Anne Bickerton was the sister of the diarist Thomas Bellingham who stayed for a time in Preston in 1689, accompanied by her daughter Betty, while Thomas and his wife were living there.
When Anne married Robert Bickerton in 1661 they brought together two of Ireland’s Anglo-Protestant colonial families: the Bellinghams seem to have obtained their estate for serving Cromwell in Ireland and the Bickerton estate originating in the Plantation days of James I. Neither family would be popular with their Irish Catholic neighbours, especially when those neighbours were incited to violence by Tyrconnel.
Robert Bickerton inherited the Ulster estate by way of a marriage linking his family with the Stanhowe family of Yorkshire. A William Stanhowe, of Beddington, Yorkshire, travelled to Ireland in 1610 to take possession of a 1,500-acre called Shanagoolan in Armagh, then returned to England and rather neglected the estate for the next few years while he was occupied with business in England. In 1640 a Robert Bickerton of St Martin in the Fields, London, received a portion of the estate, named Clontylew, as part of a marriage settlement and settled there briefly before returning to England on the outbreak of the Civil War. He died in 1647 and the estate was managed by an agent until his son, also Robert, came of age. On reaching 21 in 1661 Robert sailed for Ireland to take possession of his property and to marry Anne Bellingham.
The couple had two children, a son Henry and a daughter Elizabeth, known as Betty. In 1671 Anne Bickerton went to law as guardian of her son, which suggests that her husband had by then died. He was certainly dead by 1676 for in the will of her father, Henry Bellingham, dated that year, Anne is described as widowed. 
Paterson, on whose article the above account is based, states that Robert and Anne had a daughter Jane, who married Henry 6th Duke of Norfolk. However, this cannot be right since the Jane Bickerton who married the Duke of Norfolk was the brother of Robert, not his daughter, as the DNB makes clear. 
 T. G. F. Paterson, ‘County Armagh in 1622 a Plantation Survey’, Seanchas Ardmhacha: Journal of the Armagh Diocesan Historical Society 4, no. 1 (1960): 124–26.
 R. O. Miller, ‘Howard, Henry, Sixth Duke of Norfolk (1628–1684)’, in The Oxford Dictionary of National Biography, ed. Lawrence Goldman (online), online, 2004, http://www.oxforddnb.com/view/article/2429.