Air carrier on approach to KTN and on KTN radio's frequency elects to change from RNAV to ILS approach ending in a missed approach procedure prompting questions regarding clearance procedure change authority.

Narrative

Cleared by ANC ATC for RNAV 11 approach into KTN. While on downwind; a fog bank began rolling over the airport as reported by KTN radio. During the base turn we could see the approach end of the runway; but just barely. During the turn to final; we decided to change to the ILS and expressed our desire to KTN radio. They cleared us for it; and we engaged the approach mode of the autopilot. We had briefed it during the decent as well just in case. I put it into the FMC and we continued to the DH. We could see the approach lights but not the runway; so we executed a missed. Realizing we were on a different misssed approach in direction of hold; we made sure we told ANC ATC when we contacted them that we were on the missed approach from the ILS; not the RNAV. Just before entering the hold; ANC ATC informed us of an altitude limit that they had that prevented them from giving us a late clearance for the ILS; but we thought KTN radios approval of the approach was good enough. The ANC controller said it was pretty much for information only; but providing this report may help other crews in the future. Cause; misunderstanding of who can clear us for approaches. We thought KTN radio could change us from the RNAV to the ILS. Suggestions; fly the approach you are last cleared for; do the missed approach if required; then try another approach.