Melee Damage

Melee Damage depends on 4 values: the Melee Definition; the Attacker's Level; and the Melee Damage and Roid Damage bonuses.
Like with weapons or items, the Melee Definition defines the exact formula used to get the final melee damage. Most of these have the same standard formula, but some are slightly different
Standard1 Melee Bonus × 2.5 × (8 × β + Roid Bonus)
Krieg Melee Bonus × (16 × β × (0.02 × Level + 1) + 2.5 × Roid Bonus)
Nisha2 Melee Bonus × 1.875 × (8 × β + Roid Bonus)
Buzz Axe Rampage Melee Bonus × (80 × β × (0.02 × Level + 1) + 2.5 × Roid Bonus)
Deathtrap Melee Bonus × (8 × β + 3.5 × Roid Bonus)
Deathtrap Frenzy3 Melee Bonus × 0.62 × (8 × β + 3.5 × Roid Bonus)
Execute Melee Bonus × 5 × (8 × β + Roid Bonus)
Fist Full of Hurt Melee Bonus × 10 × (8 × β + Roid Bonus)
Backhand to Remember Melee Bonus × 96 × β × (0.02 × Level + 1)
Blood Rush Melee Bonus × 3.75 × (8 × β + Roid Bonus)
Power Fist Melee Bonus × 80 × β × (0.02 × Level + 1)
1 Melee weapon accessories force using this definition even if the character has others.
2 Does not include Thunder Crackdown, which uses the standard formula.
3 Does not include the "Frenzy End" Melee Definition, which uses the standard Deathtrap formula.

A Note on Roid Shields

When following these formulas, it would make sense to plug in the value straight from your shield's item cards right? Well this is Gearbox we're talking about.
These formulas use the calculated roid damage, not the displayed damage. The displayed roid damage is 2.5× the calculated damage.
In practice, these formulas almost always end up also multiplying the calculated damage by 2.5, so it is possible to rearrange them to take the displayed value instead. Roid Shields are the only vanilla items to give this bonus type, so this should always still line up.