It would be possible to test, but not by looking at changing ranks. The problem here is that there are so many (hundreds at least) of factors affecting rank, and it is very difficult to isolate one factor when looking at changing ranks, if not impossible.
I would propose to test the theory this way:
1) Think of a new keyword that yields zero results on Google, a random string of letters should do it. E.g. shfbwuigbv Name it something unrelated like testerpage1.html
2) Create a new page with some junk content, but it must not contain your random new keyword, this is vital.
3) Include the following code on one of your reasonably well ranked pages:
<a href="testerpage1.html" title="shfbwuigbv"><img src="img.jpg" /></a>
4) Wait for testerpage1.html to be indexed - should be reasonably quick if the page you're using to link to it is well ranked.
5) Search Google for shfbwuigbv
Now the only single place that "shfbwuigbv" is mentioned is in the title of the anchor tag - nowhere else in Google's index does that phrase appear, nowhere on testerpage1.html does it occur. Therefore, if searching Google for shfbwuigbv brings up testerpage1.html as a result, we can categorically state that Google is using the title tag from the anchor as a means of determining the content of testerpage1. From this we deduce (although we can't be certain it will hold true in all situations), that the title of an anchor tag is taken to bear some relevance, and therefore some SEO benefit, on the page being linked to.
Of course, we must also be prepared to accept the contrary: if after a reasonable amount of time testerpage1 does not appear in the results for shfbwuigbv then the title tag cannot be conferring relevance onto the linked to page.
We can also look in other SEs to see if testerpage1 appears in their results for the new word, and use the same logic to determine if <a title attributes make any difference in their algorithms.
p.s. shfbwuigbv shouldn't be used for testing now - make sure it's a brand new unique word with no results in Google.
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