I searched the 1930 census for an Arthur Herrick with a wife named Myrtle and a daughter named Patricia to no avail. However, if your grandmother, Patricia Herrick, was born after 01 Aoril 1930 I may have found her parents and siblings enumerated in the third Ward of the City of Sedalia, Pettis County, Missouri as follows.
Arthur Herrick, Head male, white, age 30, married at age 22, B. MO, parents b. OH.
Myrtle A. Herrick, Wife, female, white, age 25, married at age 17, b. MO, father b. KY, mother b. MO.
Jack I. Herrick, Son, Male, White, age 7 single, b. MO.
Dwight B. Herrick, Son, Male, White, age 5, single, b. MO.
Billy Edward Herrick, Son, Male, White, age 3, b. MO.
Marjorie Dean Herrick, Daughter, female, Whiteage 1, single, b. MO.
Hensie Herrick, Brother, male, white, age 24, single, b. MO, parents b. OH.
Is this a possibility for your great-hgrandparents?
Richard L. Herrick