It seems that "Yelizabetgrad" was the Yiddish variant town name, and today it is called Kirovohrad, Ukraine.
Here are some databases to search for your family, if you haven't already:
JewishGen Ukraine Database:
I suggest doing a soundex search on the surname, as names get spelled various ways in the records.
This site has old digitized Eastern European city and business directories. You can check if they have anything for the town you are searching.
Hope this helps,