Interesting how this compares to ほど. It looks like they both have “to the extent that” meaning, and the only thing changing seems to be the order of the reason and the result clauses. Very convenient 
苦しさの余り、大きな声を出してしまった。
大きな声を出してしまうほど、苦しかった。
These two seem to mean the same thing to me.