In Roman Society families were ruled by the paterfamilias, which is just the father of the family, only he had ABSOLUTE authority over his entire family. Like the Godfather in the mafia. Also, he didn't just rule over his household, but his close reletives, possibly his sibling's households, his children's households, and his wife. Now the paterfamilias could do what ever he wished, we have an example of a paterfamilias telling his son not to attack a local enemy and the son disobeying him. In this example the son won a great victory over the enemy, but when he got home, his fater had him tortured and killed for insabordination. As for child birth, when a child was born, he/she was placed on the floor in front of the paterfamilias (note the paterfamilias may not be the father, he may be a grandfather or uncle), if the paterfamilias picked the child up, he/she was given a name. If the paterfamilias didn't pick the child up, he/she was exposed, meaning suffocated or simply left out in the woods, alone.
The Spartans had a similer system, however it wasn't run by the family, it was run by the state. When a child was born, state officials inspected the child for defects, or clues to his strength, and if he passed, he/she was allowed to live. If the new born didn't pass, he/she was exposed.
Many pre-industrial societies practice such a selection process. As soon as a child was able, he/she was expected to help out around the house. In the beginning, helping his mother with household chores, but later helping his father with family politics and farming. The the child was in any way incapable of contrubuting to the wellfare of the family, he/she was just a mouth to feed in a society where starvation was very common. So it hard for modern thinkers to imagine why these people did this, but it wasn't because they were mean, it was because often the family couldn't afford a child that couldn't pull her/her own weight. Such a child would bring down the entire family.