How do you decide which clause is "independent" and which clause is "dependent"/"subordinate"?

Two clauses can be recast as complete sentences by leaving out the conjunction.

My brother felt sick. He ate too many chocolates.

My brother felt sick because he ate too many chocolates.

In the second case, because he ate too many chocolates is a dependent clause. 'Because' is a subordinating conjunction; it introduces a cause which is less important (subordinate to) the first clause.

A basic definition is that, a dependent clause is vital to the sentence meaning; an independent clause is not.

My brother felt sick because he ate too many chocolates.

Compare that with this:

My brother felt sick, which was unusual for him, because he ate too many chocolates.

"which was unusual for him" is an independent clause. It provides additional information or comment, but can be left out without changing the fundamental meaning of the sentence.

My brother felt sick for some reason.

This sentence contains one independent clause. The simple subject is "brother", and the simple predicate is "felt". The prepositional phrase "for that reason" is one part of that one independent clause.

My brother felt sick because of the chocolates.

Even with the word "because", this sentence also contains only one independent clause. The phrase "because of the chocolates" does the same job in this sentence as "for that reason" does in the first example.

My brother felt sick because he ate too many chocolates.

In this sentence, we have two clauses. There is the clause with brother/sick as its subject/verb pairing, and another clause with the pairing he/ate. In spite of that, the entire structure "because he ate too many chocolates" still performs the same function. In all three of these examples, the italicized constituents act as modifiers of the adjective "sick".

 

Simple substring processing is not a good model for analyzing English. The substring "My brother felt sick" appears in all three example sentences, and yet the clause "my brother felt sick" does not appear in any. The independent clause in each example includes the entire contents of its sentence.

There's a better label than independent clause for the third example. It is a matrix clause -- a clause that contains another clause. The dependent clause is subordinate to its matrix clause.

 
If you need a computational model, don't look at operators. Look at factories. In the first example, the preposition "for" takes "some reason" as its argument and produces a general modifier, the prepositional phrase "for some reason". That produced constituent (an object, in the OOP sense) is then able to attach to a suitable modificand, such as the preceding adjective.

If we think of "because" as a preposition, as a modifier factory, then we see it has more than one signature. In the second example, its argument is a prepositional phrase. In the third, it's an entire independent clause. We can also see that what it generates not as general-purpose as the constituent produced by "for". It doesn't attach as readily to a nominative construction.

If we're too old-fashioned to think of "because" as a preposition, then we're left with treating "because of" as a compound preposition, taking a grammatical object as its argument. We're also left with labeling the third example's "because" as a subordinating conjunction.

Even under that old-fashioned labeling, this subordinating conjunction is a factory. It produces an adverbial subordinate clause.

My bother felt sick [because he ate too many chocolates].

Traditional grammar treats "because" as a subordinator (or subordinating conjunction in trad terms), and hence the bracketed element is a subordinate clause, making this a so-called 'complex' sentence. Note that "because" functions as a 'marker' of subordination and hence is part of the subordinate clause.

By contrast, some modern grammars analyse "because" as a preposition, and in their terms the bracketed element is thus a preposition phrase. However, the expression "he ate too many chocolates" is a dependent in the PP, i.e. a subordinate clause serving as complement of "because".

"My brother felt sick because" cannot be called an independent clause, since it cannot stand alone as a sentence.

In other words, whichever way we choose to analyse "because", the sentence is still a complex one.

Further, some grammars (wrongly) analyse "my brother felt sick" as a main clause simply simply because it could stand alone as a sentence, but in fact it's just part of a larger clause: in this case the sentence as a whole is the main clause.

Incidentally, grammarians rarely use the terms 'simple', 'complex', 'compound' etc, since they are pretty meaningless when analysing the structure of sentences.

We can agree that both

My brother felt sick.

And

He ate too many chocolates.

Are both independent clauses and that “because” “joins” them. So why is “because he ate too many chocolates” analysed as a clause and not “my brother felt sick because”?

Conjunctions and complementizers in English are typically analysed to attach in front of a clause, transforming an independent clause into a subordinate one, this is in reverse of, say, Japanese, where they tend to attach at the back of clauses — why is this analysis favored?

Firstly, the order can be reversed:

Because he ate too many chocolates, my brother felt sick.

This is already a strong argument that “because” attaches to “he ate too many chocolates” and the entire subordinate clause then joins with the independent clause.

Another is how that an inquiry in English can be answered simply with “Because he ate too many chocolates.”, this is also how sentences are typically broken up and feel natural to native speakers, if a sentence were spread out with a pause, this would feel unnatural:

He fell sick because

he ate too many chocolates.

“because” would almost always be moved over the pause, which is what feels more natural to English native speakers.

This is how conjunctions in English are generally analysed. “you and I” in English would similarly be broken down. That they attach to the front rather than the back can also be seen in such constructs as “either this or that” or “both you and I”.

Edit: After giving it some further thought, methinks English has two fundamentally different classes of conjunctions, one functioning as complementizers grammatically, and the other not. “and”, “or”, and “therefore” are grammatically fundamentally different from “because”, “since”, and “so that”.

Namely, sentences such as:

I, because I need to eat, sell my body for a living.

My dog, since he is hungry, is barking.

Are all possible, complementive clauses created by attaching the first group before a sentence can be inserted relatively freely inside of the matrix clause, as such one can far more easily argue that they are truly subordinate to said clause.

However with “or” or “therefore” it becomes more difficult:

*I, therefore I am, think.

*I, or took a bath, had something to eat.

Clearly “or”, “and”, and “therefore” do not create similar complementive subordinate clauses that can be inserted inside of their matrix clause, but only after it.

However, even with those the argument remains that “Or I had something to eat.” feels as a natural, complete utterance, whereas “I took a bath or.” feels as if it be wanting for a conclusion, an unfinished thought, therefore I would still argue that “or”, “therefore” and “and” attach to firstly to the second part of the conjunction, and the result of that then attaches behind the first independent clause to create a conjunction of two clauses.