Bitcoins value does indeed affect the price of all altcoins since BTC/AltCoin trading pairs are the most commonly used and is usually the standard go to exchange pair on all cryptocurrency exchanges. So can Altcoin break all the relationship from bitcoin and run as a independent currency.
all alt coins are connected by bitcoin because bitcoin is the parent of this cryptocurrency, it is very difficult if the alt coin wants to be independent because it must beat the bitcoin itself which is already there first and more famous
it takes a long time to rival the bitcoin itself