The private pilot was conducting touch and go landings and full stop landings at his home base airport. The airplane touched down on the runway and started to veer to the left on landing roll out. No crosswind condition was present. The pilot stated, "I let the plane follow the path to the left, not wanting to over control it and having it tilt sideways or nose first into the grass. It struck the 3,000 foot sign on the side of the runway as it exited the runway, rolled out on the grass and came to a stop." The pilot informed the NTSB investigator during a telephone interview that he lost directional control of the airplane. Examination of the airplane by a Federal Aviation Administration Inspector and maintenance inspector from a local fixed base operator revealed the left horizontal stabilizer received structural damage. No anomalies were noted with the brakes or steering systems of the airplane.