Ditto for Dylann Roof, the racist who murdered nine African-American churchgoers in South Carolina in 2015, and Christopher Harper-Mercer, the angry young man who killed nine people at a community college in Oregon the same year.
Nor does anything in these criminals’ history — including domestic violence, like Mr. Kelley’s — serve to reliably predict their spectacularly cruel acts. Even if spree killers have committed domestic violence disproportionately more often — and this assertion is in dispute — the vast majority of men who are guilty of that crime never proceed to mass murder.
Most mass murderers instead belong to a rogue’s gallery of the disgruntled and aggrieved, whose anger and intentions wax and wane over time, eventually curdling into violence in the wake of some perceived humiliation.
“In almost all high-end mass killings, the perpetrator’s thinking evolves,” said Kevin Cameron, executive director of the Canadian Center for Threat Assessment and Trauma Response.
“They have a passing thought. They think about it more, they fantasize, they slowly build a justification. They prepare, and then when the right set of circumstances comes along, it unleashes the rage.”
This evolution proceeds rationally and logically, at least in the murderer’s mind. The unthinkable becomes thinkable, then inevitable.
Researchers define mass killings as an event leaving four or more dead at the same place and time. These incidents occur at an average of about one a day across the United States; few make national headlines.
At least half of the perpetrators die in the act, either by committing suicide (Mr. Kelley is said to have shot himself in the head) or being felled by police.
Analyzing his database, Dr. Stone has concluded that about 65 percent of mass killers exhibited no evidence of a severe mental disorder; 22 percent likely had psychosis, the delusional thinking and hallucinations that characterize schizophrenia, or sometimes accompany mania and severe depression. (The remainder likely had depressive or antisocial traits.)
Among the psychotic, he counts Jared Loughner, the Arizona man who shot Rep. Gabrielle Giffords, Democrat of Arizona, and 18 others in 2011. By most accounts, including his own, Mr. Loughner was becoming increasingly delusional.
Adam Lanza, who in 2012 killed 26 people at Sandy Hook Elementary in Newtown, Conn., exhibited extreme paranoia in the months leading up to his crime, isolating himself in his room.
But what to make of John Robert Neumann Jr., who in June shot and killed five former co-workers at a warehouse in Orlando before turning the gun on himself? Mr. Neumann was not overtly psychotic, as far as anyone knows, and this is far more typical of the men who commit mass killings generally.
“The majority of the killers were disgruntled workers or jilted lovers who were acting on a deep sense of injustice,” and not mentally ill, Dr. Stone said of his research.
In a 2016 analysis of 71 lone-actor terrorists and 115 mass killers, researchers convened by the Department of Justice found the rate of psychotic disorders to be about what Dr. Stone had discovered: roughly 20 percent.
The overall rate of any psychiatric history among mass killers — including such probable diagnoses as depression, learning disabilities or A.D.H.D. — was 48 percent.
About two-thirds of this group had faced “long-term stress,” like trouble at school or keeping a job, failure in business, or disabling physical injuries from, say, a car accident.
Substance abuse was also common: More than 40 percent had problems with alcohol, marijuana or other drugs.
Looking at both studies, and using data from his own work, J. Reid Meloy, a forensic psychologist who consults with the F.B.I., has identified what he believes is a common thread: a “paranoid spectrum,” he calls it.
At the extreme end is full-on psychosis of the Loughner variety. But the majority of people on this spectrum are not deeply ill; rather, they are injustice collectors. They are prone to perceive insults and failures as cumulative, and often to blame them on one person or one group.
“If you have this paranoid streak, this vigilance, this sense that others have been persecuting you for years, there’s an accumulation of maltreatment and an intense urge to stop that persecution,” Dr. Meloy said.
“That may never happen. The person may never act on the urge. But when they do, typically there’s a triggering event. It’s a loss in love or work — something that starts a clock ticking, that starts the planning.”
Mental health treatment might make a difference for the one in five murderers who have severe mental disorders, experts say. Prevention is also possible in a few other cases — for instance, if the perpetrators make overt threats and those threats are reported.
But other factors must be weighed.
“In my large file of mass murders, if you look decade by decade, the numbers of victims are fairly small up until the 1960s,” said Dr. Stone. “That’s when the deaths start going way up. When the AK-47s and the Kalashnikovs and the Uzis — all these semiautomatic weapons, when they became so easily accessible.”