Fifty-eight people are dead from the worst mass shooting in recent U.S. history. As happened after Omar Mateen killed 49 people at a nightclub with a gun, or after Dylann Roof killed nine African Americans with a gun, or after Adam Lanza killed 26 children and teachers with a gun, or after James Holmes killed 12 moviegoers with a gun, the call for action from some policy makers has centered on one commonality between these events: All of the killers had brains.

“Mental-health reform is the critical ingredient to making sure that we can try and prevent some of these things that have happened in the past,” House Speaker Paul Ryan said Tuesday in response to reporter questions about mass shooters. (President Obama also proposed better mental-health care last year, when recalling the mass shooting at Sandy Hook Elementary School in 2012.)

It’s worth noting that investigators and reporters have so far uncovered no psychiatric diagnoses in the background of the Las Vegas suspect, Stephen Paddock. His brother, Eric Paddock, has said to reporters that Stephen had “not a bit” of mental-illness history. But even if he did, better treatment access might not have deterred him.

While improving access to mental-health care might help lots of suffering Americans, researchers who study mass shootings doubt it would do much to curb tragedies like these. According to their work, the sorts of individuals who commit mass murder often are either not mentally ill or do not recognize themselves as such. Because they blame the outside world for their problems, mass murderers would likely resist therapies that ask them to look inside themselves or to change their behavior.

The connection between mental illness and mass shootings is weak, at best, because while mentally ill people can sometimes be a danger to themselves or others, very little violence is actually caused by mentally ill people. When the assailants are mentally ill, the anecdotes tend to overshadow the statistics. Both Jared Loughner, who shot and severely injured Representative Gabrielle Giffords, and the Aurora, Colorado, shooter James Holmes, for example, had histories of mood disorders. But a study of convicted murderers in Indiana found that just 18 percent had a serious mental-illness diagnosis. Killers with severe mental illnesses, in that study, were actually less likely to target strangers or use guns as their weapon, and they were no more likely than the mentally healthy to have killed multiple people.

“If we were able to magically cure schizophrenia, bipolar disorder, and major depression, that would be wonderful,” Jeffrey Swanson, a professor of psychiatry and behavioral sciences at the Duke University School of Medicine, told ProPublica. “But overall violence would go down by only about 4 percent.”

One review paper published in 2014 found that though “a history of childhood abuse, binge drinking, and male gender” are all linked to serious violence, mental illness was not, unless the person was also a drug addict. According to the National Center for Health Statistics, fewer than 5 percent of the 120,000 gun-related killings in the United States between 2001 and 2010 were perpetrated by people diagnosed with a mental illness. A 2001 study of teen mass murders found that only one out of four was mentally ill.

As Northeastern University criminologist James Alan Fox has written, in a database of indiscriminate mass shootings—defined as those with four or more victims—compiled by the Stanford Geospatial Center, just 15 percent of the assailants had a psychotic disorder, and 11 percent had paranoid schizophrenia. (Other studies have come to a higher estimate, suggesting about 23 percent of mass killers are mentally ill.)

Certainly, getting those 15 or 23 percent into treatment might chip away at their pathological thinking—and thus their potential future acts of violence. But as Fox argues, linking psychopathic killers with the mental-health system is no easy task. After studying mass shooters for decades, he’s concluded that the killers have more mundane motivations: revenge, money, power, a sense of loyalty, and a desire to foment terror.

He explained further in a 2013 paper in the journal Homicide Studies:

Revenge motivation is, by far, the most commonplace. Mass murderers often see themselves as victims—victims of injustice. They seek payback for what they perceive to be unfair treatment by targeting those they hold responsible for their misfortunes. Most often, the ones to be punished are family members (e.g., an unfaithful wife and all her children) or coworkers (e.g., an overbearing boss and all his employees).

“The thing about mass killers is that they externalize blame,” Fox told me. “All the disappointments, all the failures, the broken relationships, are because other people treated them wrong. They don’t see themselves as being inadequate and flawed.” Indeed, a recent paper similarly concluded, “very few of persons [sic] in the risky category of having anger traits combined with gun access had ever been hospitalized for a mental-health problem.” That could be because they didn’t think they needed help.

Other experts have echoed Fox’s view. Michael Stone, a forensic psychiatrist at the Columbia College of Physicians and Surgeons and author of The Anatomy of Evil, on the personalities of murderers, recently conducted a study that found that a fifth of mass killers had a serious mental illness. “The rest had personality or antisocial disorders or were disgruntled, jilted, humiliated, or full of intense rage,” as The Washington Post’s Michael S. Rosenwald wrote last year. “They were unlikely to be identified or helped by the mental-health system.”

Fox acknowledges that some mass killers are mentally ill. The problem is, many don’t realize it or seek treatment. In his remarks to the press about mental-health care, Ryan appeared to be referring to the 21st-Century Cures Act, which contained provisions aimed at increasing the number of mental-health providers and strengthening insurance reimbursements for mental-health care.

One thing the law does is expand Assisted Outpatient Treatment, or court-ordered mental-health care, which could help rope the mentally ill into treatment. But this route is usually only available to those with a prior history of psychiatric hospitalization or arrest.

Meanwhile, as Fox notes, mass killers tend to share a few characteristics—“depression, resentment, social isolation, the tendency to externalize blame, fascination with graphically violent entertainment, and a keen interest in weaponry”—that are common in the general population. Attempting to flag so many angsty, un-self-aware young males as potential future killers might push them closer toward violence, rather than away from it.

Finally, Fox argues that if there were some sort of law in which therapists could report their threatening patients to gun registries—as exists in California—people who wished to own guns could, in that case, simply avoid attending therapy.

Instead, a better way of predicting whether someone might be predisposed to violence is if they have a history of violence, as Swanson told ProPublica. For example, Spencer Hight, who killed his ex-wife and seven others at a football-watching party in Plano, Texas, earlier this month, had been violent at least twice, reportedly slamming his wife’s face against a wall.

Compared to those with no criminal record, handgun purchasers who have at least one misdemeanor conviction are seven times more likely to be charged with a new offense after they buy their gun. Right now, only 23 states restrict people with a history of violent misdemeanors from owning firearms.