Hi again Margaret,
Yes, I had looked at your other post on this family with the details of the 1851 census.Just curious about Alexander not being with the family in 1841 (which I had looked up on FreeCEN) and had wondered whether you had found him, and whether his whereabouts might have given any clues to his family.
This family is a bit like looking for a needle in a haystack
I'll try to clarify my own thinking.
Problems include:
(a) apparently Helen (and thus any family members who might have been with her) does not appear in the 1861 census - a major problem
(b) surname and given names are both common (which makes using SP very expensive)
(c) you might be searching for records which don't exist - another major problem
(i) the only people whom you know for sure survived into the time of statutory registration (which started in 1855) are mother Helen and sons John and Andrew
(ii) pre-1855 you are relying on church records and this family doesn't seem to show up (perhaps they attended a church the records of which did not survive, or have survived but have not been indexed) - if they died pre-1855 there will be no record on SP, and if any married pre-1855, there are quite likely to be no records to be found in the IGI or on SP.
Going through the family, here is what I was thinking:
*Husband Alexander ANDERSON, b ca 1807 (from 1851 cen) in BathgateHe could have died at any time between ca 1852 (when Andrew arrived and 1868 (when Helen died as a widow). Of course, if he died before 1855, there will be no record of him on SP to be found - let's hope he did not, and that you eventually find him.
*
Wife Helen ANDERSON, b ca 1810 (from 1851) BathgateDied, a widow, in 1868. Parents listed as Thomas RAMSAY and Margaret MASTERSON. Her birth in Bathgate in 1807 found on the IGI (but I can't see her parents' marriage or births of other children to them). Where was she living when she died?
Might be worth trying the 1871 census to see who is living at the address where she was living when she died in 1868?*
Daur Margaret ANDERSON, b ca 1831 (from 1841 census) WLNNot with the family in 1851. Might have died (there will be no record) or might be working away from home. Or she might have married (and if she married pre-1855, there may well be no record). If she survived, she might be almost anywhere in 1861.
*
Daur Janet ANDERSON, b ca 1834 (from 1841 and 1851) BathgateShe was recorded as Jane in the 1841 census - so might be known under all sorts of names such as Janet, Jane, Jean, Jeannie, Jessie.
If she survived, she might well have married pre-1855 (so there may well be no record). And might be anywhere by 1861 and later.
*
Son Thomas ANDERSON, b ca 1836 (from 1841 and 1851) BathgateIf he survived, he could be almost anywhere in 1861 and later.
*
Son James ANDERSON, b ca 1838 (from 1841) WLNNot with the family in the 1851 census. Probably did not survive (there will be no record) - or he might, at age 13, be away working somewhere though this seems unlikely given that Thomas is still at home.
*
Daur Mary ANDERSON, b ca 1840 (from 1841 and 1851) BathgateIf she survived, she might be married and almost anywhere by 1861 and later.
*
Son John ANDERSON, b ca 1842 (from 1851) BathgateYou definitely know he survived as he was the informant on his mother's death cert in 1868 and was living in Shotts.
Might be worth trying to find him in 1871? Though it's a very common name.
*
Daur Helen ANDERSON, b ca 1846 (from 1851) BathgateIf she survived, she might still have been with Helen in 1861 - but subsequently (married or not) could be anywhere.
*Son Alexander ANDERSON, b 1851 (from 1851) BathgateIf he survived, would probably still have been with Helen in 1861 - but subsequently could be anywhere.
Son Andrew ANDERSON, b ca 1852 presumably in BathgateYou know about Andrew. Would probably still have been with Helen in 1861.
Any clues in the family with whom he was lodging in 1871?*
Possible further children?Not known though unlikely given's Helen's age. Given the 1807 birth record of Helen, she would have been about 45 when Andrew arrived - quite late for those days.
Returning to the father, Alexander ANDERSON.Given Scottish naming patterns one might expect his parents to be Thomas and Janet/Jane (or, of course, Margaret which - as the name of Helen's mother - had already been used for the first daughter).
It is interesting that, in 1841, the following family (all born in West Lothian) is living right next door to Helen and the children:
Thomas ANDERSON 60 Cotton Hand Loom Weaver
Jean ANDERSON 60
Thomas ANDERSON 20 Wright
Jean ANDERSON 20
Possibly Alexander's parents and siblings? The only match in the IGI, as you say, is Alexander bap 17 Jul 1804 Bathgate, parents Thomas ANDERSON and Jean COCHRAN - but, of course, Alexander might well not have been christened.
Extracted entries in the IGI for baptisms of children to Thomas ANDERSON and Jean COCHRAN don't help - James 1800, Christian 1802, Alexander 1804, Andrew 1807, Marrion 1810, Elisabeth 1813 i.e. no sign of a later Thomas and Jean.
It seems you will need a lot of luck -
or a lottery win so you can look at the certificate for every Alexander ANDERSON death on SP 
Very best of luck,
JAP
