Single pilot reported an NMAC in the traffic pattern occurred during departure. The pilot of the other aircraft reportedly did not announce their position and stated the reporter had almost hit him.

Narrative

Date; about XA:00. DED; weather clear; winds calm. Before takeoff; waited at hold short for what appeared to be a student pilot on approach to Runway 5. Student aborted landing and went to do what I guessed was another landing attempt; staying in the pattern. Once the student was clear; I announced my intention to take off on Runway 5 and continue in a left pattern; exiting the pattern to the west. After takeoff; I announced my left-crosswind turn at about 400 ft. AGL. I did not see any traffic. Soon after; at approximately 600 ft.; I announced my left downwind turn. Soon after making the downwind turn; another airplane contacted me on the radio asking if I saw him and stated that I nearly hit him. I replied that I did not see him and apologized for the incident. Although I never saw the other plane; it was close enough for the other pilot to see my tail number. I continued on my flight as planned.Prior to the incident; I do not recall the other pilot announcing position in the pattern; although he may have and I just didn't hear it. In retrospect; I should have enquired about the other pilot's position before initiating any turns in the pattern. Once notified of the near miss; it would have been appropriate for me to ask the other pilot if he wanted to meet me on the ground to discuss.