Formula for raising vs calling when HU on the river, in position: LHE
 

Let's assume that you're HU on the river, in position, and your opponent bets into you. You have a strong enough hand to call, and you're considering raising. We're playing LHE and each player has an unlimited stack. Here's a formula that will tell you the difference between the two actions:

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Let:
A) chance that you're ahead and he calls
B) chance that you're behind and he calls
C) chance that you're ahead and he raises and you call
D) chance that you're behind and he raises and you call
E) chance that you both have the same hand and he folds
F) chance that you're behind and he folds
P) size of the pot

Notice no analysis of him having same hand and calling, or you being ahead and he folds to a raise.

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EVr = 1A - 1B + 2C - 2D + EP/2 + FP

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Two things:

1) The EP/2 section may be slightly off... I'll have to think about it when I'm not so tired.

2) If someone wants to make a formula for betting vs checking when out of position on the river heads-up, I'd be glad to help iron it out once we've got some stuff posted on it. That formula should be a little more complex than this one.

Caveat: It's basically up to player judgement as to what to assign to each chance. If you were an awesome dataminer with a good PTDB, you could probably crunch some numbers and figure it out, but it'd be pretty tough.