One of the witnesses at Martha Harrison's marriage to Jospeh Bailey in September 1843 was an Elijah Cartlide so these records may also have some relevance:
Mary Harrison, daughter of Joseph Harrison to Elijah Cartlidge, son of Richard on 1st October 1838 in Keele. A copy of the marriage register shows that Mary is the daughter of Joseph Harrison, hatter. She was of full age and living at Knutton Heath at the time of her marriage. Elijah was a potter, also of full age and from Knutton Heath. John Harrison (made his mark) was a witness. This suggests that Elijah Cartledge was Martha's brother in law.
From 1851 to 1881, Martha aged 27-57 is living with her hatter husband Joseph Bailey. From 1871 at the latest they are living at 44 Sutton Street.
In 1891 Martha is still living at 44 Sutton Street but is now widowed. Living with her are her daughter Annie, born Atherstone, Warwickshire and son in law Arthur Foster from Nantwich, Cheshire. RG12/2150/60 p40. This links Martha and Joseph as the correct parents for Elizabeth who married Daniel Cooper because Elizabeth’s daughter Mary Cooper was at one time living with Ann and Arthur Foster, giving relationship of niece.
Other pssoble connections? The marriage of Joseph Harrison (a boatman of Bagnall) and Hannah Cartlich is recorded on 27th January 1812 in Stoke on Trent (Pallot’s marriage index and IGI M009961) also marriage of Thomas Cartlidge married Elizabeth Plant at Keele on 16th November 1841 (M009951). They later lived in Penkhull.