Hi Billy,
I had a look on the British Vital Records to see if there were any more children listed to John Anderson & Janet Binning but unfortunately there weren't any others apart from the ones you have already.
The good thing about searching the BVR's is that you can leave the search area blank and only look for births using the names of the father and mother.
There were 6 William Andersons born in W. Lothian between 1807 - 1823:
William Anderson, born in Uphall, 31st January 1807 to James Anderson and Margaret Douglas
William Waddle Anderson, born in Bathgate, 13th October 1811 to William Anderson and Ann Lawrence
William Kay Anderson, born in bathgate, 1st April 1817 to Jean (? John ?) Waddle Anderson and Lilian Kay
William Anderson, born in Ecclesmachan, 16th July 1820 to James Anderson and Elisabeth Moodie
William Anderson, born in Linlithgow, 12th April 1821 to John Anderson and Marion Dumbreck
William Anderson, born in Bathgate, 18th May 1823 to John Anderson and Agnes Mills.
. . . . and there would probably have been a few more who's parents didn't bother to register them!
Anne