategy when it knows it cannot win both House seats (the answer is yes). The next question is: once the minority party has a House seat, should it run that incumbent by themself or with a running-mate. I should note that the following analysis includes candidates elected by both strategies, I'm only focusing on how they run for reelection.
There are good reasons to suspect that my model (sorry, you have to wait for the thesis, I'm not going to explain it here) should not fit this case, most notably the obvious differences between running as an incumbent and running as a challenger. In short, my model says that if you run a single-shot: 1) you will unite all Democratic voters behind your one candidate (whereas with two some might only vote once and for the weaker one); 2) you will give intentional ticket-splitters only one Democratic option to unite behind (whereas they also may have voted instead for the weaker of two Democrats); 3) enough Republicans will only vote once that you can steal a seat. At the same time, I acknowledge that there may be Democratic-leaning voters that want to use both of their votes, and they will vote for on of the Republicans. The statistics simply show that the positive effects outweigh that one negative.
An incumbent shouldn't worry as much about #1 and #2, because as the better-known of the two Democrats, they should get the large majority of the votes from people who only vote for one Democrat. Thus, the positive effects of a single-shot might be neglible when the minority party has an incumbent, while the negative effect (Democratic voters voting for a Republican in order to use both votes), is still felt.
As it turns out, history provides ambivalent evidence. I went back as far as 1994 to find cases of minority party challengers winning House seats. I found 11 that went on to run for reelection. I didn't count Jack Brown's return to the House in 2004, because he jumped down from the Senate. Mark Thompson was excluded as well, because in his 2004 reelection campaign he was not the strongest candidate from his own party (Knaperek was).
That only leaves 9 incumbents who ran for reelection, and only two who ran for a third term. Both of those who ran without running-mates lost. Some of the 7 who ran for reelection with running-mates did win (3), but the majority did not (4). That is perhaps an indication that a running-mate helps, but there simply is not enough data for meaningful analysis. Only 9 data points hardly tells us anything. And perhaps it should only be 8. I included Meg Burton Cahill because when she first ran in 2000, LD27 was probably still a Republican district. By 2002 and especially 2004, it may not have been. Cahill and Jennifer Burns are the two incumbents who ran for a third term. Both ran with running-mates and both won, but this likely has more to do with 4 years of incumbency than with their running-mates.
UPDATE: I came up with a rough measure of party strength in each of these districts, averaging Senate and House results over a few elections. The method is not too important as I don't mean this to be definitive. But, the correlation between party strength and reelection is MUCH clearer than any correlation with running-mates.
The following chart shows the year of a candidates original election (and district), rough measure of the partisan breakdown of their district, and whether they won reelection or not. Camarot and Poelstra at the bottom are in italics because they are the only two than ran without running-mates. This may be a case of adverse selection. That is, potential running-mates would be far less likely to run in a district where they understand that their party is a small minority than in a more closely divided district.
2000 (27) Meg Burton Cahill (D) D 50%-50% Won
1996 (8) Gail Griffin (R) D 52%-48% Won
2002 (25) Jennifer Burns (R) D 53%-47% Won
2006 (10) Jackie Thrasher (D) R 54%-45% Lost
2004 (17) Laura Knaperek (R) D 55%-43% Lost
1994 (20) Robert Blendu (R) D 56%-41% Lost
1994 (25) Robert Updike (R) D 56%-39% Lost
2000 (1) Henry Camarot (D) R 60%-40% Lost
2000 (14) Ed Poelstra (R) No measure Lost
Next, I will work on that measure of party strength to provide estimates for our incumbents running for reelection in Republican districts. Remember this is JUST an ESTIMATE, but it is based on recent election results.
Members in Districts of Opposite Party
Jack Brown (D-5) R 43.9%-56.1%
Eric Meyer (D-11) R 42.4%-55.9%
Rae Waters (D-20) R 45.4%-54.6%
David Stevens (R-25) D 53.7%-46.3%
Frank Pratt (R-23) D 53.5%-46.5%
Nancy Young Wright (D-26) R 46.8%-53.2%
Russ Jones (R-24) D 52.0%-48.0%
Compare the two charts. I would guess that everybody from Rae Waters on up is in a real danger zone. Stevens and Pratt are in a lesser degree of danger. Young Wright and Jones should be fine.