The family claimed Mr Stevens had been granted a 15-minute "unescorted leave" period from the hospital. But they claimed that it was "at least 90 minutes" before it was discovered he was missing.
"Had Nicholas been on escorted leave, as his family stated was the only safe option at that time, he would almost certainly have been alive today," the family said.
"Our son's death was preventable, had proper and common sense procedures been followed."
Police have referred the death to the coroner.
Waikato DHB communications director Mary Anne Gill said the internal inquiry is part of a serious adverse event review carried out by all DHBs in such instances, as laid out by the Health and Quality and Safety Commission.
The hospital's most senior staff member is working on the case, she said.
"We are profoundly sorry for the family's loss and are available to meet with them at any time to talk about the treatment we provided leading up to his disappearance. We will do all we can to ensure we involve them in an appropriate way," said Ms Gill.
Ms Gill also stressed that as well as their own internal inquiry, the DHB will cooperate with the coroner's independent inquiry.
A farewell service for Mr Stevens will be held at Hamilton Gardens Pavilion on Wednesday at 11am.