Missouri Death Certificate
www.sos.mo.gov/images/archives/deathcerts/1911/1911_00041897.
pdf

Name-Mrs. Elsie Birmingham
Place of death-Webb City, MO
Date of death-2 Dec 1911
Cause of death-Apoplexy
Age at death-62 years 6 months 27 days
Date of birth-5 May 1849
Place of birth-Pottawattomie Co., Iowa
Occupation-housewife
Name of father-John Bay
Birthplace of father-Ireland
Maiden name of mother-unknown
Birthplace of mother-unknown
Informant-Ellen Johns Webb City, MO
Date of burial-3 Dec 1911

Note that the gravestone has Elsie's birthday May 16, 1847, while her death certificate has May 5, 1849 - a conflict of two years. The death certificate conflicts within itself as it has age 62y 6m 27d. This would indicate a birth date of May 16, 1849. The year of 1847 is considered to be the correct year as Elsie's older brother, Thomas Benton Bay, was born January 27, 1849. For all intents and purposes the date on the gravestone is the one used for the birth of Elsie Kathryn (Bay) Birmingham.

We know James married Elsie Kathryn Bay, but there seems to be two marriage dates: The Pottawattamie County Genealogical Society marriage index, provided by a researcher, gives the marriage date for "I. Bermingham and Elsie Bay" as February 14, 1867. FamilySearch.org has a marriage record that has been transcribed to read: "I Be?ingham and Eley Bay" married February 14, 1867. These two records appear to be for James Birmingham and Elsie Bay and agree as to the date. The other marriage date comes from records of a Bay Family member that says they were married February 28, 1866 in Council Bluffs, Pottawattamie County, Iowa and the marriage was witnessed by Humphrey and Martha Bay. The early date would certainly fit with the birth of a child, Mary Ann, who was born February 22, 1867 and died in March 1867 - that is if her birth date is correct.