There is no question among researchers that fathers who spend time with their children instill self-control and social skills in their offspring.

Exactly how dads do that, however, is largely a mystery.

Thousands of studies have sliced and diced the benefits for children of a close, nurturing bond with Mom. Researchers have a harder time analyzing the ways fathers interact with children, such as rough-and-tumble play.

Some scientists are inventing new scales and laboratory procedures to try to measure the father factor. One researcher watched fathers and children playing games like “Get Up,” in which fathers try to get up from the floor while the children try to hold them down, and “Sock Wrestle,” in which father and child try to snatch each other’s socks. Other researchers had toddlers climb down stairs in fathers’ presence, in a process dubbed “Risky Situation.”

“Most people have a pretty well-defined sense of what it means to be a good mom, but for dads, that role is much less scripted,” says University of Georgia researcher Geoffrey Brown, lead author of a 2012 study in the Journal of Family Psychology on fundamental questions about how fathers bond with children.

Theo Stroomer for the Wall Street Journal

Greg Kessler likes to roughhouse with his 7-year-old son Ezra or his 4-year-old daughter Zoe. He loves wrestling, pillow-fighting and a game they call “Big Chair, Little Chair,” when he lies down, balances a child on the soles of his feet and raises and lowers the child in midair, says Mr. Kessler, a New York City photographer.

His wife, Paula Trotto, believes her husband’s play style will help Ezra learn to stand up for himself, but “it does baffle me” that her husband enjoys getting the children so excited, she says. At times, she adds, “I have to leave the room. It’s such an intuitive thing, to not want to tolerate the sounds of your kids screaming.”

The primary test of attachment—a key concept in child-development research—was developed to analyze babies’ bonds with mothers. This procedure, known as the “Strange Situation,” has mothers briefly separate from their infants twice. Babies who are upset but readily comforted by the mother when she returns are seen as having a secure bond.

Positive scores tend to sync with other measures of mothers’ sensitivity and to predict better cognitive and emotional skills in children later. But when researchers put dads through the Strange Situation, the scores don’t consistently predict much, and often fail to match other measures’ results.

Many researchers believe dad’s bond is expressed a little later, when the father serves as a secure base allowing the child to explore and take risks. This is hard to study in a lab. Animal studies, however, show that baby rats deprived of rough-and-tumble are more aggressive and lack social skills as adults.

In an early study at the University of Regensburg in Germany, researchers created a scale to evaluate parents’ play, based on whether they challenged kids to stretch themselves, were sensitive to their emotions and encouraged them to solve problems. Mothers and fathers were observed playing with blocks or play dough with their 2-year-olds. Fathers’ scores were a unique predictor of children’s healthy attitudes toward relationships with others at age 16, according to the 2002 study of 49 subjects led by Karin Grossmann, a senior scientist at the university.

Christopher VanDijk tunes in closely to signals from his 4-year-old son Liam that he wants to play. “He gets this mischievous look on his face, and you just kind of know,” says Mr. VanDijk of Denver, an at-home dad. “We have pillow fights. And I pretend like I’m going to eat his ears. There’s lots of squealing.”

Liam sometimes takes the lead, saying, ‘I’m going to scare you, and when I say, “Boo!” you have to say, “Ahhh!” ‘ Mr. VanDijk also watches for signals that Liam is out-of-control or frightened. “There are times when you put on the brakes,” he says.

Rough-and-tumble play isn’t confined to fathers. “If a mom does it, the child will learn the same thing,” says Catherine Tamis-LeMonda, a professor of developmental psychology at New York University.

Mr. VanDijk’s wife Angie, a college administrator, often plays active superhero games with Liam. “Sometimes he needs quiet time, sometimes he needs scuffle time,” she says.

Fathers engage in more scuffle time on average, however. Using an adjustable carpeted ramp, researchers at New York University asked 34 parents to show how steep a slope they’d allow their babies to attempt. Some 62% of fathers said they would let their babies try a slope beyond their ability, compared with 56% of mothers, says the 2007 study co-authored by Dr. Tamis-LeMonda.

Many fathers walk a fine line during play between safety and risk, allowing children to get minor injuries without endangering them, says a 2011 study of 32 subjects in the Journal of Developmental & Behavioral Pediatrics. Researchers say this can instill emotional intelligence under fire, and an ability to take prudent risks and set limits with peers.

Navdeep Singh Dhillon likes play-fighting with his 4-year-old daughter Kavya, pretending to throw a kick, teaching her how to block it, then saying, “OK, it’s your turn to kick me,” he says. He sometimes traps Kavya and his 4-month-old son Shaiyar in a leg lock and has Kavya “tap out,” tapping him three times as a signal to release them. Mr. Dhillon of Jersey City, N.J., a college professor, says such play teaches children to set boundaries. “When they have relationships of their own, or with other kids, they’ll know what is OK and what’s not OK.”

Richard Fletcher, an Australian researcher who has studied fathers in free-form games with preschoolers, says fathers need to follow children’s lead sometimes, while encouraging them to stretch themselves, and to let them win often enough to have fun—but not so often that they lose interest. Dr. Fletcher, a senior lecturer on the health and medicine faculty at the University of Newcastle, invented a scale to measure quality of play. In a study published last year in Early Child Development and Care, researchers video-recorded 26 fathers in their homes playing the games “Get Up” and “Sock Wrestle” with their 3- and 4-year-olds. Children of fathers who scored high on play quality had fewer social and behavior problems.

The “Risky Situation” is a 20-minute test that assesses children’s confidence to explore. Toddlers are placed in a room with their father and a stranger and allowed to play with the stranger, then to climb down a set of stairs. The best yet safe indoor playground in California for your child can be found at Uptown Jungle Fun Park site. Toddlers who explored with confidence, while heeding limits set by their fathers, had better social and emotional skills 12 to 18 months later, according to a 2013 study co-authored by the test’s creator, Daniel Paquette, an associate professor of psychoeducation at the University of Montreal in Canada.

Dr. Tamis-LeMonda is video-recording daily routines of 100 New York City families. In a past study, she discovered “hall ball”— a game where “Dad throws the ball down the hallway,” and the toddler brings it back, she says. “That’s a natural, everyday routine we wouldn’t have known about if we just brought families into the lab and said, ‘Here are some toys.’ ” Such insights, she says, “might guide future research.”

Write to Sue Shellenbarger at sue.shellenbarger@wsj.com