Start with the HTTP status of the source page. A 200 means the page is served normally. A 404 or 410 means the placement is gone along with the page. A 301 or 302 needs following, because a redirect to a relevant new URL that still contains your link is fine, while a redirect to a homepage or a category page usually means your placement was collateral damage in a migration.
A placement that switched to nofollow is the most common form of value loss that looks like nothing happened.
Second, link presence. Does your target URL actually appear in the returned HTML. This is the check that catches deliberate removals and quiet editorial cleanups, and it is the one most people mean when they ask whether a link is still live.
Third, anchor text. Compare what is published against what you agreed. Drift happens more than you would think, particularly with insertions, where an editor rewrites the sentence around your link and the anchor becomes click here or the publisher's own phrasing. Not fatal, worth knowing, and much easier to get corrected in the first month than the sixth.
Fourth, the rel attribute. Dofollow, nofollow, sponsored, or ugc. A placement that silently switched to nofollow is the most common form of value loss that looks like nothing happened, because the link is still there and a casual glance finds no problem at all.
Fifth, page level noindex. If the source page carries a noindex meta tag or header, the page is excluded from the index and your link passes nothing. Sites do this to thin content, tag archives, and old campaign pages, sometimes years after publishing.
Sixth, the canonical URL. If the page canonicals to a different URL, its signals consolidate there. Worth checking whether your link also exists on the canonical target, because sometimes it does and you are fine, and sometimes it does not.
Seventh, does your target URL still resolve on your own side. This one is embarrassing when it fails, and it fails regularly: you restructured your site, the page your partner linked to now redirects twice or 404s, and you have wasted a placement through your own housekeeping.
Eighth, log the timestamp. Not a check exactly, but the thing that turns checks into a record. Without a last checked timestamp you cannot tell the difference between a link that passed yesterday and one nobody has looked at since spring, and you cannot report survival rate at all.