One where the relationship between the parents falls apart, or where they fail to fulfil their basic parental duties (whether it is due to alcoholism, drugs, being in prison, or whatever).
The correlation is between negligent/poor parenting and/or an unhappy household on the one hand, and children becoming involved in bad things on the other. As to your last question there, I am not suggesting that hapless marriages should not end in divorce, or that staying together would be good for their children. I am simply saying that an unhealthy relationship between the parents will often have some sort of effect on the children.
Well hopefully I explained what I meant there above. tbh, I never had a specific study in mind. If you just google things like "relationship between parenting/divorce and delinquency/anti-social behaviour, there's and endless supply of studies that, at a glance, all seem to indicate at least some degree of correlation. They also tend to discuss the issue of causation.
Of course these ways can work, but I see them more as variations on the nuclear family take (in the sense that the bonds are still essentially biological), rather than something else entirely. Grandparents, aunts and uncles etc can of course have varying roles of involvement in bringing up children, and they can also provide a safety net should something happen to the parents. I've never heard someone seriously argue that foster care was the best way to raise a kid.
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