Just the opposite, Devil. If he has some "guilty secret," he definitely wouldn't want people to think his daughter ran away from him -- reputation must be preserved.
If people decided they thought Elizabeth had run away, that doesn't mean they'd automatically think it was her father she ran away from. Anyone who has ever had teenagers--or been a teenager--knows that they can be very unreasonable. Sometimes--it's almost a cliche--they think they live under the most horrible conditions imaginable, only to find (when they are out from under mean old Mom and Dad) that they actually had a really good thing. Most people know this about teenagers, and would not automatically blame a parent b/c his/her teenager ran away. Teenagers have a variety of reasons for doing such a thing; being a victim of a father's abuse is only one of many reasons they might do this, and the public knows this, and I believe Ed is smart enough to KNOW that the public knows this.