How strongly related is altcoin value to bitcoin price?

It is strongly correlated from what you see in the data trends. Let’s explain the basics then throw some examples.

Simple math, ALT/BTC ratio would decrease if:

• ALT coin’s value drops (i.e. people are selling the ALT coin or trading it for BTC or other ALT coins).
  and/or
• BTC value increases (i.e., more demand for BTC from various accounts including USD and ALT coin conversion)

Similarly, ALT/BTC ratio would increase if:

• ALT coin’s value increases (i.e. people are buying an ALT coin by trading BTC or other ALT coins).
  and/or
• BTC value decreases (i.e., BTC is cashed out for USD or other currency or traded for ALT coins)

Example 1: Recently you may have noticed that the increase of ALT coin’s value is highly correlated with the increase of BTC and vice-versa. Let’s add the USD to the ratio where ALT/BTC is also ALT/USD / BTC/USD. If people are buying more bitcoins then they are increasing the BTC/USD value. But since ALT is also rising, it means that BTC is also traded to buy ALT resulting in an increase of ALT/BTC which can also be translated to an increase of ALT/USD. The opposite is also correct, the decrease of ALT/BTC and BTC/USD together is a result of a quick sell of both BTC and ALT.

Example 2: You notice an increase of ALT/BTC and a decrease of BTC/USD. Remember that ALT/BTC is also “ALT/USD / BTC/USD”, so if BTC/USD is decreasing and ALT/BTC is increasing, that means that a good sum of traders are either holding on to their ALT coins and/or trading BTC for ALT while others are shorting BTC for USD in the market.