The question is struggling with what school grammar (the type of grammar that many of us were taught in middle school) considers to be a clause as opposed to what theories of syntax consider to be a clause. According to school grammar, a complex sentence consists of two or more clauses, where a clause is generally understood to contain at least a subject and a predicate, e.g.
When one considers the reactions of the dogs, the cats showed more interest.
This sentence has two clauses, each of these clauses containing a subject and predicate. The first clause, i.e. when one considers the reactions of the dogs, contains the subject one and the predicate considers the reactions of the dogs, and the second clause, i.e. the cats showed more interest, contains the subject the cats and the predicate showed more interest. The latter clause is the main clause, and the former clause the subordinate clause.
What one should be aware of in this area is that school grammar simplifies and bunches concepts so that they can be presented to young minds more easily. When one studies these distinctions in some detail, one quickly sees that the traditional definitions do not work very well. The question has touched on just such an aspect of clauses where one can see that the traditional definitions do not help so much.
The clause concept is much more extensive than what the question assumes. In othe words, the question is proceeding from too narrow of a definition of the clause. The example sentence contains what can be construed as a nonfinite clause. Further examples of nonfinite clauses follow:
a. Disgusted with the situation, he stormed out of the room.
b. She worked all night, having no choice.
c. Being required to work on Wikipedia articles upset the students.
These sentences contain the constituents disgusted with the situation, having no choice, and being required to work on Wikipedia articles, whereby these constituents can be interpreted as the predicates of clauses. The subjects of these clauses are implied, i.e. they are easily understood from the context. For sentence a, "he" is the one who was disgusted with the situation; for sentence b, "she" is the one who has no choice; and for sentence c, "the students" are the ones who are required to work on Wikipedia articles. Like most languages, English has mechanisms in place that allow it to express predicates without also having to state the subjects, since the subject is easily retrieved from the context.
If one desires a formal linguistic concept for the issue at hand, it is control. The theory of control explores when the subject of a predicate has to be expressed overtly and when it can be implied from the context. In any case, one is dealing with clauses because the subject is present (either overtly or covertly).
Finally, the notion that a subordinate clause must be introduced by a subordinator (= subordinate conjunction) is wrong. While it is true that many subordinate clauses are indeed introduced by a subordinator, there are types of subordinate clauses that are not introduced by a word that is taken to be a subordinator, e.g. relative clauses and indirect questions.
