Initiative is a contest to see who gets to act first. It's not technically a contested check in the usual sense, but it *is* the only standard situation in the game where people roll against each other to determine a winner.
That's close enough as far as I'm concerned.
>And when someone is stealthing around they use stealth instead of perception to set their initiative order.
>
>To stay undetected/unnoticed their initial initiative (based on stealth) is used against the others perception DC or when they use the seek action.
No, that's the thing, RAW you *do not stay unnoticed*, only undetected, which means the other side knows you exist. If you beat their DC, they don't know where you are, but they know *that* you are.
This is incongruent with how avoiding notice works outside of initiative rolls. That's the point, and that's what I think is a Bad Experience, Actually.
>I don’t know how to handle the secret nature of a stealth check in roll for initiative scenarios though.
You don't. As buffman mentioned, Stealth-based initiative rolls are open. But secret rolls are also one of the most common things to be ejected from the game, so a lot of people outside of PFS have nothing to rectify here.