Why SERP WHY??
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This is probably the most basic of SEO questions which no SEO I've yet met can answer. Analyzing a keyword for a client with the Basic SERP report reveals a lot but at the same time not enough. The term Logo Design (yes bad I know) shows at least 50 companies spending hard on SEO but there is no too in MOZ that can actually tell you why someone with less PA, less links, less DA actually ranks higher than someone else.
Surely you guys out there face this quandary all the time. The why? When you've exhausted everything what do you tell a client? Your numbers and your authority is better than the competition but they still rank higher!
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I hear your frustration...been there, done that, bought the t-shirt
It's important to come to terms with the idea that PA/DA isn't everything when it comes to ranking. I've worked on sites where the design and conversion strategy is terrible and people were not hanging on the site too long. These sites with higher bounce rates won't rank as well. Another factor I've seen is the continuous updating and adding content (such as in a blog) helps push rankings upward. One of my sites consistently ranks higher than our competitors with higher PA/DA and I'm certain it's these factors that's doing it.
There's probably a more techy nerdy answer out there, but this is just based on my experience. If sites aren't ranking well for me, I usually suggest a site redesign and redevelopment. Doing an overhaul and clean up can go a long way.
Good luck!