If you have joint custody of your 16 year old daughter when can she decide not to visit her father?

If you live in the US, she gets to decide when she's 18. Until that age if the parent has court-ordered visitation it has to be followed. If there is a valid reason (and it must be a very valid reason) the child does not want to visit with the other parent, you need to petition the court to modify the visitation order.