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.
At time goes on altcoins might have an independent measurement in price relationship from bitcoin, however presently every altcoins depends on the market price of bitcoin because it's the base currency which is use for measurement.