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Presumably the presence of the keyword (or lack thereof) on the page does not matter. The only thing that matters in this specific search is do the keywords show up in links pointing to a page.
Here is the question: How does Google rank the results for this search?
One simple way, would be to rank highest the pages with the highest number of inbound links containing the keywords. So for example, the #1 page for...
allinanchor: chicken soup recipes
...might be the page which has those 3 words present in the link text in that order the most # of times.
BUT, what if Google factors PageRank into the allinanchor search?
Perhaps the SERP is ordered not by who has the most links, but by a PageRank weighted product of those links.
Each link has a certain amount of PageRank it gives to a target page. What if the PR values were totaled for each incoming link with the keywords present to produce a score for the allinanchor search?
The SERP ranking would be very different. A page with 3 high PR inbound links containing the words "chicken soup recipes" could beat out a page with 50 low PR inbound links containing those words.
So which do you think Google uses to rank the allinanchor search: quantity of links with the keywords only, or the sum of the inbound PR from links with the keywords in them?
WHY THIS MATTERS
If Google factors PR into this search, then allinanchor searches can be used as a proxy to measure how well a page ranks just based on the links you have pointing to that page. And as we all know, inbound links with your keyword phrase is one of the more important elements in Google's algorithm.
If PR is a factor, then you can compare the allinanchor SERP rank of a page to the real rank of a page, and know whether you have done a good job of on-page optimization or not. If your real results were lower than your allinanchor results, then you would know that optimizing your page better is more important to increasing your rank than getting more links that page.
On the other hand, if quantity of links with the keywords is the only criteria for allinanchor SERPs, then this search cannot be used as a proxy to determine whether you need more links or better onpage optimization.
Why? Because in the real Google results, inbound links containing the keywords searched upon are weighted by the PR flowing through each of those inbound links.
As another example, I have a page that has ranked number 1 in the allinanchor search for months, yet is currently #17 in the real Google results (for a competitive 2 word phrase with 2.5M results).
I have assumed that allinanchor was weighted with the PR. But after a few attempts at major onpage changes to little effect on my ranking, I am beginning to think that PR is not factored into this search. I think that this search only looks at the quantity of links with the keyword phrase present.
Quantity only would better explain to me why I am #1 for allinanchor, as I have many low PR pages pointing to this page with my keywords in them. I don't know though which is why I am starting this discussion.
So how do think the allinanchor SERPS are scored: quantity of links only, or PR weighted quantity of links?
If PR is a factor, then you can compare the allinanchor SERP rank of a page to the real rank of a page, and know whether you have done a good job of on-page optimization or not.
I'm going to have to study this for a while. Preliminary look:
2 keyword phrase dominated by high PR Gov sites: Same results as regular search.
3 Keyword Phrase: My sites aren't there, replaced by competitors (In contrast to the regular search: My sites are in the top 5).
I think that the general searches may be difficult to analyze. It's hard to tell because Inbound Links and PR are bound together. It's like the old theological debate about the distinction between the Soul and the Spirit: Where does the soul end and the spirit begin? In this case, is it the PR of the gov sites, or is it the number of links, that is getting them to the top of the allinanchor search?
The 3 keyword phrase is interesting. The competitors who are at the top of an allinanchor search have concentrated much of their SEO efforts on anchor text, so it's no surprise to see them at the top of an allinanchor search. Of these results, the most aggressive anchor manipulator is on top, despite their lower pr.
I don't think PR is having an effect in the more specific search.
Based on my preliminary view of the 3 keyword phrase, I'm tending to believe that PR doesn't play a part.
Toolbar PR is not that useful except as a very gross indicator of the power of an individual page.
IMO, lower PR pages beat higher PR pages partially because they have more PR with respect to the keywords of a given search (i.e. keyword weighted PR). This is my opinion, and as always, I could be wrong. But such an approach would make a lot of sense for Google to do.
To answer my own question, I am tending towards believing that PR is not a factor in allinanchor searches. If anyone wants my keyword phrase where I am rank #1 for allinanchor:keywordphrase versus #17 for keywordphrase, just stickymail me and I'll send it to you to play with.