My guess is that children who are not spanked at all are likely to be poorly disciplined and, therefore, likely to have problems in later life. On the other hand, children spanked too often are likely to be abused or to be "bad seeds" (trouble makers from the start) and unlikely in either case to be successful.So, taking the two studies together at face value, there appears to be both an upper and lower statistical bound for how often one should hit their kids.