"Because", "since", and "as" can be used interchangeably in some cases, because they can all be used for the same purpose: to answer the question, "Why?". In other words, the sentence will make sense, regardless of which word you use, although it may not sound the most natural. For example:
1. I didn't go for a walk today because it was raining.
2. I didn't go for a walk today as it was raining.
3. I didn't go for a walk today, since it was raining.
Most people would probably use "because". You don't see "as" used as much this way nowadays, although I use it rather frequently in my writing. If you ask me, "since it was raining" sounds awkward at the end of a sentence, like it's hanging off. "As" and "since" are usually used at the beginning of the sentence, but this is definitely not a rule.
Here is a blog that gives more details on the usage of these words:
I hope that helped. :)