If your family lived in Ohio, then you would be considered an Ohio resident.
https://www.tax.ohio.gov/ohio_individual/individual/residency_status.aspx
Ohio Residency Statuses
Resident: You are an Ohio resident for income tax purposes if you are domiciled in Ohio. Thus, under Ohio law, the terms “domiciled” and “resident” mean the same thing. Generally, any individual with an abode in Ohio is presumed to be a resident. The abode can be either owned or rented. Temporary absence from your Ohio abode, no matter how long, does not change your residency status. Thus, if you live in Ohio, the presumption is that you are an Ohio resident. Ohio residents are eligible for the resident credit on any non-Ohio income if they were subject to, and paid tax on, that income in another state.
You do owe Ohio tax, however as an Ohio resident working in MI, there is a reciprocal agreement between MI and Ohio and hopefully, you can receive a refund from MI for taxes paid to that state. (If you are a resident of Ohio and work in MI, you only need to file an Ohio return.
https://www.michigan.gov/documents/treasury/RAB_2017-13_IIT_-_Reciprocal_Agreements_572757_7.pdf
If you also paid taxes in DC and NJ, if you had filed an Ohio return, you would have received a credit on your Ohio return for the taxes you paid as a nonresident of DC and NJ.
You may wish to have a tax accountant (not the one you used last year) look at the Ohio tax calculation in the notice from Ohio tax authorities for 2017 and see if it correctly gives you credit for tax paid in other states.
I wish you well.
