BEFORE DAWN, WITNESSES (WHO WERE PREFLIGHTING AN ACFT) SAW N732QP TAXIING TO RWY 4. THEY HEARD AN ENG RUN-UP; HOWEVER, THEY WERE UNABLE TO SEE THE ACFT TAKEOFF DUE TO HEAVY FOG THAT HAD JUST MOVED OVER THE ARPT. (THE WITNESSES DECIDED TO CANCEL THEIR FLT DUE TO DETERIORATING WX.) SHORTLY AFTER N732QP TOOK OFF, THE WITNESSES HEARD THE ENG SOUND CHANGE TO A HIGH PITCH SCREAM. THIS WAS FOLLOWED BY A THUMP, THEN THERE WAS SILENCE. AFTER THE ACDNT, THE VISIBILITY CONTINUED TO DETERIORATE, HAMPERING A SEARCH FOR THE ACFT. AS SOON AS DAWN ARRIVED, THE ACFT WAS FND WHERE IT HAD CRASHED ON ARPT PROPERTY. NO PREIMPACT MECHANICAL FAILURE OR MALFUNCTION OF THE ACFT WAS FOUND. THE PLT WAS NOT INSTRUMENT RATED. HE HAD RCVD 2.8 HRS OF DUAL INSTRUMENT TRAINING IN THE FALL OF 1981, BUT NO RECORD OF SUBSEQUENT INSTRUMENT FLT TRAINING WAS FND. ONE WITNESS RPRTD THAT AT 0630, THE WX HAD DETERIORATED TO 200' OBSCURED, 1/8 MI VISIBILITY WITH FOG.