WASHINGTON — The key variable in public tolerance of U.S. military deaths in combat is whether people believe that victory is likely, according to a new poll and study of U.S. public opinion on casualties in Iraq and in other military actions.

If the Bush administration "can persuade the public that ‘victory’ is likely in Iraq, then public support will endure," says the study, which was conducted by three Duke University political scientists.

Their report concludes that President Bush has already lost most of the "soft supporters" due to his handling of Iraq — essentially those who supported an invasion and an occupation only if they were nearly cost-free — and it predicts that a small but steady trickle of combat fatalities would not cause a collapse in support. The report estimates that, aside from confirmed hawks and doves on Iraq, about 20 percent of the public is "casualty-phobic" in that it supported the president’s actions in Iraq until casualties started to rise. It is the backing of that group that the president lost in recent months, said Peter Feaver, one of the three researchers who conducted the study.

The study notes that a precipitous drop in support could occur if the public comes to believe that victory is unlikely.

Feaver briefed White House and other administration officials on the findings on Friday. The study points to three recommendations for Bush, he said. "First, worry less about persuading the American people he really did the right thing, and more about ensuring that the mission is going to be successful — and persuading the American people of that," he said.

Also, he said, the administration needs to develop valid and convincing measures of success in Iraq, "so he himself knows whether he is winning."

Finally, Feaver said, the administration should worry less about communicating the strength of its resolve and more about "how their behind-the-scenes actions undercut their rhetoric."

In a similar survey, CBS News found that half of Americans believe the U.S. effort in Iraq is going badly, the highest proportion since Bush declared more than six months ago that major combat had ended.

But Feaver said asking about whether things are going well or badly tends to produce "very volatile" results. The question that better illuminates long-term trends, he said, is: "Do you think we’re likely to succeed?"

The Duke study is based on a survey of 1,203 American adults in late September and early October, before the latest upsurge in violence in Iraq. It was funded by the Carnegie Corp. It comes as the nation is experiencing the first sustained ground combat in four decades, since the Vietnam War. Its results challenge some of the prevailing academic wisdom about public opinion and casualties, such as the long-held view that public reaction to casualties turns on whether the public understands and supports the mission in which they occur.

The Duke study also suggests that a major reason for Bush or any president to woo foreign support is that it can shore up domestic public opinion. For example, it says that support for any military action is heavily affected by whether the president’s policies are endorsed by other major institutional players, such as Congress, NATO or the United Nations. If all those entities support the president, it says, then 68 percent of the public would likely support his policy. But if Congress opposes it, or NATO and the United Nations line up against it, support would fall to about 40 percent. And if all three go against the president, then support would drop to 22 percent.

The study arrived at some new conclusions about the relationship between the volume of media coverage and public support for the president’s handling of a situation. As casualties increased during the conventional war this spring, both support and coverage increased. "Those casualties had no significant impact" on the public’s approval of the president, the study notes.

But as casualties and coverage increased in the guerrilla fighting in late summer, support for the president declined. In that case, it says: "As coverage goes up, approval goes down."

The reason for the diverging effects of intense media coverage, Feaver said, is the nature of what is being covered. While the public has confidence in the U.S. military’s ability to wage conventional war, he noted, it is less sure of the military’s ability to prevail against an insurgency. "When the public hears ‘insurgency,’ I think they hear ‘something we’re not good at,’ and so that raises doubts about whether we can win, which our study finds is key," he said.