Since the tragic death of Google's PageRank, the SEO metrics & signals industry is flourishing, many companies try to fill the gap and create the "ultimate metric" for evaluating SEO value of URLs and domains. People love these metrics, they simplify the world around us and "help" us save time, but it is not without a cost! And what a cost it is. What do you really know about these numbers? Where do they come from? What are their limitations? How accurate they are? Are they reliable? When can we actually use them?
If you haven't asked those questions, you are either an amateur in SEO analyzing or a very naive person, because the answers to these questions reveal the truth about metrics (how non-optimal they actually are).
Before we start talking about the problems with metrics, first let's define and distinguish between macro (domain) and micro (backlink) metrics.
Micro metrics are metrics that given to specific backlinks/URLs and should represent the individual value of the URL.
The most common micro metrics are TF/CF (Majestic) PA (MOZ) & PR (Google).
Macro metrics try to measure the value of a domain as a whole and they usually kind of summation of all the individual backlinks and their micro metrics. Like micro metrics, macro metrics have similar names, and the most common macro metrics are TF/CF (Majestic) DA (MOZ) & PR (Google).
All the metrics, without exceptional are based on backlinks - the building blocks of SEO value. It's very important to remember this simple truth - we use these metrics to measure the specific and overall SEO value of backlinks. As we'll further explain and demonstrate, these metrics often mislead us and not doing their "job".
We'll start with explaining why metrics (micro and macro) are problematic, and after that we'll give tips about when and how we can still use them to our advantage.

• The problem starts, actually, with the micro metrics (as macro metrics are mostly derivatives of the micro metrics).
  And the main problem is with the basic algorithms - the ones that set the values for each URL. When you try to write an algorithm that should work on any URL in the World Wide Web, you can't do it perfectly, no one can (not even Google with their PR algorithm - although it's, IMHO, the best micro metric for older URLs), and you end up with an algorithm that has many compromises and built-in flaws & inaccuracies. This is why you can very often find amazing backlinks (from authorized & natural webpages) with very low micro metrics and vice versa, you can often find spammy backlinks or backlinks with ZERO SEO value that are getting very high micro metric values.
  
  This alone should be enough to conclude that metrics aren't good at evaluating SEO value.
• Macro metrics are kind of average of all the micro metrics. So not only the micro metrics are not accurate, the macro metric takes them all into account, even those which are very un-accurate and have very big bias.
  Another issue with this fact is that macro metrics can be pushed down a lot by irrelevant backlinks. Any domain usually has some or many irrelevant backlinks and random backlinks - i.e. backlinks that wasn't created naturally, or without any value (like spammy backlinks that are auto-generated by scripts, backlinks from comments etc.). Well, these backlinks are all part of the calculation of the macro metric, which once again increase its bias/inaccuracy.
• Here is another example of why these metrics makes no sense:
  
  SEO wise, it's very unlikely that an inner-page of a website will have a greater value than the homepage/domain itself, for the very basic reason that homepage gets more backlinks + it even gets value from all the backlinks of the inner pages, because the inner pages are part of the domain. But as we see, metrics have their own rational and it's very often that inner pages have greater metric values (micro) than the home-page's or domain's value (macro).
• Another problem is that these metric companies that are totally established on backlinks actually not familiar with all the backlinks, or even worse, considering deleted/broken/duplicate backlinks as normal active backlinks which even deepen the bias & inaccuracy of these metrics.
  Here is a simple example: apisq-qcga.ca is a Canadian domain related to Quebec, and it got very valuable backlinks from the Quebec Governance/State website - QC.ca:
  
  
  We can safely say that most of the SEO value of this domain comes from these .qc.ca backlinks.
  If we check the domain with Majestic.com, all these backlinks suddenly disappeared:
  
  
  (there are no further pages).
  So even if Majestic's TF/CF micro metrics are perfect (which as I explained earlier are very far from that), how can Majestic calculate the macro TF/CF without these highly valuable backlinks?
  
  The answer is, they can't. And this is yet another reason why you shouldn't rely on these metrics.
• To understand the problematic with these metric, lets continuing with the example above and let's imagine a fictional scenario in which Majestic are familiar with these missing backlinks.
  In this scenario, what will be the macro TF/CF, will it be higher or lower? Logically, since we know these backlinks are valuable we know that they should, but would they?!
  The answer is that we can't predict - since Majestic wasn't familiar with these backlinks at first place, it's very likely it wasn't familiar with similar URLs in the networks surrounding those .qc.ca websites or at least Majestic won't be able to properly analyze them with their TF/CF algorithms and therefore it's very realistic to assume that these backlinks would have even lower TF/CF compared to the other backlinks Majestic was able to find and analyze. Therefore, the macro metrics of TF/CF could actually be lower in a fictional scenario in which Majestic is familiar with the valuable .qc.ca backlinks.

Here are some more evidences for the inaccuracy of those metric:
jailbreakyourphone.net - a pure spam domain with spammy backlinks getting relatively high DA/PA/TF/CF:

On the other hand, adomaly.com - great domain that's also related to iphone with great authorized backlinks from forbes.com, appleinsider.com etc. getting very low DA/PA:

texasspecialedlaw.com - spammy domain with spammy and low quality backlinks getting relatively high TF/CF/DA/PA:

g8-g20-youth-summits.org - a very authorized domain with backlinks from places like ed.ac.uk, huffingtonpost etc. with very low TF

In conclusion, we've seen many defects & problems with macro and micro metrics including:
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Even if you don't "buy" or believe everything I wrote, but only partially, it's still enough to understand why these metrics are bias and therefore misleading, not reliable and without much of a correlation to SEO value.

What's the right way?
Google doesn't familiar with these metrics, only with the backlinks, this is why the key for properly analyzing a domain (for SEO purposes ) starts and ends with the backlinks themselves.
Macro metrics shouldn't be a factor at all when choosing a domain for SEO. The only time you can have some advantage by using macro metrics is when you filter a huge list of random/unknown domains, in such cases you probably want a quick way to filter out most of the really bad ones and to finally have only few domains to manually analyze. But still you will end up with a list of domains with minimum metrics which tells nothing about the SEO value and in that list still hiding many spammy and low quality domains.
As for micro metrics, we can definitely use them, but it has to be wisely and combined with manual check as well. PR is the most reliable micro metric and has the most correlation with actual SEO value, however it hasn't been updated for very long time (Since Dec 2013), so it's usually been used to assess older backlinks. We can also combine other micro metrics (like PA/TF and also OBL) to get a feeling about the backlink's quality and value, but eventually only when it combined with manual analysis by professional and with an healthy common sense, you can really evaluate the quality of each backlink and therefore the total SEO value.