A B757-200 Captain reported that after takeoff from LAS Runway 25R on the STAAV SID; the LNAV Flight Director function commanded a left turn on track while the EHSI and FMC PROG 2 page showed flight left of track.

Narrative

We were assigned and were flying the STAAV SID from Runway 25R at LAS. All waypoints were verified on the legs page prior to take off and the shift for the full length departure off the runway (it has a displaced threshold) was entered in the take off page of the FMC. Almost immediately after LNAV was selected on the MCP; the Flight Director began commanding a left turn even though the EHSI track showed that the turn would take us left of the SID flight path. The First Officer; who was flying; followed the Flight Director as we discussed the issue and I switched to PROG page 2 to see if any course deviation was showing. PROG page 2 was showing a left track deviation as the First Officer continued to follow the Flight Director commands. I finally told him to turn back right as the EHSI track showed the Flight Director commands were taking us off course and the PROG page 2 confirmed this. PROG page 2 showed a 0.3 NM left deviation at the furthest point. Before beginning the right turn to the northeast per the SID; the Flight Director vertical bar slowly drifted back to course alignment. ATC did not say anything to us about being off track. Again we reviewed the fixes on the LEGS page of the FMC prior to pushback and we had the correct runway selected for the SID and the threshold shift was entered into the TAKE OFF page of the FMC.