Dependent or Independent clause?

"Way" is the subject. "(That) people write" is a dependent clause. "Has changed" is the verb. There is no object.

This clause is independent.

"The way people write" is the subject. "has changed" is the verb.

To highlight the subject and verb, we can reduce this sentence to "way" and "changed":

The way changed.

Now this sentence does sound weird. Normally, there would be something after it in normal usage. Here's some examples that expand on the sentence and make it sound better:

The way people write has changed since the 1900s.
The way people write has changed after the invention of the computer.
The way people write has changed in recent times.

First of all I would like to say that clause is a verb along with its subject and their modifiers. That is, any clause has only one subject and only one associated with this subject verb (you should understand that a verb can be compound, that is, it can be composed from some verbs by a formula:

auxillary verb(s) + main verb

but still there is only one subject and only one associated with it compound verb in any clause).

Now let's take a look at your example:

The way people write has changed.

You should understand that this sentence is slightly shortened. The "full version" of this sentence would be:

The way that people write has changed.

So, I will talk about this second version, because it is more understandable and everything that I will tell about it is absolutely true for your shortened version too. It is neither independent nor dependent clause. This fragment cannot be called clause just because it has two pairs of subjects and their corresponding verbs, these pairs are: {the way - subject, has changed - compound verb} and {people - subject, write - verb}. It is rather an example of a complex sentence.

Complex sentence consists from one or more independent clauses and zero or more dependent clauses. In your example the complex sentence has one independent clause ("The way has changed") and one dependent clause ("people write"). To be more precise, this dependent clause is called relative clause, and there is a rule that describes why the word "that", which connects the dependent clause to the independent clause, can be omitted from this relative clause, but I won't write it here since it isn't directly connected to the question.