There is a small but reasonably persistent gap between President Obama’s net approval ratings and his head-to-head polls against Mitt Romney. Whereas Mr. Obama’s approval ratings have been almost exactly breaking even for most of the past few months — in fact, they’re very slightly underwater now according to the Real Clear Politics average — he has more often than not enjoyed a slight lead in head-to-head polls against Mr. Romney.

Liberals and conservatives tend to interpret this evidence in different ways. For liberals, it may be taken as a sign that Mr. Romney is an especially weak candidate — enough so that many voters who are on the fence about Mr. Obama’s job performance, and even a few who disapprove of it, will be willing to vote for Mr. Obama if Mr. Romney is the alternative. The claim has often been made in recent weeks, for instance (in my view, based on relatively speculative evidence), that Mr. Obama’s attacks on Mr. Romney’s tenure at Bain Capital may have damaged Mr. Romney.

Conservatives instead sometimes argue that this is a sign that Mr. Obama has some softness in his numbers. As voters become more engaged with the campaign, Mr. Romney may have a bit of wind at his back if he is able to keep the focus on Mr. Obama’s job performance; the head-to-head polls could fall more in line with Mr. Obama’s approval ratings.

Neither of these views is irrational. In support of the liberal position, my research suggests that there can be some predictable-seeming differences between a president’s net approval rating and the result he actually realizes on Election Day, based in part on “candidate quality” factors related to his opponent. For instance, opponents who have especially “extreme” ideologies (who are perceived as being very liberal or very conservative) may allow a president to over-perform his approval ratings, while challengers who are viewed as moderates may get the benefit of the doubt from voters. Elections are perhaps mostly a referendum on the incumbent, but they are not purely so.

On the other hand, in support of the conservative position, my research suggests that approval ratings may have some predictive power even once you are also accounting for head-to-head polls, especially early in the campaign.

Both of these views, however, leave aside an intriguing piece of evidence. That evidence is Mr. Obama’s favorability ratings. They are net-positive right now, as they have been throughout most of his presidency.

Right now, the Real Clear Politics average shows 50.6 percent of Americans with a favorable view of Mr. Obama, versus 45.1 percent with an unfavorable one. That contrasts with his approval ratings, which now show 46.8 percent as approving his job performance and 48.7 percent as disapproving it.

In other words, there is a small slice of the electorate, about 4 percent, that has a favorable view of Mr. Obama, but does not approve of his job performance. Given how close the election is, the way they behave in November could be decisive. If the election is a referendum on Mr. Obama based on his approval ratings, it’s going to be very close. He may be a slight underdog, especially since some of the approval ratings polls are of adults or registered voters, which are generally a point or two more favorable to the Democratic candidate than those of likely voters. However, if it’s a referendum based on Mr. Obama’s favorability ratings, his net-positive score (plus 5.5) makes him look like the favorite.

Is there evidence on whether approval ratings or favorability ratings are a better indicator of a president’s re-election chances? Actually, there’s not very much of it. Favorability ratings have received much less academic study than approval ratings. There are a couple of reasons for this. First, the two sets of ratings are normally extremely close to each other. Unless you need extreme precision for something, your choice just won’t matter that much. And second, the historical record of approval ratings is a little richer. Pollsters take them somewhat more often than favorability ratings, using somewhat more consistent question wording, and these ratings are archived more completely than the favorability ratings are. So approval ratings tend to be used as a default in studies of the presidency.

What I’ll offer here is only some cursory evidence: a comparison of approval ratings and favorability ratings for the past five incumbent presidents who were running for re-election, as taken from the CBS News poll database (most CBS News polls were conducted in conjunction with The New York Times). Five data points isn’t very many, but using both approval ratings and favorability ratings from the same polling organization at least allows for the comparisons to be a bit more apples-to-apples.

In the table below, I’ve listed the average net approval ratings and net favorability ratings for each of these presidents, in CBS News polls conducted from Sept. 1 of the election year through Election Day. These are listed alongside the results of the election.

Three of these presidents — Ronald Reagan, George H. W. Bush and George W. Bush — had nearly identical approval ratings and favorability ratings. But they diverged by a material amount in two other cases.

