Hi Brit,
Found this Mark Rice
1901 civil parish Kirkdale, reg district West Derby, sub reg district Kirkdale
address 53 Miranda Road
Mark Rice head born abt 1876 occ gas works lab born Liverpool
Mary Rice wife born abt 1874 Liverpool
John son age 11mths born Liverpool
There are 2 Alice Rices on the 1901 census
Civil parish Everton
Alice R Rice age 12 born Liverpool
Ann Rice age 52 born Liverpool
John Rice age 54 born Liverpool
The other one is aged 13 with siblings and parents James age 34 and Elizabeth age 33 these seem a bit to young to be parents of Arthur and Mark.
Theres no Arthur showing with a dob 1873 put in 2 y ears either side will have another look in case he has been mistranscribed.
these seem a bit to young