I see Lizzie is shown as immigrating in 1903. I looked for her in passenger lists and this matches.
New York Passenger Lists, 1820-1957 about Lizzie Chambers
Name: Lizzie Chambers
Arrival Date: 9 Apr 1903
Birth Year: abt 1898
Ethnicity/Race/Nationality: Scottish (Scotish)
Port of Departure: Glasgow, Scotland
Port of Arrival: New York, New York
Ship Name: Anchoria
The image shows:
Mrs.age 40, going to husband - then it says Christofer, Box. 13, So. Manchester - she has $30.00 with her
Thomas, age 18, male
Annie, age 16, female
Mary , age 14, female
Robert, age 11, male
Maggie, age 9, female
Lizzie, age 5, female
Christine, age 3, female
Cena , age 1, female
all are shown last residence Ayrshire, Scotland, destination: Hartford, CT
children shown going to father
I looked for this family in 1910 to see if Lizzie is with them. I found a Christine who is in Enfield, Hartford as a lodger
As they came in 1903 - it was 7 years later so some of the children could be married- boys should still have same surname butAnnie, Mary and Maggie could certainly have husbands by then.
I did not find a Cena or any name close to that - her name in passenger list was not that clear - I went by what the indexed entry said.
With the mother being shown as Mrs. in passenger list that does not help to find her.
I did not find Thomas but there is a Robert W. Chambers, age 19 living in Manchester, Hartford - he is a boarder, shown born Ireland but that could be a mistake if the person who gave the info didn't know.
So this is really not a lot of help to you, except it does seem likely the family who I gave you the passenger list for is the right one. It looks something may have happened to the parents to leave the kids on their own.
I took a look at the 1901 Scottish census to see if I could find them - as they were shown as from Ayrshire I looked there first.
1901 PATNA, DALMELLINGTON, AYRSHIRE, SCOTLAND
Christopher, age 40, pit engineman, born Ireland
Mary, age 39, wife, born Dalry, Aryshire
Thomas, age 20, son, pit fireman, born Dalmellington, Ayrshire
Robert, age 12, son, scholar worker, born Dalmellington, Ayrshire
Maggie, age 10, daughter, scholar, born Dalmellington, Ayrshire
Eliza, age 4, daughter, scholar, born Dalmellington, Aryshire
Christina, age 2, daughter, born Dalmellington, Ayrshire
ages are off a bit for some but that is not unusual with both census and passenger lists.
Annie and Mary are not listed so I looked for them to see if they were with relatives or ??I did not find either of them, but they are with the family in 1891 census. Let me know if you would like that census info.
I found Christopher in passenger list for 1902 - he travelled alone and is shown going to his brother Henderson Chambers. Box. 13, S. Manchester (same address as family shows in their passenger list) Christopher is listed as an engineer