Andrew rejected the offer of a new deal with League Two Grimsby Town in order to join newly promoted Rovers on a two-year contract.
In a statement to supporters of Grimsby - where he was recently named player of the season - laid out the reasons for his choice.
He said: "My move was based solely on my personal aspirations to play at the highest level possible, with Doncaster Rovers being a step up on the footballing ladder.
"I can assure the fans that had there have been another League Two club interested, there would have been no doubt in my mind that I would have stayed at Grimsby Town.
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"But I had to take the opportunity to step back up to League One."
"I would like to thank the fans for their unbelievable support throughout the season both home and away, the atmosphere you create is something I will never forget.
"I would also like to thank all those that voted for me for Player of the Year last season, you have been incredible to me since the day I arrived."
"Thank-you to everybody connected to Grimsby Town Football Club, I wish you all the best in the future and hope you push for promotion this coming season."