Initiative based on other abilities?

Would it be engaging for your initiative roll to be based not exclusively on dexterity but on the ability most germane to your action? So if you are heaving a two handed sword for which the to hit roll is based on strength then your initiative is modified by strength. The stronger you are the quicker you will be able to swing that big piece of metal, right? If you are casting a wizard spell than your mental quickness, your intelligence, determines if you are able to get that spell launched before another opponent gets the drop on you. I can imagine the idea that for long chase runs and perhaps say swimming it could be your constitution that determines if you are able act quicker than others. And then for all the other miscellaneous actions such as use object it seems to make sense to simply fall back to dexterity. This feels like it could be fun and bring interesting possibilities to the table but I am not sure. Reigning in dexterity from being the "super stat" certainly seems positive though.