Arthur Miller didn't write The Crucible based on his childhood; however, he did write it as a direct response to his experience in the McCarthy trials in the 1950s. He was horrified at the hysteria which grew out of an environment of paranoia, suspicion, and fear--very little of which could be justified by fact. His play about the Salem Witch Trials was a reminder that history does repeat itself despite our best intentions, perhaps, to ensure it doesn't.
David Mamet has had a prolific career as playwright, screenwriter, producer, director--you name it, he seems to have done it. Some of his works include: The Postman Always Rings Twice, The Verdict, Glengarry Glen Ross, and Wag the Dog to name a few. He also did some writing as Richard Weisz.
Lorraine Hansberry (who would have most certainly had more of an influence if she had lived longer) wrote A Raisin in the Sun. Although she did not grow up in the impoverished environment that the Youngers live in, she did experience living in a segregated environment and being able to talk to young African-American philosophers who visited her father (similar to Asagai's theories in the play).
In thinking of a common theme you might want to examine the plays of Arthur Miller and consider the common themes that seem to exist in his works - they all seem to centre on a dominant male who finds tragedy by either not being able to fit into society or standing up against it. Consider how Eddie Carbone and Willy Loman, for example, model American modern tragedies in their fate. John Proctor likewise has a tragic end but one through which he finds himself and re-affirms his character, rather than recognising his innate weakness and failing, as in the case of Miller's other protagonists.
Tennessee Williams is the first that comes to my mind. The Glass Menagerie and Cat on a Hot Tin Roof are two of his famous works that have been described as semi-autobiographical. Eugene O'Neill is another; Desire Under the Elms is one of his works you might reference, along with Night of the Iguana.
Enotes has online study guides for both of these American playwrights.