php - Get stock quantity of entire category in Magento

I've got the code to get stock quantity for a specific product, but I'm looking to get the total stock quantity of an entire category (to only list categories that have inventory).

so instead of

$stock = Mage::getModel('cataloginventory/stock_item')->loadByProduct($product);
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I'd like something like

$stock = Mage::getModel('cataloginventory/stock_item')->loadByCategory($category_id);
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(Which isn't a real thing of course.)

Or any other method to list only those categories containing items in stock?

We're on Magento v1.7.0.2.

Thanks...