Please try this:

$date = "2013-01-07";
$earlier = strtotime("-1 week", strtotime($date));
// should result in 31 Dec 2012, which is a Monday
echo date('r', $earlier); // to make sure we have the right date
echo "\n";
echo date('W Y', $earlier); // what you have
echo "\n";
echo date('W o', $earlier); // what I think you want
echo "\n";

I ran it and got this:

Mon, 31 Dec 2012 00:00:00 +0000
01 2012
01 2013

Indeed you're assumption is correct.
Reading the info regarding the 'o' gives meening why I'm seeing what I'm seeing.

Thanks

Thank you for taking the time to write to us, but this is not
a bug. Please double-check the documentation available at
http://www.php.net/manual/ and the instructions on how to report
a bug at http://bugs.php.net/how-to-report.php

As the other commenter said, for the ISO Year, you need the "o".