Three comments:
  1. If there is no right censoring (and it appears not), I would use
lmer on the awakening times, glmer on the FullyOriented variable.  That
is, I agree with Burt.
  Another option is GEE models

 2. If you want to use a Cox model, then you can 
   a. Add "+ cluster(id)" to the model statement.  This adds a robust
variance, and is closely related to GEE.
   b. Use coxme to fit a mixed effects model.

Terry Therneau

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