[{quoted}](name=Project Riven,realm=NA,application-id=3ErqAdtq,discussion-id=pi29fhpT,comment-id=0002000000010000,timestamp=2015-09-07T12:33:30.819+0000)
IMO, as Ashe, Shiv is a better choice.
The reason why PD is not better is because Ashe already has an ability to increase attack speed. Giving up the 10% AS to get Shiv for its passive is a better choice.
It's a 4-second steroid with an 18-second cooldown, and it's a very small AS increase.
Another reason is because Ashe's attacks are basically all crits after the first shot. So having crit chance more than 1% only benefits the first shot, rendering the 15% more crit chance on PD not as beneficial for Ashe.
Actually, her first attack can't benefit from Crit Chance; that's not the reason you build it on her.
Ashe's Crit Damage scales on her Crit Chance. I don't know if you've noticed this, but the passive specifically states that her modified crits deal 110% (+% Crit Chance) damage.
Shiv makes her crits 130% damage. Phantom Dancers make her crits 145% damage.
The reason why Shiv is better is because the passive can critically hit. The damage is also modified by items like IE. As Ashe, if you cast Volley on your target than auto attack, you deal an insane amount of damage.
The damage is also modified by Ashe's passive, making Shiv's crit significantly weaker if you don't have enough supplemental Crit Chance.
Another point entirely overlooked in this comparison is that Ashe is one of the few champions granted the lowest base movement speed in the game. Being able to walk through minions is a huge benefit for her.