William, Sarah and Lydia as well as other children (Twins: Mary and Sarah b 1855, Nancy b 1858, Benjamin b 1859) are all listed in Dwelling 733 in the 1860 Blue Twshp, Jackson Co, Missouri Census. Post office is listed as Independence MO. Occupation Carpenter. William and Sarah are listed as being from Ohio, all the children from Missouri.
The family is indexed as Millet though that is a transcription error when looking at the image it is written as Mellotte.
So the family is in an area of turmoil for sure.