In this paper, we study position-dependent timing shifts and timing resolution in position sensitive avalanche photodiodes (PSAPDs) and their effects on the coincidence window used in positron emission tomography (PET) systems using these devices. There is a delay in PSAPD signals that increases as the excitation position moves from the corner to the center of the device and the timing resolution concurrently worsens. The difference in timing between the center and the corner can be up to 30.7 ns for a 14 x 14 mm(2) area PSAPD. This means that a PSAPD-based PET system could require a very wide coincidence timing window (>60 ns) if this effect is not corrected, although the individual crystal pairs still have full-width half-maximum (FWHM) timing resolutions better than 7.4 ns. In addition to characterizing the timing properties of PSAPDs, two correction methods were developed and applied to data from a pair of PSAPD detectors. These two timing offset corrections reduced the timing shift of a crystal pair from 52.4 ns to 9.7 ns or 1.3 ns, improved the FWHM timing resolution of the detector pair from 24.6 ns to 9.5 ns or 6.0 ns and reduced the timing window (sufficient to cover at least twice the FWHM for all crystal pairs) from 65.1 ns to 22.0 ns or 15.2 ns, respectively. A two-step timing alignment method is proposed for a PET system consisting of multiple PSAPDs. Lastly, the effect of PSAPD size on the timing performance was also evaluated.