In 1996, Bill Clinton’s approval ratings (roughly plus 25) were quite a bit stronger than his favorability ratings (roughly plus 14). He won a perfectly solid victory over Bob Dole, but the 8.5-point margin was somewhat more consistent with his pretty good favorability ratings than his very good approval ratings.

Consider that Mr. Clinton in 1996 had roughly identical approval ratings to Ronald Reagan in 1984 — but Mr. Reagan’s margin over Walter Mondale was much larger than Mr. Clinton’s over Mr. Dole, perhaps because Mr. Reagan’s favorability ratings matched his lofty approval ratings while Mr. Clinton’s did not.

In 1980, Jimmy Carter had bad approval ratings (negative 14 net), but somewhat more forgivable favorability ratings (negative 6). His 10-point loss to Mr. Reagan roughly split the difference between them.

There are a few other wrinkles to consider that complicate the analysis. For instance, presidents with very strong or very weak approval ratings tend to have election results that are a little closer to the mean. A president with a plus-40 approval rating isn’t likely to win the election by as many as 40 points. When his ratings are that strong, many of the voters approving him will be of the opposite party, and some of those opposite-party voters will wind up for their party’s own nominee out of partisan loyalty.

Ordinarily, the more robust way to analyze this data would be with regression analysis. In this case — with two highly correlated measures tested upon just five data points — regression analysis is not a very powerful tool. Still, I ran those numbers for fun and got an interesting result:

Note that the regression coefficient on the approval ratings and favorability ratings is almost exactly the same. What that means is that the best policy in these past elections would simply have been to weight them equally.

If you take an average of Mr. Obama’s approval ratings and his favorability ratings right now, based on the Real Clear Politics numbers, you will get a positive rating (approve or favorable) from 48.7 percent of voters, and a negative one (disapprove or unfavorable) from 46.9 percent.

The net rating — plus 1.8 percent — almost exactly matches his current standing against Mr. Romney. The Real Clear Politics average of head-to-head polls has Mr. Obama with a 2-point lead on Mr. Romney, while our “now-cast” (which is based only on the polls and does not look at economic factors) has Mr. Obama projected to a 2.1-point lead.

Let me emphasize, again, that although this analysis produces a neat-looking result, it’s based on some relatively thin evidence — really just two presidential elections (1980 and 1996) where the two sets of ratings diverged to any appreciable degree. Check back in 2040 or so, and we might be able to answer more definitively which set of ratings has more predictive power, or whether the method of averaging them together is the most sensible choice.

Still, it seems as though the small set of voters who take a favorable view of Mr. Obama but do not approve of his job performance are very much worth fighting over for the campaigns. The split between the two sets of ratings may reflect a sensible enough reaction from voters, who have ample reason to be dissatisfied with the direction of the country, but may be more sympathetic to Mr. Obama as some of the problems began before his tenure.

If you’re part of Mr. Obama’s campaign, there could be risk in taking a negative tack that might reduce that sympathy factor. At the same time, these results suggest that Mr. Obama doesn’t necessarily need to damage Mr. Romney to win. If voters are judging Mr. Obama based on a mix of his personal qualities and their perception of his job performance, his numbers might be just strong enough to win as it is, without their factoring much about Mr. Romney into their decision.

Mr. Romney’s campaign in Boston faces an equally interesting set of choices.

Do you explicitly try to appeal to the set of voters who like Mr. Obama personally but take a neutral or negative view of his job performance? There have been times when Mr. Romney’s campaign seemed to adopt this strategy. At times on the campaign trail, Mr. Romney has told voters that he thinks of Mr. Obama as a good man, but that he’s just not up to the job of being president.

Or do you try to bring down Mr. Obama’s favorability ratings by a couple of points? This is more in line with the advertisement that Mr. Romney’s campaign released on Thursday, which accuses Mr. Obama of dishonesty in how he portrayed Mr. Romney’s tenure at Bain Capital.

That strategy seems a bit higher risk — especially as Mr. Romney’s favorability ratings are rather tepid, meaning that he does not necessarily want to give voters an excuse to think of the election as a popularity contest.

However, Mr. Obama may be slightly stronger than his approval ratings imply, if not as strong as his favorability ratings suggest. In that case, a higher-risk strategy might be called for, particularly if the small lead Mr. Obama has in head-to-head polls seems to persist for another several weeks.