If President Kennedy had used the US Navy to back up the CIA-trained exiles who invaded Cuba in '62 at the Bay of Pigs, the Cuban Missile Crisis would never have occurred. Castro would have fallen and there would be a government friendly to the U.S. in Havana. All of the Havana-Moscow dynamics that were part of Khrushchev's decision to send offensive nuclear missiles to Cuba would have been upset.
In the Madman Theory, Nixon is president instead of JFK and RN uses a different strategy when he launches what we know as the Bay of Pigs invasion. (For one thing, the landing site is not changed to the Bay of Pigs from Trinidad, as it was under Kennedy's presidency.) Yet, circumstances are not sufficiently different from those that pertained under Kennedy to change Khrushchev's calculation.